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arXiv:0707.0940v2 [math.DS] 12 Jul 2007 SOBOLEV REGULARITY OF SOLUTIONS OF THE COHOMOLOGICAL EQUATION GIOVANNI FORNI CONTENTS 1. Introduction 1 2. Fractional weighted Sobolev spaces 9 2.1. Weighted Sobolev spaces 9 2.2. Fractional Sobolev norms 14 2.3. Local analysis 20 2.4. Smoothing operators 24 3. The cohomological equation 37 3.1. Distributional solutions 37 3.2. Invariant distributions and basic currents 43 3.3. Basic cohomology 47 4. Cocycles over the Teichmüller flow 55 4.1. The Kontsevich-Zorich cocycle. 55 4.2. Distributional cocycles 71 4.3. Lyapunov exponents 79 5. Smooth solutions 92 5.1. The general case 92 5.2. Ergodic integrals 96 5.3. The generic case 105 References 117 1. I NTRODUCTION We prove the sharpest results available on the loss of regularity for so- lutions of the cohomological equation for translation flows. For any given translation surface and for the directional flow in almost all directions the Date: February 1, 2008. 1991 Mathematics Subject Classification. 37A20, 37C10, 37C15. Key words and phrases. Invariant distributions, cohomological equation, Teich ¨ muller flow, Kontsevich-Zorich cocycle, distributional cocycles. The author gratefully acknowledges support from National Science Foundation grant DMS-0244463. 1
Transcript
Page 1: Sobolev Regularity

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SOBOLEV REGULARITY OF SOLUTIONSOF THE COHOMOLOGICAL EQUATION

GIOVANNI FORNI †

CONTENTS

1. Introduction 12. Fractional weighted Sobolev spaces 92.1. Weighted Sobolev spaces 92.2. Fractional Sobolev norms 142.3. Local analysis 202.4. Smoothing operators 243. The cohomological equation 373.1. Distributional solutions 373.2. Invariant distributions and basic currents 433.3. Basic cohomology 474. Cocycles over the Teichmüller flow 554.1. The Kontsevich-Zorich cocycle. 554.2. Distributional cocycles 714.3. Lyapunov exponents 795. Smooth solutions 925.1. The general case 925.2. Ergodic integrals 965.3. The generic case 105References 117

1. INTRODUCTION

We prove the sharpest results available on the loss of regularity for so-lutions of the cohomological equation for translation flows. For any giventranslation surfaceand for the directional flow in almost all directions the

Date: February 1, 2008.1991Mathematics Subject Classification.37A20, 37C10, 37C15.Key words and phrases.Invariant distributions, cohomological equation, Teichmuller

flow, Kontsevich-Zorich cocycle, distributional cocycles.† The author gratefully acknowledges support from National Science Foundation grant

DMS-0244463.1

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2 GIOVANNI FORNI †

smallest loss Sobolev regularity available to the Fourier analysis methodsdeveloped in [For97] is essentially3 + ǫ (for any ǫ > 0). We remark thatthis is the best result available for the flow of rational polygonal billiardsin almost all directions. The motivation for improving the estimate on theloss of regularity obtained in [For97] was provided by a question of Marmi,Moussa and Yoccoz [MMY05]. We also remark that their results, on therelated problem of solutions of the cohomological equationfor almost allinterval exchange transformations (IET’s), do not apply torational billiardsfor the well-known reason that IET’s induced by rational billiard flows forma zero measure set in the space of all IET’s.

For almost all translation surfacesin every stratum of the moduli space,we prove the refined, optimal result that the loss of Sobolev regularity forthe directional flow in almost all directions is1 + ǫ (for any ǫ > 0). Inthis case, in fact we prove that for any function of Sobolev order s > 1,the solution and its derivatives up to orderk < s − 1 areL∞ functionson the surface. We also determine precisely the Sobolev orders of the dis-tributional obstructions to the existence of solutions (first constructed in[For97]) in terms of the Kontsevich-Zorich Lyapunov exponents [Zor96],[Kon97]. As a consequence we are able to determine the exact codimen-sion of coboundaries for every Sobolev regularity class of the transfer func-tion. For instance, the codimension of coboundaries with square-integrabletransfer functions (in the space of functions of Sobolev order s > 1) is ex-actly equal to the genus of the surface. For such coboundaries the transferfunction is actually inL∞.

These results implies quite immediately corresponding results for intervalexchange transformations, which improve on the loss of regularity estab-lished in [MMY05]. We should point out that in that paper the authors aremostly concerned withDiophantine conditionson interval exchange trans-formations for which the cohomological equation admits smooth solution,while we have not investigated this question at all. The reason is that thefull measure sets of systems for which our results (as well as[MMY05])are determined by several conditions which always includeOseledec regu-larity (or rather a weakercoherenceproperty) with respect to Kontsevich-Zorich renormalization cocycle. The Oseledec’s theorem isinvoked to en-sure that the set of regular (coherent) IET’s has full measure. Any sub-stantial progress over [MMY05] would have to succeed in characterizingexplicitly a full measure set of regular (coherent) points without relyingon the Oseledec’s theorem. To the best of our knowledge this goal is stillbeyond reach.

There are several motivations for this work. The study of cohomologicalequations is a relevant part of the theory of (smooth) dynamical systemsdirectly connected to basic questions such astriviality of time-changesfor

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SOBOLEV REGULARITY OF SOLUTIONS 3

flows, asymptotic of ergodic averagesand thesmooth conjugacy problemvia linearization and Nash-Moser implicit function theorem. In the hyper-bolic case (for dynamical systems with exponential divergence of nearbyorbits) such there are extensive, deep results on the cohomological equa-tion pioneered in the work of Livsic [Liv71], later developed by several au-thors (see [GK80], [CEG84], [dlLMM86]). The completely different caseof (Diophantine) linear flows on the torus is well-known, since the cohomo-logical equation for such systems is closely related to the linearized equa-tion in the classical KAM theory for Hamiltonian flows. This in an exampleof elliptic dynamics (no divergence of nearby orbits) whichcan be studiedto a great extent by the classical theory of Fourier series. It is characterizedby the ‘small divisors’ appearing in the Fourier coefficients of solutions,which lead to a loss of regularity. It is not difficult to see that the optimalloss of Sobolev regularity for the full measure set of Roth flows is1+ ǫ andthat for such flows any zero average function of Sobolev orders > 1 is anL2 coboundary. It can be proved by our methods (and by the Gottschalk-Hedlund theorem [GH55]) that the transfer function is in fact continuous.We have not been able to locate this result in the literature,however it iswell within reach of the methods of [Her83], Chap. VI, §3. However, onlythe measure zero case of rotation numbers ofconstant typeseems to havebeen explicitly considered there.

For systems with intermediate behavior, that is, for elliptic systems withsingularities or for parabolic systems (characterized by polynomial diver-gence of nearby orbits) much less is known. The author discovered in[For97] that the cohomological equation for generic translation flows (orequivalently for generic IET’s) has finitely smooth solutions for sufficientlysmooth data under finitely many distributional conditions.In other terms,on one hand the problem shares a typical feature of ‘small divisors’ prob-lems, namely the finite loss of regularity of solutions with respect to thedata; on the other hand, a new phenomenon appears: the existence of in-finitely many independent distributional obstructions (ofincreasing order)which are not given by invariant measures. In [For97] only a rough esti-mate for the loss of derivatives is explicitly obtained (≤ 9). Our goal in thispaper is to improve such estimate as much as possible. In joint papers withL. Flaminio the authors have investigated the existence of smooth solutionsof the cohomological equation for horocycle flows (on surfaces constantnegative curvature) [FF03], for generic nilflows on quotients of the Heisen-berg group [FF06] and generic nilflows on general nilmanifolds [FF07]. Inall cases the fundamental features of finite loss derivatives and of the exis-tence of infinitely many independent distributional obstructions have beenestablished (although the structure of the space of invariant distributions is

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4 GIOVANNI FORNI †

significantly different for IET’s, horocycle flows and nilflows). For horo-cycle flows and for Heisenberg nilflows it was possible to estimate that theloss of Sobolev regularity is1 + ǫ (for anyǫ > 0) and to establish the con-jectural relation that the Sobolev order of the distributional obstructions berelated to the Lyapunov exponents of the distribution underthe appropri-ate renormalization dynamics. In this paper we prove analogous results forgeneric translation flows. We should point out that for generic nilflows ongeneral nilmanifolds the loss of regularity and the regularity of the distri-butional obstructions seem to depend on the depth and rank ofthe nilpotentgroup considered, although no lower bound was established in [FF07].

Let q be aholomorphic orientable quadratic differentialon a RiemannsurfaceM of genusg ≥ 1. The horizontal and vertical measured foliations(in the Thurston’s sense) associated to a holomorphic quadratic differen-tial q on M are defined asFq = ℑ(q1/2) = 0 (the horizontal foliation)andFq = ℜ(q1/2) = 0. Such foliations are well-defined even in thecase that there is no globally defined square root of the quadratic differen-tial. The horizontal foliation is endowed with the transverse measure givenby |ℑ(q1/2)|, the vertical foliation is endowed with the transverse measuregiven by|ℜ(q1/2)|. The quadratic differential is called orientable if the hori-zontal and vertical foliations are both orientable. Orientability is equivalentto the condition that the quadratic differential is globally the square of aholomorphic (abelian) differential. The structure induced by an orientableholomorphic quadratic differential (or by a holomorphic abelian differen-tial) can also be described as follows. There is a flat metricRq associatedwith any quadratic differentialq onM . Such a metric has conical singular-ities as the finite setΣq = p ∈ M |q(p) = 0. If q is orientable there existsa (positively oriented) parallel orthonormal frameSq, Tq of the tangentbundleTM |M \ Σq such thatSq is tangent to the horizontal foliationFq

andTq is tangent to the vertical foliationF−q everywhere onM \ Σq. Inother terms,the flat metricRq has trivial holonomy. In another equivalentformulation, any orientable holomorphic quadratic differential determinesa translation structureon M , that is, an equivalence class of atlases withtransition functions given by translation of the euclideanplane (see for in-stance the excellent survey [MT05], §1.8. For a given orientable quadraticdifferentialq on a Riemann surfaceM , we will consider the one-parameterfamily of vector fields onM \ Σq defined as

(1.1) Sθ := cos θ Sq + sin θ Tq , θ ∈ S1.

The vector fieldSθ is a parallel normalized vector field in the direction atangleθ ∈ S1 with the horizontal. We remark that it is not defined as thesingular setΣq of the flat metric. hence the flow it generates is defined(almost everywhere) on the complement of the union of all separatrices

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SOBOLEV REGULARITY OF SOLUTIONS 5

of the orbit foliation (a measure foliation). The singularities of the orbitfoliation are all saddle-like, but the saddles are degenerate if the order ofzero of the quadratic differential at the singularity is strictly greater than2.In fact, since the quadratic differential is supposed to be orientable it haszeroes of even order and the orbit foliations of the vector fields (1.1) hasmstable andm unstable separatrices at any zero of order2m.

Our goal is to investigate the loss of (Sobolev) regularity of solutions of thecohomological equationSθu = f for Lebesgue almost allθ ∈ S1. Theauthor proved in [For97] that if the functionf is sufficiently regular, satis-fies a finite number of independent distributional conditions (which includeconditions on the jets at the singularities) then there exists a finitely smoothsolution (unique up to additive constants). The loss of regularity was esti-mated in that paper to be no more than9 derivatives in the Sobolev sense.If q is any orientable quadratic differential, the regularity of functions on thetranslation surface(M, q) is expressed in terms of a familyHs

q (M)|s ∈ Rof weighted Sobolev spaces. Such spaces were introduced in [For97] forall s ∈ Z as follows. Letωq be the standard (degenerate) volume formon M of the flat metricRq. The spaceH0

q (M) is the spaceL2(M, ωq) ofsquare-integrable functions. Fork ∈ N, the spaceHk

q (M) is the subspaceof functionsf ∈ H0

q (M) such that the weak derivativesSiqT

jq f ∈ H0

q (M)

[and T iqS

jqf ∈ H0

q (M)] for all i + j ≤ k and the spaceH−kq (M) is the

dual Hilbert spaceHkq (M)∗. In §2 of this paper we introduce weighted

Sobolev spaces with arbitrary (real) exponentsby methods of interpolationtheory. Although the Sobolev norms we construct do not form an inter-polation family in the sense of (holomorphic) interpolation theory, they dosatisfy a standard interpolation inequality. The weightedSobolev spacescombine standard Sobolev smoothness conditions onM \ Σq with restric-tions on the jet of the functions at the singular setΣq ⊂ M .As discovered in [For97], for functionsf ∈ C∞

0 (M \ Σq) the space of alldistributional obstructions to the existence of a solutionu ∈ C∞

0 (M \ Σq)of the cohomological equationSθu = f coincides for almost allθ ∈ S1

with the infinite dimensional space of allSθ-invariant distributions:

(1.2) Iq,θ(M \ Σq) := D ∈ D′(M \ Σq) |SθD = 0 in D′(M \ Σq) .

For dataf ∈ Hsq (M) of finite Sobolev differentiability, a complete set of

obstructions is given for almost allθ ∈ S1 by the finite dimensional sub-space of invariant distributions

(1.3) Isq,θ(M) := D ∈ H−s

q (M) |SθD = 0 in H−sq (M) .

The goal of this paper is to proveoptimal estimates on the Sobolev reg-ularity of solutions of the cohomological equationSθu = f and on the

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6 GIOVANNI FORNI †

dimension of the spacesIsq,θ(M) of invariant distributions for alls > 0. In

§5.1, Theorem 5.1, we prove

Theorem A1. Let q be any orientable holomorphic quadratic differential.Let k ∈ N be any integer such thatk ≥ 3 and lets > k and r < k − 3.For almost allθ ∈ S1 (with respect to the Lebesgue measure), there existsa constantCr,s(θ) > 0 such that the following holds. Iff ∈ Hs

q (M) is suchthat D(f) = 0 for all D ∈ Is

q,θ(M), the cohomological equationSθu = fhas a solutionu ∈ Hr

q (M) satisfying the following estimate:

(1.4) |u|r ≤ Cr,s(θ) |f |s .

The dimensions of the spaces of invariant distributions canbe estimated asfollows (see §3.3, Corollary 3.20 and Theorem 3.21):

Theorem A2. Let q be any orientable holomorphic quadratic differential.Letk ∈ N be any integer such thatk ≥ 3 and letk < s ≤ k +1. For almostall θ ∈ S1 (with respect to the Lebesgue measure),

(1.5) 1 + 2(k − 2)(g − 1) ≤ dimIsq,θ(M) ≤ 1 + (2k − 1)(g − 1) .

The proof of the above results is essentially based on the harmonic analysismethods developed in [For97]. We remark that no other methods are knownfor the case of anarbitrary orientable quadratic differential.We prove much sharper results foralmost allorientable quadratic differ-entials. The moduli space of orientable holomorphic quadratic differen-tials q on some Riemann surfaceMq with a given pattern of zeroes, thatis, with zeroes of (even) multiplicitiesκ = (k1, . . . , kσ) at a finite setΣq = p1, . . . , pσ ⊂ Mq is a stratumMκ of the moduli spaceMg ofall holomorphic quadratic differential. LetM(1)

κ ⊂ M(1)g be the subsets

of quadratic differential of total area equal to1. It was proved by H. Ma-sur [Mas82] and W. Veech [Vee86] that each stratumM

(1)κ carries an ab-

solutely continuous probability measureµ(1)κ , invariant under the action of

the Teichmüller geodesic flow, which is ergodic when restricted to eachconnected component ofM(1)

κ (the connected components of strata of ori-entable quadratic differentials were classified in [KZ03]). In fact, there isnatural action of the groupSL(2, R) on the moduli spaceM(1)

g such that theTeichmüller geodesic flow corresponds to the action of the diagonal sub-group ofSL(2, R) and the measureµ(1)

κ is SL(2, R) invariant.In [Kon97] M. Kontsevich introduced a renormalization cocycle for trans-lation flows, inspired to the Rauzy-Veech-Zorich cocycle for interval ex-change transformations. The Kontsevich-Zorich cocycle isa dynamicalsystem on an orbifold vector bundle overM

(1)κ with fiber the first cohomol-

ogyH1(Mq, R) of the Riemann surface carrying the orientable holomorphic

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SOBOLEV REGULARITY OF SOLUTIONS 7

quadratic differentialq ∈ M(1)κ . The action of such a dynamical system is

(by definition of a cocycle) linear on the fibers and projects onto the Teich-müller geodesic flow on the baseM(1)

κ . Since the cocycle is symplectic, forany probability measureµ on a stratumM

(1)κ , theLyapunov spectrumof the

Kontsevich-Zorich cocycle takes the form:

(1.6) λµ1 ≥ · · · ≥ λµ

g ≥ 0 ≥ λµg+1 = −λµ

g ≥ · · · ≥ λµ2g = −λµ

1 .

In addition, it is not difficult to prove thatλµ1 = 1. A probability measure

µ on a stratumM(1)κ , invariant under the Teichmüller geodesic flow, will be

called(a) SO(2, R)-absolutely continuousif it induces absolutely contin-uous measures on every orbit of the circle groupSO(2, R) ⊂ SL(2, R);(b) KZ-hyperbolicif all the Lyapunov exponents in (1.6) are non-zero. Itis immediate that allSL(2, R)-invariant measures areSO(2, R)-absolutelycontinuous. It was first proved in [For02] that the measureµ

(1)κ is KZ-

hyperbolic. A different proof that also reaches the stronger conclusion thatthe exponents (1.6) are all distinct has been given more recently by A. Avilaand M. Viana [AV05] who have thus completed the proof of the Zorich-Kontsevich conjectures [Zor96], [Kon97] on the Lyapunov spectrum of theKontsevich-Zorich cocycle (and its discrete counterparts).

Our sharpest results are proved foralmost allquadratic differentials with re-spect to anySO(2, R)-absolutely continuous, KZ-hyperbolic, Teichmüllerinvariant, probability measure on any stratumM(1)

κ of orientable quadraticdifferentials. The smoothness informations on the solutions is strongerthan just SobolevL2 regularity and it is naturally encoded by the followingspaces. For anyk ∈ N, letBk

q (M) be the space of all functionsu ∈ Hkq (M)

such thatSiqT

jq u = T i

qSjqu ∈ L∞(M) for all pairs of integers(i, j) such that

0 ≤ i + j ≤ k. The spaceBkq (M) is endowed with the norm defined as

follows: for anyu ∈ Bkq (M),

(1.7) |u|k,∞ :=

[∑

i+j≤k

|SiqT

jq u|2∞

]1/2

=

[∑

i+j≤k

|T iqS

jqu|

2∞

]1/2

.

Fors ∈ [k, k +1), letBsq(M) := Bk

q (M)∩Hsq (M) endowed with the norm

defined as follows: for anyu ∈ Bsq(M),

(1.8) |u|s,∞ :=(|u|2k,∞ + |u|2s

)1/2.

In §5.3, Theorem 5.19, we prove the following:

Theorem B1. Letµ be anySO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)

κ of orientable quadratic differen-tials. Let s > 1 and letr < s − 1. For µ-almost allq ∈ M

(1)κ and for

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8 GIOVANNI FORNI †

almost allθ ∈ S1 (with respect to the Lebesgue measure), there exists aconstantCr,s(θ) > 0 such that the following holds. Iff ∈ Hs

q (M) is suchthat D(f) = 0 for all D ∈ Is

q,θ(M), the cohomological equationSθu = fhas a solutionu ∈ Br

q (M) satisfying the following estimate:

(1.9) |u|r,∞ ≤ Cr,s(θ) |f |s .

The regularity of invariant distributions can be preciselydetermined as fol-lows (see §4.3, Corollary 4.33, and §5.3, Theorem 5.19). Foranyq ∈ M

(1)κ

and any distributionD ∈ D′(M \ Σq), theweighted Sobolev orderis thenumber

(1.10) OHq (D) = infs ∈ R

+|D ∈ H−sq (M) .

Let Iq,θ(M) := ∪Isq,θ(M)|s ≥ 0 denote the space of allSθ-invariant dis-

tribution of finite Sobolev order and letIq,θ(M) ⊂ Iq,θ(M) be the subspaceof invariant distributions vanishing on constant functions. It follows imme-diately by the definitions thatIq,θ(M) = C ⊕ Iq,θ(M).

Theorem B2. Letµ be anySO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)

κ of orientable quadratic differen-tials. For µ-almost allq ∈ M

(1)κ and for almost allθ ∈ S1 (with respect to

the Lebesgue measure), the spaceIq,θ(M) has a basisDi,j(θ) such that

(1.11) OHq (Di,j(θ)) = λµ

i − (j +1) , i ∈ 2, . . . , 2g−1, j ∈ N∪0 .

In addition, the basisDi,j(θ) can be generated from the finite dimensionalsubsystemD2, . . . , D2g−1 by the following differential relations:

(1.12) Di,j(θ) = T jθ Di,0(θ) , i ∈ 2, . . . , 2g − 1, j ∈ N ∪ 0 .

The above Theorems B1 and B2 are proved by methods based on renor-malization, which were inspired by the work of Marmi-Moussa-Yoccoz[MMY05]. However, our approach differs from theirs since weexplicitlyassume that the Lyapunov exponents of the Kontsevich-Zorich cocycle areall non-zero. The main idea of the argument, as in [MMY05], isto proveuniform estimates for ergodic integrals of weakly differentiable functions,then apply a version of Gottshalk-Hedlund theorem. The asymptotics of er-godic averages of functions inH1(M) was studied by the author in [For02],where the Kontsevich-Zorich conjectures on the deviation of ergodic aver-ages for smooth functions (formulated in [Kon97]) were proved. The ap-proach of [For02] is based on the analysis of distributionalcocycles over theTeichmüller flow which extend the Kontsevich-Zorich cocycles. The esti-mates proved in [For02] are (barely) not strong enough to yield the requireduniform boundedness of ergodic averages under the appropriate distribu-tional conditions. In §4.2 of this paper we have recalled thedefinition of

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SOBOLEV REGULARITY OF SOLUTIONS 9

distributional cocycles, and we have strengthened the estimates proved in[For02] under the slightly stronger (and correct) assumption that the func-tions considered belong toHs

q (M) for somes > 1.Another important techical issue that separates Theorems B1 and B2 fromthe less precise Theorems A1 and A2 is related to interpolation theory inthe presence of distributional obstructions. In the general case, we have notbeen able to overcome the related difficulties, hence the lack of precisionof Theorem A1 and A2 for intermediate Sobolev regularity. Inthegenericcase of Theorems B1 and B2 we have been able to prove a remarkable linearindependence property of invariant distributions which makes interpolationpossible in the construction of solutions of the cohomological equation.

We introduce the following definition (see Definition 5.11).A finite systemD1, . . . , DJ ⊂ H−σ

q (M) of finite order distributions is calledσ-regular(with respect to the familyHs

q (M) of weighted Sobolev spaces) if forany τ ∈ (0, 1] there exists a dual systemu1(τ), . . . , uJ(τ) ⊂ Hσ

q (M)(that is, the identitiesDi(uj(τ)) = δij hold for all i, j ∈ 1, . . . , J and allτ ∈ (0, 1]) such that the following estimates hold. For all0 ≤ r ≤ σ and allǫ > 0, there exists a constantCσ

r (ǫ) > 0 such that, for alli, j ∈ 1, . . . , J,

(1.13) |uj(τ)|r ≤ Cσr (ǫ) τOH(Dj)−r−ǫ .

A finite systemD1, . . . , DJ ⊂ H−sq (M) of finite order distributions will

be calledregular if it is σ-regular for anyσ ≥ s. A finite dimensionalsubspaceI ⊂ H−s

q (M) of finite order distributions will be calledσ-regular[regular] if it admits aσ-regular [regular] basis.We have proved that the spaces of distributional obstructions for the coho-mological equation are regular in the above sense (see Theorem 5.18).

Theorem C. Letµ be anySO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)

κ of orientable quadratic differen-tials. For µ-almost allq ∈ M

(1)κ , for almost allθ ∈ S1 and for all s > 0,

the spaceIsq,θ(M) ⊂ H−s

q (M) of Sθ-invariant distributions is regular.

2. FRACTIONAL WEIGHTED SOBOLEV SPACES

In [For97] we have introduced a natural scale of weighted Sobolev spaceswith integer exponent associated with any orientable holomorphic quadraticdifferentialq on a Riemann surfaceM (of genusg ≥ 2). In this section weextend the definition of weighted Sobolev spaces to arbitrary (real) expo-nents by methods of interpolation theory.

2.1. Weighted Sobolev spaces.Let Σq := p1, . . . , pσ ⊂ M be theset of zeros of the holomorphic quadratic differentialq, of even orders(k1, . . . , kσ) respectively withk1 + · · · + kσ = 4g − 4. Let Rq := |q|1/2 be

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10 GIOVANNI FORNI †

the flat metric with cone singularities atΣq induced by the quadratic differ-entialq onM . With respect to a holomorphic local coordinatez = x + iy,the quadratic differentialq has the formq = φ(z)dz2, whereφ is a locallydefined holomorphic function, and, consequently,

(2.1) Rq = |φ(z)|1/2(dx2 + dy2)1/2 , ωq = |φ(z)| dx ∧ dy .

The metricRq is flat, it is degenerate at the finite setΣq of zeroes ofq and,if q is orientable, it has trivial holonomy, henceq induces a structure oftranslation surfaceonM .

The weightedL2 space is the standard spaceL2q(M) := L2(M, ωq) with

respect to the area elementωq of the metricRq. Hence the weightedL2

norm | · |0 are induced by the hermitian product〈·, ·〉q defined as follows:for all functionsu,v ∈ L2

q(M),

(2.2) 〈u, v〉q :=

M

u v ωq .

Let Fq be thehorizontal foliation, F−q be thevertical foliationfor the holo-morphic quadratic differentialq onM . The foliationsFq andF−q are mea-sured foliations (in the Thurston’s sense):Fq is the foliation given (locally)by the equationℑ(q1/2) = 0 endowed with the invariant transverse measure|ℑ(q1/2)|, F−q is the foliation given (locally) by the equationℜ(q1/2) = 0endowed with the invariant transverse measure|ℜ(q1/2)|. If the quadraticdifferentialq is orientable, since the metricRq is flat with trivial holonomy,there exist commuting vector fieldsSq andTq onM \ Σq such that

(1) The frameSq, Tq is a parallel orthonormal frame with respect tothe metricRq for the restriction of the tangent bundleTM to thecomplementM \ Σq of the set of cone points;

(2) the vector fieldSq is tangent to the horizontal foliationFq, the vectorfield Tq is tangent to the vertical foliationF−q onM \ Σq [For97].

In the following we will often drop the dependence of the vector fieldsSq,Tq on the quadratic differential in order to simplify the notations. We have:

(1) LSωq = LT ωq = 0 onM \Σq , that is, the area formωq is invariantwith respect to the flows generated byS andT ;

(2) ıSωq = ℜ(q1/2) andıT ωq = ℑ(q1/2), hence the1-formsηS := ıSωq,ηT := −ıT ωq are smooth and closed onM andωq = ηT ∧ ηS.

It follows from the area-preserving property(1) that the vector fieldS, Tare anti-symmetric as densely defined operators onL2

q(M), that is, for allfunctionsu, v ∈ C∞

0 (M \ Σq), (see [For97],(2.5)),

(2.3) 〈Su, v〉q = −〈u, Sv〉q , respectively 〈Tu, v〉q = −〈u, Tv〉q .

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SOBOLEV REGULARITY OF SOLUTIONS 11

In fact, by Nelson’s criterion [Nel59], Lemma 3.10, the anti-symmetric op-eratorsS, T areessentially skew-adjointon the Hilbert spaceL2

q(M).

The weighted Sobolev norms| · |k, with integer exponentk > 0, are theeuclidean norms, introduced in [For97], induced by the hermitian productdefined as follows: for all functionsu, v ∈ L2

q(M),

(2.4) 〈u, v〉k :=1

2

i+j≤k

〈SiT ju, SiT jv〉q + 〈T iSju, T iSjv〉q .

Theweighted Sobolev normswith integer exponent−k < 0 are defined tobe the dual norms.

Theweighted Sobolev spaceHkq (M), with integer exponentk ∈ Z, is the

Hilbert space obtained as the completion with respect to thenorm | · |k ofthe maximalcommon invariant domain

(2.5) H∞q (M) :=

i,j∈N

D(SiT j) ∩ D(T iSj) .

of the closuresS, T of the essentially skew-adjoint operatorsS, T onL2

q(M). The weighted Sobolev spaceH−kq (M) is isomorphic to the dual

space of the Hilbert spaceHkq (M), for all k ∈ Z.

Since the vector fieldsS, T commute (infinitesimally) onM \ Σq, the fol-lowing weak commutation identity holds onM .

Lemma 2.1. ([For97], Lemma 3.1) For all functionsu,v ∈ H1q (M),

(2.6) 〈Su, Tv〉q = 〈Tu, Sv〉q .

By the anti-symmetry property (2.3) and the commutativity property (2.6),the frameS, T yields an essentially skew-adjoint action of the Lie algebraR2 on the Hilbert spaceL2

q(M) with common domainH1q (M). If Σq 6= ∅,

the (flat) Riemannian manifold(M \ Σq, Rq) is not complete, hence itsLaplacian∆q is not essentially self-adjoint onC∞

0 (M\Σq). By a theorem ofNelson [Nel59], §9, this is equivalent to the non-integrability of the actionof R

2 as a Lie algebra (to an action ofR2 as a Lie group).

Following [For97], the Fourier analysis on the flat surfaceMq will be basedon a canonical self-adjoint extension∆F

q of the Laplacian∆q, called theFriedrichs extension, which is uniquely determined by theDirichlet hermit-ian formQ : H1

q (M)×H1q (M) → C. We recall that, for allu, v ∈ H1

q (M),

(2.7) Q(u, v) := 〈Su, Sv〉q + 〈Tu, Tv〉q .

Theorem 2.2. ([For97], Th. 2.3) The hermitian formQ on L2q(M) has the

following spectral properties:

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12 GIOVANNI FORNI †

(1) Q is positive semi-definite and the setEV(Q) of its eigenvalues is adiscrete subset of[0, +∞);

(2) Each eigenvalue has finite multiplicity, in particular0 ∈ EV(Q) issimple and the kernel ofQ consists only of constant functions;

(3) The spaceL2q(M) splits as the orthogonal sum of the eigenspaces.

In addition, all eigenfunctions areC∞ (real analytic) onM .

The Weyl asymptoticsholds for the eigenvalue spectrum of the Dirichletform . For anyΛ > 0, letNq(Λ) := cardλ ∈ EV(Q) / λ ≤ Λ, where eacheigenvalueλ ∈ EV(Q) is counted according to its multiplicity.

Theorem 2.3. ([For97], Th. 2.5) There exists a constantC > 0 such that

(2.8) limΛ→+∞

Nq(Λ)

Λ= vol(M, Rq) .

Let ∂±q := Sq ± ı Tq be theCauchy-Riemann operatorsinduced by the holo-

morphic orientable quadratic differentialq on M , introduced in [For97],§3. LetM±

q ⊂ L2q(M) be the subspaces of meromorphic, respectively anti-

meromorphic functions (with poles atΣq). By the Riemann-Roch theorem,the subspacesM±

q have the same complex dimension equal to the genusg ≥ 1 of the Riemann surfaceM . In addition,M+

q ∩ M−q = C, hence

(2.9) Hq :=(M+

q

)⊥⊕

(M−

q

)⊥= u ∈ L2

q(M) |

M

u ωq = 0 .

Let H1q := Hq ∩ H1

q (M). By Theorem 2.2, the restriction of the hermitianform to H1

q is positive definite, hence it induces a norm. By the Poincaréinequality (see [For97], Lemma 2.2 or [For02], Lemma 6.9), the Hilbertspace(H1

q , Q) is isomorphic to the Hilbert space(H1q , 〈·, ·〉1).

Proposition 2.4. ([For97], Prop. 3.2) The Cauchy-Riemann operators∂±q

are closable operators on the common domainC∞0 (M \Σq) ⊂ L2

q(M) andtheir closures (denote by the same symbols) have the following properties:

(1) the domainsD(∂±q ) = H1

q (M) and the kernelsN(∂±q ) = C;

(2) the rangesR±q := Ran(∂±

q ) =(M∓

q

)⊥are closed inL2

q(M);(3) the operators∂±

q : (H1q , Q) → (R±, 〈·, ·〉q) are isometric.

Let E = en |n ∈ N ⊂ H1q (M) ∩ C∞(M) be an orthonormal basis of the

Hilbert spaceL2q(M) of eigenfunctions of the Dirichlet form (2.7) and let

λ : N → R+ ∪ 0 be the corresponding sequence of eigenvalues:

(2.10) λn := Q(en, en) , for eachn ∈ N .

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SOBOLEV REGULARITY OF SOLUTIONS 13

TheFriedrichs weighted Sobolev norm‖ · ‖s of orders ∈ R+ is the norminduced by the hermitian product defined as follows: for allu, v ∈ L2

q(M),

(2.11) (u, v)s :=∑

n∈N

(1 + λn)s 〈u, en〉q 〈en, v〉q .

The inner products (2.4) and (2.11) induceequivalentSobolev norms onthe weighted Sobolev spaceHk

q (M), for all k ∈ Z+. In fact, the followingresult, a sharp version of Lemma 4.2 of [For97], holds:

Lemma 2.5. For eachk ∈ Z+ there exists a constantCk > 1 such that,for any orientable holomorphic quadratic differentialq on M and for allfunctionsu ∈ Hk

q (M),

(2.12) C−1k |u|k ≤ ‖u‖k ≤ Ck |u|k .

Proof. By Proposition 2.4,(3), and Lemma 2.1, for allu ∈ Hk+1q (M),

k ∈ N, the following identity holds (see(4.4) in [For97]):

(2.13) |u|2k+1 = |u|20 +∑

i+j≤k

Q(SiT ju, SiT ju) =

= |u|20 +∑

i+j≤k

〈SiT j∂±q u, SiT j∂±

q u〉q = |u|20 + |∂±q u|2k .

Hence in particularu ∈ Hk+1q (M), k ∈ N, implies∂±

q u ∈ Hkq (M). If

k ≥ 1, a second application of the identity (2.13) yields

(2.14) |u|2k+1 = |u|21 + |∂∓q ∂±

q u|2k−1 .

The statement then follows by induction onk ∈ N. Fork = 0 it is immedi-ate and fork = 1, by the identity (2.13),

(2.15) |u|21 = |u|20 + Q(u, u) =∑

n∈N

(1 + λn) |〈u, en〉q|2 .

For k > 1, by induction hypothesis we can assume that the norms| · |k−1

and‖ · ‖k−1 are equivalent, that is, there exists a constantCk−1 > 1 suchthat, for allu ∈ Hk−1

q (M),

(2.16) C−1k−1 |u|k−1 ≤ ‖u‖k−1 ≤ Ck−1 |u|k−1 .

Since by (2.3) for allu, v ∈ H1q (M), 〈∂±

q u, v〉q = −〈u, ∂∓q v〉q, the adjoint

operator(∂±q )∗ = −∂∓

q onH1q (M). By Proposition 2.4,(3), we have

(2.17) ‖∂∓q ∂±

q u‖2k−1 =

n∈N

(1 + λn)k−1 |〈∂±q u, ∂±

q en〉q|2 =

=∑

n∈N

(1 + λn)k−1 |Q(u, en)|2 =

n∈N

(1 + λn)k−1λ2n |〈u, en〉q|

2 .

Page 14: Sobolev Regularity

14 GIOVANNI FORNI †

There exists a constantCk+1 > 1 such that, for allλ ≥ 0,

(2.18) C−2k+1 (1 + λ)k+1 ≤ 1 + λ + C−2

k−1 λ2(1 + λ)k−1

≤ 1 + λ + C2k−1λ

2 (1 + λ)k−1 ≤ C2k+1 (1 + λ)k+1 .

By (2.14), (2.15), (2.16), (2.17) and (2.18), the estimate

(2.19) C−1k+1 |u|k+1 ≤ ‖u‖k+1 ≤ Ck+1 |u|k+1

follows, thereby completing the induction step.

2.2. Fractional Sobolev norms. Let q be any orientable quadratic differ-ential onM . For alls ≥ 0, let

(2.20) Hsq (M) := u ∈ L2

q(M) /∑

n∈N

(1 + λn)s|〈u, en〉q|2 < +∞,

endowed with the hermitian product given by (2.11) and, for any s > 0, letH−s

q (M) be the dual space of the Hilbert spaceHsq (M). The spacesHs

q (M)will be called theFriedrichs (fractional) weighted Sobolev spaces.

Let H1 ⊂ H2 be Hilbert spaces such thatH1 embeds continuously intoH2

with dense image. For allθ ∈ [0, 1], let [H1, H2]θ be the (holomorphic)interpolation space ofH1 ⊂ H2 in the sense of Lions-Magenes [LM68],Chap. 1, endowed with the canonical interpolation norm. By the resultsof [LM68], Chap. 1, §§2, 5, 6 and 14, we have the following:

Lemma 2.6.The Friedrichs weighted Sobolev spaces form an interpolationfamilyHs

q (M)s∈R of Hilbert spaces: for allr, s ∈ R with r < s,

(2.21) H(1−θ)r+θsq (M) ≡ [Hr

q (M), Hsq (M)]θ .

The familyHsq (M)s∈R of fractional weighted Sobolev spaceswill be de-

fined as follows. Let[s] ∈ N denote theinteger partands ∈ [0, 1) thefractional partof any real numbers ≥ 0.

Definition 2.7. (1) The fractional weighted Sobolev norm| · |s of or-ders ≥ 0 is the euclidean norm induced by the hermitian productdefined as follows: for all functionsu, v ∈ H∞

q (M),

(2.22) 〈u, v〉s :=1

2

i+j≤[s]

(SiT ju, SiT jv)s + (T iSju, T iSjv)s .

(2) The fractional weighted Sobolev norm| · |−s of order−s < 0 isdefined as the dual norm of the weighted Sobolev norm| · |s.

(3) Thefractional weighted Sobolev spaceHsq (M) of orders ∈ R is de-

fined as the completion with respect to the norm| · |s of the maximalcommon invariant domainH∞

q (M).

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SOBOLEV REGULARITY OF SOLUTIONS 15

It can be proved that the weighted Sobolev spaceH−sq (M) is isomorphic to

the dual space of the Hilbert spaceHsq (M), for all s ∈ R.

Definition 2.8. For any distributionD on M \ Σq, the weighted Sobolevorder OH

q (D) and theFriedrichs weighted Sobolev orderOHq (D) are the

real numbers defined as follows:

(2.23)OH

q (D) := infs ∈ R |D ∈ H−sq (M) ;

OHq (D) := infs ∈ R |D ∈ H−s

q (M) .

The definition of the fractional weighted Sobolev norms is motivated by thefollowing basic result.

Lemma 2.9. For all s ≥ 0, the restrictions of the Cauchy-Riemann oper-ators∂±

q : H1q (M) → L2

q(M) to the subspacesHs+1q (M) ⊂ H1

q (M) yieldbounded operators

∂±s : Hs+1

q (M) → Hsq (M)

(which do not extend to operatorsHs+1q (M) → Hs

q (M) unlessM is thetorus). On the other hand, the Laplace operator

(2.24) ∆q = ∂+q ∂−

q = ∂−q ∂+

q : H2q (M) → L2

q(M)

yields a bounded operator∆s : Hs+2q (M) → Hs

q (M), defined as the re-striction of the Friedrichs extension∆F

q : H2q (M) → L2

q(M).

Proof. The restrictions∂±s : Hs+1

q (M) → Hsq (M) of the Cauchy-Riemann

operators are well-defined and bounded for alls > 0 by definition of theSobolev spacesHs

q (M).

The operators∂±s : Hs+1

q (M) → Hsq (M) do not extend to bounded oper-

atorsHs+1q (M) → Hs

q (M) unlessMq is a flat torus. In fact, every finitecombinationf of eigenfunctions of the Dirichlet form belongs toHs

q (M),for all s ∈ R, but ∂±

q f 6∈ H1q (M) in all cases because of the presence of

obstructions in the Taylor expansion of eigenfunctions at the singular setΣq 6= ∅. In fact, if ∂+

q en ∈ H1q (M) (or ∂−

q en ∈ H1q (M)) for all n ∈ N,

then∂+q en ∈ H1

q (M) and∂−q en ∈ H1

q (M), for all n ∈ N, since the eigen-functionsen can be chosen real. It follows thaten ∈ H2

q (M), for all n ∈ N,henceH2

q (M) = H2q (M) is the domain of the Friedrichs extension∆F

q ofthe Laplacian∆q of the metricRq. Thus the Laplacian∆q is self-adjoint onthe domainH2

q (M), hence the metricRq has no singularities andM is thetorus. In fact, the spaceH2

q (M) is the domain of the closure of the Lapla-cian∆q on the common invariant domainH∞

q (M). If H2q (M) = H2

q (M),then∆q is essentially self-adjoint onH∞

q (M) and by [Nel59], Th. 5 or Cor.9.1, the action of the commutative Lie algebra spanned byS, T integratesto a Lie group action. Hence, the singularity setΣq = ∅ andM is the torus.

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16 GIOVANNI FORNI †

Finally, the Friedrichs extension∆Fq , defined onH2

q (M), has a boundedrestriction∆s : Hs+2

q (M) → Hsq (M), for all s ≥ 0. In fact, we have

(2.25) ∆Fq u =

n∈N

λn 〈u, en〉q en , for all u ∈ H2q (M) ,

hence∆Fq u ∈ Hs

q (M) if u ∈ Hs+2q (M), by definition of the Friedrichs

weighted Sobolev spacesHsq (M) (in terms of eigenfunction expansions for

the Dirichlet form).

Lemma 2.10.The fractional weighted Sobolev norms satisfy the followinginterpolation inequalities. For any0 ≤ r < s there exists a constantCr,s >0 such that, for anyθ ∈ [0, 1] and any functionu ∈ Hs

q (M),

(2.26) |u|(1−θ)r+θs ≤ Cr,s |u|1−θr |u|θs .

Proof. The argument will be carried out in three steps:(1) the open interval(r, s) does not contain integers;(2) the open interval(r, s) contains a singleinteger;(3) the general case.

In case(1) there existsk ∈ N such thatk ≤ r < s ≤ k + 1. The in-terpolation inequality follows from the definition (2.22) of the euclideanproduct which induces the fractional Sobolev norms, from the interpola-tion inequality for Friedrichs weigthed Sobolev norms (which are by def-inition interpolation norms) and from the Hölder inequality. In fact, since0 ≤ r − k ≤ s − k ≤ 1 andθ ∈ (0, 1), the fractional part

(2.27) (1 − θ)r + θs = (1 − θ)(r − k) + θ(s − k) ,

hence, by the interpolation inequality for Friedrichs norms (see for instance[LM68], Chap. 1, §2.5), for alli + j ≤ k the following estimates hold:

(2.28)|SiT ju|(1−θ)r+θs ≤ |SiT ju|1−θ

r−k |SiT ju|θs−k ;

|T iSju|(1−θ)r+θs ≤ |T iSju|1−θr−k |T

iSju|θs−k .

By the definition (2.22) of the Sobolev norms, the interpolation inequality(2.26) follows from (2.28) by Hölder inequality.

In case(2), there existsk ∈ N such thatk − 1 ≤ r < k < s ≤ k + 1. Weclaim that, for anyu ∈ Hs

q (M),

(2.29) |u|k ≤ Cr,s |u|s−ks−rr |u|

k−rs−rs .

Let us prove that step(1) and the above claim (2.29) imply step(2). Letσ = (1 − θ)r + θs. We will consider only the case whenσ ∈ (r, k) sincethe case whenσ ∈ (k, s) is similar. By step(1) we have the inequality

(2.30) |u|σ ≤ Cr |u|k−σk−rr |u|

σ−rk−r

k .

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SOBOLEV REGULARITY OF SOLUTIONS 17

By the claim (2.29) it then follows that

(2.31) |u|σ ≤ C(1)r,s |u|

k−σk−r

+ σ−rk−r

s−ks−r

r |u|σ−rk−r

k−rs−r

s .

It is immediate to verify that

k − σ

k − r+

σ − r

k − r

s − k

s − r=

s − σ

s − r.

Let us turn to the proof of the claim (2.29). Since−1 ≤ r−k < 0 < s−k ≤1 and the weighted Sobolev norm| · |s coincides with the Friedrichs norm‖ · ‖s for anys ∈ [−1, 1], by the interpolation inequality for the FriedrichsSobolev norms, the following estimates hold: for alli, j ∈ N with i+j ≤ kand for any functionu ∈ Hs

q (M),

(2.32)|SiT ju|0 ≤ C(2)

r,s |SiT ju|s−ks−r

r−k |SiT ju|k−rs−r

s−k ;

|T iSju|0 ≤ C(2)r,s |T iSju|

s−ks−r

r−k |T iSju|k−rs−r

s−k .

Since the operatorsS, T : H1q (M) → L2

q(M) are well-defined, boundedand have bounded linear extensionsL2

q(M) → H−1q (M), by the fundamen-

tal theorem of interpolation (see for instance [LM68], Chap. 1, §5.1), theoperatorsS, T : Hs

q (M) → Hs−1q (M) are well-defined and bounded for

anys ∈ [0, 1]. It follows that there exists a constantCr > 0 such that

(2.33)

i+j≤k

|SiT ju|2r−k ≤ (C ′r)

2∑

i+j≤k−1

|SiT ju|2r−(k−1) ;

i+j≤k

|T iSju|2r−k ≤ (C ′r)

2∑

i+j≤k−1

|T iSju|2r−(k−1)

The claim (2.29) then follows by Hölder inequality from (2.32) and (2.33).

In general, letk1 < k2 be positive integers such that

k1 − 1 ≤ r < k1 < k2 < s ≤ k2 + 1 .

By Lemma 2.5, since the Friedrichs norms are interpolation norms, we havethat there exists a constantCk1,k2 > 0 such that, for allk ∈ N ∩ [k1, k2],

(2.34) |u|k ≤ Ck1,k2 |u|k2−k

k2−k1k1

|u|k−k1

k2−k1k2

.

By step(2) there exists a constantC(3)r,s > 0 such that

(2.35)|u|k1 ≤ C(3)

r,s |u|1

k1+1−r

r |u|k1−r

k1+1−r

k1+1 ;

|u|k2 ≤ C(3)r,s |u|

s−k2s−k2+1

k2−1 |u|1

s−k2+1s .

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18 GIOVANNI FORNI †

The estimates in (2.35) imply, by bootstrap-type estimatesbased on (2.34)for k = k1 + 1 andk = k2 − 1, that there exists a constantC

(4)r,s > 0 such

that

(2.36)|u|k1 ≤ C(4)

r,s |u|k2−k1k2−r

r |u|k1−r

k2−r

k2;

|u|k2 ≤ C(4)r,s |u|

s−k2s−k1k1

|u|k2−k1s−k1

s .

By (2.36) and again by bootstrap, there exists a constantC(5)r,s > 0 such that

(2.37)|u|k1 ≤ C(5)

r,s |u|s−k1s−r

r |u|k1−r

s−rs ;

|u|k2 ≤ C(5)r,s |u|

s−k2s−r

r |u|k2−r

s−rs ,

Let σ ∈ (r, s). We have proved the interpolation inequality for the subcasesσ = k1 andσ = k2. Let us prove that the general case can be reduced tothese subcases. Ifσ ∈ (r, k1), by step(1) there existsC ′′

r > 0 such that

(2.38) |u|σ ≤ C ′′r |u|

k1−σ

k1−r

r |u|σ−rk1−r

k1.

The interpolation inequality in this case follows immediately from (2.37)and (2.38). Ifσ ∈ (k2, s), the argument is similar. Ifσ ∈ (k1, k2), then bystep(1), there existsC[σ] > 0 such that

(2.39) |u|σ ≤ C[σ]|u|1−σ[σ] |u|

σ[σ]+1 .

The interpolation inequality then follows from (2.34), (2.37) and (2.39).

Let Hs(M), s ∈ R, denote a family of standard Sobolev spaces on thecompact manifoldM (defined with respect to a Riemannian metric). Thecomparison lemma below clarifies to some extent the relations between thedifferent scales of fractional Sobolev spaces.

Lemma 2.11. The following continuous embedding and isomorphisms ofBanach spaces hold:

(1) Hs(M) ⊂ Hsq (M) ≡ Hs

q (M) , for 0 ≤ s < 1;(2) Hs(M) ≡ Hs

q (M) ≡ Hsq (M) , for s = 1;

(3) Hsq (M) ⊂ Hs

q (M) ⊂ Hs(M) , for s > 1.

For s ∈ [0, 1], the spaceHs(M) is dense inHsq (M) and, fors > 1, the

closure ofHsq (M) in Hs

q (M) or Hs(M) has finite codimension.

Proof. By definitionH0(M) = L2(M) andH0q (M) = H0

q (M) = L2q(M).

Since the area form induced by any quadratic differential issmooth onM ,which is a compact surface, it follows thatL2(M) ⊂ L2

q(M). The em-beddingH1

q (M) ⊂ H1q (M) follows by Lemma 2.5 and the embedding

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SOBOLEV REGULARITY OF SOLUTIONS 19

H1q (M) ⊂ H1

q (M) holds since the eigenfunctions of the Dirichlet formare inH1

q (M). The isomorphismH1(M) ≡ H1q (M) is proved in [For02],

§6.2. Hence(2) is proved and(1) follows by interpolation.Let s > 1. If [s] = 2k is even, there exists a constantAk > 0 such that, forall functionsu ∈ H∞

q (M), we have

(2.40) ‖u‖2s = ‖(I − ∆F

q )ku‖2s ≤ A2

k

i+j≤2k

‖SiT ju‖2s = C2

k |u|s .

If [s] = 2k+1 is odd, we argue as follows. The Cauchy-Riemann operators∂±

q : H1q (M) → L2

q(M) are bounded and extend by duality to boundedoperators∂±

0 : L2q(M) → H−1

q (M). Hence, by the fundamental theoremof interpolation (see [LM68], Chap. 1, §5.1), for allσ ∈ [0, 1] the Cauchy-Riemann operators have bounded restrictions

(2.41) ∂±σ : Hσ

q (M) → Hσ−1q (M) .

It follows that there exists a constantBk > 0 such that, for all functionsu ∈ H∞

q (M), we have(2.42)

‖u‖2s = ‖(I − ∆F

q )k+1u‖2s−1 ≤ B2

k

i+j≤2k+1

‖SiT ju‖2s = C2

k |u|s .

Thus the embeddingsHsq (M) ⊂ Hs

q (M) hold for alls > 1.

It was proved in [For02], §6.2, thatHkq (M) ⊂ Hk(M), for all k ∈ Z+. We

prove below the stronger statement thatHsq (M) ⊂ Hs(M), for all s ∈ R+.

Let R be a smooth Riemannian metric onM conformally equivalent to thedegenerate metricRq and letHs

R(M), s ≥ 0, denote the Sobolev spacesof the Riemannian manifold(M, R) which are defined as the domains ofthe powers of the essentially self-adjoint Laplacian∆R of the metric. SinceM is compact, the Sobolev spacesHs

R(M) ≡ Hs(M) are independent, astopological vector spaces, of the choice of the Riemannian metricR, for alls ∈ R. We claim thatH2k

q (M) ⊂ H2kR (M), for all k ∈ Z+. In fact, there

exists a smooth non-negative real-valued functionW onM (vanishing onlyat Σq) such thatW∆F

q ⊂ ∆R. Let W be the unique function such thatthe area forms of the metrics are related by the identityωq = WωR. Ifu ∈ H2

q (M), then∆Fq u ∈ L2

q(M), so that

(2.43) ∆Ru = W∆Fq u ∈ L2(M, ωR) .

Let us assume thatH2k−2q (M) ⊂ H2k−2

R (M) and letu ∈ H2kq (M). We have

(2.44) ∆kRu = ∆k−1

R W∆Fq u = [∆k−1

R , W ]∆Fq u + W∆k−1

R ∆Fq u .

Since the commutator[∆k−1R , W ] and∆k−1

R are differential operators of or-der2k − 2 on M and∆F

q u ∈ H2k−2R (M) by the induction hypothesis, the

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20 GIOVANNI FORNI †

function∆kRu ∈ L2(M, ωR). The claim is therefore proved. It follows by

interpolation thatHsq (M) ⊂ Hs

R(M), for all s ≥ 1. Thus(3) is proved.

For s ∈ [0, 1], the spaceC∞0 (M \ Σq) ⊂ Hs

q (M) is dense inHsq (M).

For s > 1, the subsetC∞(M) ∩ Hsq (M) is dense inHs

q (M), since theeigenfunctions of the Dirichlet form (hence all finite linear combinations)belong toC∞(M) and the spaceC∞(M) is dense inHs(M). Finally, thesubspaceC∞(M) ∩ Hk

q (M) ⊂ C∞(M) can be described, for anyk ∈ N,as the kernel of a finite number of distributions of finite order supported onthe finite setΣq (see [For02],(7.9)), hence for anyk > s the closure ofHk

q (M) ⊂ Hsq (M) in Hs

q (M) or in Hs(M) has finite codimension.

2.3. Local analysis. For eachp ∈ M and allk ∈ Z+, let Hkq (p), Hk

q (p),andHk(p) the spaces of germs of fuctions atp which belong toHk

q (M),Hk

q (M) andHk(M), endowed with the respective direct limit topologies.More precisely, a germ of functionf atp belongs to the spaceHk

q (p), Hkq (p)

or Hk(p) iff it can be realized by a functionF on M which belongs to thespaceHk

q (M), Hkq (M) or Hk(M) respectively and the open sets inHk

q (p),Hk

q (p) or Hk(p) are defined as the images of open sets inHkq (M), Hk

q (M)

or Hk(M) under the natural mapsHkq (M) → Hk

q (p), Hkq (M) → Hk

q (p) orHk(M) → Hk(p).By Lemma 2.11 we have the inclusions

(2.45)

H0(p) ⊂ H0q (p) ⊂ H0

q (p) ;

H1q (p) = H1

q (p) = H1(p) ;

Hkq (p) ⊂ Hk

q (p) ⊂ Hk(p) .

If p 6∈ Σq, since there is an open neighbourhoodDp of p in M isomorphicto a flat disk and the operator∆F

q is elliptic of order2 on Dp (isomorphicto the flat Laplacian), all the inclusions in (2.45) are identities. We willdescribe precisely the inclusionsHk

q (p) ⊂ Hkq (p) ⊂ Hk(p) for k > 1.

Let p ∈ Σq be a zero of (even) order2m of the (orientable) quadratic dif-ferentialq on M . There exists a unique canonical holomorphic coordinatez : Dp → C, defined on a neighbourhoodDp of p ∈ M , such thatz(p) = 0andq(z) = z2mdz2. With respect to the canonical coordinate the Cauchy-Riemann operators∂±

q can be written in the following form:

(2.46) ∂+q =

2

zm

∂zand ∂−

q =2

zm

∂z.

Page 21: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 21

Let C∞(p) be the space of germs atp ∈ M of smooth complex-valuedfunctions onM and for anyu ∈ C∞(p) let

(2.47) u(z, z) =∑

i,j∈N

aij(u, p)zizj .

be its (formal) Taylor series atp (with respect to the canonical coordinate).

Lemma 2.12. Let p ∈ Σq be a zero of (even) order2m of the (orientable)quadratic differentialq onM . For anyk ∈ N, a germ

u ∈ C∞(p) ∩ Hkq (p) ⇔ aij(u, p) = 0 , for all i + j ≤ (k − 1)(m + 1) ,

except all pairs(i, j) for which one of the following conditions holds:

(2.48)

(1) i ∈ N · (m + 1) , j ∈ N · (m + 1) ;

(2) i ∈ N · (m + 1) , j 6∈ N · (m + 1) and i < j ;

(3) i 6∈ N · (m + 1) , j ∈ N · (m + 1) and i > j .

Proof. Let u ∈ C∞(p). For anyn ∈ N, there is a local Taylor expansion

u(z, z) =∑

i+j≤n

aij(u, p)zizj + Run(z, z)

where the remainderRun is a smooth function vanishing at ordern at p.

A straightforward calculation (based on formulas (2.46) yields that anysmooth functionR vanishing atp at ordern belongs to the spaceHk

q (p) ⊂

Hkq (p) if n > (k − 1)m. It follows thatu ∈ Hk(p) iff its Taylor polynomial

of any ordern > (k − 1)m does. The argument can therefore be reduced tothe case of polynomials.

It follows from formulas (2.46) that, for allℓ ∈ N and all(i, j) ∈ N × N,there exists a complex constantcm,ℓ

ij such that

(2.49) ∆ℓq(z

izj) = cm,ℓij zi−ℓ(m+1) zj−ℓ(m+1) .

The area form of the quadratic differentialq can be written as

(2.50) ωq = |z|2mdx ∧ dy

with respect to the canonical coordinatez := x+ıy. Hence, straightforwardcomputations in polar coordinates yield that, ifcm,ℓ

ij 6= 0,

(2.51)∆ℓ

q(zizj) ∈ H0

q (p) ⇔ i + j − 2ℓ(m + 1) > −(m + 1) ;

∆ℓq(z

izj) ∈ H1q (p) ⇔ i + j − 2ℓ(m + 1) > 0 .

If cm,ℓij = 0, then eitheri ∈ N · (m+1) andi < ℓ(m+1) or j ∈ N · (m+1)

andj < ℓ(m + 1). It follows that, if i, j 6∈ N · (m + 1), then

(2.52) zizj ∈ Hkq (p) ⇔ i + j − (k − 1)(m + 1) > 0 .

Page 22: Sobolev Regularity

22 GIOVANNI FORNI †

If i ∈ N · (m + 1), i = h(m + 1), andj 6∈ N · (m + 1), then conditions(2.51) apply for allℓ ≤ h, hence (2.52) holds ifk ≤ 2h, while if k > 2h,

(2.53) zizj ∈ Hkq (p) ⇔ ∆h

q (zizj) ∈ H1

q (p) ⇔ j > i .

Similarly, if j ∈ N · (m+1), j = h(m+1), andi 6∈ N · (m+1), then (2.52)holds ifk ≤ 2h, while if k > 2h,

(2.54) zizj ∈ Hkq (p) ⇔ ∆h

q (zizj) ∈ H1

q (p) ⇔ i > j .

It follows immediately from (2.52), (2.53) and (2.54) that the conditionslisted in the statement of the lemma are sufficient. The necessity followsfrom the following argument. For anyr1 < r2, letD(r1, r2) ⊂ M be the an-nulus (centered atp) defined by the inequalitiesr1 < |z| < r2. The systemof Laurent monomialszizj |i, j ∈ Z is orthogonal inHk(Dr1,r2). In fact, acomputation in polar coordinates shows that, for all(i, j) 6= (i′, j′) ∈ Z×Z,

(2.55)∫

D(r1,r2)

∆ℓq(z

izj)∆ℓ′

q (zi′zj′) ωq = 0 ,

hence

(2.56) ‖∑

i+j≤n

aijzizj‖2

k =∑

i+j≤n

|aij|2 ‖zizj‖2

k .

It follows that only Laurent monomialszizj ∈ Hk(p) can appear in theTaylor expansion of a functionf ∈ C∞(p) ∩ Hk(p).

Lemma 2.13.Letp ∈ Σq be a zero of order2m of the quadratic differentialq onM . For anyk ∈ N, a germ

u ∈ C∞(p) ∩ Hkq (p) ⇔ aij(u, p) = 0 , for all i + j ≤ (k − 1)(m + 1) ,

except all pairs(i, j) ∈ N · (m + 1) × N · (m + 1).

Proof. The proof is similar to that of Lemma 2.12 above. In fact, formulas(2.49) are replaced by the following formulas. For allα ∈ N and alli ∈ N,there exists a complex constantcm,α

i such that

(2.57) (∂+q )α(∂−

q )βzizj = cm,αi cm,β

j zi−α(m+1) zj−β(m+1) .

As in the proof of Lemma 2.12, it follows by a straightforwardcomputationin polar coordinates that, ifcm,α

i cm,βj 6= 0,

(2.58) (∂+)α(∂−)βzizj ∈ H0q (p) ⇔ i+ j− (α+β)(m+1) > −(m+1) .

Since there existsα ∈ N such thatcm,αi = 0 iff i ∈ N · (m + 1), either

i, j ∈ N · (m + 1), in which casezizj ∈ Hkq (p), or there exists(α, β)

such thatα + β = k andcm,αi cm,β

j 6= 0, in which casezizj ∈ Hkq (p) iff

i + j > (k − 1)(m + 1).

Page 23: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 23

Let p ∈ Σq be a zero of order2mp of the orientable quadratic differentialq.Let Tp ⊂ N × N be the set of(i, j) such that

(2.59)i ∈ N · (mp + 1) , j 6∈ N · (mp + 1) and i < j or

i 6∈ N · (mp + 1) , j ∈ N · (mp + 1) and i > j,

For any(i, j) ∈ Tp, let δijp be the linear functional (distribution) onC∞(M)

defined as follows. Letu(z, z) =∑

aij(u, p)zizj denote the Taylor expan-sion of u ∈ C∞(M) at p ∈ Σq with respect to the canonical coordinatez : Dp → C for the differentialq atp. Let

(2.60) δijp (u) := aij(u, p) .

It is clear from the definition thatδijp = δji

p for all (i, j) ∈ Tp. A calculationshows that, for anyh ∈ N\N·(mp+1) we have the following representationin terms of the Cauchy principal value: for anyu ∈ C∞(p),

(2.61)δh0p (u) = −

1

4πhPV

M

∆qu

zhωq ;

δ0hp (u) = −

1

4πhPV

M

∆qu

zhωq .

(The above formulas can be derived one from the other by conjugation). Inaddition, for anyℓ ∈ N, by formulas (2.49) there exist complex constantscmp,ℓ0,h 6= 0 andc

mp,ℓh,0 6= 0 such that the following identities hold in the sense

of distributions:

(2.62)cmp,ℓ0,h δℓ(mp+1),ℓ(mp+1)+h

p = ∆ℓq

(δ0,hp

);

cmp,ℓh,0 δℓ(mp+1)+h,ℓ(mp+1)

p = ∆ℓq

(δh,0p

).

Let Tkp ⊂ Tp be the subset of(i, j) such thati + j ≤ (k − 1)(mp + 1).

Lemma 2.14. For each(i, j) ∈ Tkp , the functionalδij

p has a unique (non-trivial) continuous extension to the spaceHk

q (p) and the following holds:

(2.63) Hkq (p) = u ∈ Hk

q (p) | δijp (u) = 0 for all (i, j) ∈ Tk

p .

Proof. The functionsz−h and z−h ∈ L2q(D) for all 1 ≤ h ≤ mp and the

operator∆Fq : H2

q (p) → L2q(p) is bounded. Hence the linear functionals

δ0hp andδh0

p are continuous onH2q (p) for all 1 ≤ h ≤ mp. Similarly, the

distributions PV(z−h) and PV(z−h) ∈ H−1(Dp) for all mp < h < 2(mp +1) and the operator∆F

q : H3q (p) → H1(p) is bounded. Hence the linear

functionalsδ0hp andδh0

p are continuous onH3q (p) for all mp < h < 2(mp +

1). Since the spaceHkq (p) is equal to the closure inHk

q (p) of the subspaceC∞(p)∩Hk

q (p), the statement fork = 2, k = 3 follows from Lemmas 2.12and 2.13.

Page 24: Sobolev Regularity

24 GIOVANNI FORNI †

We complete the argument by induction onk ∈ N. The Friedrichs extensiondefines bouned operators∆F

q : Hk+1q (p) → Hk−1

q (p) and its dual∆Fq :

H−k+1q (p) → H−k−1

q (p). By the induction hypothesis, since all functionalsin Tk−1

p extend (uniquely) to bounded functionals inH−k+1q (p), it follows

that all functionals in∆Fq (Tk−1

p ) extend (uniquely) to bounded functionalsin H−k−1

q (p) and the following holds. For anyu ∈ Hk+1q (p),

(2.64) ∆Fq u ∈ Hk−1

q (p) ⇔ δijp (u) = 0 for all δij

p ∈ ∆Fq (Tk−1

p ) .

Let Ek+1q ⊂ Hk+1

q (p) the closed finite-codimensional subspace defined as

(2.65) Ek+1q := u ∈ Hk+1

q (p) | δijp (u) = 0 for all δij

p ∈ ∆Fq (Tk−1

p ) .

By formulas (2.62), any distributionδp ∈ Tk+1p \ ∆F

q (Tk−1p ) is of the form

δp = δ0hp or δp = δh0

p with 1 ≤ h ≤ k(mp + 1). By formulas (2.61), suchdistributions have a (unique) continuous extension to the subspaceEk+1

q ⊂

Hk+1q (p). In fact, the distributions PV(z−h) and PV(z−h) ∈ H−k+1

q (Dp),for all 1 ≤ h < k(mp + 1), and∆F

q (f) ∈ Hk−1q (D) for all f ∈ Ek+1

q .Hence all distributions in the setTk+1

p have a continuous extension to thespaceHk+1

q (p) and the characterization (2.63) of the subspaceHk+1q (p) ⊂

Hk+1q (p) follows by Lemmas 2.12 and 2.13.

Let Dkq be the set of all continuous extensions to the spaceHk

q (M) of thefunctionalsδij

p for all p ∈ Σq and all(i, j) ∈ Tkp . Let

(2.66) Dq :=⋃

k∈N

Dkq .

Theorem 2.15.The (closed) kernel of the systemDkq on Hk

q (M) coincideswith the subspaceHk

q (M), that is,

Hkq (M) = u ∈ Hk

q (M) | δ(u) = 0 , for all δ ∈ Dkq .

2.4. Smoothing operators. We will establish below finer results on theSobolev regularity of the distributions inDq. The key step will be to con-struct smoothing operators for the scale of Sobolev spacesHk

q (p) | k ∈ N.For anyp ∈ Σq, let z : Dp → C be a canonical coordinate for the (ori-entable) quadratic differentialq such thatp ∈ Dp andz(p) = 0. For any(i, j) ∈ N × N, let Z ij

p ∈ C∞(M) be a function such that

(2.67) Z ijp (z) ≡ zizj on Dp .

Lemma 2.16. Let p ∈ Σq be a zero of order2m of the quadratic differen-tial q on M . There exists a one-parameter familyKp(τ) | τ ∈ (0, 1] ofbounded operatorsKp(τ) : C∞(M) → H∞

q (M) such that the following

Page 25: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 25

estimates hold. For eachk ∈ N, there exists a constantCk > 0 such that,for any(i, j) ∈ N × N and for all τ ∈ (0, 1]:

(2.68)

|Kp(τ)(Z ijp ) − Z ij

p |k ≤ Ck τ 1+ i+jm+1

−k , for k < 1 +i + j

m + 1;

|Kp(τ)(Z ijp )|k ≤ Ck | log τ |1/2 , for k = 1 +

i + j

m + 1;

|Kp(τ)(Z ijp )|k ≤ Ck τ−[k−(1+ i+j

m+1)] , for k > 1 +

i + j

m + 1.

Proof. Let z : Dp → C be a canonical coordinate atp ∈ Σq defined on anopen neighbourhoodDp ⊂ M such thatDp∩Σq = p. There existsr1 > 0such thatD(r1) ⊂⊂ z(Dp), whereD(r1) is the euclidean disk centered atthe origin of radiusr1 > 0 . Let 0 < r2 < r1 and letφ : C → R beany non-negative smooth function identically zero on the closure ofD(r2)and identically equal to1 outsideD(r1). Let D′

p ⊂⊂ Dp be any relativelycompact neighbourhood ofp in Dp such thatD(r1) ⊂⊂ z(D′

p). For anyτ ∈ (0, 1], let us define, for allF ∈ C∞(M),

(2.69) Kp(τ) (F ) (x) :=

φ(τ− 1

m+1 z(x)) F (z(x)) , for x ∈ Dp ;

F (x) , for x 6∈ D′p .

Clearly, the definition (2.69) is well-posed for allτ ∈ (0, 1] and the func-tions Kp(τ) (F ) ∈ H∞

q (M), since the rescaled functionsφτ : C → R

defined asφτ (z) := φ(τ− 1

m+1 z) , for z ∈ C ,

are smooth and have support away from the origin. Since the functionsφτ

are bounded uniformly with respect to the parameterτ ∈ (0, 1], for anyfunctionF ∈ C∞(M) ⊂ L2

q(M), by the dominated convergence theorem,

(2.70) |Kp(τ) (F ) − F |0 → 0 , asτ → 0+ .

Let us denote for convenience, for any(a, b) ∈ N × N,

φabτ (z) := [(∂+

q )a(∂−q )bφ](τ− 1

m+1 z) , for z ∈ C .

The functionsφabτ are smooth and bounded onC, uniformly with respect to

τ > 0. For any(a, b) such that(a, b) 6= (0, 0), the functionφabτ has compact

support contained in the euclidean annulus centered at the origin of innerradiusr1 τ

1m+1 and outer radiusr2τ

1m+1 . For a = b = 0, φab

τ = φτ hassupport outside the euclidean disk of radiusr1 τ

1m+1 and is identically equal

to 1 outside the disk of radiusr2 τ1

m+1 .

For any(i, j) ∈ N × N, let Kijτ := Kp(τ)

(Z ij

p

)∈ H∞

q (M). A calculationshows that the following formulas hold. For allm, i andα ∈ N and all

Page 26: Sobolev Regularity

26 GIOVANNI FORNI †

a ∈ N such that0 ≤ a ≤ α, let

(2.71) Cm,iα,a =

(αa

) α−a∏

ℓ=1

[i − ℓ(m + 1)] .

For any(α, β) ∈ N × N, the derivative(∂+q )α(∂−

q )β (Kijτ ) (z) is given on

Dp \ p by the sum

(2.72)α∑

a=0

β∑

b=0

Cm,iα,a Cm,j

β,b φabτ (z)τ−(a+b) zi−(α−a)(m+1) zj−(β−b)(m+1) .

If i + j > (α + β − 1)(m + 1), since the functions

zi−(α−a)(m+1) zj−(β−b)(m+1) ∈ L2q(Dp) ,

for all 0 ≤ a ≤ α and 0 ≤ b ≤ β, andφabτ is bounded, by change of

variables we obtain that for(a, b) 6= (0, 0) there exists a constantCa,b > 0such that

(2.73) |φabτ (z)τ−(a+b) zi−(α−a)(m+1) zj−(β−b)(m+1)|0 ≤ Ca,b τ 1+ i+j

m+1−(α+β) ,

and that similarly, fora = b = 0, there exists a constantC0 > 0

(2.74) |(φτ (z) − 1) zi−α(m+1)zj−β(m+1)|0 ≤ C0 τ 1+ i+jm+1

−(α+β) .

If i + j < [(α − a) + (β − b) − 1](m + 1), there exists a constantC ′a,b > 0

such that

(2.75) |φabτ (z)τ−(a+b) zi−(α−a)(m+1) zj−(β−b)(m+1)|0 ≤ C ′

a,b τ 1+ i+jm+1

−(α+β) ,

since the functionφabτ is bounded and supported outside the euclidean disk

of radiusr1 τ1

m+1 centered at the origin.

If i + j = [(α − a) + (β − b) − 1](m + 1), a similar calculation yields(2.76)

|φabτ (z)τ−(a+b) zi−(α−a)(m+1) zj−(β−b)(m+1)|0

≤ C ′a,b τ 1+ i+j

m+1−(α+β) | log τ |

12 .

By formula (2.72) for the Cauchy-Riemann iterated derivatives, the requiredestimates (2.68) follow immediately from estimates (2.73), (2.74), (2.75)and (2.76).

We derive below estimates for the local smoothing familyKp(τ) con-structed in Lemma 2.16 with respect to the fractional weighted Sobolevnorms. Letp ∈ Σq be any zero of order2mp of the quadratic differential

Page 27: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 27

q on M . For each pair(i, j) ∈ N × N, let eijp : R+ → [0, 1] denote the

function defined as follows:

(2.77) eijp (s) :=

s , if [s] = 1 + i+jmp+1

;

1 − s , if [s] = i+jmp+1

;

0 , otherwise.

Theorem 2.17.The familyKp(τ) | τ ∈ (0, 1] of local smoothing oper-ators Kp(τ) : C∞(M) → H∞

q (M), defined in (2.69), has the followingproperties. For eachs ∈ R+, there exists a constantCs > 0 such that, forany pair(i, j) ∈ N × N and all τ ∈ (0, 1]:

|Kp(τ)(Z ijp ) − Z ij

p |s ≤ Csτ1+ i+j

mp+1−s

| log τ |eijp (s)

2 , for s < 1 +i + j

mp + 1;

|Kp(τ)(Z ijp )|s ≤ Cs| log τ | | log τ |

eijp (s)

2 , for s = 1 +i + j

mp + 1;

|Kp(τ)(Z ijp )|s ≤ Csτ

−[s−(1+ i+jmp+1

)]| log τ |

eijp (s)

2 , for s > 1 +i + j

mp + 1.

Proof. For anyk ∈ N, the functionZ ijp ∈ Hk

q (M), if k < 1+(i+j)/(m+1),sincezizj ∈ Hk

q (p). By Lemma 2.16, there exists a constantCk > 0 suchthat, for allτ ∈ (0, 1],

(2.78)|Kij

p (τ) − Kijp (τ/2)|k ≤ |(Kij

p (τ) − Z ijp ) − (Kij

p (τ/2) − Z ijp )|k ;

≤ |Kijp (τ) − Z ij

p |k + |Kijp (τ/2) − Z ij

p |k ≤ Ck τ 1+ i+jm+1

−k .

If k ≥ 1 + (i + j)/(m + 1),

(2.79) |Kijp (τ) − Kij

p (τ/2)|k ≤ |Kijp (τ)|k + |Kij

p (τ/2)|k

hence, by Lemma 2.16,

(2.80)|Kij

p (τ) − Kijp (τ/2)|k ≤ 2Ck | log τ |1/2, if k = 1 +

i + j

m + 1;

|Kijp (τ) − Kij

p (τ/2)|k ≤ 2Ck τ 1+ i+jm+1

−k, if k > 1 +i + j

m + 1.

As a consequence, by the interpolation inequality (Lemma 2.10), for everys ∈ R+ there exists a constantCs > 0 such that, for allτ ∈ (0, 1],

|Kijp (τ) − Kij

p (τ/2)|s ≤ Cs τ 1+ i+jm+1

−s , if [s], [s] + 1 6= 1 +i + j

m + 1;

|Kijp (τ) − Kij

p (τ/2)|s ≤ Cs τ 1+ i+jm+1

−s| log τ |1−s

2 , if [s] = 1 +i + j

m + 1;

|Kijp (τ) − Kij

p (τ/2)|s ≤ Cs τ 1+ i+jm+1

−s| log τ |s2 , if [s] =

i + j

m + 1.

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28 GIOVANNI FORNI †

If s < 1 + (i + j)/(m + 1) and[s] 6= (i + j)/(m + 1), for all n ∈ N,

(2.81) |Kijp (τ/2n) − Kij

p (τ/2n+1)|s ≤ Cs τ 1+ i+jm+1

−s2−n(1+ i+jm+1

−s) ,

hence, for any fixedτ ∈ (0, 1], the sequenceKijp (τ/2n)n∈N is Cauchy,

and therefore convergent, in the Hilbert spaceHsq (M). By Lemma 2.16

Kijp (τ/2n)n∈N converges toZ ij

p in H[s]q (M). SinceH

[s]q (M) ⊂ Hs

q (M),by uniqueness of the limitZ ij

p ∈ Hsq (M) andKij

p (τ/2n)n∈N converges toZ ij

p in Hsq (M). The estimate (2.81) also implies that

(2.82)|Kij

p (τ) − Z ijp |s ≤

n∈N

|Kijp (τ/2n) − Kij

p (τ/2n+1)|s ≤ C ′s τ 1+ i+j

m+1−s .

If s < 1+(i+ j)/(m+1) and[s] = (i+ j)/(m+1), by a similar argumentwe again get thatZ ij

p ∈ Hsq (M) and

(2.83) |Kijp (τ) − Z ij

p |s ≤ C ′s τ 1+ i+j

m+1−s | log τ |

s2 .

If s ≥ 1 + (i + j)/(m + 1) and[s], [s] + 1 6= 1 + (i + j)/(m + 1) we argueas follows. For eachτ ≤ 1/2, we have

(2.84) |Kijp (2τ) − Kij

p (τ)|s ≤ Cs (2τ)1+ i+jm+1

−s ,

hence ifτ ≤ 2−n, for all 0 ≤ k < n,

(2.85) |Kijp (2k+1τ) − Kij

p (2kτ)|s ≤ Cs 2(k+1)(1+ i+jm+1

−s)τ 1+ i+jm+1

−s .

It follows that, there exists a constantC ′s > 0 such that

(2.86) |Kijp (2nτ) − Kij

p (τ)|s ≤ C ′s 2n(1+ i+j

m+1−s)τ 1+ i+j

m+1−s .

For everyτ ∈ (0, 1], let n(τ) be the maximumn ∈ N such that2nτ ≤ 1.By this definition it follows that1/2 < 2n(τ)τ ≤ 1. Since

sup1/2≤τ≤1

|Kijp (τ)|s ≤ sup

1/2≤τ≤1

|Kijp (τ)|[s]+1 < +∞ ,

it follows that, there exists a constantC ′′s > 0 such that

(2.87)|Kij

p (τ)|s ≤ C ′′s τ 1+ i+j

m+1−s , if s > 1 +

i + j

m + 1;

|Kijp (τ)|s ≤ C ′′

s | log τ | , if s = 1 +i + j

m + 1.

By a similar argument, fors > 1 + (i + j)/(m + 1) we have

(2.88)|Kij

p (τ)|s ≤ C ′′s τ 1+ i+j

m+1−s| log τ |

1−s2 , if [s] = 1 +

i + j

m + 1;

|Kijp (τ)|s ≤ C ′′

s τ 1+ i+jm+1

−s| log τ |s2 , if [s] =

i + j

m + 1,

Page 29: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 29

while for s = 1 + (i + j)/(m + 1) we have

(2.89)|Kij

p (τ)|s ≤ C ′′s | log τ | | log τ |

1−s2 , if [s] = 1 +

i + j

m + 1;

|Kijp (τ)|s ≤ C ′′

s | log τ | | log τ |s2 , if [s] =

i + j

m + 1.

Theorem 2.17 implies in particular the following smoothness results.

Corollary 2.18. Let z : Dp → C be a canonical coordinate for an ori-entable quadratic differentialq at a zerop ∈ Σq of order 2m. For each(i, j) ∈ N × N, the function

(2.90) zizj ∈ Hsq (p) , for all s < 1 +

i + j

mp + 1.

Corollary 2.19. Letp ∈ Σq be a zero of order2mp and let(i, j) ∈ Tp. Thedistributionδij

p has the following regularity properties:

(2.91)

δijp ∈ H−s

q (p) for s > 1 +i + j

mp + 1,

δijp 6∈ H−s

q (p) for s < 1 +i + j

mp + 1.

Proof. By the formulas (2.61) and by Lemma 2.14, for anyh ∈ N \ N ·

(mp + 1) and for anyℓ ∈ N, there exist constantsCmp,ℓh0 6= 0 andC

mp,ℓ0h 6= 0

such that the following identities hold in the dual Hilbert spaceH−kq (p) for

any integerk ≥ 1 + h/(mp + 1):

(2.92)C

mp,ℓh0 δh0

p = ∆ℓ+1q

(zℓ(mp+1)−hzℓ(mp+1)

),

Cmp,ℓ0h δ0h

p = ∆ℓ+1q

(zℓ(mp+1)zℓ(mp+1)−h

).

By Corollary 2.18, ifℓ(mp + 1) − h > 0, for all s < 2ℓ + 1 − h/(mp + 1),

zℓ(mp+1)−hzℓ(mp+1) and zℓ(mp+1)zℓ(mp+1)−h ∈ Hsq (p) ⊂ Hs

q (p) .

Henceδh0p , δ0h

p ∈ H−sq (p) for all s > 1 + h/(mp + 1). By formulas (2.62) it

then follows thatδijp ∈ H−s

q (p) for all s > 1 + (i + j)/(mp + 1) as claimed.

Let (i, j) ∈ Tp ands < 1+(i+j)/(mp+1). By Corollary 2.18, the functionzizj ∈ Hs

q (p). Since by definitionH∞q (p) is dense inHs

q (p) for anys > 0,the functionalδij

p ≡ 0 onH∞q (p) andδij

p (zizj) = 1, it follows thatδijp does

not extend to a bounded functional onHsq (p).

Page 30: Sobolev Regularity

30 GIOVANNI FORNI †

Let p ∈ Σq be a zero of (even) order2mp the quadratic differentialq onM .For everys ∈ R+, let Ts

p ⊂ Tp be the subset defined as

(2.93) Tsp := (i, j) ∈ Tp | i + j < (s − 1)(mp + 1) .

Let Dsq ⊂ H−s

q (M) be the set of distributions defined as follows:

(2.94) Dsq := δij

p | p ∈ Σq and (i, j) ∈ Tsp .

Corollary 2.20. The closure of the subspaceHsq (M) in Hs

q (M) is a subsetof the (closed) kernel of the systemDs

q on Hsq (M), that is,

(2.95) Hsq (M) ⊂ u ∈ Hs

q (M) | δ(u) = 0 , for all δ ∈ Dsq

The reverse inclusion holds if the following sufficient condition is satisfied:

(2.96) s 6∈ 1 + (i + j)/(mp + 1) | p ∈ Σq and (i, j) ∈ Tp .

Proof. SinceH∞q (M) is dense inHs

q (M) ⊂ Hsq (M),

H∞q (M) ⊂ u ∈ Hs

q (M) | δ(u) = 0 , for all δ ∈ Dsq

andDsq ⊂ H−s

q (M), it follows that

Hsq (M) ⊂ u ∈ Hs

q (M) | δ(u) = 0 , for all δ ∈ Dsq .

Conversely, if condition (2.96) is satisfied, by Corollary 2.18 the subspace

u ∈ C∞(M) | δ(u) = 0 , for all δ ∈ Dsq ⊂ Hs

q (M) .

SinceC∞(M) ∩ Hsq (M) is dense inHs

q (M), the result follows.

The regularity result proved in Corollary 2.18 extends to a certain subset ofall pairs(i, j) ∈ Z × Z if the functionszizj are interpreted as distributionsin the sense of the Cauchy principal value:

(2.97) PV(zizj

)(v) := PV

M

zizj v ωq , for all v ∈ C∞(p) .

The most general regularity result for the distributions (2.97) is based on thefollowing generalization of Corollary 2.18 to include logarithmic factors.

Lemma 2.21.Let z : Dp → C be a canonical coordinate for an orientableholomorphic quadratic differentialq at a zerop ∈ Σq of order 2m. Foreach(i, j, h) ∈ N × N × N, the function

zizj logh |z| ∈ Hsq (p) , for all s < 1 +

i + j

mp + 1.

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SOBOLEV REGULARITY OF SOLUTIONS 31

Proof. Simple calculations show thatlog |z| ∈ L2q(M) and that by formulas

(2.46) the following identities hold onDp \ p:

(2.98) ∂+ log |z| =1

zm+1and ∂− log |z| =

1

zm+1.

It follows that, for each(i, j, h) ∈ N×N×N and each(α, β) ∈ N×N, thereexists a finite sequence of non-zero constantsC1, . . . , Ch, which depend on(i, j, h, α, β, m), such that the following identity holds onDp \ p:

(2.99) (∂+)α(∂−)β(zizj logh |z|

)= zi−α(m+1)zj−β(m+1)

h∑

ℓ=0

Cℓ logℓ |z| .

For all (i, j, h) ∈ N×N×N, letLijhp ∈ C∞(M \ p) be any function such

thatLijhp (z) = zizj logh |z| for all z ∈ Dp. By (2.99), the function

Lijhp ∈ Hk

q (M) , if k ∈ N and k < 1 +i + j

mp + 1

Let Kp(τ) | τ ∈ (0, 1] be the family of local smoothing operators definedby formulas (2.69). By computations similar to those carried out in theproof of Lemma 2.16, based on formulas (2.99), it is possibleto prove thatfor eachk ∈ N, there exists a constantCk > 0 such that for allτ ∈ (0, 1]:

|Kp(τ)(Lijhp ) − Lijh

p |k ≤ Ck τ 1+ i+jm+1

−k | log τ |h, for k < 1 +i + j

m + 1;

|Kp(τ)(Lijhp )|k ≤ Ck | log τ |1/2| log τ |h, for k = 1 +

i + j

m + 1;

|Kp(τ)(Lijhp )|k ≤ Ck τ−[k−(1+ i+j

m+1)] | log τ |h, for k > 1 +

i + j

m + 1.

Reasoning as in the proof of Theorem 2.17, we can derive similar estimatesfor fractional Sobolev norms. For each(i, j) ∈ N×N, let eij

p : R+ → [0, 1]be the function defined in formula (2.77). By the interpolation Lemma 2.10,for anys < 1 + (i + j)/(m + 1) there exists a constantCs > 0 such that

|Kp(τ)(Lijhp ) − Kp(τ/2)(Lijh

p )|s ≤ Cs τ 1+ i+jm+1

−s | log τ |h+eijp (s)

2 .

It follows that the sequenceKp(τ/2n)(Lijhp )n∈N is Cauchy and therefore

converges inHsq (M). By uniqueness of the limit

Lijhp ∈ Hs

q (M) , for all s < 1 +i + j

mp + 1.

Page 32: Sobolev Regularity

32 GIOVANNI FORNI †

In addition, the following estimates hold. For eachs ∈ R+ there exists aconstantC ′

s > 0 such that for allτ ∈ (0, 1]:

|Kp(τ)(Lijhp )|s ≤ C ′

s | log τ | | log τ |h+eijp (s)

2 , for s = 1 +i + j

mp + 1;

|Kp(τ)(Lijhp )|s ≤ C ′

s τ−[s−(1+ i+j

mp+1)]| log τ |h+

eijp (s)

2 , for s > 1 +i + j

mp + 1.

Theorem 2.22.Let z : Dp → C be a canonical coordinate for an ori-entable holomorphic quadratic differentialq at a zerop ∈ Σq of order2mp. For each(i, j) ∈ Z × Z such that(1) i − j 6∈ Z · (mp + 1) or (2)i > −(mp + 1) or (3) j > −(mp + 1), the distribution

(2.100) PV(zizj logh |z|

)∈ Hs

q (p) , for all s < 1 +i + j

mp + 1.

Proof. For all (i, j) ∈ Z × Z such thati − j 6∈ Z · (mp + 1) and for anyh ∈ Z the following formulas hold for all functionsv ∈ H∞

q (p):(2.101)

(a) PV∫

M

∂+(zizj logh |z|)v ωq = −PV∫

M

zizj logh |z| ∂+v ωq ;

(b) PV∫

M

∂−(zizj logh |z|)v ωq = −PV∫

M

zizj logh |z| ∂−v ωq .

Formulas (2.101) also hold in case(a) if i > −(mp +1), j ∈ Z, and in case(b) if j > −(mp + 1), i ∈ Z, for all germsv ∈ C∞(p).

By taking into account the formulas (2.46) for the Cauchy-Riemann opera-tors with respect to a canonical coordinate, it follows fromformulas (2.101)by induction onh ∈ N that if

PV(zizj logh |z|) ∈ Hsq (p) , for all h ∈ N ,

then, if i − j 6∈ Z · (mp + 1) or i > −(mp + 1) andj ∈ Z,

PV(zizj−(mp+1) logh |z|) ∈ Hs−1q (p) , for all h ∈ N ,

and, if i − j 6∈ Z · (mp + 1) or j > −(mp + 1) andi ∈ Z,

PV(zi−(mp+1)zj logh |z|) ∈ Hs−1q (p) , for all h ∈ N .

Thus, the statement of the theorem can be derived from Corollary 2.18, byan induction argument based on formulas (2.101).

Corollary 2.23. Let z : Dp → C be a canonical coordinate for an ori-entable holomorphic quadratic differentialq at a zerop ∈ Σq of order2mp.

Page 33: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 33

If (i, j) 6∈ Z · (mp + 1) × Z · (mp + 1), the distribution

(2.102) PV(zizj logh |z|

)6∈ Hs

q (p) , for s > 1 +i + j

mp + 1,

and, if i − j ∈ Z · (mp + 1) and bothi ≤ −(m + 1) and j ≤ −(m + 1),

(2.103) PV(zizj logh |z|

)6∈ H−∞

q (p) .

Proof. We argue by contradiction. Assume there exists(i, j) ∈ Z × Z suchthat (i, j) 6∈ Z · (mp + 1) × Z · (mp + 1) and PV(zizj) ∈ Hr

q (p) forsomer > 1 + (i + j)/(mp + 1). By taking Cauchy-Riemann derivativesif necessary, we can assume thati ≤ 0 andj ≤ 0. By Theorem 2.22, thedistribution

PV(z−i−(m+1)z−j−(m+1)

)∈ Hs

q (p) , for all s < −1 −i + j

mp + 1.

It follows that, for any positive smooth functionφ ∈ C∞0 (Dp) identically

equal to1 on a diskD′p ⊂⊂ Dp centered atp ∈ Σq, the principal value

PV∫

M

φ(z) zizj z−i−(m+1)z−j−(m+1) ωq

is finite, which can be proved to be false by a simple computation in (ge-odesic) polar coordinates. This contradiction proves the first part of thestatement.

If i− j ∈ Z · (mp +1) and bothi ≤ −(m+1) andj ≤ −(m+1), we argueas follows. It is not restrictive to assume thati ≥ j, hence the functionzi−j ∈ H∞

q (M). However, by a computation in polar coordinates, sincei ≤ −(mp + 1),

PV∫

M

φ(z) zizj log |z| zi−j ωq = +∞ .

It follows that PV(zizj logh |z|

)6∈ H−∞

q (p), hence the second part of thestatement is also proved.

We conclude this section with a fundamental smoothing theorem for the1-parameter family of weighted Sobolev spaces.

Theorem 2.24.For eachk ∈ N, there exists a familySk(τ) | τ ∈ (0, 1]of bounded operatorsSk(τ) : L2

q(M) → Hkq (M) such that the following

estimates hold. For anys, r ∈ [0, k] and for anyǫ > 0, there exists aconstantCk

r,s(ǫ) > 0 such that, for allu ∈ Hsq (M) and for all τ ∈ (0, 1]:

(2.104)|Sk(τ)(u) − u|r ≤ Ck

r,s(ǫ)‖u‖s τ s−r−ǫ , if s > r ;

|Sk(τ)(u)|r ≤ Ckr,s(ǫ)‖u‖s τ s−r−ǫ , if s ≤ r .

Page 34: Sobolev Regularity

34 GIOVANNI FORNI †

Proof. For eachp ∈ Σq, let z : Dp → C be a canonical coordinate definedon a diskDp (centered atp) such thatDp ∩Σq = p. For each(i, j) ∈ Tp,let Z ij

p ∈ C∞(M) be a (fixed) smooth extension, as in (2.67), of the locallydefined functionzizj ∈ C∞(p). Let P k be the linear operator defined asfollows:

(2.105) P k(f) := f −∑

p∈Σq

(i,j)∈Tkp

δijp (f) Z ij

p , for all f ∈ Hkq (M) .

The operatorPk : Hkq (M) → Hk

q (M) is well-defined and bounded. It iswell-defined by Lemma 2.14 and Theorem 2.15. It is bounded since, for allp ∈ Σq and all(i, j) ∈ Tp, the functionsZ ij

p ∈ H∞q (M) by Lemma 2.12

and, for all(i, j) ∈ Tℓp, the distributionsδij

p ∈ H−kq (M) by Corollary 2.19.

In fact, the condition(i, j) ∈ Tℓp impliesi + j < (k − 1)(mp + 1).

For eachp ∈ Σq, let Kp(τ) | τ ∈ (0, 1] be the family of local smoothingoperators constructed in Lemma 2.16. LetSk

τ |τ ∈ (0, 1] be the one-parameter family of bounded linear operatorsSk

τ : Hkq (M) → Hk

q (M)

defined as follows. For allf ∈ Hkq (M), we let

(2.106) Skτ (f) := P k(f) +

p∈Σq

(i,j)∈Tkp

δijp (f) Kp(τ)

(Z ij

p

).

By definition the following identity holds for allf ∈ Hkq (M):

Skτ (f) − f =

p∈Σq

(i,j)∈Tkp

δijp (f)

[Kp(τ)

(Z ij

p

)− Z ij

p

].

Since for allp ∈ Σq the condition(i, j) ∈ Tp impliesi+j 6∈ N·(mp+1), byLemma 2.16 the following estimate holds. For eachℓ ∈ N such thatℓ ≤ k,there exists a constantCk

ℓ > 0 such that, for anyf ∈ Hkq (M),

(2.107) ‖Skτ (f) − f‖ℓ ≤ Ck

p∈Σq

(i,j)∈Tkp

τ1+ i+j

mp+1−ℓ

|δijp (f)| .

In fact, for eachp ∈ Σq and each(i, j) ∈ Tp, sinceZ ijp ∈ Hℓ

q(M), whichimplies Kp(τ)

(Z ij

p

)− Z ij

p ∈ Hℓq(M), if ℓ < 1 + (i + j)/(mp + 1), by

Lemma 2.16, there exist constantsC ′ℓ > 0, C ′′

ℓ > 0 such that

‖Kp(τ)(Z ij

p

)− Z ij

p ‖ℓ ≤ C ′ℓ|Kp(τ)

(Z ij

p

)− Z ij

p |ℓ ≤ C ′′ℓ τ

1+ i+jmp+1

−ℓ,

while, if ℓ > 1 + (i + j)/(mp + 1), sinceZ ijp ∈ H∞

q (M),

‖Kp(τ)(Z ij

p

)− Z ij

p ‖ℓ ≤ ‖Kp(τ)(Z ij

p

)‖ℓ + ‖Z ij

p ‖ℓ ≤ C ′′ℓ τ

1+ i+jmp+1

−ℓ.

The scale of Friedrichs Sobolev spaces admits a standard family of smooth-ing operatorsTτ | τ > 0 such that the operatorTτ : L2

q(M) → H∞q (M)

Page 35: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 35

is defined, for eachτ > 0, by the following truncation of Fourier series. Leten |n ∈ N be an orthonormal basis of eigenfunctions of the FriedrichsLaplacian∆F

q and letλ : N → R+ ∪ 0 be the corresponding sequence of

eigenvalues. Then

Tτ (u) :=∑

τ2λn≤1

〈u, en〉q en , if u =∑

n∈N

〈u, en〉q en .

If u ∈ Hsq (M), thenTτ (u) → u in Hs

q (M) (asτ → 0+) and the followingestimates hold. For allr ∈ R+, there exists a constantCr,s(q) > 0 such that

(2.108)‖Tτ (u) − u‖r ≤ Cr,s(q) ‖u‖s τ s−r , if s ≥ r ;

‖Tτ (u)‖r ≤ Cr,s(q) ‖u‖s τ−(r−s) , if r ≥ s .

If p ∈ Σq and(i, j) ∈ Tp, the distributionδijp ∈ H

−sijq (M) for anysij > 0

such thati+j < (sij−1)(mp+1). Hence there exists a constantCijp (q) > 0

such that, by estimates (2.108), for allu ∈ Hsq (M),

(2.109) |δijp (Tτ (u)) | ≤ Cij

p (q) ‖u‖s max1, τ s−sij .

If u ∈ Hsq (M), thenδp

ij(u) = 0, for all p ∈ Σq and all(i, j) ∈ Tsp. Hence, if

sij ≤ s andi + j < (sij − 1)(mp + 1),

(2.110) |δijp (Tτ (u)) | = |δij

p (Tτ (u) − u) | ≤ Cijp (q) ‖u‖s τ s−sij .

The following estimates hold. Lets ∈ R+ andℓ ∈ N. For anyǫ > 0, thereexists a constantCℓ,s(ǫ) > 0 such that, for allτ ∈ (0, 1] and allu ∈ Hs

q (M),

(2.111) ‖Skτ Tτ (u) − Tτ (u)‖ℓ ≤ Cℓ,s(ǫ) ‖u‖s τ s−ℓ−ǫ .

In fact, if p ∈ Σq and(i, j) ∈ Tkp ,

(2.112) |δijp (Tτ (u)) | ≤ Cij

p (q) ‖u‖s τ s−sij ;

for anysij > 1 + (i + j)/(mp + 1) ≥ s, if (i, j) ∈ Tkp \ Ts

p, and for any1+(i+j)/(mp+1) < sij ≤ s, if (i, j) ∈ Ts

p. The claim (2.111) then followsfrom (2.107). By estimates (2.108) and (2.111), for anyǫ > 0, there existsa constantC ′

ℓ,s(ǫ) > 0 such that, for allτ ∈ (0, 1] and for allu ∈ Hsq (M),

(2.113) ‖Skτ Tτ (u) − u‖ℓ ≤ C ′

ℓ,s(ǫ) ‖u‖s τ s−ℓ−ǫ .

Let Sk(τ) | τ ∈ (0, 1] be the family of operatorsSk(τ) : L2q(M) →

Hkq (M) defined as follows: for eachτ ∈ (0, 1],

(2.114) Sk(τ) := Skτ Tτ .

By estimate (2.113), for anyǫ > 0, there exists a constantC ′′ℓ,s(ǫ) > 0 such

that, for allτ ∈ (0, 1] and for allu ∈ Hsq (M),

(2.115) ‖Sk(τ)(u) − Sk(τ/2)(u)‖ℓ ≤ C ′′ℓ,s(ǫ) ‖u‖s τ s−ℓ−ǫ .

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36 GIOVANNI FORNI †

SinceSk(τ)(u) ∈ Hkq (M) for all τ ∈ (0, 1], by the interpolation inequality

proved in Lemma 2.10 it follows that, for anyr ∈ [0, k] and for anyǫ > 0there existsCr,s(ǫ) > 0 such that, for allτ ∈ (0, 1] and for allu ∈ Hs

q (M),

(2.116) |Sk(τ)(u) − Sk(τ/2)(u)|r ≤ Cr,s(ǫ) ‖u‖s τ s−r−ǫ .

It follows that, for everyn ∈ N and for everyτ ∈ (0, 1],(2.117)

|Sk(τ/2n)(u) − Sk(τ/2n+1)(u)|r ≤ Cr,s(ǫ) ‖u‖s2−n(s−r−ǫ) τ s−r−ǫ .

If s > r and0 < ǫ < s− r, the sequenceSk(τ/2n)(u)n∈N is Cauchy andtherefore convergent to the functionu ∈ Hs

q (M) ⊂ Hrq (M) in the Hilbert

spaceHrq (M). It follows that, for allτ ∈ (0, 1] and for allu ∈ Hs

q (M),

(2.118) |Sk(τ)(u) − u|r ≤ Cr,s(ǫ) ‖u‖s τ s−r−ǫ .

If s ≤ r ands − r ≤ 0 < ǫ, we argue as follows. By estimate (2.116), foreachτ ≤ 1/2 we have

(2.119) |Sk(2τ)(u) − Sk(τ)(u)|r ≤ Cr,s(ǫ) ‖u‖s (2τ)s−r−ǫ ,

hence ifτ ≤ 2−n, for all 0 ≤ k < n,(2.120)

|Sk(2k+1τ)(u) − Sk(2kτ)(u)|r ≤ Cr,s(ǫ) ‖u‖s 2(k+1)(s−r−ǫ) τ s−r−ǫ ,

It follows that, there exists a constantC ′r,s(ǫ) > 0 such that

(2.121) |Sk(2nτ)(u) − Sk(τ)(u)|r ≤ C ′r,s(ǫ) ‖u‖s 2n(s−r−ǫ) τ s−r−ǫ ,

For everyτ ∈ (0, 1], letn(τ) be the maximumn ∈ N such that2nτ ≤ 1. Bythis definition it follows that1/2 < 2n(τ)τ ≤ 1, hence by estimate (2.113)there exists a constantCk > 0 such that, for allu ∈ Hs

q (M),

|Sk(2n(τ)τ)(u)|r ≤ sup1/2≤τ≤1

‖Sk(τ)(u)‖k < Ck‖u‖s .

It follows that there existsC ′′r,s(ǫ) > 0 such that, for allτ ∈ (0, 1] and for

all u ∈ Hsq (M),

(2.122) |Sk(τ)(u)|r ≤ C ′′r,s(ǫ) ‖u‖s τ s−r−ǫ ,

By Lemma 2.11 and Theorem 2.24, we have the following comparison es-timate for the (Friedrichs) weigthed Sobolev norms :

Corollary 2.25. For any 0 < r < s there exists constantsCr > 0 andCr,s > 0 such that, for allu ∈ Hs

q (M), the following inequalities hold:

C−1r ‖u‖r ≤ |u|r ≤ Cr,s ‖u‖s .

Page 37: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 37

Finally, we derive a crucial interpolation estimate for thedual weightedSobolev norms:

Corollary 2.26. Let 0 ≤ s1 < s2. For anys1 ≤ r < s ≤ s2 there exists aconstantCr,s > 0 such that for any distributionu ∈ H−s1

q (M) the followinginterpolation inequality holds:

(2.123) |u|−s ≤ Cr,s |u|s2−r

s2−s1−s1

|u|r−s1

s2−s1−s2

.

Proof. Let k ∈ N be any integer larger thans2 > s1 and letSk(τ) :L2

q(M) → Hkq (M) be the family of smoothing operators constructed above.

By Theorem 2.24, since0 ≤ r − s1 < s− s1 and anyr − s2 < s − s2 ≤ 0,there exists a constantCk

r,s > 0 such that the following holds: for anyu ∈ H−s1

q (M) \ 0, anyv ∈ Hsq (M) and for allτ ∈ (0, 1],

(2.124)|〈u, v〉| ≤ |u|−s1|v − Sk(τ)(v)|s1 + |u|−s2|S

k(τ)(v)|s2

≤ Ckr,s τ

r−s1|u|−s1 + τ r−s2 |u|−s2 |v|s .

The interpolation inequality (2.123) then follows by taking

τ =

(|u|−s2

|u|−s1

) 1s2−s1

∈ (0, 1] .

3. THE COHOMOLOGICAL EQUATION

3.1. Distributional solutions. In this section we give a streamlined ver-sion of the main argument of [For97] (Theorem 4.1) with the goal of estab-lishing the sharpest bound on the loss of Sobolev regularitywithin the reachof the methods of [For97]. We were initially motivated by a question ofMarmi, Moussa and Yoccoz who found foralmost allorientable quadraticdifferentials a loss of regularity of1 + BV (they find bounded solutionsfor absolutely continuous data with first derivative of bounded variation un-der finitely many independent compatibility conditions andcorrespondingresults for higher smoothness) [MMY03], [MMY05] . The results of thissection, as those of [For97], hold forall orientable quadratic differentials.

There is a natural action of the circle groupS1 ≡ SO(2, R) on the spaceQ(M) of holomorphic quadratic differentials on a Riemann surfaceM :

rθ(q) := eiθq , for all (rθ, q) ∈ SO(2, R)× Q(M) .

Let qθ denote the quadratic differentialrθ(q) and letSθ, Tθ denote theframe (introduced in §2.1) associated to the quadratic differentialqθ for any

Page 38: Sobolev Regularity

38 GIOVANNI FORNI †

θ ∈ S1. We have the following formulas:

(3.1)Sθ = cos

2

)Sq + sin

2

)Tq =

e−i θ2

2∂+

q +ei θ

2

2∂−

q ;

Tθ = − sin

2

)Sq + cos

2

)Tq =

e−i θ2

2i∂+

q −ei θ

2

2i∂−

q ;

Definition 3.1. Let q be an orientable quadratic differential. A distributionu ∈ H−r

q (M) will be called a(distributional) solutionof the cohomologicalequationSqu = f for a given functionf ∈ H−s

q (M) if

〈u, Sqv〉 = −〈f, v〉 , for all v ∈ Hr+1q (M) ∩ Hs

q (M) .

Let Hsq(M) ⊂ Hs

q (M), Hsq(M) ⊂ Hs

q (M) (for anys ∈ R) be the subspacesorthogonal to constant functions, that is

(3.2)Hs

q(M) := f ∈ Hsq (M) | 〈f, 1〉s = 0 ,

Hsq(M) := f ∈ Hs

q (M) | (f, 1)s = 0 .

The spacesHsq(M) ⊂ Hs

q (M) andHsq(M) ⊂ Hs

q (M) coincide with thesubspaces of functions of zero average fors ≥ 0, and with the subspaces ofdistributions vanishing on constant functions fors < 0.

Theorem 3.2. Let r > 2 andp ∈ (0, 1) be such that andrp > 2. Thereexists a bounded linear operator

U : H−1q (M) → Lp

(S1, H−r

q (M))

such that the following holds. For anyf ∈ H−1q (M) there exists a full

measure subsetFr(f) ⊂ S1 such thatu := U(f)(θ) ∈ H−rq (M) is a distri-

butional solution of the cohomological equationSθu = f for all θ ∈ Fr(f).

Proof. We claim that for anyr > 2, anyp ∈ (0, 1) such thatpr > 2 andanyf ∈ H−1

q (M), there exists a measurable functionAq := Aq(p, r, f) ∈Lp(S1, R+) such that the following estimates hold. Letθ ∈ S1 be such thatAq(θ) < +∞. For allv ∈ Hr+1

q (M) we have

(3.3) |〈f, v〉| ≤ Aq(θ) ‖Sθv‖r .

In addition, the following bound for theLp norm of the functionAq holds.There exists a constantBq(p) > 0 such that

(3.4) |Aq|p ≤ Bq(p) ‖f‖−1 .

Assuming the claim, we prove the statement of the theorem. Infact, by theestimate (3.3) the linear map given by

(3.5) Sθv → −〈f, v〉 , for all v ∈ Hr+1q (M) ,

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SOBOLEV REGULARITY OF SOLUTIONS 39

is well defined and extends by continuity to the closure of therangeRr(θ)of the linear operatorSθ in Hr

q (M). Let U(f)(θ) be the extension uniquelydefined by the condition thatU(f)(θ) vanishes on the orthogonal comple-ment ofRr(θ) in Hr

q (M). By construction, for almost allθ ∈ S1 the linearfunctionalu := U(f)(θ) ∈ H−r

q (M) yields a distributional solution of thecohomological equationSθu = f whose norm satisfies the bound

‖U(f)(θ)‖−r ≤ Aq(θ) .

By (3.4) theLp norm of the measurable functionU(f) : S1 → H−rq (M)

satisfies the required estimate

|U(f)|p :=

(∫

S1

‖U(f)(θ)‖p−r dθ

)1/p

≤ Bq(p) ‖f‖−1 .

We turn now to the proof of the above claim. LetR±q =

(M∓

q

)⊥be the

(closed) ranges of the Cauchy-Riemann operators∂±q : H1

q (M) → L2q(M)

(see Proposition 2.4). Following [For97], we introduce thelinear operatorUq : R−

q → R+q defined as

(3.6) Uq := ∂−q (∂+

q )−1 .

By Proposition 2.4,(3), the operatorUq is a partial isometry onL2q(M),

hence by the standard theory of partial isometries on Hilbert spaces, it hasa family of unitary extensionsUJ : L2

q(M) → L2q(M) parametrized by

isometriesJ : M+q → M−

q (see formulas(3.10)-(3.12) in [For97]). Bydefinition the following identities hold onH1

q (M) (see formulas(3.13) in[For97]) :

(3.7) Sθ =ei θ

2

2

(UJ + e−iθ

)∂+

q =e−i θ

2

2

(U−1

J + eiθ)∂−

q .

The proof of estimate (3.3) is going to be based on propertiesof the re-solvent of the operatorUJ . In fact, the proof of (3.3) is based on the re-sults, summarized in [For97], Corollary 3.4, concerning the non-tangentialboundary behaviour of the resolvent of a unitary operator ona Hilbertspace, applied to the operatorsUJ , U−1

J on L2q(M). The Fourier analysis

of [For97], §2, also plays a relevant role through Lemma 4.2 in [For97] andthe Weyl’s asymptotic formula (Theorem 2.3).

Following [For97], Prop. 4.6A, or [For02], Lemma 7.3, we prove that thereexists a constantCq > 0 such that the following holds. For any distributionf ∈ H−1

q (M) there exist (weak) solutionsF± ∈ L2q(M) of the equations

∂±q F± = f such that

(3.8) |F±|0 ≤ Cq ‖f‖−1 .

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40 GIOVANNI FORNI †

In fact, the maps given by

(3.9) ∂±q v → −〈f, v〉 , for all v ∈ H1

q (M) ,

are bounded linear functionals on the (closed) rangesR±q ⊂ L2

q(M) (of theCauchy-Riemann operator∂±

q : H1q (M) → L2

q(M). In fact, the functionalsare well-defined sincef vanishes on constant functions, that is, on the ker-nel of the Cauchy-Riemann operators, and it is bounded sinceby Poincaréinequality (see [For97], Lemma 2.2 or [For02], Lemma 6.9) there exists aconstantCq > 0 such that, for anyv ∈ H1

q(M) ⊂ H1q (M),

(3.10) |〈f, v〉| ≤ ‖f‖−1‖v‖1 ≤ Cq ‖f‖−1 |∂±q v|0 .

Let Φ± be the unique linear extension of the linear map (3.9) toL2q(M)

which vanishes on the orthogonal complement ofR±q in L2

q(M). By (3.10),the functionalsΦ± are bounded onL2

q(M) with norm

‖Φ±‖ ≤ Cq ‖f‖−1 .

By the Riesz representation theorem, there exist two (unique) functionsF± ∈ L2

q(M) such that

〈v, F±〉q = Φ±(v) , for all v ∈ L2q(M) .

The functionsF± are by construction (weak) solutions of the equations∂±

q F± = f satisfying the required bound (3.8).

The identities (3.7) immediately imply that

(3.11)〈∂±

q v, F±〉q = 2e∓i θ2 〈R±

J (z)Sθv, F±〉q

− (z + e∓iθ)〈R±J (z)∂±

q v, F±〉q ,

whereR+J (z) andR−

J (z) denote the resolvents of the unitary operatorsUJ

andU−1J respectively, which yield holomorphic families of boundedopera-

tors on the unit diskD ⊂ C.

Let r > 2 and letp ∈ (0, 1) be such thatpr > 2. Let E = ekk∈N be theorthonormal Fourier basis of the Hilbert spaceL2

q(M) described in §2. ByCorollary 3.4 in [For97] all holomorphic functions

(3.12) R±k (z) := 〈R±

J (z)ek, F±〉q , k ∈ N ,

belong to the Hardy spaceHp(D), for any0 < p < 1. The correspondingnon-tangential maximal functionsN±

k (over cones of arbitrary fixed aper-ture0 < α < 1) belong to the spaceLp(S1, dθ) and for all0 < p < 1 thereexists a constantAα,p > 0 such that the following inequalities hold:

(3.13) |N±k |p ≤ Aα,p|ek|0 |F

±|0 = Aα,p |F±|0 ≤ Aα,p Cq ‖f‖−1 .

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SOBOLEV REGULARITY OF SOLUTIONS 41

Let λkk∈N be the sequence of the eigenvalues of the Dirichlet formQ

introduced in §2. Letw ∈ Hrq (M). We have

(3.14) 〈R±J (z)w, F±〉q =

∞∑

k=0

〈w, ek〉q R±k (z) ,

hence, by the Cauchy-Schwarz inequality,

(3.15) |〈R±J (z)w, F±〉q| ≤

( ∞∑

k=0

|R±k (z)|2

(1 + λk)r

)1/2

‖w‖r ,

Let N±(θ) be the functions defined as

(3.16) N±(θ) :=( ∞∑

k=0

|N±k (θ)|2

(1 + λk)r

)1/2

.

Let N±(w) be the non-tangential maximal function for the holomorphicfunction〈R±

J (z)w, F±〉q. By formulas (3.15) and (3.16), for allθ ∈ S1 andall functionsw ∈ Hr

q (M), we have

(3.17) N±(w)(θ) ≤ N±(θ) ‖w‖r .

The functionsN± ∈ Lp(S1, dθ) for any 0 < p < 1. In fact, by formula(3.13) and (following a suggestion of Stephen Semmes) by the‘triangularinequality’ forLp spaces with0 < p < 1, we have

(3.18) |N±|pp ≤ (Aα,p Cq)p( ∞∑

k=0

1

(1 + λk)pr/2

)‖f‖p

−1 < +∞ .

The series in formula (3.18) is convergent by the Weyl asymptotics (Theo-rem 2.3) sincepr/2 > 1.

By taking the non-tangential limit asz → −e∓iθ in the identity (3.11),formula (3.17) implies that, for allθ ∈ S1 such thatN±(π ∓ θ) < +∞,

|〈∂±v, F±〉q| ≤ N±(π ∓ θ) ‖Sθv‖r ,

hence the required estimates (3.3) and (3.4) are proved wih the choice ofthe functionAq(θ) := N+(π − θ) or Aq(θ) := N−(π + θ) for all θ ∈ S1.Since the claim is proved the result follows.

Theorem 3.3. Let r > 2. For almost all θ ∈ S1 (with respect to theLebesgue measure), there exists a constantCr(θ) > 0 such that, for allf ∈ Hr−1

q (M) such that∫

Mf ωq = 0, the cohomological equationSθu = f

has a distributional solutionu ∈ H−rq (M) satisfying the following estimate:

‖u‖−r ≤ Cr(θ) ‖f‖r−1 .

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42 GIOVANNI FORNI †

Proof. Let E = ekk∈N be the orthonormal Fourier basis of the HilbertspaceL2

q(M) described in §2. Letr > 2 and p ∈ (0, 1) be such thatpr > 2. By Theorem 3.2, for anyk ∈ N \ 0 there exists a functionwith distributional valuesuk := U(ek) ∈ Lp

(S1, H−r

q (M))

such that thefollowing holds. There exists a constantCq := Cq(p, r) > 0 such that

(3.19)

(∫

S1

‖uk(θ)‖p−r dθ

)1/p

≤ Cq ‖ek‖−1 = Cq (1 + λk)−1/2 .

In addition, for anyk ∈ N \ 0, there exists a full measure setFk ⊂ S1

such that, for allθ ∈ Fk, the distributionu := uk(θ) ∈ H−rq (M) is a

(distributional) solution of the cohomological equationSθ u = ek.

Any functionf ∈ Hr−1q (M) such that

∫M

f ωq = 0 has a Fourier decompo-sition inL2

q(M):

f =∑

k∈N\0

〈f, ek〉q ek .

A (formal) solution of the cohomological equationSθu = f is thereforegiven by the series

(3.20) uθ :=∑

k∈N\0

〈f, ek〉q uk(θ) .

By the triangular inequality inH−rq (M) and by Hölder inequality, we have

‖uθ‖−r ≤

k∈N\0

‖uk(θ)‖2−r

(1 + λk)r−1

1/2

‖f‖r−1 ,

hence by the ‘triangular inequality’ forLp spaces (with0 < p < 1) and bythe estimate (3.19),

(3.21)∫

S1

‖uθ‖p−r dθ ≤ Cp

q

k∈N\0

1

(1 + λk)pr/2

‖f‖pr−1 .

Sincepr/2 > 1 the series in (3.21) is convergent, hence by Chebyshevinequality for the spaceLp(S1), there exists a full measure setB ⊂ S1

such that, for allθ ∈ B, formula (3.20) yields a well-defined distributionuθ ∈ H−r

q (M) and there exists a constantCq(θ) > 0 such that

(3.22) ‖uθ‖−r ≤ Cq(θ) ‖f‖r−1 .

The setF = ∩Fk∩B has full measure and for allθ ∈ F, for all k ∈ N\0,the distributionuk(θ) ∈ H−r

q (M) is a solution of the equationSθu = ek.It follows thatuθ ∈ H−r

q (M) is a solution of the cohomological equationSθu = f which satisfies the required bound (3.22).

Page 43: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 43

We finally derive a result on distributional solutions of thecohomologicalequation for distributional data of arbitrary regularity:

Corollary 3.4. For any s ∈ R there existsr > 0 such that the followingholds. For almost allθ ∈ S1 (with respect to the Lebesgue measure), thereexists a constantCr,s(θ) > 0 such that, for allF ∈ Hs

q (M) orthogonal toconstant functions, the cohomological equationSθU = F has a distribu-tional solutionU ∈ H−r

q (M) satisfying the following estimate:

‖U‖−r ≤ Cr,s(θ) ‖F‖s .

Proof. SinceF ∈ Hsq (M) is orthogonal to constant functions, for every

k ∈ N there existsfk ∈ Hs+2kq (M), orthogonal to constant functions, such

that (I − ∆Fq )kfk = F . In fact, the family of Friedrichs Sobolev spaces

Hsq (M)|s ∈ R is defined in terms of the Friedrichs extension∆F

q of theLaplace operator∆q of the flat metric determined by the quadratic differ-ential. Letn ∈ N be the minimum integer such thatσ := s + 2n + 1 > 2.By Theorem 3.3, for almost allθ ∈ S1 (with respect to the Lebesgue mea-sure) the cohomological equationSθu = fn has a distributional solutionuθ ∈ H−σ

q (M) satisfying the following estimate:

(3.23) ‖uθ‖−σ ≤ Cσ(θ) ‖fn‖σ−1 .

Let Uθ := (I − ∆Fq )nuθ ∈ H−σ−2n

q (M). It follows immediately from theestimate (3.23) and from the definitions that

‖Uθ‖−σ−2n = ‖un‖−σ ≤ Cσ(θ) ‖fn‖σ−1 = Cσ(θ) ‖F‖s .

Finally Uθ ∈ H−σ−2nq (M) is a distributional solution of the cohomological

equationSθU = F , for almost allθ ∈ S1. In fact, for anyv ∈ Hσ+2n+1q (M),

the functionSθv ∈ Hσ+2nq (M), hence(I −∆q)

nSθv = Sθ(I −∆q)nv and,

since the distributionuθ ∈ H−σq (M) is a solution of the cohomological

equationSθu = fn, for almost allθ ∈ S1, and(I − ∆Fq )nv ∈ Hσ+1

q (M),

〈Uθ, Sθv〉 = 〈(I − ∆Fq )nuθ, Sθv〉 = 〈uθ, Sθ(I − ∆q)

nv〉

= −〈fn, (I − ∆Fq )nv〉 = −〈(I − ∆F

q )nfn, v〉 = −〈F, v〉 ,

as required by the definition of distributional solution of the cohomologicalequation (Definition 3.1).

3.2. Invariant distributions and basic currents. Invariant distributionsyield obstructions to the existence of smooth solutions of the cohomolog-ical equation. We derive below from Theorem 3.2 a sharp version of themain results of [For02], §6, about the Sobolev regularity ofinvariant distri-butions. We then recall the structure theorem proved in thatpaper on thespace of invariant distributions (see [For02], Th. 7.7).

Page 44: Sobolev Regularity

44 GIOVANNI FORNI †

Invariant distributions for the horizontal [respectivelyvertical] vector fieldof an orientable quadratic differentialq are closely related tobasic currents(of dimension and degree equal to1) for the horizontal [vertical] foliationFq

[ F−q]. The notion of a basic current for a measured foliation on a Riemannsurface has been studied in detail in [For02], §6, in the context of weightedSobolev spaces with integer exponent. We outline below someof the basicconstructions and results on basic currents and invariant distributions whichcarry over without modifications to the more general contextof fractionalweighted Sobolev spaces. Finally, we derive from Theorem 3.2 a result onthe Sobolev regularity of basic currents (or invariant distributions) whichimproves upon a similar result proved in [For02] (see Theorem 7.1 (i)).

Let Σ ⊂ M be a finite subset. The spaceD(M \Σ) will denote the standardspace of de Rham currents on the open manifoldM \ Σ, that is the dual ofthe Fréchet spaceΩc(M \ Σ) of differential forms with compact support inM \ Σ. A homogenous current of dimensiond ∈ N (and degree2 − d)on M \ Σ is a continuous linear functional on the subspaceΩd

c(M \ Σ) ofdiferential forms of degreed. The subspace of homogeneous currents ofdimensiond onM \ Σ will be denote byDd(M \ Σ).

Let q be an orientable quadratic differential on a Riemann surface M . LetΣq be the (finite) set of its zeroes. In [For02] we have introduced the fol-lowing spaceΩq(M) of smooth test forms onM .

Definition 3.5. For anyp ∈ M of (even) orderk = 2m ∈ N (m = 0 if p 6∈Σq), let z : Up → C be a canonical complex coordinate on a neighbourhoodUp of p ∈ M , that is a complex coordinate such thatz(p) = 0 andq = zkdz2

onUp. Let πp : Up → C be the (local) covering map defined by

πp(z) :=zm+1

m + 1, z ∈ C .

The spaceΩq(M) is defined as the space of smooth formsα on M suchthat the following holds: for allp ∈ M , there exists a smooth formλp ona neighbourhood of0 ∈ C such thatα = π∗

p(λp) on U′p ⊂ Up. The space

Ωq(M) is the direct sum of the subspacesΩdq(M) of homogeneous forms of

degreed ∈ 0, 1, 2. The spacesΩdq(M), for anyd ∈ 0, 1, 2, andΩq(M)

can be endowed with a natural Fréchet topology modeled on thesmoothtopology in every coordinate neighbourhood.

Lemma 3.6.For any orientable quadratic differentialq ∈ Q(M), the spaceof functionsΩ0

q(M) is dense in the spaceH∞q (M) endowed with the inverse

limit Fréchet topology induced by the family of weighted Sobolev norms.

Proof. By definition, the MacLaurin series of anyf ∈ Ωq(M) with respectto a canonical complex coordinatez for q at everyp ∈ Σq (of order2mp)

Page 45: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 45

has the following form:

(3.24) f(z) =∑

h,k∈N

fhkzh(mp+1)zk(mp+1) .

By Lemmas 2.12 and 2.13,f ∈ H∞q (M). ThusΩq(M) ⊂ H∞

q (M).

Let F ∈ H∞q (M). By Lemma 2.11 the functionF ∈ C∞(M) and by

Lemmas 2.12 and 2.13 its MacLaurin series has the form (3.24)at everyp ∈ Σq (of order2mp). By Borel’s thereorem and by a partition of unityargument, there exists a functionf ∈ Ωq(M) such thatF − f ∈ C∞(M)vanishes at infinite order atΣq. Let Uτ be the open neighbourhood ofΣq

which is the union of a finite number ofdisjoint geodesic disksDτ (p) ofradiusτ ∈ (0, τ0), each centered at a pointp ∈ Σq. Let φτ : M → [0, 1] bea smooth function such that(a)φτ ∈ C∞

0 (M \ Σq), (b) φτ ≡ 1 onM \ Uτ

and(c) for each(i, j) ∈ N × N there exists a constantCij > 0 such that,for all τ ∈ (0, τ0),

maxx∈M

|SiT jφτ (x)| ≤Cij

τ i+j.

If can be proved that, sinceF − f vanishes at infinite order atΣq,

f + φτ (F − f) → F in H∞q (M) , as τ → 0+ ,

which implies, since by constructionf + φτ (F − f) ∈ Ω0q(M), that F

belongs to the closure ofΩ0q(M) in H∞

q (M).

Definition 3.7. The spaceSq(M) ⊂ D(M \ Σq) of q-tempered currents(introduced in [For02], §6.1) is the dual space of the Fréchet spaceΩq(M).A homogeneousq-tempered current of dimensiond (and degree2 − d) is acontinuous functional on the subspaceΩd

q(M) ⊂ Ωq(M) of homogeneousforms of degreed ∈ 0, 1, 2. The space of homogeneous currents of di-mensiond (and degree2 − d) will be denoted bySd

q(M).

For any quadratic differentialq on M , there is a natural operator∗, whichmaps the spaceD0(M \ Σq) of currents of dimension0 and degree2 onthe non-compact manifoldM \ Σq (which is naturally identified with thespace of distributions onM \Σq) bijectively onto the spaceD2(M \Σq) ofcurrents of dimension2 and degree0 onM \ Σq. The operator

∗ : D0(M \ Σq) → D2(M \ Σq)

is defined as follows. Letωq be the smooth area form associated with the(orientable) quadratic differentialq onM . It is a standard fact in the theoryof currents that any distributionU on the2-dimensional surfaceM \Σq canbe written asU = U∗ωq for a unique currentU∗ of dimension2 and degree0. Sinceωq ∈ Ω2

q(M), the map∗ extends to a bijective map∗ : S0

q(M) → S2q(M) .

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46 GIOVANNI FORNI †

Definition 3.8. A distributionD ∈ D2(M \Σq) is horizontally [vertically]quasi-invariantif SD = 0 [TD = 0] in D2(M \ Σq). A distributionD ∈ S2

q(M) is horizontally [vertically] invariant if SD = 0 [TD = 0] inS2

q(M). The space of horizontally [vertically] quasi-invariant distributionswill be denoted byIq(M \Σq) [Iq(M \Σq)] and the subspace of horizontally[vertically] invariant distributions will be denoted byIq(M) [I−q(M)].

Definition 3.9. For anys ∈ R+, let

(3.25)Is±q(M \ Σq) := I±q(M \ Σq) ∩ Hs

q (M) ;

Is±q(M) := I±q(M) ∩ Hs

q (M) .

The subspacesIs±q(M) ⊂ H−s

q (M) of horizontally [vertically] invariantdistributions can also be defined as follows:

(3.26)Isq(M) := D ∈ Hs

q (M) |SD = 0 in H−s−1q (M) ;

[Is−q(M) := D ∈ Hs

q (M) | TD = 0 in H−s−1q (M)] .

The subspaces of horizontally [vertically] invariant distributions which canbe extended to bounded functionals on Friedrichs weighted Sobolev spaceswill be denoted by

(3.27)Is±q(M \ Σq) := I±q(M \ Σq) ∩ H−s

q (M) ;

Is±q(M) := I±q(M) ∩ H−s

q (M) .

Let Vq(M) be the space of vector fieldsX onM \Σq such that the contrac-tion ıXα and the Lie derivativeLXα ∈ Ωq(M) for all α ∈ Ωq(M).

Definition 3.10. A currentC ∈ D1(M \ Σq) is horizontally [vertically]quasi-basic, that is basic forFq [F−q] in the standard sense onM \ Σq, ifthe identities

(3.28) ıXC = LXC = 0

hold inD(M \ Σq) for all smooth vector fieldsX tangent toFq [F−q] withcompact support onM \ Σq. A q-tempered currentC ∈ S1

q(M) is horizon-tally [vertically] basic if the identities (3.28) holds inSq(M) for all vectorfields X ∈ Vq(M), tangent toFq [F−q] on M \ Σq. The vector spacesof horizontally [vertically] quasi-basic (real) currentswill be denoted byBq(M \ Σq) [B−q(M \ Σq)] and the subspace of horizontally [vertically]basic (real) currents will be denoted byBq(M) [B−q(M)].

Definition 3.11. For anys ∈ R, the Friedrichs weighted Sobolev space of1-currentsW s

q (M) ⊂ Sq(M) and the weighted Sobolev space of1-currentsW s

q (M) ⊂ Sq(M) are defined as follows:

(3.29)W s

q (M) := α ∈ Sq(M) | (ıSα, ıT α) ∈ Hsq (M) × Hs

q (M) ;

W sq (M) := α ∈ Sq(M) | (ıSα, ıT α) ∈ Hs

q (M) × Hsq (M) .

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SOBOLEV REGULARITY OF SOLUTIONS 47

Definition 3.12. For any1-currentC ∈ D(M \ Σq), theweighted Sobolevorder OW

q (C) and theFriedrichs weighted Sobolev orderOWq (C) are the

real numbers defined as follows:

(3.30)OW

q (C) := infs ∈ R |D ∈ W−sq (M) ;

OWq (C) := infs ∈ R |C ∈ W−s

q (M) .

Definition 3.13. For anys ∈ R, let

(3.31)Bs

±q(M \ Σq) := B±q(M \ Σq) ∩ W sq (M) ;

Bs±q(M) := B±q(M) ∩ W s

q (M) .

The subspacesBs±q(M) ⊂ W−s

q (M) of horizontally [vertically] basic cur-rentscan also be defined as follows:

(3.32)ıSC = 0 in H−s

q (M) and LSC = 0 in W−s−1q (M) ;

[ıT C = 0 in H−sq (M) and LT C = 0 in W−s−1

q (M)] .

The subspaces of basic currents which can be extended to bounded func-tionals on Friedrichs weighted Sobolev spaces will be denoted by

(3.33)Bs

±q(M \ Σq) := B±q(M \ Σq) ∩ W−sq (M) ;

Bs±q(M) := B±q(M) ∩ W−s

q (M) .

According to Lemma 6.5 of [For02], the notions of invariant distributionsand basic currents are related (see also Lemma 6.6 in [For02]):

Lemma 3.14. A currentC ∈ Bsq(M \ Σq) [C ∈ Bs

q(M)] if and only ifthe distributionC ∧ ℜ(q1/2) ∈ Is

q(M \ Σq) [C ∧ ℜ(q1/2) ∈ Isq(M)]. A

currentC ∈ Bs−q(M \ Σq) [C ∈ Bs

−q(M)] if and only if the distributionC ∧ ℑ(q1/2) ∈ Is

q(M \ Σq) [C ∧ ℑ(q1/2) ∈ Isq(M)]. In addition, the map

(3.34)Dq : C → −C ∧ ℜ(q1/2) ;

[D−q : C → C ∧ ℑ(q1/2)] ;

is a bijection from the spaceBsq(M \ Σq) [Bs

−q(M \ Σq)] onto the spaceIsq(M \ Σq) [Is

−q(M \ Σq)], which maps the subspaceBsq(M) [ Bs

−q(M)]onto the subspaceIs

q(M) [Is−q(M)]. The map (3.34) also maps the space

Bsq(M) [Bs

−q(M)] onto Isq(M) [ Is

−q(M)].

3.3. Basic cohomology.Let Z(M \Σ) ⊂ D1(M \Σ) denote the subspaceof all (real) closed currents, that is, the space of all (real) de Rham currentsC ∈ D1(M \Σ) such that the exterior derivativedC = 0 in D(M \Σ). LetZq(M) ⊂ S1

q(M) be the subspace of all (real) closedq-tempered currents,that is, the space of allq-tempered (real) currentsC such thatdC = 0 in

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48 GIOVANNI FORNI †

Sq(M). It was proved in [For02], Lemma 6.2, that the natural de Rhamcohomology map

(3.35) jq : Z(M \ Σq) → H1(M \ Σq, R)

has the property that the subspace of closedq-tempered currents is mappedonto the absolute real cohomology of the surface, that is,

(3.36) jq : Zq(M) → H1(M, R) ⊂ H1(M \ Σq, R) .

It was also proved in [For02], Lemma 6.2’, that quasi-basic and basic cur-rents are closed, in the sense that the following inclusionshold:

(3.37)B±q(M \ Σq) ⊂ Z(M \ Σq) ,

B±q(M) ⊂ Zq(M) .

The images of the restrictions of the natural cohomology mapto the variousspaces of basic currents are called thehorizontal [vertical] basic cohomolo-gies, namely the spaces

(3.38)

H1±q(M \ Σq, R) := jq (B±q(M \ Σq)) ⊂ H1(M \ Σq, R) ;

H1,s±q (M \ Σq, R) := jq

(Bs

±q(M \ Σq))⊂ H1

±q(M \ Σq, R) ;

H1±q(M, R) := jq (B±q(M)) ⊂ H1(M, R) ;

H1,s±q (M, R) := jq

(Bs

±q(M))⊂ H1

±q(M, R) .

Following [For02], Theorem 7.1, we give below a descriptionof the hori-zontal [vertical] basic cohomologies for the orientable quadratic differentialqθ, for any orientable holomorphic quadratic differentialq onM and for al-most allθ ∈ S1. The result we obtain below is stronger than Theorem 7.1of [For02] since it requires weaker Sobolev regularity assumptions.

(Absolute) real cohomology classes onM can be represented in terms ofmeromorphic (or anti-meromorphic) functions inL2

q(M) (see [For02], §2).In fact, by the Hodge theory on Riemann surfaces [FK92], III.2, all realcohomology classes can be represented as the real (or imaginary) part ofa holomorphic (or anti-holomorphic) differential onM . In turn, any ori-entable holomorphic quadratic differential induces an isomorphism betweenthe space Hol+(M) [Hol−(M)] of holomorphic [anti-holomorphic] differ-entials and the space of square-integrable meromorphic [anti-meromorphic]functions. LetM+

q [M−q ] be the space of meromorphic [anti-meromorphic]

functions onM which belong to the Hilbert spaceL2q(M) (see Proposi-

tion 2.4). Such spaces can be characterized as the spaces of all meromor-phic [anti-meromorphic] functions with poles atΣq = q = 0 of ordersbounded in terms of the multiplicity of the pointsp ∈ Σq as zeroes of thequadratic differentialq. In fact, if p ∈ Σq is a zero ofq of order2m, thatpis a pole of order at mostm for anym± ∈ M±

q .

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SOBOLEV REGULARITY OF SOLUTIONS 49

Let q1/2 be a holomorphic square root ofq on M . Holomorphic [anti-holomorphic] differentialsh+ [h−] on M can be written in terms of mero-morphic [anti-meromorphic] functions inL2

q(M) as follows:

(3.39)h+ := m+q1/2 , m+ ∈ M+

q ;

h− := m−q1/2 , m− ∈ M−q .

The followingrepresentations of real cohomology classestherefore hold:

(3.40)c ∈ H1(M, R) ⇐⇒ c = [ℜ(m+q1/2)] , m+ ∈ M+

q ;

c ∈ H1(M, R) ⇐⇒ c = [ℜ(m−q1/2)] , m− ∈ M−q .

The mapsc±q : M±q → H1(M, R) given by the representations (3.40) are

bijective and it is in factisometricif the spacesM±q are endowed with the

euclidean structure induced byL2q(M) andH1(M, R) with theHodge prod-

uct relative to the complex structure of the Riemann surfaceM . In fact, theHodge norm‖c‖2

H of a cohomology classc ∈ H1(M, R) is defined as fol-lows:

(3.41) ‖c‖2H :=

ı

2

M

h± ∧ h± if c = [ℜ(h±)] , h± ∈ Hol±(M) .

We remark that the Hodge norm is defined in terms of the complexstructureof the Riemann surfaceM (carrying a holomorphic quadratic differentialq ∈ Q(M)) but does not depend on the quadratic differential. Ifq ∈ Q(M)is any orientable quadratic differential onM , by the representation (3.40),we can also write:

(3.42)‖c+

q (m+)‖2H :=

M

|m+|2 ωq , for all m+ ∈ M+q ;

‖c−q (m−)‖2H :=

M

|m−|2 ωq , for all m− ∈ M−q .

The representation (3.39)-(3.40) can be extended to the punctured cohomol-ogyH1(M \Σq, R) as follows. For any finite setΣ ⊂ M , let Hol+(M \Σ)[Hol−(M \ Σ)] be the space of meromorphic [anti-meromorphic] differen-tials with at most simple poles atΣ. By Riemann surface theory, any realcohomology classc ∈ H1(M \ Σ, R) can be represented as the real (orimaginary) part of a differentialh+ ∈ Hol+(M \Σ) or h− ∈ Hol−(M \Σ).Let Σ ⊂ M be a finite set and letM+(Σ) [M+(Σ)] be the space of allmeromorphic [anti-meromorphic] functions which are holomorphic [anti-holomorphic] onM \ Σ. The spacesM±(Σ) can be identified with a sub-space of the distributional spaceD2(M \ Σ). In fact, if q is any orientableholomorphic quadratic differential onM , the spacesM±(Σq) identify withsubspaces of the spaceS2

q(M) of q-tempered distributions. The distributiondetermined by a functionφ ∈ M+(Σq) or M−(Σq) is defined by integration

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50 GIOVANNI FORNI †

(in the standard way) as a linear functional onC∞0 (M \ Σq), which can be

extended to the spaceΩ0q(M) as follows:

φ(v) := PV∫

M

φ v ωq , for all v ∈ Ω0q(M) .

The Sobolev regularity of a distributionsφ ∈ M±(Σq) depend on the orderof its poles. In fact, by Theorem 2.22 we have the following:

Lemma 3.15. Let φ ∈ M+(Σq) [ φ ∈ M+(Σq) ] be a meromorphic [anti-meromorphic] function with poles atΣq. For any s ∈ R, the associateddistributionφ ∈ H−s

q (M) if at everyp ∈ Σq of order2mp the functionφhas a pole of order< (mp + 1)(s + 1).

We introduce the following notation: for alls > 0,

(3.43) M±s (Σq) := M±(Σq) ∩ H−s

q (M) .

There exist natural mapsφ±q : Hol±(M\Σq) → M±(Σq) defined as follows:

for all h± ∈ Hol±(M \ Σq),

φ+q (h+) = h+/q1/2 [φ−

q (h−) = h−/q1/2] .

By Lemma 3.15 the range of the mapsφ±q is contained in the weighted

Sobolev spaceH−sq (M) for all s > 0, hence there are well-defined maps

(3.44) φ±q : Hol±(M \ Σq) → M±

s (Σq) for all s > 0 .

The maps (3.44) are clearly injective and by Corollary 2.23 there existssq > 0 such that, for anys ∈ (0, sq), they are also surjective. Let

(3.45) M±∗ (Σq) =

s>0

M±s (Σq) = M±

s (Σq) , for anys ∈ (0, sq) .

The representation (3.40) of the absolute real cohomology generalizes tothe punctured real cohomology as follows.

(3.46)c ∈ H1(M \ Σq, R) ⇐⇒ c = [ℜ(m+q1/2)] , m+ ∈ M+

∗ (Σq) ;

c ∈ H1(M \ Σq, R) ⇐⇒ c = [ℜ(m−q1/2)] , m− ∈ M−∗ (Σq) .

The following lemma, proved in [For02], Lemma 7.6, for weighted Sobolevspaces with integer exponent, holds:

Lemma 3.16. Let s ∈ R+. Let C ∈ W−sq (M) be any real current of di-

mension (and degree) equal to1, closed in the spaceD(M \Σq) of currentsonM \Σq. There exists a distributionU ∈ H−s+1

q (M) and a meromorphicdifferentialh+ ∈ Hol+(M \ Σq) such that

(3.47) dU∗ = ℜ(h+) − C in W−sq (M) .

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SOBOLEV REGULARITY OF SOLUTIONS 51

If C is closed in the spaceSq(M) of q-tempered currents there exists a dis-tribution U ∈ H−s+1

q (M) and a holomorphic differentialh+ ∈ Hol+(M)such that the identity (3.47) holds.

The argument given in [For02], Lemma 7.6, in the case of integer orders ∈ N extends the general case of orders ∈ R+. In fact, it follows from thedistributional identity (3.47) inSq(M) that the currentU∗ ∈ H−s+1

q (M) ifand only if the currentC ∈ W−s

q (M), for anys ∈ R+. Hence, Lemma 3.16follows immediately from [For02], Lemma 7.6.

The construction of basic currents (or, equivalently, of invariant distribu-tions) is based on the following method.

Lemma 3.17.Let q be an orientable holomorphic quadratic differential ona Riemann surfaceM . Letm+ ∈ M+

s (Σq) be a meromorphic function withpoles atΣq ⊂ M . A distributionU ∈ H−s+1

q (M) is a (distributional)solution inD(M \ Σq) of the cohomological equation

(3.48) SU = ℜ(m+) [TU = −ℑ(m+)] in D(M \ Σq) ,

if and only if the currentC ∈ W−sq (M) uniquely determined by the identity

(3.49) dU∗ = ℜ(m+q1/2) + C

is horizontally [vertically] quasi-basic. Ifℜ(m+q1/2) ∈ Hol+(M), thedistributionU ∈ H−s+1

q (M) is a solution of the cohomological equation(3.48) in the spaceSq(M) of q-tempered currents if and only if the cur-rent C ∈ W−s

q (M) uniquely determined by formula (3.49) is horizontally[vertically] basic.

Proof. If formula (3.49) holds inD(M \ Σq), thenC is closed inD(M \Σq). If the differentialℜ(m+q1/2) is holomorphic and formula (3.49) holdsin Sq(M), thenC is closed inSq(M). The standard formula for the Liederivative of a current,

(3.50) LXC = ıXdC + dıXC = 0 ,

holds inD(M \Σq) for any vector fieldX with compact support containedin M \Σq and it holds inSq(M) for any vector fieldX ∈ Vq(M) It followsthat a currentC ∈ D(M \ Σq) is horizontally [vertically] quasi-basic ifand only if it is closed andıSC = 0 [ıT C = 0] in D(M \ Σq) and itis horizontally [vertically] basic if and only ifC ∈ Sq(M) is closed andıSC = 0 [ıT C = 0] in Sq(M). The distributionU ∈ H−s

q (M) in formula(3.49) is a solution of the cohomological equation (3.48) inD(M \ Σq)or Sq(M) if and only if ıSC = 0 [ıT C = 0] in D(M \ Σq) or Sq(M)respectively. As a consequence, the lemma is proved.

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52 GIOVANNI FORNI †

Let q be an orientable holomorphic quadratic differential on a Riemann sur-faceM (of genusg ≥ 1). Let Π±q(M \ Σq, R) ⊂ H1(M \ Σq, R) be thecodimension1 subspaces defined as follows:

(3.51)Π1

q(M \ Σq, R) := c ∈ H1(M \ Σq, R) | c ∧ [ℑ(q1/2)] = 0 ;

Π1−q(M \ Σq, R) := c ∈ H1(M \ Σq, R) | c ∧ [ℜ(q1/2)] = 0 .

Since the absolute cohomology can be regarded as a subspace of the punc-tured cohomology it also is possible to define the subspaces

(3.52)Π1

q(M, R) := Π1q(M \ Σq, R) ∩ H1(M, R) ;

Π1−q(M, R) := Π1

−q(M \ Σq, R) ∩ H1(M, R) .

Theorem 3.18.For anys > 3 there exists a full measure setFs ⊂ S1 suchthat the following holds. For anyθ ∈ Fs, the following inclusions hold

(3.53)Π1

±qθ(M \ Σq, R) ⊂ H1,s

±qθ(M \ Σq, R) ,

Π1±qθ

(M, R) ⊂ H1,s±qθ

(M, R) .

Proof. Let m+ ∈ M+−1(Σq) be any meromorphic function such that the

induced distribution PV(m+) ∈ H−1q (M). A computation shows that, for

all θ ∈ S1,

(3.54) PV∫

M

ℜ(m+) ωq = 0 ⇐⇒ [ℜ(m+q1/2θ )] ∈ Π1

qθ(M \ Σq, R) .

Under the zero-average condition (3.54), by Theorem 3.2 forany r > 2there exists a full measure setFr(m

+) ⊂ S1 such that, for allθ ∈ S1,the cohomological equationSθU = ℜ(m+) has a distributional solutionUθ(m

+) ∈ H−rq (M). Let U∗

θ (m+) be the current of dimension2 corre-sponding to the distributionUθ(m

+). Let Cθ(m+) ∈ W−r−1

q (M) be the1-dimensional current determined by the identity

(3.55) dU∗θ (m+) = ℜ(m+q

1/2θ ) − Cθ(m

+) .

By Lemma 3.17, we have thus proved that, for all meromorphic functionsm+ ∈ M+

−1(Σq) and for allθ ∈ Fr(m+), there exists a quasi-basic current

Cθ(m+) ∈ Br+1

qθ(M \ Σq) such that

[Cθ(m+)] = [ℜ(m+q

1/2θ )] ∈ Π1

qθ(M \ Σq, R) ;

in addition, wheneverm+ ∈ M+q , the currentCθ(m

+) ∈ Br+1qθ

(M) is basicand has a cohomology class

[Cθ(m+)] = [ℜ(m+q

1/2θ )] ∈ Π1

qθ(M, R) .

Let σ ∈ N be the cardinality of the setΣq ⊂ M and let

m+1 , . . . , m+

2g−1, . . . , m+2g+σ−1

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SOBOLEV REGULARITY OF SOLUTIONS 53

be a basis (overR) of the real subspace ofM+∗ (Σq) defined by the zero-

average condition (3.54). For anys > 3, let

F+s :=

2g+σ−1⋂

i=1

Fs−1(m+i ) .

Clearly he setF+s has full Lebesgue measure. We claim that for allθ ∈ F+

s

the following inclusions hold:

(3.56)Π1

qθ(M \ Σq, R) ⊂ H1,s

qθ(M \ Σq, R) ,

Π1qθ

(M, R) ⊂ H1,sqθ

(M \ R) .

The claim is proved as follows. For anyc ∈ H1(M \ Σq, R) there exists aunique meromorphic functionm+ ∈ M+

∗ (Σq) such thatc = [ℜ(m+q1/2θ )].

The functionm+ ∈ M+q for any c ∈ H1(M, R). If c ∈ Π1

qθ(M, R), the

distributionℜ(m+) vanishes on constant functions as in (3.54), hence forall θ ∈ F+

s , there exists a solutionU ∈ H−s+1q (M) of the cohomological

equationSθU = ℜ(m+). The currentC ∈ Bsqθ

(M \Σq) such that[C] = c ∈Π1

qθ(M \Σq, R) is then given by the identity (3.55). By the above discussion

the currentC ∈ Bsqθ

(M) for all c ∈ Π1qθ

(M, R).

By a similar argument it is possible to construct a full measure setF−s such

that, for allθ ∈ F−s , the following inclusions hold:

(3.57)Π1

−qθ(M \ Σq, R) ⊂ H1,s

−qθ(M \ Σq, R) ,

Π1−qθ

(M, R) ⊂ H1,s−qθ

(M, R) .

Thus the setFs := F+s ∩ F−

s has the required properties since it has fullmeasure and the inclusions (3.53) hold.

By Lemma 3.14 and Theorem 3.18 the following holds:

Corollary 3.19. For any s > 3 there exists a full measure setFs ⊂ S1

such that, for allθ ∈ Fs, the spacesIs±qθ

(M) ⊂ H−sq (M) of horizontally or

vertically quasi-invariant distributions have dimensionat least2g + σ − 1and the spacesIs

±qθ(M) ⊂ H−s

q (M) of horizontally or vertically invariantdistributions have dimension at least2g − 1.

Corollary 3.20. For anys > 3 and for almost allθ ∈ S1,

(3.58) H1,s±qθ

(M, R) = Π1±qθ

(M, R) .

For anys > 4 and for almost allθ ∈ S1,

(3.59) H1,s±qθ

(M \ Σq, R) = H1(M \ Σq, R) .

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54 GIOVANNI FORNI †

Proof. The inclusionsH1,s±q (M, R) ⊂ Π1

±q(M, R) hold for any orientablequadratic differentialq onM and for anys > 0. In fact,

(3.60)[C ∧ ℑ(q1/2)](1) = ıSC(ωq) = 0 , if C ∈ Bq(M) ;

[C ∧ ℜ(q1/2)](1) = −ıT C(ωq) = 0 , if C ∈ B−q(M) .

Thus, the identity (3.58) follows immediately from Theorem3.18.

By Theorem 3.18, in order to prove the identity (3.59) it is enough to provethat for almost allθ ∈ S1 the cohomology class[ℜ(qθ)] ∈ Bqθ

(M \ Σq)and the cohomology class[ℑ(qθ)] ∈ B−qθ

(M \ Σq). By Lemma 3.17 theargument is therefore reduced to the construction, for anys > 3 and for al-most allθ ∈ S1, of a solutionU ∈ H−s

q (M) of the cohomological equationSθU = 1 [TθU = 1] in D(M \ Σq). Such a construction can be carried outas follows. Letδp be the Dirac mass at any pointp ∈ Σq. The distributionF = 1 − δp ∈ H−s(M) ⊂ H−s

q (M) for any s > 1. We claim that forany s > 3 and for almost allθ ∈ S1 there exists a distributional solutionU ∈ H−s

q (M) of the cohomological equationSθU = F [TθU = F ] inH−s−1

q (M). It follows thatU is a solution of the cohomological equationSθU = 1 [TθU = 1] in D(M \ Σq). The above claim is proved as follows.By [For97], Prop. 4.6, or [For02], Lemma 7.3, sinceF ∈ H−2

q (M) van-ishes on constant functions, there exists a distributionf ∈ H−1

q (M) suchthat∂+

q f = F (as well as a distributionf ′ ∈ H−1q (M) such that∂−

q f ′ = F ).By Theorem 3.2, for almost allθ ∈ S1 and for alls > 2, there exists a so-lution u ∈ H−s

q (M) of the equationSθu = f [Tθu = f ]. The distributionU := ∂+

q u ∈ H−sq (M) for anys > 3 and solves the cohomological equation

SθU = F [TθU = F ] in H−s−1q (M).

The structure of the space of basic currents with vanishing cohomologyclass with respect to the filtration induced by weighted Sobolev spaces withinteger exponent was described in [For02], §7. We extend below the resultsof [For02] to fractional weighted Sobolev spaces.

Let δ±q : B±q(M \ Σq) → B±q(M \ Σq) be the linear maps defined asfollows (see [For02], formula(7.18′)):

(3.61)δq(C) := − d

(C ∧ ℜ(q1/2)

)⋆, for C ∈ Bq(M \ Σq) ;

δ−q(C) := d(C ∧ ℑ(q1/2)

)⋆, for C ∈ B−q(M \ Σq) .

It can be proved by Lemma 3.14 and by the definition of the weightedSobolev spacesHs

q (M) andW sq (M) that the above formulas (3.61) define,

for all s ∈ R+, bounded linear maps

(3.62)δs±q : Bs−1

±q (M \ Σq) → Bs±q(M \ Σq) ;

δs±q : Bs−1

±q (M) → Bs±q(M) .

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SOBOLEV REGULARITY OF SOLUTIONS 55

We remark that a similar statement is false in general for theFriedrichsSobolev spaces of currents. The following result extends Theorem 7.7 of[For02] to fractional weighted Sobolev spaces.

Theorem 3.21.For all s ∈ R+ there exist exact sequences

(3.63)0 → R → Bs−1

±q (M \ Σq)δs±q

−−−−→Bs±q(M \ Σq)

jq−−−→H1(M \ Σq, R) ;

0 → R → Bs−1±q (M)

δs±q

−−−−→Bs±q(M)

jq−−−→H1(M, R) .

Proof. The mapsi±q : R → Bs−1±q (M) ⊂ Bs−1

±q (M \ Σq) defined as

iq(τ) = τηS and i−q(τ) = τηT , for all τ ∈ R ,

are clearly injective and the kernels Ker(δs±q) = i±q(R), for all s ∈ R+. In

fact, if a currentC ∈ Ker(δs+q), then the distribution

(C ∧ ℜ(q1/2)

)⋆∈ R,

henceC ∧ ℜ(q1/2) ∈ R · ωq and C ∈ R · ℜ(q1/2) + R · ℑ(q1/2). It followsthat C ∈ R · ℑ(q1/2) sinceC ∈ Bq(M \ Σq). Similarly, if the currentC ∈ Ker(δs

−q), it follows thatC ∧ ℑ(q1/2) ∈ R · ωq andC ∈ R · ℜ(q1/2)sinceC ∈ B−q(M \ Σq). This proves the inclusions Ker(δs

±q) ⊂ i±q(R).The opposite inclusions are immediate.

By Lemma 3.16 a currentC ∈ Bs±q(M \ Σq) has zero cohomology class,

that is, it is in the kernel of the cohomology map, if and only if there existsa distributionUC ∈ H−s+1

q (M) such thatdU∗C = C in D(M \ Σq) and

this identity holds inSq(M), hence inW−sq (M), if C ∈ Bs

±q(M). It isimmediate to verify thatUC ∈ Is−1

±q (M\Σq) if and only ifC ∈ Bs±q(M\Σq)

and thatUC ∈ Is−1±q (M) if and only if C ∈ Bs

±q(M). By Lemma 3.14 wehave thus proved that the mapjq : Bs

±q(M \ Σq) → H1(M \ Σq, R) haskernel equal to the rangeδs

±q

(Bs−1

±q (M))

and that the mapjq : Bs±q(M\) →

H1(M, R) has kernel equal to the rangeδs±q

(Bs−1

±q (M)).

Finer results on invariant distributions and on smooth solutions of the co-homological equation for directional flows can be obtained by combiningthe results of this section with the renormalization methodbased on theTeichmüller flow and related cocycles, such as the Kontsevich-Zorich co-cycle. Our goal is to improve upon the results of Marmi, Moussa and Yoc-coz [MMY05] who have studied the cohomological equation forintervalexchange transformations solely by methods based on the renormalizationdynamics (the Rauzy-Veech-Zorich induction).

4. COCYCLES OVER THETEICHMÜLLER FLOW

4.1. The Kontsevich-Zorich cocycle.The Kontsevich-Zorich cocycle isa multiplicative cocycle over the Teichmüller geodesic flowon the moduli

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56 GIOVANNI FORNI †

space of (orientable) holomorphic quadratic differentials on compact Rie-mann surfaces. This cocycle appears in the study of the dynamics of intervalexchange transformations and of (translation) flows on surfaces for whichit represents a renormalization dynamics and of the Teichmüller flow itself.In fact, the study tangent cocycle of the Teichmüller flow canbe reduced tothat of the Kontsevich-Zorich cocycle.

Let Tg andQg be respectively theTeichmüller spacesof complex (confor-mal) structures and of holomorphic quadratic differentials on a surface ofgenusg ≥ 1. We recall that the spacesTg andQg can be described as fol-lows. Let Diff+0 (M) is the group of orientation preserving diffeomorphismsof the surfaceM which are isotopic to the identity (equivalently, it is theconnected component of the identity in the Lie group of all orientation pre-serving diffeomorphisms ofM). By definition

(4.1)Tg := complex (conformal) structures/Diff +

0 (M) ,

Qg := holomorphic quadratic differentials/Diff +0 (M) .

A theorem of L. Ahlfors, L. Bers and S. Wolpert states thatTg has a complexstructure holomorphically equivalent to that of a Stein (strongly pseudo-convex) domain inC3g−3 [Ber74], §6, or [Nag88], Chap. 3, 4 and Appendix§6. The spaceQg of holomorphic quadratic differentials is a complex vectorbundle overTg which can be identified to the cotangent bundle ofTg. LetΓg := Diff +(M)/Diff +

0 (M) be themapping class groupand letRg, Mg

be respectively themoduli spacesof complex (conformal) structures andof holomorphic quadratic differentials on a surface of genus g ≥ 1. ThespacesRg andMg can be described as the quotient spaces:

(4.2) Rg := Tg/Γg , Mg := Qg/Γg ,

In caseg = 1, the Teichmüller spaceT1 of elliptic curves (complex struc-tures onT 2) is isomorphic to the upper half planeC+ and the TeichmüllerspaceQ1 of holomorphic quadratic differentials on elliptic curvesis a com-plex line bundle overT1 (see [Nag88], Ex. 2.1.8). The mapping class groupcan be identified with the latticeSL(2, Z) which acts on the upper half planeC+ in the standard way. The moduli spaceR1 := C+/SL(2, Z) is a non-compact finite volume surface with constant negative curvature, called themodular surface. The moduli spaceM1 can be identified to the cotangentbundle of the modular surface.

TheTeichmüller (geodesic) flowonMg can be defined as the geodesic flowfor a natural metric onRg called theTeichmüller metric. Such a metricmeasures the amount ofquasi-conformal distorsionbetween two differ-ent (equivalent classes of) complex structures inRg. In the higher genuscase, the Teichmüller metric is not Riemannian, but onlyFinsler (that is,

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SOBOLEV REGULARITY OF SOLUTIONS 57

the norm on each tangent space does not come from an euclideanproduct)and, as H. Masur proved, does not have negative curvature in any reason-able sense [Ber74], §3 (E). Ifg = 1, the Teichmüller metric coincides withthe Poincaré metric on the modular surfaceR1 [Nag88], 2.6.5, in particularit is Riemannian with constant negative curvature.There is a natural action of the Lie groupGL(2, R) onQg (see also [HS05],§1.4 or [Mas05], §3) which is defined as follows. The map

q → (Fq, F−q) , q ∈ Qg ,

is a bijection between the spaceQg and the space of all pairs of transversemeasured foliations. We recall that transversality for a pair (F,F⊥) of mea-sured foliations is taken in the sense thatF andF⊥ have a common setΣ of(saddle) singularities, have the same index at each singularity and are trans-verse in the standard sense onM \ Σ. The setΣ of common singularitiescoincides with the setΣq of zeroes of the holomorphic quadratic differen-tial q ≡ (F, F⊥). Any pair of transverse measured foliations is determinedlocally by a pair(η, η⊥) of (locally defined) transverse real-valued closed1-forms. The groupGL(2, R) acts naturally by left multiplication on thespace of (locally defined) pairs of transverse real-valued closed1-forms,hence it acts on the space of all pairs of transverse measuredfoliations anon the space ofQg of holomorphic quadratic differentials. Such an actionis equivariant with respect to the action of the mapping class group, henceit passes to the quotientMg. The Teichmüller flowGtt∈R is given bythe action of the diagonal subgroupdiag(e−t, et)t∈R on Qg (on Mg). Inother terms, if we identify holomorphic quadratic differentials with pairs oftransverse measured foliations as explained above, we have:

(4.3) Gt(Fq, F−q) := (e−tFq, etF−q) .

In geometric terms, the action of the Teichmüller flow on quadratic differ-entials induces a one-parameter family of deformations of the conformalstructure which consist in contracting along vertical leaves (with respect tothe horizontal length) and expanding along horizontal leaves (with respectto the vertical length) by reciprocal (exponential) factors. The reader cancompare the definition in terms of theSL(2, R)-action with the analogousdescription of the geodesic flow on a surface of constant negative curvature(such as the modular surface). In fact, in caseg = 1 the above definitionreduces to the standard Lie group presentation of the geodesic flow on themodular surface: the unit sub-bundleM

(1)1 ⊂ M1 of all holomorphic qua-

dratic differentials of unit total area on elliptic curves can be identified withthe homogeneous spaceSL(2, R)/SL(2, Z) and the geodesic flow on themodular surface is then identified with the action of the diagonal subgroupof SL(2, R) onSL(2, R)/SL(2, Z).

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58 GIOVANNI FORNI †

We list below, following [Vee90], [Kon97], the main structures carried bythe moduli spaceMg of quadratic differentials:

(1) the moduli spaceMg is a stratified analytic orbifold; each stra-tum Mκ ⊂ Mg (determined by the multiplicitiesκ = (k1, . . . , kσ)of the zeroesp1, . . . , pσ of quadratic differentials) isGL(2, R)-invariant, hence in particularGt-invariant;

(2) The total area functionA : Mg → R+,

A(q) :=

M

|q| ,

is SL(2, R)-invariant; hence theunit bundleM(1)g := A−1(1) and

its strataM(1)κ := Mκ ∩ M

(1)g areSL(2, R)-invariant and, in partic-

ular,Gt-invariant.

Let Mκ be a stratum oforientablequadratic differentials, that is,quadratic differentials which are squares of holomorphic1-forms.In this case, the natural numbers(k1, . . . , kσ) are all even.

(3) The stratum of squaresMκ has a locally affine structure modeledon the affine spaceH1(M, Σκ; C), with Σκ := p1, . . . , pσ. Localcharts are given by the period mapq → [q1/2] ∈ H1(M, Σκ; C).

(4) The Lebesgue measure on the euclidean spaceH1(M, Σκ; C), nor-malized so that the quotient torus

H1(M, Σκ; C) / H1(M, Σκ; Z ⊕ iZ)

has volume1, induces an absolutely continuousSL(2, R)-invariantmeasureµκ on Mκ. The conditional measureµ(1)

κ induced on thestratumM

(1)κ is SL(2, R)-invariant, henceGt-invariant.

All the above structures (the stratification, the area function, the locallyaffine structure on the strata of squares) lift to corresponding structures atthe level of the Teichmüller space of quadratic differentials, equivariant un-der the action of the mapping class group.It was discovered by W. Veech [Vee90] thatM

(1)κ has in general several

connected components. The connected components for the strata of abeliandifferentials (or equivalently of orientable quadratic differentials) have beenclassified completely by M. Kontsevich and A. Zorich [KZ03].A similarclassification for the case of strata of non-orientable holomorphic quadraticdifferentials has been recently obtained by E. Lanneau in his thesis [Lan03].Taking this phenomenon into account, the following result holds:

Theorem 4.1. [Mas82], [Vee86] The total volume of the measureµ(1)κ on

M(1)κ is finite and the Teichmüller geodesic flowGtt∈R is ergodic on each

connected component ofM(1)κ .

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SOBOLEV REGULARITY OF SOLUTIONS 59

We will describe below several results about the Lyapunov structure of var-ious coycles over the Teichmüller geodesic flow (including the tangent co-cycle). We refer the reader to the recent and excellent survey [BP05] byL. Barreira and Ya. Pesin (and references therein), which covers all therelevant results on thetheory of Lyapunov exponents, including Oseledec’smultiplicative ergodic theorem and the Oseledec-Pesin reduction theorem.The Lyapunov spectrum of the Teichmüller flow, with respect to any ergodicinvariant probability measureµ on the moduli space, has symmetries. Infact, there exists non-negative numbersλµ

1 = 1 ≥ λµ2 ≥ · · · ≥ λµ

g such thatthe Lyapunov spectrum of the Teichmüller flow has the following form (see§5 in [Zor96], §7 in [Kon97] or §2.3 in [Zor99]):

(4.4)

2 ≥ (1 + λµ2) ≥ · · · ≥ (1 + λµ

g ) ≥

#(Σκ)−1︷ ︸︸ ︷1 = · · · = 1 ≥ (1 − λµ

g ) ≥

≥ · · · ≥ (1 − λµ2 ) ≥ 0 = 0 ≥ −(1 − λµ

2 ) ≥ · · · ≥ −(1 − λµg ) ≥

≥ −1 = · · · = −1︸ ︷︷ ︸#(Σκ)−1

≥ −(1 + λµg ) ≥ · · · ≥ −(1 + λµ

2) ≥ −2 .

In [Vee86] Veech proved that the Teichmüller flow isnon-uniformly hy-perbolic, in the sense that all of its Lyapunov exponents, except one cor-responding to the flow direction, are non-zero. By formulas (4.4) Veech’stheorem can be formulated as follows:

Theorem 4.2. [Vee86]The inequality

(4.5) λµ2 < λµ

1 = 1 .

holds ifµ is the absolutely continuousSL(2, R)-invariant ergodic proba-bility measure on any connected component of a stratumM

(1)κ ⊂ M

(1)g of

orientable quadratic differentials.

M. Kontsevich and A. Zorich have interpreted the non-negative numbers

(4.6) λµ1 = 1 ≥ λµ

2 ≥ · · · ≥ λµg

as the non-negative Lyapunov exponents of a symplectic cocycle over theTeichmüller flow, which we now describe.The Kontsevich-Zorich cocycleΦtt∈R is a cocycle over the Teichmüllerflow Gtt∈R on the moduli spaceMg, defined as the projection of the trivialcocycle

(4.7) Gt × id : Qg × H1(M, R) → Qg × H1(M, R)

onto the orbifold vector bundleH1g(M, R) overMg defined as

(4.8) H1g(M, R) :=

(Qg × H1(M, R)

)/Γg .

The mapping class groupΓg acts naturally on the cohomology by pull-back.

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60 GIOVANNI FORNI †

The Kontsevich-Zorich cocycle was introduced in [Kon97] asa continuous-time version of the Zorich cocycle. The Zorich cocycle was introduced ear-lier by A. Zorich [Zor96], [Zor97] in order to explain polynomial deviationsin the homological asymptotic behavior of typical leaves oforientable mea-sured foliations on compact surfaces, a phenomenon he had discovered innumerical experiments [Zor94]. We recall that the real homologyH1(M, R)and the real cohomologyH1(M, R) of an orientable closed surfaceM are(symplectically) isomorphic by Poincaré duality.Let Mκ ⊂ Mg be a stratum of orientable holomorphic quadratic differen-tials andQκ ⊂ Qg the pull-back of the stratumMκ to the Teichmüller spaceof quadratic differentials. Therelative Kontsevich-Zorich coycleΦtt∈R

is defined as the projection of the trivial cocycle

(4.9) Gt × id :⋃

q∈Qκ

q × H1(M, Σq; R) →⋃

q∈Qκ

q × H1(M, Σq; R)

on the orbifold vector bundleH1κ(M, Σκ; R) overMκ defined as

(4.10) H1κ(M, Σκ; R) :=

( ⋃

q∈Qκ

q × H1(M, Σq; R))/Γg .

By a similar construction it is possible to define apuncturedKontsevich-Zorich coycleΨtt∈R defined as the projection of the trivial cocycle

(4.11) Gt × id :⋃

q∈Qκ

q×H1(M \Σq; R) →⋃

q∈Qκ

q×H1(M \Σq; R)

onto the orbifold vector bundleH1κ(M \ Σκ; R) overMκ defined as

(4.12) H1κ(M \ Σκ; R) :=

( ⋃

q∈Qκ

q × H1(M \ Σq; R))/Γg .

However, by Poincaré-Lefschetz duality, there exists a natural isomorphism

(4.13) H1(M, Σκ; R) ≡ H1(M \ Σκ; R)∗ ,

hence the punctured Kontsevich-Zorich cocycleΨtt∈R is isomorphic tothe dualΦ∗

tt∈R of the relative Kontsevich-Zorich cocycle.

LetH1κ(M, R) be the restriction to the stratumM(1)

κ of the bundleH1g(M, R)

defined in (4.8). LetK1κ(M, Σκ; R) be the bundle defined as the kernel of

the natural bundle mapH1κ(M, Σκ; R) → H1

κ(M, R).

Lemma 4.3. The sub-bundleK1κ(M, Σκ; R) is invariant under the rela-

tive Kontsevich-Zorich cocycleΦtt∈R. The Lyapunov spectrum of therestrictionΦt|K

1κ(M, Σκ; R) consists of the single exponent0 with mul-

tiplicity #(Σκ) − 1. In fact, there exists a Lyapunov norm on the bundleK1

κ(M, Σκ; R) for which the cocycle is isometric.

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SOBOLEV REGULARITY OF SOLUTIONS 61

Proof. By de Rham theorem for the relative cohomology, the relativeco-homology complexH∗(M, Σq; R) is isomorphic to the cohomology of thecomplex ofrelativedifferential forms, which are defined as all differentialforms vanishing atΣq. The mapH∗(M, Σq; R) → H∗(M, R) is naturallydefined on the de Rham cohomology since every closed, exact relative formis also a closed, respectively exact form onM . It follows that any classc ∈ H1(M, Σq; R) which belongs to the kernelK1(M, Σq; R) of the mapH1(M, Σq; R) → H1(M, R) can be represented by a differential1-formexact onM . Hence for anyc ∈ K1(M, Σq; R) there exists a smooth func-tion fc : M → R, uniquely determined up to the addition of any functionvanishing atΣq, such thatc = [dfc] as a relative de Rham cohomology class.

As a consequence of the above discussion, the following formula yields awell-defined euclidean norm‖ · ‖K onK1(M, Σq; R):

(4.14) ‖c‖2K :=

p1,p2∈Σq

|fc(p1) − fc(p2)|2 , c = [dfc] ∈ K1(M, Σq; R) .

The norm (4.14) induces an euclidean norm on the bundleK1κ(M, Σκ; R)

which is invariant under the relative Kontsevich-Zorich cocycle.

Since the vector bundleH1g(M, R) has a symplectic structure, given by the

intersection form on its fibers, which are isomorphic to the cohomologyH1(M, R), the Lyapunov spectrum of the cocycleΦtt∈R with respect toanyGt-invariant ergodic probability measureµ onM

(1)g is symmetric:

(4.15) λµ1 ≥ · · · ≥ λµ

g ≥ 0 ≥ λµg+1 = −λµ

g ≥ · · · ≥ λµ2g = −λµ

1 .

The non-negative Kontsevich-Zorich exponents coincide with the numbers(4.6) which appear in the Lyapunov spectrum (4.4) of the Teichmüller flow.This relation is explained for instance in [For05].

Zorich conjectured (see [Zor96]) that the exponents (4.15)are all distinctand different from zero whenµ is the canonical absolutely continuous er-godic invariant probability measure on any connected component of a stra-tum M

(1)κ of orientable quadratic differentials. In other terms he conjec-

tured that the canonical measures areKZ-hyperbolic and andKZ-simple,according to the following:

Definition 4.4. A Gt-invariant ergodic probability measureµ on a stra-tum of orientable quadratic differentials will be calledKZ-hyperbolicif theKontsevich-Zorich cocycle(Φt, µ) is non-uniformly hyperbolic, in thesense that its Lyapunov exponents satisfy the inequalities

(4.16) λµ1 = 1 ≥ λµ

2 ≥ · · · ≥ λµg > 0 .

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62 GIOVANNI FORNI †

A KZ-hyperbolic measureµ on a stratum of orientable quadratic differen-tials will be calledKZ-simpleif the Kontsevich-Zorich cocycle(Φt, µ) issimple, in the sense that all inequalities (4.16) are strict.

In [For02] we have proved the following result:

Theorem 4.5. ([For02], Th. 8.5) The absolutely continuous,SL(2, R)-invariant, ergodic probability measure on any connected component of astratumM

(1)κ ⊂ M

(1)g of orientable quadratic differentials is KZ-hyperbolic.

In [For05], §7, we have given an example of aSL(2, R)-invariant measureµ on M3, supported on the closedSL(2, R)-orbit of a particular genus3branched cover of the2-torus, such thatλµ

2 = λµ3 = 0.

A proof of the simplicity of the Zorich and Kontsevich-Zorich cocycles,which yields in particular a new independent proof of Theorem 4.5, hasbeen recently obtained by A. Avila and M. Viana [AV05] by methods com-pletely different from ours.

Theorem 4.6. [AV05] The absolutely continuous,SL(2, R)-invariant, er-godic probability measure on any connected component of a stratumM

(1)κ ⊂

M(1)g of orientable quadratic differentials is KZ-simple.

The results of this paper do not depend in any way on the simplicity of theKontsevich-Zorich cocycle, while the non-uniform hyperbolicity is crucialto the sharp estimates on the regularity of solutions of the cohomologicalequation proven in §5.3. However, our results can be refined by taking intoaccount that the Kontsevich-Zorich exponents (4.16) are all distinct.

Our proof of Theorem 4.5 in [For02] yields in particular a newindependentproof of a strong version of Veech’s Theorem 4.2. In fact, we have provedthat the strict inequality (4.5) holds for an arbitrary probability Gt-invariantmeasure on any stratum of orientable quadratic differentials. By combiningour methods with a recent result of J. Athreya [Ath06] on large deviationsof the Teichmüller flow, it is possible to prove a similar strict upper boundfor the second exponent for Lebesgue almost all quadratic differentials inany orbit of the circle groupSO(2, R) on any stratumMκ of orientablequadratic differentials.

We recall below the variational formulas for the evolution of Hodge normof absolute cohomology classes under the Kontsevich-Zorich cocycle. Fol-lowing §2 in [For02], such formulas can be written in terms ofa naturalR-linear extensionUq of the partial isometryUq, defined in formula (3.6),which plays a crucial role in the construction of distributional solutions ofthe cohomological equation. LetUq : L2

q(M) → L2q(M) be theR-linear

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SOBOLEV REGULARITY OF SOLUTIONS 63

isometry defined as follows in terms of the partial isometryUq and of theorthogonal projectionsπ±

q : L2q(M) → M±

q :

(4.17) Uq := Uq (I − π−q ) − π−

q .

Let qtt∈R be the orbit of an orientable quadratic differentialq ∈ Q(1)κ un-

der the Teichmüller flowGtt∈R. We remark that by the definition of theTeichmüller flowGtt∈R, the area formωt of the metric induced by thequadratic differentialqt is constantequal toωq for all t ∈ R. Hence theHilbert spaceL2

q(M) is invariant under the action of the Teichmüller flow(in fact, it is invariant under the fullSL(2, R) action). For eacht ∈ R,let Mt the Riemann surface carryingqt ∈ Q

(1)κ and letM±

t ⊂ L2q(M) be

the space of meromorphic, respectively anti-meromorphic,functions on theRiemann surfaceMt. Such spaces are respectively the kernels of the ad-joints of the Cauchy-Riemann operators∂∓

t , associated to the holomorphicquadratic differentialqt. The dimension ofM±

t is constant equal to thegenusg ≥ 1 of M (see §2.1) and it can be proved thatM±

t t∈R are smoothfamilies ofg-dimensional subspaces of the fixed Hilbert spaceL2

q(M).

By (3.40) there exists a one-parameter familym+t t∈R ⊂ M+

t such that

(4.18) ct = c+qt(m+

t ) := [ℜ(m+t q

1/2t )] ∈ H1(Mt, R) .

Lemma 4.7. ([For02], Lemma 2.1) The ordinary differential equation

(4.19) u′ = Uqt(u)

is well defined inL2q(M) and satifies the following properties:

(1) Solutions of the Cauchy problem for (4.19) exist for all times andare uniquely determined by the initial condition;

(2) If ut ∈ L2q(M) is any solution of (4.19) such that the initial condi-

tion u0 ∈ M+q , thenut ∈ M+

t for all t ∈ R.(3) Letm+

t ∈ M+t be the unique solution of (4.19) with initial condition

m+0 = m+ ∈ M+

q . For all t ∈ R, we have

(4.20) Φt

(c+q (m+)

)= c+

qt(m+

t ) .

It follows immediately from Proposition 2.4 that, for everyu ∈ L2q(M),

there exist functionsv± ∈ H1q (M) such that

(4.21) u = ∂+q v+ + π−

q (u) = ∂−q v− + π+

q (u) .

The O. D. E. in formula (4.19) can be written explicitly, in terms of theorthogonal decompositions (4.21), as follows. Letπ±

t : L2q(M) → M±

t

denote the orthogonal projections in the (fixed) Hilbert spaceL2q(M). By

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64 GIOVANNI FORNI †

definition, the projectionsπ±t coincide with the projectionsπ±

q for q = qt,for anyt ∈ R. A functionu ∈ C1

(R, L2

q(M))

satisfies equation (4.19) iff

(4.22)

ut = ∂+

t vt + π−t (ut) ;

ddt

ut = ∂−t vt − π−

t (ut) .

An immediate consequence of Lemma 4.7 is the following result on thevariation of the Hodge norm of cohomology classes under the action of theKontsevich-Zorich cocycle. LetBq : L2

q(M)×L2q(M) → C be the complex

bilinear form given by

(4.23) Bq(u, v) :=

M

u v ωq , for all u, v ∈ L2q(M) .

Corollary 4.8. ([For02], Lemma 2.1’) The variation of the Hodge norm‖ct‖H , which coincides with theL2

q-norm |m+t |0 under the identification

(4.18), is given by the following formulas:

(4.24)

(a)d

dt|m+

t |20 = −2ℜ[Bq(m

+t )] = −2ℜ

[∫

M

(m+t )2ωq

];

(b)d2

dt2|m+

t |20 = 4

|π−

t (m+t )|20 − ℜ

[∫

M

(∂+t vt) (∂−

t vt) ωq

].

The second order variational formula (4.24),(b), is crucial in our proof oflower boundsfor the Kontsevich-Zorich exponents (see Theorem 4.5). Thefirst order variational formula (4.24),(a), implies quite immediately an ef-fective upper bound for the second exponent, which yields inparticular theaverage spectral gapresult proved in [For02], Corollary 2.2 (a generaliza-tion of Veech’s Theorem 4.2 to arbitraryGt-invariant ergodic probabilitymeasures):

Theorem 4.9. ([For02], Corollary 2.2) The inequality

(4.25) λµ2 < λµ

1 = 1 .

holds for anyGt-invariant ergodic probability measureµ on any connectedcomponent of any stratumM(1)

κ of orientable quadratic differentials.

The above spectral gap result implies the following unique ergodicity theo-rem for measured foliations:

Corollary 4.10. For any stratumM(1)κ of orientable quadratic differentials,

the set of quadratic differentialsq ∈ M(1)κ with minimal but not uniquely

ergodic horizontal [vertical] foliation has zero measure with respect to anyGt-invariant probability measureµ onM

(1)κ .

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SOBOLEV REGULARITY OF SOLUTIONS 65

In the remainder of this section we prove apointwise spectral gapresultwhich holds for almost all quadratic differentials inany orbit of the circlegroupSO(2, R) on any stratumM

(1)κ of orientable quadratic differentials.

The argument is based on formula (4.24),(a) and, as mentioned above, ona result of J. Athreya [Ath06] on large deviations of the Teichmüller flow.

The upper second Lyapunov exponent of the Kontsevich-Zorich cocycle atany (orientable) quadratic differentialsq ∈ M

(1)κ is defined as follows. Let

Iq(M, R) ⊂ H1(M, R) be the subspace of real dimension2 defined as

(4.26) Iq(M, R) := R · ℜ(q1/2) + R · ℑ(q1/2)

and letI⊥q (M, R) be the symplectic orthogonal ofIq(M, R) in H1(M, R),

with respect to the symplectic structure induced by the intersection form:

(4.27) I⊥q (M, R) := c ∈ H1(M, R)|c ∧ [q1/2] = 0 .

The complementary sub-bundlesIκ(M, R) and I⊥κ (M, R) ⊂ H1(M, R),

with fibers at anyq ∈ M(1)κ respectively equal toIq(M, R) andI⊥

q (M, R),are invariant under the Kontsevich-Zorich cocycle. In fact, it is immediateto verify that the sub-bundleIκ(M, R) is invariant under the Kontsevich-Zorich cocycle and that the Lyapunov spectrum of the restriction of theKontsevich-Zorich cocycle toIκ(M, R) equals1,−1 (both exponentswith multiplicity 1). Since the Kontsevich-Zorich cocycle is symplectic,the symplectic orthogonal bundleI⊥

κ (M, R) is also invariant. In addition, itis not difficult to verify that1 is the top (upper) exponent for the cocycle onthe full cohomology bundleH1

κ(M, R).

Thesecond upper (forward) exponentof the Kontsevich-Zorich cocycle isthe top upper (forward) Lyapunov exponent at any quadratic differentialq ∈ M

(1)κ of the restriction of the cocycle to the sub-bundleI⊥

κ (M, R):

(4.28) λ+2 (q) := lim sup

t→+∞

1

tlog ‖Φt|I

⊥q (M, R)‖H .

Theorem 4.11.For any stratumM(1)κ of orientable quadratic differentials,

there exists a measurable functionLκ : M(1)κ → [0, 1) such that for any

(orientable) quadratic differentialq ∈ M(1)κ ,

(4.29) λ+2 (qθ) ≤ Lκ(q) < 1 , for almost all θ ∈ S1 .

Proof. The argument follows closely the proof of Corollary 2.2 in [For02].Under the isomorphism (3.40), the vector spaceI⊥

q (M, R) is representedby meromorphic functions withzero average(orthogonal to constant func-tions). It can be seen that the subspace of zero average meromorphic func-tions is invariant under the flow of equation (4.20) or, equivalently (4.22).

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66 GIOVANNI FORNI †

By formula(a) in (4.24),

(4.30)d

dtlog |m+

t |20 = −2

ℜBq(m+t )

|m+t |

20

.

Following [For02], we introduce a continuous functionΛ+κ : M

(1)κ → R

+

defined as follows: for anyq ∈ M(1)κ ,

(4.31) Λ+κ (q) := max

|Bq(m+)|

|m+|20|m+ ∈ M+

q \ 0 ,

M

m+ ωq = 0 .

Since by the Schwarz inequality,

(4.32) |Bq(m+t )| = |(m+

t , m+t )q| ≤ |m+

t |20 ,

the image of the functionΛ+κ is contained in the interval[0, 1]. We claim that

Λ+κ (q) < 1 for all q ∈ M

(1)κ . In fact,Λ+(q) = 1 if and only if there exists

a non-zeromeromorphic function with zero averagem+ ∈ M+q such that

|(m+, m+)q| = |m+|20. A well-known property of the Schwarz inequalitythen implies that there existsu ∈ C such thatm+ = u m+. However,it cannot be so, sincem+ would be meromorphic and anti-meromorphic,hence constant, and by the zero average condition it would bezero.

It follows from formula (4.30) that, for anyq ∈ M(1)κ ,

(4.33)1

tlog ‖Φt|I

⊥q (M, R)‖H ≤

1

t

∫ t

0

Λ+κ (Gs(q)) ds .

Let q0 ∈ M(1)κ . By the large deviation result of J. Athreya (see [Ath06],

Corollary 2.4) the following holds. For anyλ < 1 there exists a compactsetK ⊂ M

(1)κ such that, for almost allq ∈ SO(2, R) · q0,

(4.34) lim supt→∞

1

t|0 ≤ s ≤ t |Gs(q) 6∈ K| ≤ λ .

Let ΛK := maxΛκ(q) | q ∈ K and, for any(t, q) ∈ R+ × M(1)κ , let

EK(t, q) := |0 ≤ s ≤ t |Gs(q) 6∈ K| .

Since the functionΛ+κ is continuous andΛ+

κ (q) < 1 for all q ∈ M(1)κ , its

maximum on any compact set is< 1, in particularΛK < 1. The followingimmediate inequality holds:

(4.35)∫ t

0

Λ+κ (Gs(q)) ds ≤ (1 − ΛK) EK(t, q) + t ΛK .

Page 67: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 67

It follows from (4.33), (4.34) and (4.35) that, for almost all q ∈ SO(2, R) ·q0,

(4.36) lim supt→+∞

1

tlog ‖Φt|I

⊥q (M, R)‖H ≤ (1 − ΛK)λ + ΛK < 1 .

The functionLκ : M(1)κ → [0, 1) can be defined for everyq0 ∈ M

(1)κ as

the essential supremum of the second upper Lyapunov exponent over theorbit SO(2, R) · q0 of the circle group. Such a function is measurable bydefinition and it is everywhere< 1 by the above argument.

For any Oseledec regular pointq ∈ Q(1)κ of the Kontsevich-Zorich cocycle,

letE+q (M, R)[E−

q (M, R)] ⊂ H1(M, R) be the unstable [stable] subspace ofthe Kontsevich-Zorich cocycle. Homology cycles which are Poincaré dualsto cohomology classes inE±

q (M, R) are calledZorich cyclesfor the foli-ationF±q. It follows from Theorem 4.5 and from the symplectic propertyof the cocycle thatE±

q are transverse Lagrangian subspaces (with respectto the intersection form), as conjectured by Zorich in [Zor96]. In [For02],Theorem 8.3 (see also [For05], Theorem 8.2) we have proved the followingrepresentation theorem:

Theorem 4.12.For Lebesgue almost allq ∈ M(1)κ , we have

(4.37) E+q (M, R) = H1,1

q (M, R) , E−q (M, R) = H1,1

−q (M, R) .

(The Poincaré duals of Zorich cycles for a generic orientable measuredfoliation F are represented by basic currents forF of Sobolev order≤ 1).

We prove below a conjectural relation between Lyapunov exponents of acohomology class under the Kontsevich-Zorich cocycle and the Sobolevregularity of the basic current representing the cohomology class. This re-sult answers in the affirmative a question posed by the authorin [For02](Question 9.9) and, independently by M. Kontsevich (personal communi-cation). In order to formulate our results in the greatest possible generality,we introduce the following class of measures on the moduli space.

Definition 4.13. A measureµ on the moduli spaceMg of holomorphic qua-dratic differentials will be calledSO(2, R)-absolutely continousif inducesabsolutely continous conditional measures onπ∗(µ)-almost every fiber ofthe fibrationπ : Mg → Mg/SO(2, R) (with respect to the Haar/Lebesguemeasure class on each fiber).

It is immediate that anySO(2, R)-invariant, hencea fortiori anySL(2, R)-invariant measure isSO(2, R)-absolutely continous.

Page 68: Sobolev Regularity

68 GIOVANNI FORNI †

Lemma 4.14. Let µ be anySO(2, R)-absolutely continous,Gt-invariantergodic probability measure on a stratumM(1)

κ of orientable quadratic dif-ferentials. Forµ-almost allq ∈ M

(1)κ , the unique basic currentC ∈ B1

q(M)

[C ∈ B1−q(M)] which represents a cohomology classc ∈ E+

q (M, R)[c ∈ E−

q (M, R)] of Lyapunov exponentλ(c) > 0 [λ(c) < 0] under theKontsevich-Zorich cocycle, has the following Sobolev regularity:

C ∈ W−sq (M) , for all s > 1 − |λ(c)| .

Proof. The argument follows the proof of Lemma 8.2 in [For02]. The in-terpolation inequality (2.26) for fractional weighted Sobolev spaces allowsus to estimate the Sobolev regularity of the basic current constructed there.

Let q ∈ M(1)κ be any Oseledec regular point of the Kontsevich-Zorich co-

cycle and letc ∈ H1(M, R) be a cohomology class. Letqtt∈R ⊂ Q(1)κ

denotes the lift to the Teichmüller space of the orbitGt(q)t∈R of q underthe Teichmüller flow. LetM+

t ⊂ L2q(M) be the space of meromorphic

functions on the Riemann surfaceMt carrying the quadratic differentialqt ∈ Q

(1)κ . According to the representation formula (3.40), there exists a

(smooth) familym+t t∈R ⊂ M+

t such that, for eacht ∈ R,

(4.38) Φt(c) = ℜ[m+t q

1/2t ] .

By the variational formulas (4.20) and (4.22), there existsa (smooth) familyvt∈R ⊂ H1(M) of zero-average functionssuch that

(4.39)

m+

t = ∂+t vt + π−

t (m+t ) ,

ddt

m+t = ∂−

t vt − π−t (m+

t ) .

SinceΦt(c) ≡ c ∈ H1(M, R) (by the definition (4.7) of the Kontsevich-Zorich cocycleΦt over the Teichmüller space), for eacht ∈ R thereexists a unique zero average functionUt ∈ L2

q(M) such that

(4.40) dUt = ℜ[m+t q

1/2t ] − ℜ[m+

0 q1/2] .

It follows that the familyUtt∈R is smooth and satisfies, by the variationalformulas (4.39), the following Cauchy problem inL2

q(M):

(4.41)

ddt

Ut = 2ℜ(vt) ,U0 = 0 ,

We claim that, ifc ∈ E+q (M, R) [c ∈ E−q(M, R)] has Lyapunov exponent

λ(c) > 0 [λ(c) < 0], the setUt | t ≤ 0 [ Ut | t ≥ 0] is a bounded subsetof the Hilbert spaceHs

q (M) for anys < |λ(c)|.

By Oseledec’s theorem, for any0 < λ < |λ(c)|, there exist a measurablefunctionKλ > 0 onM

(1)κ and such that

(4.42) ‖Φt(c)‖H = |m+t |0 ≤ Kλ(q)|m

+0 |0 exp(−λ|t|) , t ≤ 0 [t ≥ 0] .

Page 69: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 69

For any (orientable) quadratic differentialq ∈ Q(1)κ , let ‖q‖ denote the

length of the shortest geodesic segment with endpoints in the setΣq (ofzeroes ofq) with respect to the induced metric. By the Poincaré inequalityproved in [For02], Lemma 6.9, there exists a constantKg,σ > 0 (depend-ing on the genusg ≥ 2 of the Riemann surfaceMq and on the cardinalityσ := #(Σq) of anyq ∈ Q

(1)κ ) such that

(4.43) |v −

M

v ωq|0 ≤Kg,σ

‖q‖Q(v, v) , for all v ∈ H1

q (M) .

By the commutativity property (2.6) of the horizontal and vertical vectorfields, the Dirichlet formQ of the quadratic differential can be written as

Q(v, v) = |∂±q v|20 , for all v ∈ H1

q (M)

(where∂±q are the Cauchy-Riemann operators introduced in §2).

By the Poincaré inequality and by the orthogonality of the decompositionsin (4.39) with respect to the invariant euclidean structureon L2

q(M), wehave

(4.44) |vt|0 ≤ Kg,σ‖qt‖−1|∂+

t vt|0 ≤ Kg,σ‖qt‖−1|m+

t |0 .

It follows by (4.41), (4.42) and (4.44) that there exists a measurable functionK ′

λ > 0 onM(1)κ such that ifc ∈ E+

q (M, R) [c ∈ E−q (M, R)] the following

inequality holds for allt ≤ 0 [ t ≥ 0]:

(4.45) |d

dtUt|0 ≤ 2 |vt|0 ≤ K ′

λ(q) |m+0 |0 ‖qt‖

−1 e−λ|t| .

By (4.39) and (4.41) we also have the following straightforward estimatefor the norm of the funciond

dtUt in the fixed Hilbert spaceH1

q (M):

(4.46) |d

dtUt|1 ≤ e|t| |m+

t |0 ≤ Kλ(q)|m+0 |0 e(1−λ)|t| .

It follows from the inequalities (4.45) and (4.46), by the interpolation in-equality proved in Lemma 2.10, that for anys ∈ (0, 1) there exists a mea-surable functionKs

λ > 0 on the stratumQ(1)κ such that, ifc ∈ E+

q (M, R)[c ∈ E−

q (M, R)] the following inequality holds for allt ≤ 0 [ t ≥ 0]:

(4.47) |d

dtUt|s ≤ e|t| |m+

t |0 ≤ Ksλ(q)|m

+0 |0 e(s−λ)|t| .

SinceU0 = 0, by Minkowski’s integral inequality we finally obtain theestimate:

(4.48) |Ut|s ≤ Ksλ(q)|m

+0 |0

∫ |t|

0

e(s−λ)τ‖qτ‖−1 dτ .

Page 70: Sobolev Regularity

70 GIOVANNI FORNI †

By the logarithmic law for the Teichmüller geodesic flow on the modulispace, proved by H. Masur in [Mas93], the following estimateholds foralmost all quadratic differentialsq ∈ M

(1)κ (see [Mas93], Prop. 1.2):

(4.49) lim supτ→±∞

− log ‖qτ‖

log |τ |≤

1

2.

As a consequence, forµ-almost allq ∈ M(1)κ and for anys < λ, the integral

in formula (4.48) is uniformly bounded fort ≤ 0 [t ≥ 0]. Since for anys < |λ(c)|, there existsλ ∈ (s, |λ(c)|), it follows that the family of func-tionsUt | t ≤ 0 [Ut | t ≥ 0] is uniformy bounded in the Sobolev spaceHs

q (M) for anys < |λ(c)|, as claimed.

For anys < |λ(c)|, let U+s ∈ Hs

q (M) [U−s ∈ Hs

q (M)] be any weak limit ofthe familyUt ast → −∞ [ast → −∞], which exists since all boundedsubsets of the separable Hilbert spaceHs

q (M) are sequentially weakly com-pact. Since the functionsUt have zero average for allt ∈ R and the sub-space of zero average functions is closed inHs

q (M) for all s > 0, the weaklimit U+

s [U−s ] has zero average. By contraction of the identity (4.40) with

the horizontal vector fieldS [with the vertical vector fieldT ] we have:

(4.50)SUt = −ℜ(m+

0 ) + et ℜ(m+t ) , t ≤ 0 ,

[ TUt = ℑ(m+0 ) + e−t ℜ(m+

t ) , t ≥ 0 , ]

and by taking the limit ast → −∞ [ast → +∞],

(4.51)SU+

s = −ℜ(m+0 ) ,

[ TU−s = ℑ(m+

0 ) .]

Since for almost all quadratic differentialq ∈ Q(1)κ the horizontal foliation

[the vertical foliation] is ergodic, the solutionU+ ∈ L2q(M) [U− ∈ L2

q(M)]of the cohomological equation (4.51) is unique (if it exists). Hence thereexists a unique zero-average functionU+ ∈ L2

q(M) [U− ∈ L2q(M)] , which

solves the cohomological equation (4.51), such thatU+s = U+ [U−

s = U−]for all s < |λ(c)|. As a consequence,U+ ∈ Hs

q (M) [U− ∈ Hsq (M)] for

all s < |λ(c)|. The currentC+ ∈ W s−1q (M) [C− ∈ W s−1

q (M)] uniquelydetermined by the identity

(4.52)dU+ = C+ − ℜ[m+

0 q1/2] ,

[ dU− = C− − ℜ[m+0 q1/2] , ]

is basic for the horizontal [vertical] foliation by Lemma 3.17 and representsthe cohomology classc ∈ E+

q (M, R) [c ∈ E−q (M, R)] of Lyapunov expo-

nentλ(c) > 0 [λ(c) < 0].

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SOBOLEV REGULARITY OF SOLUTIONS 71

4.2. Distributional cocycles. Let Qκ(M) be the space of all orientablequadratic differentials, holomorphic with respect to somecomplex struc-ture on a closed surfaceM , with zeros of multiplicitiesκ = (k1, . . . , kσ).For anys ∈ R, there is a natural action of the group Diff+(M) of orientationpreserving diffeomorphisms on the trivial bundles

(4.53)⋃

q∈Qκ(M)

q × Hsq (M) ⊂

q∈Qκ(M)

q × Hsq (M) .

In fact, any diffeomorphismf ∈ Diff +(M) defines by pull-back an iso-morphismf ∗ : Hs

q (M) → Hsf∗(q)(M) which maps the subspaceHs

q (M) ⊂

Hsq (M) ontoHs

f∗(q)(M) ⊂ Hsf∗(q)(M). The quotient bundles

(4.54)

( ⋃

q∈Qκ(M)

q × Hsq (M)

)/Diff +

0 (M) ,

( ⋃

q∈Qκ(M)

q × Hsq (M)

)/Diff +

0 (M)

are well-defined bundles over the stratumQκ of the Teichmüller space ofquadratic differentials. There is natural action of the mapping class groupΓg on the bundles (4.54) induced by the action of Diff+(M) on the bundles(4.53). The resulting quotient bundles

(4.55)

Hsκ(M) :=

( ⋃

q∈Qκ(M)

q × Hsq (M)

)/Diff +(M) ,

Hsκ(M) :=

( ⋃

q∈Qκ(M)

q × Hsq (M)

)/Diff +(M)

are well-defined bundles over the stratumMκ of the moduli space. We also

introduce bundles of1-currents over a stratumMκ as follows:

(4.56)

W sκ(M) :=

( ⋃

q∈Qκ(M)

q × W sq (M)

)/Diff +(M) ,

W sκ(M) :=

( ⋃

q∈Qκ(M)

q × W sq (M)

)/Diff +(M) .

Since, for allq ∈ Qκ(M) and for alls ∈ R, the weighted Sobolev space of1-currentsW s

q (M) is isomorphic to the tensor productR2 ⊗ Hsq (M),

(4.57) W sκ(M) ≡ R

2 ⊗ Hsκ(M) , W s

κ(M) ≡ R2 ⊗ Hs

κ(M)

Page 72: Sobolev Regularity

72 GIOVANNI FORNI †

Let Gstt∈R be the cocycle over the Teichmüller flow, defined as the pro-

jection onto the bundleH−sκ (M) of the trivial skew-product cocycle

(4.58) Gt × id :⋃

q∈Qκ(M)

q × H−sq (M) →

q∈Qκ(M)

q × H−sq (M) .

Let Φstt∈R be the cocycle over the Teichmüller flow, defined as the pro-

jection onto the bundleW−sκ (M) of the trivial skew-product cocycle:

(4.59) Gt × id :⋃

q∈Qκ(M)

q × W−sq (M) →

q∈Qκ(M)

q × W−sq (M) .

Such cocycles can be described as the cocycles obtained by parallel trans-port of distributions and1-currents with respect to the trivial connectionalong the orbits of the Teichmüller flow. The following immediate identityallows to express the cocycleΦs

tt∈R in terms of the cocycleGstt∈R:

(4.60) Φst := diag(e−t, et) ⊗ Gs

t on W−sκ (M) = R

2 ⊗ H−sκ (M) .

Lemma 4.15. For any s ∈ R, the spacesHsκ(M), Hs

κ(M), W sκ(M) and

W sκ(M) are well-defined Hilbert bundles over a stratumM(1)

κ of orientablequadratic differentials. The flowsGs

tt∈R and Φstt∈R are well-defined

smooth cocycles on the bundlesH−sκ (M) andW−s

κ (M) respectively.

Proof. Let q0 ∈ Qκ(M). There exists a neighbourhoodS0 ⊂ M of the setΣq0 and a neighbourhoodD0 ⊂ Qκ(M) in the space of quadratic differen-tials with zeros of multiplicitiesκ = (k1, . . . , kσ) such that, for allq ∈ D0,Σq ⊂ S0 and the quadratic differentialq is isotopic toq0 onS0. Thus thereexists a smooth mapf : D0 → Diff +(M) such thatq = f ∗

q (q0) onS0. ThebundleHs

κ(M) andHsκ(M) are trivialized overD0 by the map

(4.61)Hs

κ(M)|D0 → D0 × Hsq0

(M);

(q, D) →(q, (f−1

q )∗(D)),

which maps the subspaceHsκ(M)|D0 ontoD0 × Hs

q0(M). It follows that

Hsκ(M) andHs

κ(M) are well-defined Hilbert bundles, hence so areW sκ(M)

andW sκ(M) by formula (4.57). The dynamical systemGs

tt∈R concideswith the product cocycleGt × id on D0 × H−s

q0(M) with respect to the

trivialization (4.61), henceGstt∈R andΦs

tt∈R are well-defined smoothcocycles. In fact, the weighted Sobolev spacesHs

q (M) andW sq (M) are

invariant under the action of the Teichmüller flow on the space Qκ(M) (al-though their Hilbert structure is not).

We point out that for anys > 0 there is no natural extension of the distribu-tional cocyclesGs

tt∈R andΦstt∈R respectively to the bundlesH−s

κ (M)and W−s

κ (M), since the action of the Teichmüller flow onQκ does not

Page 73: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 73

respect the domain of the Friedrichs Laplacian, hence the trivial cocycleGt × id is not defined on the bundlesD0 × H−s

q0(M).

Let Isκ,+(M)[Is

κ,−(M)] ⊂ H−sκ (M) be the finite dimensional sub-bundle of

horizontally [vertically] invariant distributions. By definition, the fiber ofthe bundleIs

κ,+(M) [Isκ,−(M)] at anyq ∈ M

(1)κ coincides with the vector

spaceIs+q(M) [Is

−q(M)] of horizontally [vertically] invariant distributions.

Lemma 4.16. For anys ≥ 0 and for anyGt-invariant ergodic probabilitymeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials,

(1) the sub-bundlesIsκ,±(M) ⊂ H−s

κ (M) areGst -invariant, measurable

and of finite, almost everywhere constant rank;(2) the cocycleGs

t |Isκ,±(M) satisfies the Oseledec’s theorem.

Proof. (1) The horizontal and vertical vector fields determined by a qua-dratic differentialq ∈ Qκ(M) are rescaled under the Teichmüller geodesicflow and the spaceSq(M) of q-tempered currents is invariant. Hence thespacesI±q(M) of all horizontally, respectively vertically, invariant distri-butions are invariant. Since the Sobolev spacesH−s

q (M) are invariant, thespacesIs

±q(M) are also invariant under the Teichmüller geodesic flow.

For anyq ∈ Qκ(M), the spaceI−s+q(M) [I−s

−q(M)] can be characterized asthe perpendicular of the closure inHs

q (M) of the range of the Lie derivativeLSq

[LTq] as a linear operator defined on the spaceHs+1

q (M):

(4.62)I−sq (M) = Squ | u ∈ Hs+1

q (M)⊥

;

[I−s−q(M) = Tqu | u ∈ Hs+1

q (M)⊥] .

Let q0 ∈ Qκ(M) and letf : D0 → Diff +(M) be a map defined on aneighbourhoodD0 ⊂ Qκ(M) of q0 which trivializes the bundlesH−s

κ (M)as in formula (4.61). For any fixedv ∈ Hs+1

q0(M) the maps

(4.63) q → (f−1q )∗Sqf

∗q (v) and q → (f−1

q )∗Tqf∗q (v)

are well-defined and continuous onD0 with values inHsq0

(M). Since theHilbert spacesHs

q (M) are separable, the sub-spaces

(f−1q )∗I−s

±q(M) ⊂ H−sq0

(M)

are measurable functions of the quadratic differentialsq ∈ Qκ(M). In fact,the orthogonal projectionsI±q on (f−1

q )∗I−s±q(M) can be constructed as fol-

lows. LetBs+10 be an orthonormal basis forHs+1

q0(M). If the horizontal

[vertical] foliation of q ∈ D0 is minimal, the subset ofHsq (M) defined as

(4.64) Bsq := Sqf

∗q (v)|v ∈ Bs+1

0 [ Bs−q := Tqf

∗q (v)|v ∈ Bs+1

0 ]

Page 74: Sobolev Regularity

74 GIOVANNI FORNI †

is linerarly independent. Since the set ofq ∈ D0 with non-minimal hori-zontal or vertical foliation has measure zero, the Gram-Schmidt orthonor-malization algorithm applied to the system(f−1

q )∗Bsq [ (f−1

q )∗Bs−q] yields

an orthonormal basisu+k (q)k∈N [u−

k (q)k∈N ] in Hsq0

(M) of the subspace

(4.65) (f−1q )∗Sqv | v ∈ Hs+1

q (M) [ (f−1q )∗Tqv | v ∈ Hs+1

q (M) ]

such that, for allk ∈ N, the functionsu±k : D0 → Hs

q0(M) are defined

µ-almost everywhere and are continuous on their domain of definition bythe continuity of the maps (4.63). LetD±

k (q)k∈N ⊂ H−sq0

(M) be the (or-thonormal) system dual to the orthonormal systemu±

k (q)k∈N ⊂ Hsq0

(M).Then we can write

(4.66) I±q(D) = D −∑

k∈N

〈D, D±k (q)〉−s D±

k (q) , for all D ∈ H−sq0

(M).

It is immediate to verify that formula (4.66) yields the orthogonal projec-tions onto the subspaces(f−1

q )∗I−s±q(M) for anyq ∈ D0. Such projections

can therefore be obtained as a limit of operators which depend continuouslyon q ∈ Qκ(M) in the complement of the subsets of quadratic differentialwith non-minimal horizontal [vertical] foliation. Since such a set has mea-sure zero with respect to anyGt-invariant ergodic probability measure, themeasurability of the sub-bundles of invariant distributions is proved.

The rank of the bundlesIsκ,±(M) is finite by Lemma 3.14 and Theorem

3.21 and it is almost everywhere constant with respect to anyergodicGt-invariant probability measure by definition of ergodicity.

(2) It follows from the Definition 2.7 of the weighted Sobolev norms andfrom the fundamental theorem of interpolation (see [LM68],Chap. 1, §5.1)applied to the interpolation family of Hilbert spacesHs

q (M)|s ∈ [0, 1]that, for anys ∈ R and for anyq ∈ Qκ(M), the following estimates hold:

(4.67) ‖Gst : H−s

q (M) → H−sGt(q)

(M)‖ ≤ e|s| |t| .

Since the bundlesIsκ,±(M) are finite dimensional and measurable, the Os-

eledec’s theorem applies to the cocycleGst |I

sκ,±(M) with respect to any

ergodicGt-invariant probability measure.

The measurable dependence of the spaces of invariant distributions on thequadratic differentials can also holds for distributionalsolutions of the co-homological equation with arbitrary data. In order to formulate the result,which will be relevant later on, we introduce the following:

Page 75: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 75

Definition 4.17. For anys ∈ R and anyr > 0, the range (inH−sq (M))

of the horizontal [vertical] Lie derivative operator (onH−rq (M)) is the sub-

space

(4.68)Rr,s

q (M) := SqU ∈ H−sq (M) |U ∈ H−r

q (M)

[Rr,s−q(M) := TqU ∈ H−s

q (M) |U ∈ H−rq (M)]

The Green operatorsUr,s±q : Rr,s

±q(M) → H−rq (M) are defined as follows:

for any distributionF ∈ Rr,sq (M) [ F ∈ Rr,s

−q(M)] :

(4.69)Ur,s

q (F ) := U , if SqU = F and U ∈ Irq(M)⊥ ⊂ H−r

q (M) ;

[Ur,s−q(F ) := U , if TqU = F and U ∈ Ir

−q(M)⊥ ⊂ H−rq (M)] .

By Corollary 3.4, for anys ∈ R there existsr > 0 such that, for almostall quadratic differentials in every circle orbit in a stratum M

(1)κ , the Green

operators areboundedoperators defined on the codimension1 subspaceH−s

q (M) ⊂ H−sq (M) of distributions vanishing on constant functions.

For anys ∈ R, let Hsκ(M) ⊂ Hs

κ(M) be the continuous sub-bundle ofdistributions vanishing on constant functions:

Hsκ(M) := (q, D) ∈ Hs

κ(M) | 〈D, 1〉s = 0 .

Lemma 4.18. Let µ be anySO(2, R)-absolutely continous,Gt-invariantergodic probability measure on a stratumM(1)

κ of orientable quadratic dif-ferentials. For anys ∈ R, there existsr > 0 such that the the Green op-erator U

r,s±q yield a measurable bundle mapUr,s

κ,± : H−sκ (M) → H−r

κ (M).In particular, the operator norm‖Ur,s

±q‖ yields a well-defined measurable

real-valued function of the quadratic differentialq ∈ M(1)κ .

Proof. Let q0 ∈ Qκ(M) and letf : D0 → Diff +(M) be a map defined on aneighbourhoodD0 ⊂ Qκ(M) of q0 which trivializes the bundlesH−s

κ (M)as in formula (4.61). The argument is similar to the one givenin the proofof the measurability of the sub-bundles of invariant distributions in Lemma4.16. LetBr+1

0 ⊂ H∞q0

(M) be a basis for the Hilbert spaceHr+1q0

(M). Thesubset ofHr

q (M) defined as

(4.70) Brq := Sqf

∗q (v)|v ∈ Br+1

0 [ Br−q := Tqf

∗q (v)|v ∈ Br+1

0 ]

is linerarly independent if the horizontal [vertical] foliation of q ∈ D0 isminimal, Hence, by the continuity of the maps (4.63), the Gram-Schmidtorthonormalization applied to the systemBr

q [ Br−q] yields an orthonormal

basisu+k (q)k∈N [u−

k (q)k∈N ] in Hrq (M) of the subspace

(4.71) Sqv | v ∈ Hr+1q (M) [Tqv | v ∈ Hr+1

q (M) ]

Page 76: Sobolev Regularity

76 GIOVANNI FORNI †

such that, for allk ∈ N, the functions(f−1)∗ u±k : D0 → Hr

q0(M) are

definedµ-almost everywhere and continuous on their domain of defini-tion. In fact, there exists basesv±

k k∈N ⊂ H∞q0

(M) of the Hilbert spaceHr+1

q0(M) such that the following holds. For eachk ∈ N there exists a

functionv±k : D0 → span(v±

1 , . . . , v±k ), definedµ-almost everywhere and

continuous on its domain of definition, such that forµ-almost allq ∈ D0

the setv±k (q)k∈N is a basis of the Hilbert spaceHr+1

q0(M) and

u+k (q) = Sq(f

∗ v+k )(q) [u−

k (q) = Tq(f∗ v−

k )(q)] .

Let D±k (q)k∈N ⊂ H−r

q0(M) be the system dual to the linearly independent

system(f−1q )∗u±

k (q)k∈N ⊂ Hrq0

(M). Such systems are in general notorthonormal. Then for anyF ∈ H−s

q0(M), the following formula holds:

(4.72) (f−1q )∗ U

r,s±q f ∗

q (F ) :=∑

k∈N

〈F, v±k (q)〉D±

k (q) .

In fact, let U±k (q) be the series on the right hand side of formula (4.72).

Since〈f ∗q D±

k (q), u±k (q)〉 = δkh for anyk, h ∈ N, it follows that

(4.73)〈f ∗

q U+k (q), Sqf

∗q v+

k (q)〉 = 〈F, v+k (q)〉 = 〈f ∗

q (F ), f ∗q v+

k (q)〉

[〈f ∗q U−

k (q), Tqf∗q v−

k (q)〉 = 〈F, v−k (q)〉 = 〈f ∗

q (F ), f ∗q v−

k (q)〉]

which implies thatf ∗q U+

k (q) [f ∗q U−

k (q)] is a distributional solution of theequationSqu = f ∗

q (F ) [Tqu = f ∗q (F )]. Finally, since all the distributions

f ∗q D+

k (q) [f ∗q D−

k (q)] vanish on the orthogonal complement of the spaceSqv | v ∈ Hr+1

q (M) [Tqv | v ∈ Hr+1q (M)] in Hr

q (M), it follows thatf ∗

q U±k (q) is orthogonal to the spaceIr

±q(M) of invariant distributions, henceU

r,s±q f ∗

q (F ) = f ∗q U±

k (q). It is also immediate from formula (4.72) that, foranyF ∈ H−s

q0(M), the distribution(f−1

q )∗ Ur,s±q f ∗

q (F ) is aµ-measurablefunction (defined almost everywhere) of the quadratic differential q ∈ D0

with values in the Hilbert spaceH−rq0

(M). In fact, by construction the func-tionsD±

k : D0 → H−rq0

(M) are definedµ-almost everywhere and continu-ous on their domain of definition.

Let Zsκ(M) ⊂ W−s

κ (M) be the infinite dimensional sub-bundle ofclosedcurrentsover M

(1)κ . By definition, the fiber of the bundleZs

κ(M) at anyq ∈ M

(1)κ coincides with the vector space ofclosed currents:

(4.74) Zsq(M) := Zq(M) ∩ W−s

q (M) .

The bundleZsκ(M) and the sub-bundleEs

κ(M) ⊂ Zsκ(M) of exactcurrents

are smooth,Φst -invariant sub-bundles of the bundleW−s

q (M). The quotientcocycle, defined on theH−s de Rham cohomology bundle, is isomorphic

Page 77: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 77

to the Kontsevich-Zorich cocycle. The latter isomorphism is the essentialmotivation for the formulas (4.58) and (4.60) which define, respectively, thecocyclesGs

tt∈R andΦstt∈R. In fact, let

(4.75) jκ : Zsκ(M) → H1

κ(M, R)

be the natural de Rham cohomology map onto the cohomology bundleH1

κ(M, R), defined as the restriction to the stratumM(1)κ of the cohomol-

ogy bundleH1g(M, R) introduced in formula (4.8).

Let Bsκ,±(M) ⊂ Zs

κ(M) be the sub-bundles with fiber atq ∈ M(1)κ given by

the vector spacesBs±q(M) of F±q-basic currents (defined in (3.31)).

Lemma 4.19. For anys ≥ 0 and for anyGt-invariant ergodic probabilitymeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials,

(1) the identityjκ Φst = Φt jκ holds everywhere onZs

κ(M);(2) the sub-bundlesBs

κ,±(M) ⊂ Zsκ(M) are Φs

t -invariant, measurableand of finite, almost everywhere constant rank;

(3) the cocycleΦst |B

sκ,±(M) satisfies the Oseledec’s theorem.

Proof. (1) It is an immediate consequence of the definition of the cocycleΦs

tt∈R on the bundleZsκ(M) of closed currents.

(2) The horizontal and vertical measured foliations of a quadratic differen-tial q ∈ Qκ(M) are projectively invariant under the Teichmüller geodesicflow and the spaceSq(M) of q-tempered currents is invariant. As a conse-quence, the spacesB+q(M) andB−q(M) of horizontally, respectively ver-tically, basic currents are invariant. The Sobolev spacesW−s

q (M) are alsoinvariant (although the Hilbert structure is not). Hence the spacesBs

±q(M)are invariant under the Teichmüller geodesic flow.

By Lemma 3.14, the measurability of the bundlesBsκ,±(M) ⊂ W−s

κ (M) ofbasic currents is equivalent to the measurability of the bundlesIs

κ,±(M) ⊂H−s

q (M), proved in Lemma 4.16.

The rank of the bundlesBsκ,±(M) is finite by Theorem 3.21 and it is almost

everywhere constant with respect to anyergodicGt-invariant probabilitymeasure by definition of ergodicity.

(3) It follows immediately from the identities (4.60) and from the bound(4.67) that, for anys ∈ R and for anyq ∈ Qκ(M), the following estimateshold:

(4.76) ‖Φst : W−s

q (M) → W−sGt(q)

(M)‖ ≤ e(|s|+1)|t| .

Since the bundlesBsκ,±(M) are finite dimensional and measurable, the Os-

eledec’s theorem applies to the cocycleΦst |B

sκ,±(M) with respect to any

ergodicGt-invariant probability measure.

Page 78: Sobolev Regularity

78 GIOVANNI FORNI †

Lemma 4.16 can be generalized to the bundle of quasi-invariant distribu-tions. LetIs

κ,±(M \Σκ) ⊂ H−sκ (M) be the bundle overM(1)

κ defined as fol-

lows: its fiber at eachq ∈ M(1)κ is the vector spaceIs

±q(M \Σq) ⊂ H−sq (M)

of quasi-invariant distributions. An argument analogous to the proof ofLemma 4.16 proves the following:

Lemma 4.20. For anys ≥ 0 and for anyGt-invariant ergodic probabilitymeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials,

(1) the sub-bundlesIsκ,±(M \ Σκ) ⊂ H−s

κ (M) are Gst -invariant, mea-

surable and of finite, almost everywhere constant rank;(2) the cocycleGs

t |Isκ,±(M \ Σκ) satisfies the Oseledec’s theorem.

Lemma 4.19 can be generalized to the bundle of currents closed on the com-plement of the singular set and to the sub-bundle of quasi-basic currents.Let Zs

κ(M \ Σκ) ⊂ W−sκ (M) be the bundle overM(1)

κ defined as follows:its fiber at eachq ∈ M

(1)κ is the vector spaceZs(M \ Σq) ⊂ W−s

q (M) ofclosed currents on the open manifoldM \Σq. The bundleZs

κ(M \Σκ) andthe sub-bundleEs

κ(M \ Σκ) ⊂ Zsκ(M \ Σκ) of exactcurrents are smooth,

Φst -invariant sub-bundles of the bundleW−s

q (M). The quotient cocycle, de-fined on theH−s de Rham punctured cohomology bundle, is isomorphic tothe punctured Kontsevich-Zorich cocycle. Let

(4.77) jκ : Zsκ(M \ Σκ) → H1

κ(M \ Σκ, R)

be the natural de Rham cohomology map onto the punctured cohomologybundleH1

κ(M \ Σκ, R) introduced in formula (4.12).

Let Bsκ,±(M \Σκ) ⊂ Zs

κ(M \Σκ) be the sub-bundles with fiber atq ∈ M(1)κ

given by the vector spacesBs±q(M \ Σq) of quasi-basic currents for the

measured foliationF±q (defined in (3.31)). An argument analogous to theproof of Lemma 4.19 proves the following:

Lemma 4.21. For anys ≥ 0 and for anyGt-invariant ergodic probabilitymeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials,

(1) the identityjκ Φst = Ψt jκ holds everywhere onZs

κ(M \ Σκ);(2) the sub-bundlesBs

κ,±(M \ Σκ) ⊂ Zsκ(M \ Σκ) are Φs

t -invariant,measurable and of finite, almost everywhere constant rank;

(3) the cocycleΦst |B

sκ,±(M \Σκ)t∈R satisfies the Oseledec’s theorem.

It follows immediately from the definitions and from Lemma 3.14 that thefollowing cocycle isomorphisms hold:

(4.78)Φs

t |Bsκ,±(M) ≡ e±t Gs

t |Isκ,±(M) ;

Φst |B

sκ,±(M \ Σκ) ≡ e±t Gs

t |Isκ,±(M \ Σκ) .

Page 79: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 79

As a consequence of such an isomorphism parts(2) and(3) in Lemma 4.19and Lemma 4.21 can be immediately derived from Lemma 4.16 andLemma4.20 respectively. In addition, the Lyapunov spectra and the Oseledec’sdecomposition of the cocyclesΦs

t |Bsκ,±(M) [Φs

t |Bsκ,±(M \ Σκ)] and

Gst |I

sκ,±(M) [Gs

t |Isκ,±(M \ Σκ)] can be immediately derived from one

another. By part(1) in Lemma 4.19 and Lemma 4.21, by Corollary 3.20and by the structure theorem for basic currents (Theorem 3.21), informa-tions on the Lyapunov spectrum of the cocyclesΦs

t |Bsκ,±(M \ Σκ) and

Φst |B

sκ,±(M) can be derived from that of the (punctured) Kontsevich-

Zorich cocycle.

Fors = 1, by [For02], Lemma 8.1, and by the representation Lemma 4.14,in the non-uniformly hyperbolic case it is possible to give aquite completedescription of the distributional cocycle on the bundle of basic currents.

Lemma 4.22.Letµ be anySO(2, R)-absolutely continuous, KZ-hyperbolicmeasure on a stratumM(1)

κ of orientable quadratic differentials.

(1) The cocycleΦ1tt∈R has strictly positive [strictly negative] Lya-

punov spectrum, with respect to the measureµ on M(1)κ , on the in-

variant sub-bundleB1κ,+(M) [B1

κ,−(M)].

(2) The sumB1κ(M) := B1

κ,+(M) + B1κ,−(M) of bundles overM(1)

κ isdirect and the restriction of the mapjκ to the sub-bundleB1

κ(M) isµ-almost everywhere injective.

(3) The coycleΦ1t |B

1κ(M)t∈R is isomorphic to the Kontsevich-Zorich

cocycleΦtt∈R on the real cohomology bundleH1κ(M, R), hence

it has the same Lyapunov spectrum.

4.3. Lyapunov exponents. We prove below that the the spaces of all hor-izontally and vertically (quasi)-basic currents and invariant distributionshave well-defined Oseledec decompositions at almost all point of any stra-tum of the moduli space (with respect to anyGt-invariant ergodic measure).We establish a fundamental relation between the Lyapunov exponents andthe Sobolev order of basic currents or distributions in eachOseledec sub-space. We conclude with a crucial ‘spectral gap’ result for the distributionalcocycle on the bundle of exact currents which is the basis forsharp es-timates on the growth of ergodic averages, hence for the construction ofsquare-integrable solutions of the cohomological equation in §5.2.

We introduce the following:

Definition 4.23. A non-zero currentC± ∈ Bsκ,±(M \ Σκ) will be called

(Oseledec) simpleif it belongs to an Oseledec subspace of the cocycleΦs

t |Bsκ,±(M \ Σκ). A non-zero invariant distributionD± ∈ Is

κ(M \ Σκ)

Page 80: Sobolev Regularity

80 GIOVANNI FORNI †

will be called (Oseledec) simpleif it belongs to an Oseledec subspace ofthe cocycleGs

t |Isκ(M \ Σκ).

The above definition is well-posed since for anyr ≤ s the cocyclesΦrt

[Grt] are the restrictions of the cocyclesΦs

t [Gst]. It is also immedi-

ate to prove that the image of any Oseledec simple basic current under theisomorphismD±q : Bs

±q(M \ Σq) → Is±q(M \ Σq) introduced in formula

(3.34) is an Oseledec simple invariant distribution.

Lemma 4.24. Let µ be anyGt-invariant probability measure onM(1)κ and

let Rµ ⊂ M(1)κ be the set of all holomorphic orientable quadratic differ-

entials which, for alls ≥ 0, are Oseledec regular point for the cocy-cles Φs

t |Bsκ,+(M \ Σκ) and Φs

t |Bsκ,−(M \ Σκ) over the Teichmüller

flow (Gt, µ). The setRµ has full measure and there exist measurablefunctionsL±

µ : Bκ,±(M \ Σκ) → R (with almost everywhere constantrange) such that, for anyq ∈ Rµ, the numberL±

µ (C±) is equal to theLyapunov exponent of the Oseledec simple currentC± ∈ B±q(M \ Σq)of Sobolev orderOW

q (C±) ≥ 0 (see Definition3.12), with respect to thecocycleΦs

t |Bsκ,±(M \ Σκ) over the flow(Gt, µ) for anys > OW

q (C±).

Proof. For anyr ≤ s, the embeddingsBrκ,±(M \Σκ) ⊂ Bs

κ,±(M \Σκ) holdand the cocycleΦr

tt∈R on the bundleW−rκ (M) coincides with restriction

of the cocycleΦstt∈R, defined on the bundleW−s

κ (M). The Oseledec’stheorem holds for the cocyclesΦs

t |Bsκ,±(M \ Σκ)t∈R on the measurable,

finite dimensional sub-bundlesBsκ,±(M \ Σκ) ⊂ W−s

κ (M) for all s ≥ 0.

Let Rµ ⊂ M(1)κ be the set of points which are Oseledec regular for the co-

cyclesΦkt |B

kκ,±(M \ Σκ)t∈R for all k ∈ N. By the Oseledec theorem the

setRµ has full measure and, for anyq ∈ Rµ and for alls ≥ r, the Lya-punov exponents of any Oseledec simple currentC± ∈ Br

κ,±(M \Σκ) withrespect to the cocyclesΦr

tt∈R andΦstt∈R are well-defined and coincide.

The common valueL±µ (C±) of all Lyapunov exponents with respect to the

cocyclesΦstt∈R for s > OW

q (C±) is therefore well-defined.

By the isomorphisms (4.78) an analogous statement holds forthe cocycleson the bundles of invariant distributions:

Lemma 4.25. Let µ be aGt-invariant probability measure onM(1)κ . There

exist measurable functionsl±µ : Iκ,±(M \Σκ) → R (with almost everywhereconstant range) such that, for any quadratic differentialq ∈ Rµ, the numberl±µ (D±) is equal to the Lyapunov exponent of the Oseledec simple invariantdistributionD± ∈ I±q(M \ Σq) of Sobolev orderOH

q (D±) ≥ 0 (see Def-inition 2.8), with respect to the cocycleGs

t |Isκ,±(M \ Σκ) over the flow

(Gt, µ) for anys > OHq (D±). LetD±q : B±q(M \ Σq) → I±q(M \ Σq)

Page 81: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 81

be the isomorphism introduced in formula (3.34), the following identitieshold:

(4.79) l±µ D±q = L±µ ∓ 1 , on B±q(M \ Σq) .

Lyapunov exponents of basic currents impose restrictions on their Sobolevregularity. In fact, we have:

Lemma 4.26.The following inequalities hold for any quadratic differentialq ∈ Rµ ⊂ M

(1)κ . For any Oseledec simple basic currentC± ∈ B±q(M \Σq)

and any Oseledec simple invariant distributionD± ∈ I±q(M \ Σq) theSobolev order functions satisfy the following lower bounds:

(4.80) OWq (C±) ≥ |L±

µ (C±) ∓ 1| and OHq (D±) ≥ |l±µ (D±)| .

Proof. SinceC± ∈ B±q(M \Σq), by the identities (4.60) and by the bound(4.67) the following inequalities hold for anys ≥ r:

(4.81) |Φst (C

±)|−s ≤ |Φrt (C

±)|−r ≤ er|t|±t|C±|−r , for all t ∈ R .

On the other hand, by the Oseledec’s theorem, for anyq ∈ Rµ and anyǫ > 0there exists a constantKǫ(q) > 0 such that

(4.82)|Φs

t (C±)|−s ≥ Kǫ(q) e(L±

µ (C±)−ǫ) t |C±|−s , for t ≥ 0 ;

|Φst (C

±)|−s ≥ Kǫ(q) e(L±µ (C±)+ǫ) t |C±|−s , for t < 0 ;

A comparison of the inequalities (4.81) and (4.82) proves the statement forthe case of basic currents. The statement for invariant distributions followsimmediately by Lemma 4.25 since the following identity holds:

OHq D±q = OW

q on B±q(M \ Σq) .

A similar argument, based on the spectral gap for almost all quadratic dif-ferentials in every circle orbit (see Theorem 4.11), provesthe following:

Theorem 4.27.For any quadratic differentialq ∈ M(1)κ there exists a real

numbers(q) > 0 such that, for almost allθ ∈ S1 and for alls < s(q)

(4.83)dimBs

qθ(M \ Σq) = dimBs

−qθ(M \ Σq) = 1 ;

dimIsqθ

(M \ Σq) = dimIs−qθ

(M \ Σq) = 1 .

Proof. If there existss ∈ (0, 1) such that (4.83) does not hold on a positivemeasure subset of the circle, it follows that there exists a positive measuresetEs ⊂ S1 such that dimBs

−qθ(M \Σq) > 1. Hence, for anyθ ∈ Es, there

exists anon-vanishingvertically (quasi)-basic currentCθ ∈ Bs−qθ

(M \ Σq)

such thatCθ ∧ q1/2θ = 0. We claim that for almost allθ ∈ Es, the currentCθ

has non-zero cohomology classcθ ∈ H1(M \ Σq, R). In fact, by Theorem

Page 82: Sobolev Regularity

82 GIOVANNI FORNI †

3.21, ifCθ has vanishing cohomology class, then there exists a basic currentCθ ∈ Bs−1

−qθ(M \ Σq) such thatδ1

−qθ(Cθ) = Cθ. Sinces < 1, if the vertical

foliation Fqθis ergodic, then the currentCθ ∈ R · ℜ(qθ) which implies

Cθ = 0, a contradiction. Since by the Keane conjecture (see [Mas82] or[Vee82]) the vertical foliationFqθ

is uniquely ergodic for almost allθ ∈ S1,the claim is proved. As in formula (4.81), we have

(4.84) |Φst (Cθ)|−s ≤ es|t|−t|Cθ|−s , for all t ∈ R .

Let Ψtt∈R be the punctured Kontsevich-Zorich cocycle, introduced in§4.1. By Lemma 4.19 the following identity holds:

[Φst (Cθ)] = Ψt(cθ) ∈ H1(M \ Σq, R) , for all t ∈ R .

The de Rham punctured cohomology bundleH1κ(M \ Σκ, R) can be en-

dowed, for anys ≥ 0, with the quotient norm| · |sκ induced by the Sobolevnorm on the bundleW−s

κ (M). It follows by the above discussion and bythe estimate (4.84) that, for eachθ ∈ Es, there exists a non-vanishing coho-mology classcθ ∈ H1

qθ(M \ Σq, R) such that

(4.85)cθ ∧ [ℜ(q

1/2θ )] = cθ ∧ [ℑ(q

1/2θ )] = 0 ,

lim supt→+∞

1

|t|log |Ψt(cθ)|

sκ ≤ s − 1 .

By Lemma 4.3 and by the Poincaré-Lefschetz duality (4.13) between rel-ative and punctured cohomology, the cohomology bundleH1

κ(M, R) ad-mits a complement in the punctured cohomology bundleH1

κ(M \Σκ, R) onwhich the punctured Kontsevich-Zorich cocycle is isometric (with respectto a continuous norm). By the estimate in (4.85) on the upper Lyapunov ex-ponent, it follows that, sinces < 1, the cohomology classcθ ∈ H1

qθ(M, R)

for all θ ∈ Es. The restriction of the punctured Kontsevich-Zorich cocycleΨtt∈R to the bundleH1

κ(M, R) concides with the Kontsevich-Zorich co-cycle Φtt∈R. Since the latter is symplectic, it follows from (4.85) thatits upper Lyapunov exponent on the symplectic subspaceI⊥

qθ(M, R) ⊂

H1qθ

(M, R) (see formulas (4.26) and (4.27)) satisfies the estimate:

(4.86) λ+2 (qθ) ≥ 1 − s , for all θ ∈ Es .

By Theorem 4.11 the inequalityλ+2 (qθ) ≤ Lκ(q) < 1 holds for allq ∈ M

(1)κ

and for almost allθ ∈ S1. Since the setEs has positive measure, it followsthats > 1 − Lκ(q) > 0.

By Lemma 4.14, Lemma 4.22 and Lemma 4.26 we derive a fundamentalrelation between the weighted Sobolev order and the Lyapunov exponentsof Oseledec simple basic currents and invariant distributions.

Page 83: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 83

Definition 4.28. An Oseledec basisfor the spaceIs±q(M) [ Bs

±q(M)] ofinvariant distributions [of basic currents] is a basis contained in the union ofall Oseledec subspaces for the cocycleGs

t |Isκ,±(M) [ Φs

t |Bsκ,±(M)] at

any Oseledec regular pointq ∈ M(1)κ . Since the cocyclesΦs

t |Isκ,±(M) and

Φst |B

sκ,±(M) are isomorphic via the mapsD±q : Bs

±q(M) → Is±q(M)

introduced in (3.34), any basisβs±q ⊂ Is

±q(M) is Oseledec if and only if thebasisD±q(β

s±q) ⊂ Bs

±q(M) is Oseledec as well.

Theorem 4.29. For any SO(2, R)-absolutely continuous, KZ-hyperbolicmeasureµ on a stratumM

(1)κ of orientable quadratic differentials, forµ-

almost allq ∈ M(1)κ and for any Oseledec basisC±

1 , . . . , C±g ⊂ B1

±q(M):

(4.87) OWq (C±

i ) = 1 − |L±µ (C±

i )| = 1 − λµi , for all i ∈ 1, . . . , g .

Consequently, for any Oseledec basisD±1 , . . . , D±

g ⊂ I1±q(M),

(4.88) OHq (D±

i ) = |l±µ (D±i )| = 1 − λµ

i , for all i ∈ 1, . . . , g .

The restriction of the cocycleΦ1tt∈R to the invariant sub-bundleZ1

κ(M) isdescribed by the following Oseledec-type theorem, which isa straightfor-ward generalization of [For02], Theorem 8.7, based on Lemma4.22.

Theorem 4.30. For any SO(2, R)-absolutely continuous, KZ-hyperbolicmeasure on a stratumM(1)

κ of orientable quadratic differentials, there existsa measurableΦ1

t -invariant splitting:

(4.89) Z1κ(M) = B1

κ,+(M) ⊕ B1κ,−(M) ⊕ E1

κ(M) .

(1) The restriction of the cocycleΦ1tt∈R to the bundleB1

κ(M) :=B1

κ,+(M)⊕B1κ,−(M) is (measurably) isomorphic to the Kontsevich-

Zorich cocycle, hence it has Lyapunov spectrum (4.15). The sub-bundleB1

κ,+(M) corresponds to the strictly positive exponents andthe sub-bundleB1

κ,−(M) to the strictly negative exponents.(2) The Lyapunov spectrum of restriction of the cocycleΦ1

tt∈R to theinfinite dimensional bundleE1

κ(M) of exact currents is reduced tothe single Lyapunov exponents0 (in fact, the cocycle is isometricwith respect to a suitable continuous norm onE1

κ(M)).

Informations on the Lyapunov structure of the restrictionsof the cocyclesΦs

tt∈R to the sub-bundlesBsκ,±(M) ⊂ Bs

κ,±(M \Σκ) ⊂ W−sκ (M), for any

s ≥ 1, can be derived from Theorem 4.30 and from the structure theoremfor basic currents (see Theorem 3.21) combined with the following result:

Lemma 4.31. Let µ be anyGt-invariant ergodic probability measure ona stratumM

(1)κ of orientable quadratic differentials. For anyq ∈ Rµ, the

image under the mapδ±q : B±q(M \ Σq) → B±q(M \ Σq) defined by

Page 84: Sobolev Regularity

84 GIOVANNI FORNI †

formulas (3.61) of any simple currentC± ∈ Bs±q(M \ Σq) is a simple

current δ±q(C±) ∈ Bs+1

±q (M \ Σq). The following identities hold for theSobolev order map and the Lyapunov exponent map onB±q(M \ Σq):

(4.90)OW

q δ±q = OWq − 1 , for all q ∈ M(1)

κ ;

L±µ δ±q = L±

µ ∓ 1 , for all q ∈ Rµ ⊂ M(1)κ .

Proof. For anyC± ∈ B±q(M \ Σq), it follows immediately by the defi-nitions of the Sobolev spaces and of the mapsδ±q on B±q(M \ Σq) thatδq(C

+) ∈ W−s−1q (M) if and only if C+ ∧ℜ(q1/2) ∈ H−s

q (M), hence if andonly if C+ ∈ W−s

q (M). Similarly, δ−q(C−) ∈ W−s−1

q (M) if and only ifC− ∧ ℑ(q1/2) ∈ H−s

q (M), hence if and only ifC− ∈ W−sq (M). The first

identity in (4.90) is therefore proved. In fact, we have proved that the mapsδs±q sendBs

±q(M \Σq) onto the subspace of cohomologically trivial currentsin Bs+1

±q (M \ Σq) while the spaceBs±q(M) is mapped onto the subspace of

cohomologically trivial currents inBs+1±q (M).

The following identity follows immediately from the definitions: for anyquadratic differentialq ∈ M

(1)κ , for eachs ≥ 0 and allt ∈ R, we have

(4.91) δs±Gt(q) Φs

t = e±t (Φs+1t δs

±q) on Bs±q(M \ Σq) .

Since the mapsδs±q : Bs

±q(M \ Σq) → Bs+1±q (M \ Σq) are embeddings, it

follows from the identity (4.91) and from the Oseledec’s theorem that thecurrentδs

±q(C±) ∈ Bs+1

±q (M \ Σq) is simple ifC± ∈ Bs±q(M \ Σq) is. It

also follows that the second indentity in (4.90) holds.

An analogous statement for invariant distributions can be derived. For anyq ∈ M

(1)κ , let L±q : Sq(M) → Sq(M) denote the Lie derivative operators

on the spaceSq(M) of all q-tempered currents: for anyC ∈ Sq(M),

(4.92) Lq(C) := LSq(C) and L−q(C) := LTq

(C) .

Lemma 4.32. For anyq ∈ M(1)κ , the operatorsL±q : I±q(M) → I±q(M)

are well-defined and injective. Letµ be anyGt-invariant ergodic probabil-ity measure onM(1)

κ . The following identities hold on the spacesI±q(M):

(4.93)OH

q L±q = OHq − 1 , for all q ∈ M(1)

κ ;

l±µ L±q = l±µ ∓ 1 , for all q ∈ Rµ ⊂ M(1)κ .

Corollary 4.33. Letµ be anyGt-invariant ergodic probability measure ona stratumM

(1)κ of orientable holomorphic quadratic differentials. For any

s ≥ 0 the (finite) Lyapunov spectrum of the cocycleΦst |B

sκ,±(M) is a

finite subset of the countable set

(4.94) ±λµ1 ∪ ±λµ

i ∓ j |1 < i < 2g, j ∈ N ;

Page 85: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 85

hence the Lyapunov spectrum of the cocycleGst |I

sκ,±(M) is a finite subset

of the countable set

(4.95) 0 ∪ ±λµi ∓ (j + 1) |1 < i < 2g, j ∈ N

(each element of the sets (4.94) and (4.95) is taken with multiplicity one).

Proof. Let Π1κ,+(M, R) [Π1

κ,−(M, R)] be the continuous sub-bundles of the

cohomology bundleH1κ(M, R) overM(1)

κ which fibers at any quadratic dif-ferential q ∈ M

(1)κ are given by the spacesΠ1

+q(M, R) [Π1−q(M, R)] de-

fined in formula (3.52). The sub-bundlesΠ1κ,±(M, R) are invariant under

the Kontsevich-Zorich cocycleΦtt∈R and the Lyapunov spectrum of therestrictionΦt|Π

1κ,±(M, R) consists of the set±λµ

i |1 ≤ i ≤ 2g − 1. Byformulas (3.60), the image of the bundleBs

κ,±(M) under the cohomologymapjκ : Z1

κ(M) → H1κ(M, R) is a sub-bundle ofΠ1

κ,±(M, R) which isinvariant under the Kontsevich-Zorich cocycle. By Lemma 4.19 the cocy-cleΦs

t |Bsκ,±(M) is mapped under the cohomology map onto a restriction

of the Kontsevich-Zorich cocycle. Letδsκ,± : Bs−1

κ,±(M) → Bsκ,±(M) the

measurable bundle maps defined fiber-wise forµ-almost allq ∈ M(1)κ as the

mapsδs±q : Bs−1

±q (M) → Bs±q(M), defined in formula (3.61). The kernel

of the cohomology mapjκ onBsκ,±(M) is aΦs

t -invariant sub-bundle whichcoincides with the range of the mapδs

κ,±. By Lemma 4.31, the Lyapunovspectrum (4.94) ofΦs

t |Bsκ,±(M) can therefore be derived by induction on

[s] ∈ N. The Lyapunov spectrum of (4.95) ofGst |I

sκ,±(M) can then be

derived by the isomorphism (4.78).

By Corollary 3.20 for anys > 3 the cohomology map is surjective foralmost all quadratic differentials in any circle orbit. Theabove result can berefined as follows:

Corollary 4.34. Letµ be anySO(2, R)-absolutely continous,Gt-invariantergodic probability measure on a stratumM(1)

κ ⊂ M(1)g of orientable holo-

morphic quadratic differentials. For anys > 3, there exists an integervectorhs := (hs

2, . . . , hs2g−1) ∈ N2g−2 such that the Lyapunov spectrum of

the cocycleΦst |B

sκ,±(M) is the (finite) set

(4.96) ±λµ1 ∪ ±λµ

i ∓ j |1 < i < 2g, 0 ≤ j ≤ hsi ;

hence the Lyapunov spectrum of the cocycleGst |I

sκ,±(M) is the (finite) set

(4.97) 0 ∪ ±λµi ∓ (j + 1) |1 < i < 2g, 0 ≤ j ≤ hs

i

(each element of the sets (4.96) and (4.97) is taken with multiplicity one).

The integer vectorhs depends on the Sobolev regularity of basic currents inthe Oseledec’s spaces related to the Lyapunov exponentsλµ

2 , . . . , λµ2g−1

Page 86: Sobolev Regularity

86 GIOVANNI FORNI †

which come from the Kontsevich-Zorich cocycle. Hence, in particular, incaseµ is the unique absolutely continuous,SL(2, R)-invariant ergodic mea-sure on a connected component of a stratum, by Theorem 4.29 the followingestimate for the numbers(hs

2, . . . , hg) holds: for alli ∈ 2, . . . , g,

hsi = maxh|1 − λµ

i + h < s , if s 6∈ N − λµi .

The Oseledec-Pesin theory of the cocyclesGst |I

sκ,±(M) has crucial im-

plications for the theory of the cohomological equation. Inparticular, itimplies that (ifµ is a KZ-hyperbolic measure) forµ-almost allq ∈ M

(1)κ ,

Oseledec bases of the spacesIs±q(M) of invariant distributions have strong,

quantitative linear independence properties.

Theorem 4.35.Let µ be anyGt-invariant ergodic probability measure ona stratumM

(1)κ of orientable holomorphic quadratic differentials. For any

s > 0 and for µ-almost allq ∈ M(1)κ , let bs

±q = D±1 , . . . , D±

J±(s) be aOseledec basis of the spaceIs

±q(M) of invariant distributions and let

(4.98) (l±1 , . . . , l±J±(s)) :=(l±µ (D±

1 ), . . . , l±µ (D±J±(s))

)

denote the Lyapunov spectrum of the cocycleGst |I

sκ,±(M) over(Gt, µ).

For any ǫ > 0, there exists a measurable functionKsǫ : M

(1)κ → R+ such

that the following holds. For everyτ ∈ (0, 1], there exist linearly indepen-dent systems of smooth functionsu±

1 (τ), . . . , u±J±(s)(τ) ⊂ H∞

q (M) suchthat, for all i, j ∈ 1, . . . , J±(s) and for all0 ≤ r ≤ s,

(4.99) D±i

(u±

j (τ))

= δij and |u±j (τ)|r ≤ Kǫ(q) τ |l±j |−r−ǫ .

Proof. By Corollary 4.33, for anyGt-invariant ergodic probability measureµ on M

(1)κ and anys > 0, the Lyapunov exponents of the distributional

cocycleGst |I

sκ,±(M) are all of the same sign, namely

(4.100) l±1 , . . . , l±J±(s) ⊂ ∓R+ ∪ 0 .

It follows from the Oseledec’s theorem that, for anyǫ > 0, there exists astrictly positive measurable functionC(1)

ǫ : M(1)κ → R+ such that, for every

i ∈ 1, . . . , J±(s), for µ-almost allq ∈ M(1)κ and for allt ≥ 0,

(4.101) |Gs∓t(D

±i )|−s ≥ C(1)

ǫ (q) e(|l±i |−ǫ)t .

It also follows from the Oseledec’s theorem that for anys > 0, for µ-almostall q ∈ M

(1)κ and any Oseledec basisbs

±q ⊂ Is±q(M), the distorsion of the

(Oseledec) basisGst (b

s±q) ⊂ Hs

Gt(q)(M) grows subexponentially in time.

Page 87: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 87

Thedistorsionof a basisbs±q := D±

1 , . . . , D±J ⊂ Is

±q(M) is the number

(4.102) dq(bs±q) := sup

∑J±(s)i=1 |ci| |D

±i |H−s

q (M)

|∑J±(s)

i=1 ci D±i |H−s

q (M)

| c ∈ CJ±(s) .

The Oseledec’s theorem implies that, ifbs±q ⊂ Is

±q(M) is an Oseledec basis,

then for anyǫ > 0, there exists a measurable functionC(2)ǫ : M

(1)κ → R

+

such that forµ-almost allq ∈ M(1)κ and for allt ∈ R,

(4.103) dGt(q)

(Gs

t (bs±q)

)≤ C(2)

ǫ (q) eǫ|t| .

By the estimates (4.101) and (4.103), it follows that, for any ǫ > 0, thereexists a measurable functionC(3)

ǫ : M(1)κ → R+ such that the following

holds. For everyt ≥ 0, sinceH∞q (M) is dense inHs

q (M) for anys ∈ R,there exists a systemv±

1 (t), . . . , v±J±(s)(t) ⊂ Hs

q (M) such that, forµ-

almost allq ∈ M(1)κ and for alli, j ∈ 1, . . . , J±(s),

(4.104) D±i

(v±

j (t))

= δij and |v±j (t)|Hs

G∓t(q)(M) ≤ C(3)

ǫ (q) e−(|l±j |−ǫ)t .

By the bound (4.67) on the norm‖Gst : Hs

q (M) → HsGt(q)

(M)‖, it followsthat, for any0 ≤ r ≤ s and for allj ∈ 1, . . . , J±(s),

(4.105) |v±j (t)|Hr

q (M) ≤ C(3)ǫ (q) e(r−|l±j |+ǫ)t , for any t ≥ 0 .

It follows that the systemu±1 (τ), . . . , u±

J±(s)(τ) ⊂ H∞q (M), defined for

anyτ ∈ (0, 1] by the identities

(4.106) u±j (τ) := v±

j (− log τ) , j ∈ 1, . . . , J±(s) ,

satisisfies the required properties (4.99).

The following theorem, which can be interpreted as spectralgap result forthe cocyclesG0

tt∈R on the bundleH0κ(M) of square-integrable functions,

is the main technical result of the paper.

For any(σ, l) ∈ R2, we introduce theupper (forward) Lyapunov normofa distributionU ∈ H−σ

q (M) at a quadratic differentialq ∈ M(1)κ as the

non-negative extended real number

(4.107) Nσ,lq (U) := sup

t→+∞e−lt |Gσ

t (U)|−σ

Theorem 4.36.Let µ be anySO(2, R)-absolutely continous,Gt-invariantergodic probability measure on a stratumM(1)

κ of orientable quadratic dif-ferentials. For any anyσ > 0 and anyl < 1, there exist a real number

Page 88: Sobolev Regularity

88 GIOVANNI FORNI †

ǫ := ǫ(σ, l) > 0 and a measurable functionCσ,l : M(1)κ → R+ such that, for

µ-almost allq ∈ M(1)κ and all functionsU ∈ L2

q(M) of zero average,

(4.108) Nσ,−ǫq (U) < Cσ,l(q) |U |0 + N1,l

q (SqU) .

Proof. The outline of the argument is as follows. From the results onthe cohomological equation and on the Lyapunov spectrum of the coyclesGr

tt∈R on the bundles of invariant distributions we will derive (exponen-tial) estimates on the norms|Gr

t (U)|−r for a sufficiently larger > 0, along asequence of suitable visiting times. Since|G0

t (U)|0 is constant, as a conse-quence of the invariance of theL2

q(M) norm under the Teichmüller flow, theinterpolation inequality for dual weighted Sobolev norms (Corollary 2.26)implies the required (exponential) estimates on|Gσ

t (U)|−σ for anyσ > 0.

Let q ∈ M(1)κ andU ∈ L2

q(M) be a function of zero average. Let us as-sume that there existl0 < 1 andN ∈ R+ such that the Lyapunov normN1,l0

q (SqU) ≤ N . By definition (4.107) of Lyapunov norms, for anyt ≥ 0,

(4.109) |G1t (SqU)|−1 ≤ N el0t .

By the spectral gap Theorem 4.9 and by Corollary 4.33, there exist C1 > 0,l1 > 0 and a positive measure setP1 ⊂ M

(1)κ such that, for allq ∈ P1 and

all Sq-invariant distributionsD ∈ H−rq (M) ∩ Ir

q(M), the following holds:

(4.110) |Grt (D)|−r ≤ C1 e−l1t |D|−r .

By Lemma 4.18 on the measurability of the Green operators forthe coho-mological equation, there existsr > 0 such that the following holds. Thereexists a constantC2 > 0 and a setP2 ⊂ P1 of positive measure such that

(4.111) ‖Ur,1q : H−1

q (M) → H−rq (M)‖ ≤ C2 , for all q ∈ P2 .

Forµ-almost allq ∈ M(1)κ , there exists a sequence(tn)n∈N of visiting times

of the forward orbitGt(q) | t ≥ 0 to the positive measure setP2 such that,for eachn ∈ N, the functiontn : M

(1)κ → R

+ is measurable and

(4.112) limn→+∞

tn(q)

n:= p >

log C1

l1.

For eachn ∈ N and anyr ∈ R, let us introduce the notations:

Gn := Gtn , qn := Gn(q) , Sn, Tn := Sqn, Tqn

,Gr

n := Grtn U r

n := Grn(U) , Fn := G1

n(SqU) ,Hr

n(M) := Hrqn

(M) , Hrn(M) := Hr

qn(M) , Ir

n(M) := Irqn

(M) .

By the definition of the Teichmüller flowGt, the orthonormal frameSn, Tn = e−tnSq, e

tnTq for all n ∈ N. Hence the following coho-mological equation holds:

(4.113) SnUrn = e−tnFn ∈ H−1

n (M) .

Page 89: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 89

For eachn ∈ N and anyr ∈ R let | · |r,n denote the weighted Sobolev normon the spaceHr

n(M). It follows from equation (4.109) that

(4.114) |SnU rn|−1,n ≤ N e−(1−l0)tn .

By the definition of the Green operators and by equation (4.111), there ex-ists a solutionGn ∈ H−r

n (M), orthogonal toIrn(M), of the cohomological

equationSnGn = SnU rn ∈ H−1

n (M) such that

(4.115) |Gn|−r,n ≤ C2 N e−(1−l0)tn .

It follows that there existsDn ∈ Irn(M)∩H−r

n (M) such thatUn ∈ H0n(M)

has the following orthogonal decomposition in the Hilbert spaceH−rn (M):

(4.116) U rn = Gn + Dn .

For eachn ∈ N, let πn : H−rn (M) → Ir

n(M) be the orthogonal projectiononto the subspace ofSn-invariant distributions and letτn := tn+1−tn. Sinceby the definitionsU r

n+1 = Grτn

(U rn), for all n ∈ N, and

Grτn

(Irn(M) ∩ H−r

n (M))⊂ Ir

n+1(M) ∩ H−rn+1(M)

the following recursive identity holds:

(4.117) Dn+1 = Grτn

(Dn) + πn+1 Grτn

(Gn) .

Since|Grτn

(Gn)|−r,n+1 ≤ erτn |Gn|−r,n by the bound (4.67) on the norm ofGr

t : H−rt (M) → H−r

Gt(q)(M), it follows from (4.110) and (4.115) that

(4.118) |Dn+1|−r,n+1 ≤ C1 e−l1τn |Dn|−r,n + C2 N erτn−(1−l0)tn ,

which implies by induction a bound of the form

(4.119) |Dn+1|−r,n+1 ≤ Cn1 e−l1(tn+1−t0) |D0|−r,0 + C2 N

n−1∑

j=0

Cj1 esn,j

with the sequencesn,j|n ∈ N, 0 ≤ j ≤ n given by the identity

(4.120) sn,j := −l1(tn+1 − tn−j+1) + rτn−j − (1 − l0)tn−j .

SinceU ∈ L2q(M) andGr

n(U) = G0n(U) ∈ H0

n(M) = L2q(M) ⊂ H−r

n (M),the following bound holds:

(4.121) |Grn(U)|0,n ≤ |U |0 , for all n ∈ N .

It follows in particular from the decomposition (4.116) that

(4.122) |D0|−r,0 ≤ |U r0 |−r,0 ≤ |Gr

0(U)|0,0 ≤ |U |0 .

Page 90: Sobolev Regularity

90 GIOVANNI FORNI †

The main step in the argument is the proof of the following claim. Thereexist a positive measurable functionC3 : P2 → R+ and a real numberl3 := l3(l0, l1) > 0 such that

(4.123)n−1∑

j=0

Cj1 esn,j ≤ C3(q) e−l3n , for all n ∈ N .

Let ω1 < ω < ω2 be positive real numbers such that

(4.124)(a) (1 − l0)ω1 − (l1 + r)(ω2 − ω1) > 0 ,

(b) l1ω2 − l1(ω2 − ω1) > log C2 .

By condition (4.112) on the sequence(tn)n∈N there exists a measurablefunctionn0 : M

(1)κ → N such that

(4.125) ω1 n ≤ tn(q) ≤ ω2 n , for all n ≥ n0(q) .

It follows that, for anyq ∈ M(1)κ , all n ≥ n0(q) and allj ≤ n,

(4.126)sn,j ≤ [(l1 + r)(ω2 − ω1) − (1 − l0)ω1] n

+ [(1 − l0)ω1 − r(ω2 − ω1) − l1ω2] j + l1(ω2 − ω1) + rω2 .

Let K := C2 e(1−l0)ω1−r(ω2−ω1)−l1ω2 . There are two cases to consider:

(4.127) (a) K ≤ 1 (b) K > 1 .

In case(a) we immediately obtain that, for alln ≥ n0(q),

(4.128)n−1∑

j=n0

Cj1 esn,j ≤ n el1(ω2−ω1)+rω2 e−[(1−l0)ω1−(l1+r)(ω2−ω1)] n ,

in case(b) we obtain instead that

(4.129)n−1∑

j=n0

Cj1 esn,j ≤

el1(ω2−ω1)+rω2

K − 1Cn

1 e−[l1ω2−l1(ω2−ω1)] n .

It follows that there exist constantsA > 0 andα := α(l0, l1) > 0 such that

(4.130)n−1∑

j=n0

Cj1 esn,j ≤ A e−α n , for all n ≥ n0(q) .

By condition (4.112) on the sequence(tn)n∈N and by definition (4.120)

(4.131)sn,j(q)

n→ −(1 − l2)ω , as n → +∞ ,

Page 91: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 91

for µ-almost allq ∈ M(1)κ , uniformly with respect toj ∈ 0, . . . , n0(q)−1.

Hence there exists a measurable functionn1 : M(1)κ → N with n1 ≥ n0 and

positive constantsB > 0 andβ := β(l0, l1) > 0 such that

(4.132)n0−1∑

j=0

Cj1 esn,j ≤ B e−β n , for all n ≥ n1(q) .

The claim (4.123) then follows from the estimates (4.130) and (4.132).

It then follows from the orthogonal decomposition (4.116),from the claim(4.123), proved above, together with the upper bounds (4.115), (4.119),(4.122) and the lower bound for visiting times in (4.125), that there exist ameasurable functionC4 : M

(1)κ → R+ and a constantl4 := l4(l0, l1) > 0

such that

(4.133) |Grn(U)|−r,n ≤ C4(q)|U |0 + N e−l4 n , for all n ≥ n1(q) .

By theinterpolation inequalityfor the scale of dual weighted Sobolev norms(see Corollary 2.26), from the upper bounds (4.121) and (4.133) it followsthat for anyσ > 0 there exist a measurable functionCσ : M

(1)κ → R+ and a

constantlσ := l(σ, l0, l1) > 0 such that

(4.134) |Gσn(U)|−σ,n ≤ Cσ(q)|U |0 + N e−lσ n , for all n ≥ n1(q) .

It remains to prove that the latter bound implies the statement of the theo-rem. Forµ-almost allq ∈ M

(1)κ and for allt ≥ t0(q), there exists a unique

n(t, q) ∈ N such that

(4.135) tn(t,q)(q) ≤ t < tn(t,q)+1(q) .

Let σ > 0 be fixed and letω(σ)1 < ω < ω

(σ)2 be positive real numbers

such thatlσ − σ(ω(σ)2 − ω

(σ)1 ) > 0. There exists a measurable function

n(σ)2 : M

(1)κ → R such thatn(s)

2 ≥ n1 and, forµ-almostq ∈ M(1)κ ,

(4.136) ω(σ)1 n ≤ tn(q) ≤ ω

(σ)2 n , for all n ≥ n

(σ)2 (q) .

Let t(σ)1 : M

(1)κ → R+ be a measurable function such thatt

(σ)1 ≥ t0 and

n(t, q) ≥ n(σ)2 (q) if t ≥ t

(σ)1 (q), for µ-almost allq ∈ M

(1)κ . It follows from

(4.135) and (4.136) that, for anyt ≥ t(σ)1 (q),

(4.137) n(t, q) + 1 ≥tn(t,q)+1(q)

ω(σ)2

>t

ω(σ)2

.

It follows by (4.136) and (4.137), forµ-almostq ∈ M(1)κ and allt ≥ t

(σ)1 (q),

(4.138) τn(q,t) ≤ (ω(σ)2 − ω

(σ)1 ) (n(q, t) + 1) + ω

(σ)1 .

Page 92: Sobolev Regularity

92 GIOVANNI FORNI †

Since, for anyn ∈ N and anytn ≤ t < tn+1,

(4.139)|Gσ

t (U)|−σ = |Gσt−tn Gt1(U)|−σ

≤ eσ(t−tn) |Gσn(U)|−σ ≤ eστn |Gσ

n(U)|−σ ,

by the upper bounds (4.134), (4.137) and (4.138) the following estimateholds. Letǫ := ǫ(σ, l0, l1) be the real number defined as follows:

(4.140) ǫ :=lσ − σ(ω

(σ)2 − ω

(σ)1 )

ω(σ)2

> 0 .

Forµ-almost allq ∈ M(1)κ and allt ≥ t

(σ)1 (q)

(4.141) |Gσt (U)|−σ ≤ Cσ(q) eσω

(σ)1 +lσ|U |0 + N e−ǫt .

Finally, for µ-almost allq ∈ M(1)κ and all0 ≤ t ≤ t

(σ)1 (q),

(4.142) |Gσt (U)|−σ ≤ |U |0 ≤ eǫt

(σ)1 (q)|U |0 e−ǫt .

The desired estimate (4.108) immediately follows from the upper bounds(4.141) and (4.142).

5. SMOOTH SOLUTIONS

In this final section we prove our main theorems on Sobolev regularity ofsmooth solutions of the cohomological equation for translation flows. Thegeneral result which we are able to prove for any translationsurface (andalmost all directions) is a direct consequence of the Fourier analysis con-struction of distributional solutions in §§3.1 and of the uniqueness resultthat follows from Theorem 4.27. The sharper result which we will prove foralmost all translation surfaces (and almost all directions) requires a deeperanalysis of ergodic averages of translation flows. In fact, the constructionof square integrable (bounded) solutions is based (as in [MMY05]) on theGottschalk-Hedlund theorem. The required bounds on the growth of er-godic integrals are derived from the Oseledec-type result (Theorem 4.30)and the spectral gap result (Theorem 4.36) for distributional cocycles, bythe methods developed in [For02] in the study of the deviation of ergodicaverages. The Oseledec-type theorem was in fact already proved in [For02],while the spectral gap theorem for distributional cocyclesis new.

5.1. The general case.In this section we prove a result on existence ofsmooth solutions of the cohomological equation for translation flows whichholds foranyorientable quadratic differential in almost all directions. Sucha result answers a question of Marmi, Moussa and Yoccoz who asked whatis the best, that is, the smallestregularity losswithin reach of the Fourier

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SOBOLEV REGULARITY OF SOLUTIONS 93

analysis methods of [For97]. As indicated in [MMY05], the answer is es-sentially that the solution loses no more than3+ǫ derivatives (for anyǫ > 0)with respect to the scale of weighted Sobolev spaces introduced in §2.1. Werecall that the regularity loss obtained in [MMY05] (which only holds foralmost allquadratic differentials) is essentially1 + BV.

Theorem 5.1. Let q ∈ M(1)κ be any quadratic differential. Letk ∈ N be

any integer such thatk ≥ 3 and lets > k and r < k − 3. For almostall θ ∈ S1 (with respect to the Lebesgue measure), there exists a constantCr,s(θ) > 0 such that the following holds. Iff ∈ Hs

q (M) is such thatD(f) = 0 for all D ∈ Is

qθ(M), the cohomological equationSθu = f has a

solutionu ∈ Hrq (M) satisfying the following estimate:

(5.1) |u|r ≤ Cr,s(θ) |f |s .

Proof. As a first step, we prove that, under finitely many distributional con-ditions onf ∈ Hs

q (M) with s > 3, the cohomological equationSθu = fhas, for almost allθ ∈ S1, a solutionu ∈ H−r

q (M) for anyr > 0, whichsatisfies an estimate such as (5.1). Since∆F

q f ∈ Hs−2q (M) with s − 2 > 1,

by Theorem 3.3, for anyr > 0 and for almost allθ ∈ S1, there exists a so-lution U ∈ H−2−r

q (M) of the cohomological equationSθU = ∆Fq f , which

vanishes on constant functions and satisfies the bound

(5.2) ‖U‖−2−r ≤ C(1)r,s (θ) ‖∆F

q f‖s−2 ≤ C(1)r,s (θ) ‖f‖s .

Letu ∈ H−rq (M) be the unique distribution vanishing on constant functions

such that∆Fq u = U . Since the commutation relationSθ∆qv = ∆qSθv holds

for anyv ∈ H3+rq (M), the following distributional equation holds:

(5.3) ∆q (Sθu − f) = 0 in H−3−rq (M) .

In addition, from the estimate (5.2) it follows immediatelythat

(5.4) |u|−r ≤ C(1)r,s (θ) |f |s .

Let N rθ ⊂ H−r

q (M) be the subspace defined as follows:

N rθ := u ∈ H−r

q (M) |∆qSθu = 0 ∈ H−3−rq (M) .

Since the kernelKr(∆q) ⊂ H−1−rq (M) (which is equal to the subspace

perpendicular to the range of the operator∆q : H3+rq (M) → H1+r

q (M)) andthe spaceIr

qθ(M) of Sθ-invariant distributions are both finite dimensional,

it follows that the subspaceN rθ is finite dimensional (for almost allθ ∈ S1).

As a consequence, the space

(5.5) Sθ(Nrθ ) := Sθu | u ∈ N r

θ ⊂ Kr(∆q)

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94 GIOVANNI FORNI †

is finite dimensional, hence closed inH−1−rq (M). We claim that there exists

a constantC(2)r,s (θ) > 0 such that the following holds. For almost allθ ∈ S1,

there exists a unique distributionUθ(f) ∈ H−rq (M) such that

(a) Uθ(f) is orthogonal to the subspaceIrqθ

(M);(b) SθUθ(f) − f ∈ Kr(∆q) is orthogonal to the subspaceSθ(N

rθ );

(c) Uθ(f) satisfies the following bound:

(5.6) |Uθ(f)|−r ≤ C(2)r,s (θ) |f |s .

In fact, letu ∈ Hrq (M) be any solution of equation (5.3) which satisfies

the bound (5.4). Letu1 ∈ H−rq (M) be the component ofu orthogonal

in H−rq (M) to the subspaceIr

qθ(M) of Sθ-invariant distributions. Since

u − u1 ∈ Irqθ

(M), the distributionu1 still satisfies the equation (5.3). Inaddition, the norm|u1|−r ≤ |u|−r, hence the bound (5.4) still holds. SinceN r

θ is finite dimensional, there existsu2 ∈ H−rq (M), a solution of (5.3),

such thatSθu2 − f is equal to the component ofSθu1 − f orthogonal toSθ(N

rθ ) in H−1−r

q (M). Let Uθ(f) be the component ofu2 orthogonal toIrqθ

(M) in H−rq (M). By constructionUθ(f) satisfies the conditions(a) and

(b) above and it is uniquely determined by them. Let us prove condition(c). The linear operatorSθ : N r

θ → Sθ(Nrθ ) is surjective and its domain is a

finite dimensional subspace, hence it has a bounded right inverse and thereexistsKr := Kr(θ) > 0 such that, for anyu ∈ N r

θ orthogonal toIrqθ

(M),

(5.7) |u|−r ≤ Kr |Sθu|−1−r .

Since by definitionUθ := Uθ(f) andu1 ∈ H−rq (M) are both orthogonal to

Irqθ

(M) andUθ − u1 ∈ N rθ , by the bound (5.7),

(5.8) |Uθ|−r ≤ |Uθ − u1|−r + |u1|−r ≤ Kr|Sθ (Uθ − u1) |−1−r + |u1|−r .

Since again by definitionSθUθ = Sθu2 andSθu2 − f is equal to the com-ponent ofSθu1 − f orthogonal toSθ(N

rθ ),

(5.9)|Sθ (Uθ − u1) |−1−r ≤ |Sθu1 − f |−1−r + |Sθu2 − f |−1−r

≤ 2|Sθu1 − f |−1−r ≤ 2|u1|−r + 2|f |−1−r .

It follows from the bounds (5.8), (5.9) and from the bound (5.4) for thedistributionu1 ∈ H−r

q (M) that the required bound (5.6) holds with

C(2)r,s (θ) := [(2Kr(θ) + 1)C(1)

r,s (θ)] + 1 .

SinceSθ(Nrθ ) ⊂ Kr(∆q) is finite dimensional, there exist a finite (linearly

independent) setΦ1, . . . , ΦK of bounded linear (real-valued) functionalson the Hilbert spaceH−1−r

q (M) such that

Kr(∆q) ∩ Sθ(Nrθ )⊥ ∩ Ker(Φ1) ∩ · · · ∩ Ker (ΦK) = 0 .

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SOBOLEV REGULARITY OF SOLUTIONS 95

Let D1, . . . , DK ⊂ H−sq (M) be the system of distributions defined as

follows: for eachj ∈ 1, . . . , K,

(5.10) Dj(f) := Φj (SθUθ(f) − f) , for all f ∈ Hsq (M) .

The systemD1, . . . , DK has by construction the following property: ifDj(f) = 0 for all j ∈ 1, . . . , K, thenUθ(f) is the required solution ofthe cohomological equationSθu = f . In fact, under such conditionsUθ(f)is by construction a solution orthogonal to constant functions. By Theorem4.27, there existss(q) > 0 such that for all0 < s < s(q) the spaceIs

qθ(M)

is 1-dimensional for almost allθ ∈ S1. Thus, if r < s(q), for almost allθ ∈ S1 the solutionu ∈ H−r

q (M) of the cohomological equationSθu = fis unique (if it exists). It follows that, for anys(q) > r > 0, the distributionUθ(f) is the unique solution inH−r

q (M) of the cohomological equation,which implies thatUθ(f) ∈ H−r

q (M) for anyr > 0.

We claim thatD1, . . . , DK ⊂ Isqθ

(M). In fact, by its definition in formula(5.10), the distributionDj ∈ H−s

q (M) for all j ∈ 1, . . . , K. In addition,if there existsv ∈ Hs+1

q (M) such thatf = Sθv , by conditions(a) and(b)on the distributionUθ(f) we have

Sθ (Uθ(f) − v) = SθUθ(f) − f ∈ Kr(∆q) ∩ Sθ(Nrθ )⊥ .

However, by definitionSθ (Uθ(f) − v) ∈ Sθ(Nrθ ), henceSθUθ(f) − f = 0

which implies thatD1(f) = · · · = DK(f) = 0. Hence by definition all thedistributions of the systemD1, . . . , DK areSθ-invariant.Finally we prove the statement of the theorem by induction onk ∈ N.For k = 3 the statement holds by the previous argument. Let us assumethat the statement holds fork ≥ 3. By the induction hypothesis, for anys > k+1 and anyr < k−2 there existsCr,s(θ) > 0 such that, for almost allθ ∈ S1 and for anyf ∈ Hs

q (M) with D(f) = 0 for all D ∈ Isqθ

(M), thereexist solutionu, uS anduT ∈ Hr−1

q (M) of the cohomological equationsSθu = f , SθuS = Sf andSθuT = Tf respectively such thatu, uS anduT

are orthogonal to constant functions and the following bound holds:(5.11)

|u|2r−1 + |uS|2r−1 + |uT |

2r−1 ≤ Cr,s(θ)

(|f |2s−1 + |Tf |2s−1 + |Sf |2s−1

).

In fact, sincef ∈ Hsq (M) is such thatD(f) = 0 for all D ∈ Is

qθ(M) it

follows immediately thatD(Sf) = D(Tf) = 0 for all D ∈ Is−1qθ

(M).SinceSθ commutes withS, T in the sense of distributions, it follows thatthe distributionsSu − uS andTu − uT ∈ Hr−1

q (M) areSθ-invariant. Lets(q) > 0 be such that for all0 < s < s(q) and for almost allθ ∈ S1

the spaceIsqθ

(M) is 1-dimensional (see Theorem 4.27). Ifs(q) > 1 − r,sinceSu − uS andTu − uT ∈ Hr−1

q (M) areSθ-invariant and orthogonalto constant functions, it follows thatSu = uS andTu = uT . The latter

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96 GIOVANNI FORNI †

identities imply thatu ∈ Hrq (M) and by (5.11) the required bound (5.1) is

satisfied. The proof of the induction step is therefore completed.

5.2. Ergodic integrals. Optimal results on the loss of regularity for almostall orientable quadratic differentials (and almost all directions) will be de-rived from bounds on ergodic integrals by the Gottschalk-Hedlund theorem.The required bounds will be proved along the lines of §9 in [For02] withthe key improvement given by the ’spectral gap’ Theorem 4.36.

The key idea of the argument given in [For02] consists in studying the dy-namics of the distributional cocycleΦ1

tt∈R on the infinite dimensional(non-closed) sub-bundlesΓ±

κ ⊂ W−1κ (M) of 1-dimensional currents gener-

ated by segments of leaves of the horizontal and vertical foliations.Let T > 0. We will denote byγT

±q a positively oriented segment of lengthT > 0 of a leaf of the measured foliationF±q respectively. By the trace the-orems for standard Sobolev spaces and by the comparison Lemma 2.11, thevector spacesΓ±

q generated by the setsγT±q(p)|(p, T) ∈ M × R are sub-

spaces of the weighted Sobolev spaceW−sq (M) of 1-dimensional currents

for anys ≥ 1 and the corresponding sub-bundlesΓ±κ are invariant under the

action of the cocycleΦstt∈R. LetFq : M ×R → M [F−q : M ×R → M ]

be the (almost everywhere defined) flow of the horizontal [vertical] vec-tor fields Sq [Tq] on M . The horizontal and vertical foliationsF±q co-incide almost everywhere with the orbit foliations of the flows F±q. LetγT±q ∈ Γ±

q be a positively oriented segment with initial pointp± ∈ M andlet α := f+ηT + f−ηS ∈ W s

q (M). The following identity holds:

(5.12) γT

±q(α) =

∫T

0

f± F±q(p±, τ) dτ ,

The ergodic averages of the functionsf± ∈ Hsq (M) (for s ≥ 1) can

therefore be understood by studying the dynamics of the ‘renormalization’cocyclesΦs

tt∈R on the sub-bundleΓ±κ . In [For02] we have proved the

following basic estimate on the weighted Sobolev norm of thecurrentsγT±q ∈ W−1

q (M). Let Rq be the flat metric with conical singularities associ-

ated to the quadratic differentialq ∈ M(1)κ and let||q|| denote theRq-length

of the shortest saddle connection of the flat surface(M, Rq). (We recall thatasaddle connectionis a segment joining conical points).

Lemma 5.2. ([For02], Lemma 9.2) There exists a constantK > 0 suchthat, for all quadratic differentialsq ∈ M

(1)κ ,

(5.13) |γT

±q|W−1q (M) ≤ K(1 +

T

||q||) .

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SOBOLEV REGULARITY OF SOLUTIONS 97

The above estimate is a ‘trivial’ bound, linear with respectto the lengthT > 0 of the orbit segment, and can be quite easily derived from theSobolevtrace theorem for rectangles in the euclidean planeR

2. The number||q|| > 0,which measures the pinching of the flat surface, gives the order of magni-tude of the edges of the largest flat rectangle which can be embedded in theflat surface(M, Rq) around an arbitrary regular point.

We recall below the terminology and the notations, introduced in §9 of[For02], concerning return trajectories of translation flows.

Definition 5.3. A point p ∈ M is regular with respect to a measured folia-tion F if it belongs to a regular (non-singular) leaf ofF. For any quadraticdifferentialq ∈ Q(M). A point p ∈ M will be said to beq-regular if it isregular with respect to the horizontal and vertical foliationsF±q.

The set ofq-regular points is of full measure and it is equivariant under theactions of the group Diff+(M) and of the Teichmüller flow onQ(M).

Definition 5.4. Let p ∈ M be aq-regular point and letI±q(p) be the vertical[horizontal] segment of length||q||/2 centered atp. A forward horizontal[vertical] return time of p ∈ M is a real numberT := T±q(p) > 0 suchthatF±q(p, T) ∈ I∓q(p). If T > 0 is a horizontal [vertical] return time fora q-regular pointp ∈ M , the horizontal [vertical] forward orbit segmentγT±q(p) will be called aforward horizontal [vertical] return trajectoryatp.

There is a natural map from the set of horizontal [vertical] return trajectoriesinto the set of homotopically non-trivial closed curves.

Definition 5.5. Theclosingof any horizontal [vertical] trajectory segmentγT±q(p) is the piece-wise smooth homotopically non-trivial closedcurve

(5.14) γT

±q(p) := γT

±q(p) ∪ γ(p, F±q(p, T)

),

obtained as the union of the trajectory segmentγT±q(p) with the shortest

geodesic segmentγ(p, F±q(p, T)

)joining its endpointsp andF±q(p, T).

Let T(1)±q (p) > 0 be theforward horizontal [vertical] first return timeof a

q-regular pointp ∈ M , defined to be the real number

(5.15) T(1)±q (p) := minT > 0 |F±q(p, T) ∈ I∓q(p) .

The corresponding forward horizontal [vertical] trajectory γ(1)±q (p) with ini-

tial point p will be called theforward horizontal [vertical] first return tra-jectoryatp. The following bounds hold for first return times:

Lemma 5.6. ([For02], Lemma 9.2’) There exists a measurable functionKr : Mκ → R+ such that, ifT(1)

±q (p) is the forward horizontal [vertical]

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98 GIOVANNI FORNI †

first return time of aq-regular pointp ∈ M , then

(5.16) ||q||/2 ≤ T(1)±q (p) ≤ Kr(q) .

The lower bound in (5.16) is an immediate consequence of the definition of||q|| as the length of the shortest saddle connection, while the upper bounddepends essentially on the fact that the first return map of a translation flowto a transverse interval is an interval exchange transformation, hence thereturn-time function is piece-wise constant (and bounded).

In §9.3 of [For02] we have constructed special sequences of horizontal [ver-tical] return times for almost all quadratic differentialsgenerated. Such spe-cial return times are related to visiting times of the Teichmüller flow, which‘renormalizes’ translation flows, to appropriate compact subsets of positivemeasure. Letq ∈ M

(1)κ be a Birkhoff generic point of the Teichmüller flow

Gtt∈R and letSκ(q) ⊂ M(1)κ be a smooth compact hypersurface of codi-

mension1, such thatq ∈ Sκ(q) andSκ(q) is transverse to the Teichmüllerflow. Let (tk) be the sequence of visiting times of the orbitGt(q) | t ∈ Rto Sκ(q). Since, by definition, for anyt ∈ R,

(5.17) (FGt(q), F−Gt(q)) = (e−tFq, etF−q) ,

if t := tk < 0 is a backwardvisiting time of the orbitGt(q)t∈R, anyforward first return trajectory of the horizontal foliationFGt(q) is a for-ward return trajectory of the horizontal foliationFq, provided|tk| is suf-ficiently large. In a similar way, ift := tk > 0 is a forward visiting timeof Gt(q)t∈R, any forward first return trajectory of the vertical foliationF−Gt(q) is a forward return trajectory of the vertical foliationF−q.

By theclosingof the return trajectories of the horizontal [vertical] foliation,as in (5.14), we obtainclosedcurrents of Sobolev order1. The evolution ofsuch currents under the action of the Teichmüller flow is therefore describedby the cocyclesΦs

tt∈R on the bundleZsκ(M) for anys ≥ 1. In the case

s = 1 the dynamics of the cocycleΦ1t |Z

1κ(M) is completely described

by Theorem 4.30. In the following we will analyze Lyapunov exponentsof closed currents given by the closing of horizontal [vertical] trajectoriesof translation flows under the cocyclesΦs

tt∈R for s > 1. A completedescription of the Lyapunov structure of the cocyclesΦs

t |Zsκ(M) for s >

1 is not relevant for our purposes and will not be attempted.

Let µ be aSO(2, R)-absolutely continuous, KZ-hyperbolic measure on astratumM

(1)κ of orientable quadratic differentials (in the sense of Definitions

4.13 and 4.4). We recall that according to [For02] all the canonical abso-lutely continuous,SL(2, R)-invariant measures on any connected compo-nent of any stratum of orientable quadratic differentials are KZ-hyperbolic.A different and stronger proof which establishes that all the afore-mentioned

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SOBOLEV REGULARITY OF SOLUTIONS 99

measures are KZ-simple has been given in [AV05] (see Theorem4.6). Thelatter theorem is not necessary for any of the results of thispaper to hold.

By Theorem 4.30, for anys ≥ 1, there exists a measurable splitting

(5.18) Zsκ(M) = B1

κ,+(M) ⊕ B1κ,−(M) ⊕ Es

κ(M) .

The Lyapunov exponents of the restriction of the distributional cocycleΦs

tt∈R to the finite dimensional sub-bundlesB1κ,+(M) [B1

κ,−(M)] areequal to the non-negative [non-positive] exponents of the Kontsevich-Zorichcocycle. It follows from the non-uniform hyperbolicity hypothesis for thelatter that all Lyapunov exponents of the cocycleΦs

t |B1κ,+(M) are strictly

positive while all Lyapunov exponents of the cocycleΦst |B

1κ,−(M) are

strictly negative. In particular, by Oseledec’s theorem there exists a mea-surable functionΛs

κ : M(1)κ → R+ and a real numberαµ > 0 such that, for

µ-almost allq ∈ M(1)κ , for all C ∈ B1

−q(M) and allt ≥ 0,

(5.19) |Φst (C)|−s ≤ Λs

κ(q)|C|−s e−αµt .

In [For02], we have proved that fors = 1 the restrictionΦst |E

sκ(M) has

0 has the unique Lyapunov exponent. In fact, this cocycle is isometric withrespect to a continuous Lyapunov norm (see Theorem 4.30). For s > 1,estimates on Lyapunov exponents and Lyapunov norms for the restrictionΦs

t |Esκ(M) will be derived from Theorem 4.36. Letσ := s − 1 > 0. For

anyq ∈ M(1)κ , anyl < 1 and anyǫ > 0, let

(5.20) Nσ,ǫ,lκ (q) := sup

U∈H0q(M)

Nσ,−ǫq (U)

|U |0 + N1,lq (SqU)

.

By Theorem 4.36 for anyl < 1 there existsǫ := ǫ(σ, l) > 0 such thatformula (5.20) defines a (measurable) functionNσ,ǫ,l

κ : M(1)κ → R+ .

Let Π±q : Z1q(M) → B1

±q(M) andEq : Z1q(M) → E1

q(M) be the projec-tions determined by the splitting (5.18). For anys ≥ 1, the restrictions ofthe projectionsΠ±q andEq to the subspaceZs

q(M) ⊂ Z1q(M) can be ex-

tended to projectionsΠs±q : W−s

q (M) → B1±q(M) andEs

q : W−sq (M) →

Esq(M), defined on the Sobolev spaceW−s

q (M) of 1-dimensional currents,by composition with the orthogonal projectionW−s

q (M) → Zsq(M) onto

the closed subspace of closed currents. Letdsκ : M

(1)κ → R+ be thedis-

torsionof the splitting (5.18), that is, the function defined forµ-almost allq ∈ M

(1)κ as

(5.21) dsκ(q) := sup

C∈Zsq(M)

|Πq(C)|−s + |Π−q(C)|−s + |Eq(C)|−s

|C|−s

.

Let s > 1 and l < 1. Let P(1)κ ⊂ M

(1)κ be a compact set satisfying the

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100 GIOVANNI FORNI†

following conditions:

(1) All q ∈ P(1)κ are Birkhoff generic points for the Teichmüller flow

Gt and Oseledec regular points for the cocycleΦst |B

1κ(M);

(2) the setP(1)κ is transverse to the Teichmüller flow and has positive

transverse measure;(3) there exists a constantK

(1)r > 0 such that, for allq ∈ P

(1)κ and all

q-regularp ∈ M , the first return timesT(1)±q (p) ≤ K

(1)r ;

(4) the real-valued functionsΛsκ, Nσ,ǫ,l

κ andδ1κ, introduced respectively

in formulas (5.19), (5.20) and (5.21), are bounded onP(1)κ .

By the ergodicity of the system(Gt, µ), by the Oseledec’s theorem forthe cocycleΦs

t |B1κ(M) and by Lemma 4.6, it follows that the union of all

setsP(1)κ with the properties(1) − (4) is a full measure subset ofM

(1)κ .

Definition 5.7. Let q ∈ M(1)κ and(tk)k∈N be the sequence of visiting times

of thebackwardorbit Gt(q) | t ≤ t1 = 0 to a compact positive measuresetP(1)

κ ⊂ M(1)κ . A principal sequence(T(k)

q (p))k∈N of forward return timesfor the horizontal foliationFq at aq-regular pointp ∈ M is the sequence

(5.22) T(k)q (p) := T

(1)Gt(q)

(p) exp |t| , t = tk .

For eachk ∈ N, a horizontal principal (forward) return trajectoryγ(k)q (p)

at aq-regular pointp ∈ M is the horizontal forward return trajectory at thepointp corresponding to a principal return timeT

(k)q (p) > 0.

We remark that a horizontal principal return trajectoryγ(k)q (p) coincides

with the horizontalfirst return trajectory atp of the quadratic differentialGt(q), t = tk < 0. A similar definition can be given for the vertical foliationF−q by considering forward visiting times of the Teichmüller flow.The following standard splitting lemma allows to reduce theanalysis ofarbitrary regular trajectories to that of principal returntrajectories.

Lemma 5.8. ([For02], Lemma 9.4) Under conditions(1) − (3) there existsa constantKP > 0 such that the following holds. Letq ∈ P

(1)κ and, for any

T > 0, let γTq (p) be a forward trajectory at aq-regular pointp ∈ M . There

exists a finite set of pointsp(k)j | 1 ≤ k ≤ n, , 1 ≤ j ≤ mk ⊂ γT

q (p), such

that the principal return trajectoriesγ(k)q (p

(k)j ) ⊂ γT

q (p) do not overlap and

(5.23)

γT

q (p) =

n∑

k=1

mk∑

j=1

γ(k)q (p

(k)j ) + bT

q (p) ,

with mk ≤ KP e|tk+1|−|tk| , for 1 ≤ k ≤ n ,

and theRq-lengthLq

(bT

q (p))≤ KP .

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SOBOLEV REGULARITY OF SOLUTIONS 101

Proof. We recall the proof given in [For02], Lemma 9.4. The argumentis based on the following estimate on principal return times. By (5.22),Lemma 5.6 and condition(3), there exists a constantKpr > 0 such that, forall q ∈ P

(1)κ , all q-regular pointsp ∈ M and allk ∈ N,

(5.24) K−1pr exp |tk| ≤ T(k)

q (p) ≤ Kpr exp |tk| .

Let n = maxk ∈ N |T(k)q (p) ≤ T . The maximum exists (finite) by

(5.24). Letp(n)1 := p. The sequence(p(k)

j ) with the properties stated in(5.23) can be constructed by a finite iteration of the following procedure.Let p(k)

j be the last point already determined in the sequence and let

(5.25) p(k)j+ := Fq

(p

(k)j , T(k)

q (p(k)j )

)∈ γT

q (p) .

Let thenk′ ∈ 1, . . . , k be the largest integer such that

(5.26) Fq

(p

(k)j+ , T(k′)

q (p(k)j+)

)∈ γT

q (p) .

If k′ = k, let p(k)j+1 := p

(k)j+ . If k′ < k, let mk := j, mh = 0 (no points) for

all k′ < h < k andp(k′)1 := p

(k)j+ . The iteration step is concluded. By (5.24)

it follows that

(5.27) K−1pr e|tk| mk ≤ T(k+1)

q (p(k)1 ) ≤ Kpr e|tk+1| .

The length of the remainderbTq (p) has to be less than any upper bound for

the length of first return times. Hence, by (5.27), Lemma 5.6 and condition(3), the estimates in (5.23) are proved and the argument is concluded.

The result below gives a fundamental uniform estimate on ergodic integrals.

Theorem 5.9.Letµ be anSO(2, R)-absolutely continuous, KZ-hyperbolicmeasure on a stratumM(1)

κ of orientable quadratic differentials. For anys > 1 there exists a measurable functionΓs

κ : M(1)κ → R+ such that, for

µ-almostq ∈ M(1)κ , for all q-regularp ∈ M and for all T > 0 ,

(5.28) |Πs−q

(γT

q (p))|−s + |Es

q

(γT

q (p))|−s ≤ Γs

κ(q) .

Proof. Let P := P(1)κ be a compact set satisfying conditions(1)− (4) listed

above. Forµ-almost allq ∈ P, there exists a sequence(tk)k∈N of back-ward return times of the Teichmüller orbitGt(q) to the setP. By Lemma5.8 the uniform estimate (5.28) can be reduced (exponential) estimates forprincipal return trajectoriesγ(k)

q (p). We claim that for everys > 1 thereexist constantsKs

P> 0 andαs > 0 such that forµ-almost allq ∈ P, for all

q-regularp ∈ M and allk ∈ N,

(5.29) |Πs−q

(γ(k)

q (p))|−s + |Es

−q

(γ(k)

q (p))|−s ≤ Ks

Pe−αs|tk| .

Page 102: Sobolev Regularity

102 GIOVANNI FORNI†

By Lemma 5.2, by conditions(3) and (4) on the setP, in particular thebound on the distorsion, there exists a constantKs

1(P) > 0 such that thefollowing holds. Forµ-almost allq ∈ P and all q-regularp ∈ M , theclosingγ

(1)q (p) of the first return horizontal trajectoryγ(1)

q (p) (see Definition5.5) satisfies the following bound:

(5.30) |Π−q

(γ(1)

q (p))|−1 + |Eq

(γ(1)

q (p))|−1 ≤ K1(P)

Since by definition of the principal return trajectories

(5.31) γ(k)q (p) = Φs

|t|

(1)Gt(q)

(p))

, for any t = tk ≤ 0 ,

it follows from the initial bound (5.30), from the invariance of the sub-bundleB1

κ,−(M) under the cocyclesΦst, from the Lyapunov bound (5.19)

on the cocycleΦst |B

1κ,−(M) and from condition(4) on the setP ⊂ M

(1)κ

that there exists a constantKs2(P) > 0 such that the following estimate

holds. Forµ-almost allq ∈ P and for allq-regularp ∈ M ,

(5.32) |Π−q

(γ(k)

q (p))|−s ≤ Ks

2(P) e−αµ|tk| , for all k ∈ N .

A similar estimate holds for the projections of principal return trajectorieson the sub-bundleE1

κ(M) of exact currents. In fact, there existKs3(P) > 0

andǫs > 0 such that, forµ-almost allq ∈ P and for allq-regularp ∈ M ,

(5.33) |Eq

(γ(k)

q (p))|−s ≤ Ks

2(P) e−ǫs|tk| , for all k ∈ N .

The estimate (5.33) is proved as follows. By definition of thebundleE1κ(M)

of exact currents, for allq ∈ M(1)κ and allq-regularp ∈ M , there exists a

unique functionU (1)q (p) ∈ H0

q(M) ⊂ L2q(M) such that

(5.34) Eq

(γ(1)

q (p))

= dU (1)q (p) .

By the definitions of the distributional cocyclesΦst andGs

t the follow-ing identity of cocycles holds for anys ≥ 1:

(5.35) Φst d = d Gs−1

t on H−s+1q (M) .

The exponential estimate (5.33) will therefore follow fromTheorem 4.36if we can prove that there exist a positive numberl < 1 and a constantK3(P) > 0 such that, forµ-almost allq ∈ P and allq-regularp ∈ M , thefollowing holds:

(5.36) |U (1)q (p)|0 + N1,l

q

[SqU

(1)q (p))

]≤ Ks

3(P) .

Let | · |κ be the norm on the bundleE1κ(M) of exact currents defined as

follows: for anyq ∈ M(1)κ andC ∈ E1

q(M),

(5.37) |C|E := |U |L2q(M) , if C = dU with U ∈ H0

q(M) .

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SOBOLEV REGULARITY OF SOLUTIONS 103

By the definitions of weighted Sobolev norms, it is immediateto prove thatthe mapd : H0

q(M) → E1q(M) is a continous bijective map of Hilbert

spaces. Hence, by definition (5.37) and by the open mapping theorem, foreachq ∈ M

(1)κ there existsK(q) > 0 such that

(5.38) K(q)| · |E ≤ | · |−1 ≤ | · |E on E1q(M) .

Since the sub-bundleH0κ(M) ⊂ H0

κ(M) and the norms| · |E, | · |−1 onthe bundleE1

κ(M) are all continuous, the functionK : M(1)κ → R

+ canbe chosen continuous (see Lemma 9.3 in [For02] for a detailedargument).Hence by the estimates (5.30) and by identity (5.34), there exists a constantC1(P) > 0 such that, forµ-almost allq ∈ P and allq-regularp ∈ M ,

(5.39) |U (1)q (p)|L2

q(M) ≤ K(q)−1|Eq

(γ(1)

q (p))|−1 ≤ C1(P) .

The proof of the estimate (5.36) is completed as follows. Sinceγ(1)q (p) is a

horizontal trajectory, by the splitting (5.18) and by the identity (5.34), thefollowing formula holds:

(5.40) SqU(1)q (p) = ıSq

[γ(1)q (p) − γ(1)

q (p)] − ıSqΠ−q[γ

(1)q (p)] .

Since the distributionıSq[γ

(1)q (p) − γ

(1)q (p)] is given by integration along a

vertical arc of unit length, by the Sobolev embedding Lemma 5.2 and by thelogarithmic law of geodesics for the Teichmüller geodesic flow on modulispaces (see [Mas93], Prop. 1.2, or §4.1, formula (4.49)) it follows that foranyl > 0 there exists a constantCl > 0 such that, forµ-almost allq ∈ M

(1)κ

and allq-regularp ∈ M , the Lyapunov norm

(5.41) N1,lq

(ıSq

[γ(1)q (p) − γ(1)

q (p)])≤ Cl .

By the cocycle isomorphism (4.78), by the Lyapunov estimate(5.19) and bycondition(4) on the setP, since the distributionıSq

Π−q[γ(1)q (p)] ∈ I1

−q(M),there exists a constantC2(P) > 0 such that, for any1 > l ≥ 1 − αµ, forµ-almost allq ∈ M

(1)κ and allq-regularp ∈ M , the Lyapunov norm

(5.42) N1,lq

(ıSq

Π−q[γ(1)q (p)]

)≤ C2(P) .

The bound (5.36) follows immediately from the bound (5.39),from theidentity (5.40) and from the bounds (5.41) and (5.42).

As hinted above, the estimate (5.36) implies the required exponential esti-mate (5.33) on the projections of principal return trajectories on the spaceof exact currents. In fact, by Theorem 4.36 and by condition(4) on the setP, for any1 > l ≥ 1 − αµ and anyσ > 0, there existKσ

4 (P) > 0 andǫs := ǫ(l, σ) > 0 such that, forµ-almost allq ∈ P and allq-regularp ∈ M ,

(5.43) Nσ,ǫq

(U (1)

q (p))≤ Ks

4(P) .

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104 GIOVANNI FORNI†

Let us introduce the following notation: for anyk ∈ N, let

(5.44) qk := Gtk(q) ∈ P and U (k)q (p) := G0

|tk|

(U (1)

qk(p)

)∈ L2

q(M) .

Since the splittings (5.18) areΦst-invariant (in particular fors = 1), by

the ddefinition of principal return trajectories (5.31), bythe identity (5.34)and the cocycle identity (5.35), the following holds:

(5.45)Eq

(γ(k)

q (p))

= Eq Φ1|tk|

(γ(1)

qk(p)

)=

= Φ1|tk |

(dU (1)

qk(p)

)= dU (k)

q (p)

Let s > 1 andσ := s − 1. It follows from (5.43) (by the definition (4.107)of Lyapunov norms) that, forµ-almost allq ∈ P and allq-regularp ∈ M ,

(5.46) |Eq

(γ(k)

q (p))|−s ≤ |G0

|tk|

(U (1)

qk(p)

)|−σ ≤ Ks

4(P) e−ǫs|tk| .

The crucial exponential estimate (5.29) on projections of principal returntrajectories, claimed above, can now be derived from estimates (5.32) and(5.33) together with the remark that, for anyl > 0, there exists a constantCl > 0 such that, forµ-almost allq ∈ P and allq-regularp ∈ M ,

(5.47) |γ(k)q (p) − γ(k)

q (p)|−1 ≤ Cl e−l|tk| .

By the definition of principal return trajectories and closing, the latter esti-mate follows immediately from (5.41), sinceγ(k)

q (p) − γ(k)q (p) are currents

of integration along a verticalq-regular arc.

The required estimate (5.28) finally follows from estimate (5.29) by thetrajectory splitting Lemma 5.8. In fact, for anys > 1 there exist positivereal numbersω(1)

s < ω(2)s and a measurable mapks : P → N such that

αsω(1)s > ω

(2)s − ω

(1)s and, forµ-almost allq ∈ P,

(5.48) ω(1)s k ≤ |tk(q)| ≤ ω(2)

s k , for all k ≥ ks(q) .

Let p(k)j |1 ≤ k ≤ n , 1 ≤ j ≤ mk ⊂ γT(p) be the sequence ofq-regular

points constructed in Lemma 5.8. Sinceαsω(1)s − (ω

(2)s − ω

(1)s ) > 0, by

estimates (5.29) and (5.48) , there exists a constantKs5(P) > 0 such that,

for µ-almost allq ∈ P, all q-regularp ∈ M and allT > 0,

(5.49)

n∑

k=ks(q)

e|tk+1|−|tk| |Πs−q

(γ(k)

q (p(k)j )

)|−s ≤ Ks

5(P) ;

n∑

k=ks(q)

e|tk+1|−|tk| |Esq

(γ(k)

q (p(k)j )

)|−s ≤ Ks

5(P) .

Page 105: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 105

In addition, by Lemma 5.8 there exists a constantK6(P) > 0 such that

(5.50) Lq

[ks(q)∑

k=1

mk∑

j=1

γ(k)q (p

(j)k )

]≤ K6(P)

ks(q)∑

k=1

e|tk+1(q)| ,

hence the estimate (5.28) follow from estimates (5.49), from the trajectorysplitting Lemma 5.8, from estimate (5.50) and finally from Lemma 5.2,which yields a bound on weighted Sobolev norms of (horizontal and verti-cal) trajectories in terms of theirRq-length.

5.3. The generic case.The above Theorem 5.9 implies, by a standardGottschalk-Hedlund argument, the following sharp result on the existenceof aGreen operatorfor the (horizontal) cohomological equation in the caseof generic orientable quadratic differentials.

Theorem 5.10. For any SO(2, R)-absolutely continuous, KZ-hyperbolicmeasureµ on a stratumM

(1)κ of orientable quadratic differentials and for

µ-almost allq ∈ M(1)κ , there exists a densely defined, anti-symmetric Green

operatorUq : H0q(M) → H0

q(M) for the horizontal cohomological equa-tion Squ = f . For anys > 1, the maximal domain of the Green operatorUq on the Hilbert spaceH0

q(M) ⊂ L2q(M) contains the dense subspace

(5.51) [Isq(M)]⊥ := f ∈ Hs

q (M)|D(f) = 0 for all D ∈ Isq(M) ,

and there exists a measurable functionCsκ : M

(1)κ → R+ such that the fol-

lowing holds. For anyf ∈ [Isq(M)]⊥, the Green solutionUq(f) ∈ L∞(M)

has zero-average and satisfies the estimate:

(5.52) |Uq(f)|L∞(M) ≤ Csκ(q) |f |s .

Proof. Let uTT∈R be the1-parameter family of measurable functions de-fined almost everywhere onM as follows: for allq-regularp ∈ M ,

(5.53) uT(p) :=1

T

∫T

0

∫ τ

0

f Fq(p, s) ds dτ .

By Theorem 5.9 (and Lemma 5.2) for anys > 1 there exists a measurablefunction Cs

κ : M(1)κ → R+ such that, forµ-almost allq ∈ M

(1)κ , for all

q-regularp ∈ M and allT > 0,

(5.54) |γT

q (p) − Πsq

(γT

q (p))|−s ≤ Cs

κ(q) .

We recall that the projectionΠsq : W−s

q (M) → B1q(M) is defined as the

composition of the orthogonal projectionW−sq (M) → Zs

q(M) with the pro-jectionZs

q(M) → B1q(M) determined by the splitting (5.18).

Page 106: Sobolev Regularity

106 GIOVANNI FORNI†

If f ∈ [Is(M)]⊥, the1-form αf := fηT ∈ W sq (M) is such thatC(α) = 0

for any horizontally basic currentC ∈ B1q(M). It follows from the estimate

(5.54) that, forµ-almost allq ∈ M(1)κ , for all q-regularp ∈ M and allT > 0,

(5.55) |uT(p)| ≤ |γT

q (α)| = |γT

q (p) − Πsq

(γT

q (p))(α)| ≤ Cs

κ(q) |f |s ,

hence the familyuTT∈R is uniformly bounded in the Hilbert spaceL2q(M)

for µ-almost allq ∈ M(1)κ . In addition, a computation shows that if the

horizontal flowFq is ergodic, asT → +∞,

(5.56) SquT = f −1

T

∫T

0

f Fq(·, τ) dτ → f in L2q(M) ,

hence any weak limitu ∈ L2q(M) of the familyuTT∈R is a solution of the

cohomological equationSqu = f . Since any functionf ∈ [Is(M)]⊥ haszero average, it follows by the definition (5.53) thatuT has zero average forall T > 0, hence any weak limit of the familyuTT∈R has zero average.By the ergodicity of the horizontal flowFq, the cohomological equationSqu = f has a unique zero average solution inUq(f) ∈ L2

q(M). It followsthat the operatorUq : [Is

q(M)]⊥ → H0q(M) ⊂ L2

q(M) is well-defined and

linear, forµ-almost allq ∈ M(1)κ . In addition, by the uniform bound (5.55)

the functionUq(f) ∈ L∞(M) and satisfies the required bound (5.52).

It remains to be proven that the linear operatorUq : H0q(M) → H0

q(M) isanti-symmetric. For anyf ∈ [Is

q(M)]⊥ andv ∈ Hs+1q (M) ⊂ Hs+1

q (M),sinceUq(f) ∈ L2

q(M) is a weak-solution of the equationSqu = f ,

(5.57) 〈Uq(f), Sqv〉q = −〈f, v〉q = −〈f, Uq(Sqv)〉q .

Since the subspaceSqv ∈ Hsq (M)|v ∈ Hs+1

q (M) is dense in the subspace[Is

q(M)]⊥ ⊂ Hsq (M) andUq : [Is

q(M)]⊥ → L2q(M) is continuous, it follows

from identity (5.57) by a density argument that

(5.58) 〈Uq(f), g〉q = −〈f, Uq(g)〉q , for all f, g ∈ [Isq(M)]⊥ .

Optimal results on the regularity of solutions of the cohomological equationfor intermediate ‘fractional’ regularity require smoothing and interpolationtechniques in the presence of distributional obstructions. The following def-inition appears to capture the relevant condition on the obstructions whichallow for effective smoothing and interpolation results.

Page 107: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 107

Definition 5.11. Let Hs|s ≥ 0 be a1-parameter family of normed spacessuch that the following embeddings hold:

(5.59) H∞ :=⋂

s>0

Hs ⊂ Hs ⊂ Hr ⊂ H0 , for all s ≥ r .

The order OH(D) (with respect toHs|s ≥ 0) of any linear functionalD ∈ (H∞)∗ is the non-negative real number

(5.60) OH(D) := infs ≥ 0 |D ∈ (Hs)∗ .

A finite system of linear functionalsD1, . . . , DJ ⊂ (Hσ)∗ will be calledσ-regular (with respect to the familyHs|s ≥ 0) if, for any τ ∈ (0, 1]there exists adualsystemu1(τ), . . . , uJ(τ) ⊂ Hσ such that the followingestimates hold. For all0 ≤ r ≤ σ and allǫ > 0, there exists a constantCσ

r (ǫ) > 0 such that, for alli, j ∈ 1, . . . , J,

(5.61) |uj(τ)|r ≤ Cσr (ǫ) τOH(Dj)−r−ǫ .

A finite systemD1, . . . , DJ ⊂ (Hs)∗ will be called regular if it is σ-regular for anyσ ≥ s. A finite dimensional subspaceI ⊂ (H∞)∗ will becalledσ-regular [regular] if it admits aσ-regular [regular] basis.

It was proved in §1.4, in particular in Corollary 2.19, that for anys ≥ 0 theclosureHs

q (M) ⊂ Hsq (M) of the weighted Sobolev space as a subspace

of the Friedrichs weighted Sobolev space, is equal to the perpendicularof a subspaceDs

q ⊂ H−sq (M), introduced in (2.94), of distributions sup-

ported on the singular setΣq ⊂ M . It follows by Theorem 2.17 and byCorollary 2.19 that all the spacesDs

q are regular with respect to the fam-ily Hs

q (M)|s ≥ 0 of Friedrichs weighted Sobolev spaces. This regular-ity result is the key to the existence of smoothing operatorsfor the familyHs

q (M)|s ≥ 0 of weighted Sobolev spaces. In fact, their construction inTheorem 2.24 is based on the regularity of the subspacesDs

q ⊂ H−sq (M)

and on the existence of smoothings for the family of Friedrichs weightedSobolev spaces, which can be defined by appropriate truncations of theFourier expansion.

Our goal in this section is to implement a similar strategy inorder to con-struct smoothing families with values in the perpendicularof the subspacesof invariant distributions. We prove below, under the hypothesis that theKontsevich-Zorich cocycle is non-uniformly hyperbolic, apreliminary re-sult on the regularity of the spaces of horizontal [vertical] invariant distri-butions with respect to the family of weighted Sobolev spaces. The basiccriterion for the regularity of spaces in invariant distributions is based onTheorem 4.35 and the following notion:

Page 108: Sobolev Regularity

108 GIOVANNI FORNI†

Definition 5.12. Let µ be anyGt-invariant ergodic probability measure onany stratumM

(1)κ ⊂ M

(1)g of orientable holomorphic quadratic differentials.

For anyq ∈ Rµ, a simple invariant distributionD± ∈ I±q(M \ Σq) will becalledcoherent(with respect to the familyHs

q (M)|s ≥ 0) if its weightedSobolev order and its Lyapunov exponent are related:

OHq (D±) = |l±µ (D±)| .

A finite dimensional space of invariant distributions will be called coherent(with respect to the familyHs

q (M)|s ≥ 0) if it has a basis of simplecoherent elements.

By the definitions of coherence and regularity, the following result followsimmediately from Theorem 4.35:

Lemma 5.13. Let µ be anyGt-invariant ergodic probability measure onany stratumM

(1)κ ⊂ M

(1)g of orientable holomorphic quadratic differen-

tials. For anyq ∈ Rµ, any coherent finite dimensional space of invariantdistributions is regular (with respect to the familyHs

q (M)|s ≥ 0).

For anyq ∈ M(1)κ and for anyk ∈ N\0, letJk

q (M) ⊂ Ikq (M) be the space

of horizontally invariant distributions defined as follows:

(5.62) Jkq (M) := ⊕k−1

j=0 LjTq

[I1q(M)

].

Corollary 5.14. For any SO(2, R)-absolutely continuous, KZ-hyperbolicmeasureµ on any stratumM

(1)κ ⊂ M

(1)g of orientable quadratic differen-

tials, for µ-almost all orientable quadratic differentialq ∈ M(1)κ and for

any k ∈ N, the spaceJkq (M) ⊂ Ik

q (M) is coherent, hence regular, withrespect to the familyHs

q (M)|s ≥ 0 of weighted Sobolev spaces.

Proof. It follows immediately from Theorem 4.29 for the casek = 0 andfrom Theorem 4.29 and Lemma 4.32 in the general case.

It was proved in Theorem 2.24 that the familyHsq (M)|s ≥ 0 of weighted

Sobolev spaces admits a family of smoothing operators. The existence ofa smoothing family together with the regularity of the distributional ob-structions are the key elements of the interpolation theoryfor solutions ofthe cohomological equation. We formalize below the basic construction ofsmoothing projectors onto the perpendicular of a regular subspace.

Definition 5.15. Let Hs|s ≥ 0 be a1-parameter family of normed spacessuch that the embeddings (5.59) hold. Asmoothing projectionof degreeσ ∈ R

+relative to a subspaceIσ ⊂ (Hσ)∗ is a familyP σ(τ)|τ ∈ (0, 1]

of linear operators such that the operatorP σ(τ) : H0 → [Iσ]⊥ ⊂ Hσ isbounded for allτ ∈ (0, 1] and the following estimates hold. For anyr,

Page 109: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 109

s ∈ [0, σ] and for anyǫ > 0, there exists a constantCσr,s(ǫ) > 0 such that,

for all u ∈ [Iσ ∩ (Hs)∗]⊥ ⊂ Hs and for allτ ∈ (0, 1],

(5.63)|P σ(τ)(u) − u|r ≤ Cσ

r,s(ǫ) |u|s τ s−r−ǫ , if s > r ;

|P σ(τ)(u)|r ≤ Cσr,s(ǫ) |u|s τ s−r−ǫ , if s ≤ r .

A smoothing projection relative to the trivial subspaceIσ = 0 ⊂ Hσ willbe called asmoothingof degreeσ ∈ R

+for the familyHs|s ≥ 0.

The following result is a straightforward generalization of Theorem 2.24,which implies the existence of smoothings of any finite degree for the familyHs

q (M)|s ≥ 0 of weighted Sobolev spaces.

Theorem 5.16.LetHs|s ≥ 0 be a1-parameter family of normed spacessuch that the embeddings (5.59) hold. If the familyHs|s ≥ 0 has asmoothing of degreeσ ∈ R+ and the subspaceIσ ⊂ (Hσ)∗ is σ-regular,there exists a smoothing projection of degreeσ ∈ R+ relative to Iσ .

Proof. Let Sσ(τ)|τ ∈ (0, 1] be a smoothing of degreeσ > 0 for the fam-ily Hs|s ≥ 0 and letD1, . . .DJ be aσ-regular basis for the subspaceIσ ⊂ (Hσ)∗. By the Definition 5.11 of regularity, for anyτ ∈ (0, 1], thereexists a dual basisu1(τ), . . . , uJ(τ) ⊂ Hσ such that the estimates (5.61)hold for all0 ≤ r ≤ σ and allǫ > 0. For anyu ∈ H0, we define

(5.64) P σ(τ)(u) := Sσ(τ)(u) −J∑

j=1

Dj (Sσ(τ)(u))uj(τ) .

We claim that the familyP σ(τ)|τ ∈ (0, 1] is a smoothing projection (ofdegreeσ > 0) relative to the subspaceIσ ⊂ (Hσ)∗. It follows immediatelyfrom the definition thatP σ(τ)(u) ∈ [Iσ]⊥ ⊂ Hσ for any u ∈ H0, hencethe operatorsP σ(τ) : H0 → [Iσ]⊥ ⊂ Hσ are well-defined, linear andbounded. We claim that, for anys ∈ [0, σ], for any j ∈ 1, . . . , J andfor any ǫ > 0, there exists a constantCs

j (ǫ) > 0 such that, for all vectorsu ∈ [Iσ ∩ (Hs)∗]⊥ ⊂ Hs, the following estimate holds:

(5.65) |Dj (Sσ(τ)(u)) | ≤ Csj (ǫ) |u|s τ s−OH(Dj)−ǫ , for all τ ∈ (0, 1] .

In fact, if the orderOH(Dj) ≥ s, sinceDj ∈ (Hrj)∗ for anyrj > OH(Dj)and the familySσ(τ)|τ ∈ (0, 1] is a smoothing forHs|s ≥ 0, for anyǫ > 0 there exists a constantAs

j(ǫ) > 0 such that for allu ∈ Hs and allτ ∈ (0, 1],

(5.66) |Dj (Sσ(τ)(u)) | ≤ |Dj|∗rj|Sσ(τ)(u)|rj

≤ Asj(ǫ) |u|s τ s−rj−ǫ/2 .

If OH(Dj) < s, sinceDj ∈ (Hrj)∗ ⊂ (Hs)∗, for anyOH(Dj) < rj < sand the familySσ(τ)|τ ∈ (0, 1] is a smoothing forHs|s ≥ 0, there

Page 110: Sobolev Regularity

110 GIOVANNI FORNI†

existsBsj (ǫ) > 0 such that, for allu ∈ [Iσ ∩ (Hs)∗]⊥ and allτ ∈ (0, 1],

(5.67)|Dj (Sσ(τ)(u)) | = |Dj (Sσ(τ)(u) − u) |

≤ |Dj|∗rj|Sσ(τ)(u) − u|rj

≤ Bsj (ǫ) |u|s τ s−rj−ǫ/2 .

The estimate (5.65) follows immediately from estimates (5.66) and (5.67)by takingrj ∈ (OH(Dj), O

H(Dj) + ǫ/2) for all j ∈ 1, . . . , J in bothcases.

By estimate (5.65) (just proved) and by the estimates onu1(τ), . . . , uJ(τ)in formula (5.61), for anyr ∈ [0, σ] and for anyǫ > 0, there exists a constantC

(1)r,s (ǫ) > 0 such that, for allu ∈ [Iσ ∩ (Hs)∗]⊥ and allτ ∈ (0, 1],

(5.68)J∑

j=1

|Dj (Sσ(τ)(u)) | |uj(τ)|r ≤ C(1)r,s (ǫ) |u|s τ s−r−ǫ .

Finally, the required estimates (5.63) for the familyP σ(τ)|τ ∈ (0, 1], de-fined in (5.64), follow from (5.68), sinceSσ(τ)|τ ∈ (0, 1] is a smoothing(of degreeσ > 0) for the familyHs|s ≥ 0.

The smoothness of the solutions of the cohomological equation is best ex-pressed with respect to the following uniform norms.

Definition 5.17. For anyk ∈ N, let Bkq (M) be the space of all functions

u ∈ Hkq (M) such thatSi

qTjq u = T i

qSjqu ∈ L∞(M) for all pairs of integers

(i, j) such that0 ≤ i + j ≤ k. The spaceBkq (M) is endowed with the norm

defined as follows: for anyu ∈ Bkq (M),

(5.69) |u|k,∞ :=

[∑

i+j≤k

|SiqT

jq u|2∞

]1/2

=

[∑

i+j≤k

|T iqS

jqu|

2∞

]1/2

.

Fors ∈ [k, k +1), letBsq(M) := Bk

q (M)∩Hsq (M) endowed with the norm

defined as follows: for anyu ∈ Bsq(M),

(5.70) |u|s,∞ :=(|u|2k,∞ + |u|2s

)1/2.

Theorem 5.18.Letµ be aSO(2, R)-absolutely continuous, KZ-hyperbolicmeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials.

For µ-almost allq ∈ M(1)κ and for anyk ∈ N, the spaceIk

q (M) is coherent,hence regular (with respect to the familyHs

q (M)|s ≥ 0) and, for any

s > k, there exists a measurable functionCk,sκ : M

(1)κ → R+ such that

the following holds. For any functionf ∈ [Ikq (M)]⊥ ∩ Hs

q (M) the Green

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SOBOLEV REGULARITY OF SOLUTIONS 111

solution Uq(f) of the cohomological equationSqu = f belongs to thespaceBk−1

q (M) and satisfies the estimates:

(5.71) |Uq(f)|k−1 ≤ |Uq(f)|k−1,∞ ≤ Ck,sκ (q) |f |s

Proof. For k = 1 the statement reduces to Theorem 5.10 and Corollary5.14. Sincef ∈ [J2

q(M)]⊥ implies by definition thatf andTqf ∈ [I1q(M)]⊥,

by Theorem 5.10, forµ-almost allq ∈ M(1)κ and for anys > k = 2, the

cohomological equationsSqu = f and SquT = Tqf have Green (zeroaverage) solutionsu(f) anduT (f) ∈ L∞(M) ⊂ L2

q(M) which satisfy thebounds,

(5.72)|u(f)|∞ ≤ Cs−1

κ (q) |f |s−1 ,

|uT (f)|∞ ≤ Cs−1κ (q) |f |s .

It follows that the distributionuT (f) − Tqu(f) ∈ H−1q (M) is horizontally

invariant. LetD1, . . . , Dg ⊂ I1q(M) be any regular basis such that

(5.73) OHq (Dj) = 1 − λµ

j , for all j ∈ 1, . . . , g .

It is no restrictive to assume thatD1 is the average. Since Green solutionshave zero average by definition, there existsF2(f), . . . Fg(f) ∈ C such that

(5.74) uT (f) − Tqu(f) =

g∑

j=2

Fj(f) Dj .

It follows from the bounds (5.72) that the mapsFj : [J2q(M)]⊥ → C are

linear bounded functionals, for allj ∈ 2, . . . , g, defined on the closedsubspace[J2

q(M)]⊥ ⊂ Hsq (M). In fact, letu1(τ), . . . , ug(τ) ⊂ H1

q (M)be any dual basis of the regular basisD1, . . . , Dg as in Definition 5.11.By Theorem 5.10, for anys > 2 and for any(τ2, . . . , τg) ∈ (0, 1]g−1,

(5.75)|Fj(f)| = |〈uT (f), uj(τj)〉q| + |〈u(f), Tquj(τj)〉q|

≤ Cs−1κ (q) |uj(τj)|0 |f |s + |uj(τj)|1 |f |s−1 .

We claim that, for eachj ∈ 2, . . . , g, the linear functionalFj extends toa horizontally invariant distributionΦj ∈ H−s

q (M) such that

(5.76) OHq (Φj) ≤ 1 + λµ

j , for all j ∈ 2, . . . , g .

In fact, since by Corollary 5.14 the spaceJ2q(M) is regular, by Theorem 5.16

there exists a smoothing projectionP σJ (τ)|τ ∈ (0, 1] of degreeσ > 2

relative to the subspaceJ2q(M) ⊂ H−2

q (M). Hence by definition, for anyr,s ∈ [0, σ] and anyǫ > 0, there exists a constantCσ

r,s(ǫ) > 0 such that, for

Page 112: Sobolev Regularity

112 GIOVANNI FORNI†

all f ∈ [J2q(M)]⊥ ∩ Hs

q (M) and for allτ ∈ (0, 1],

(5.77)|P σ

J (τ)(f) − f |r ≤ Cσr,s(ǫ) |u|s τ s−r−ǫ , if s > r ;

|P σJ (τ)(f)|r ≤ Cσ

r,s(ǫ) |f |s τ s−r−ǫ , if s ≤ r .

Let (fn)n∈N be the sequence of functions defined as follows:

(5.78) fn := P σJ (2−n)(f) ∈ [J2

q(M)]⊥ ∩ Hσq (M) , for all n ∈ N .

It follows from estimates (5.77) that, ifs − 1 < 1 + λµj < sj < s, for any

ǫ > 0, there exists a constantCσsj ,s(ǫ) > 0 such that

(5.79)|fn+1 − fn|s−1 ≤ Cσ

sj ,s(ǫ) |f |sj2−n(sj−s+1−ǫ) ;

|fn+1 − fn|s ≤ Cσsj ,s(ǫ) |f |sj

2n(s−sj+ǫ) .

Let u(n)1 , . . . , u

(n)g ⊂ H1

q (M) be the sequence of dual basis of the regularbasisD1, . . . , Dg ⊂ I1

q(M) defined as follows: for eachj ∈ 1, . . . , g,

(5.80) u(n)j := uj(2

−n) , for all n ∈ N .

By the estimate (5.61) in Definition (5.11) and by the identities (5.73), foranyǫ > 0 there exists a constantC(ǫ) > 0 such that, for allj ∈ 1, . . . , g,

(5.81)|u

(n)j |0 ≤ C(ǫ) 2−n(1−λµ

j −ǫ) ;

|u(n)j |1 ≤ C(ǫ) 2n(λµ

j +ǫ) .

For anysj > 1 + λj , there existss ∈ (2, 2 + λµj ) andǫ > 0 such that

(5.82) sj + 1 − s − λµj − 2ǫ := αj > 0 .

Hence by the estimate (5.61) and (5.75), forµ-almostq ∈ M(1)κ , there exists

a constantCjq (ǫ) := Cq(ǫ, s, sj, αj) > 0 such that

(5.83) |Fj(fn+1 − fn)| ≤ Cjq (ǫ)2

−αjn |f |sj.

By (5.77) and (5.78) the sequencefn → f in Hsq (M) for any s < σ.

Since (5.75) holds for anys > 2, it follows from (5.83) that, forµ-almostq ∈ M

(1)κ ,

(5.84) |Fj(f)| ≤Cj

q (ǫ)

1 − 2−αj|f |sj

+ |P σJ (1)(f)|σ .

Since the operatorP σJ (1) : L2

q(M) → Hσq (M) is bounded, we conclude that

for anysj > 1 + λµj there exists a continuous extensionΦ

sj

j ∈ H−sjq (M)

of the linear functionalFj : [J2q(M)]⊥ ∩ Hs

q (M) → C. By construction,for any rj, sj > 1 + λµ

j , the extensionsΦrj

j = Φsj

j (mod. J2q(M)). Since

J2q(M) is finite dimensional, for eachj ∈ 2, . . . , g there exists a distri-

bution Φj ∈ H−2q (M), which extends the linear functionalFj , such that

Page 113: Sobolev Regularity

SOBOLEV REGULARITY OF SOLUTIONS 113

the Sobolev orderOHq (Φj) ≤ 1 + λµ

j as claimed in (5.76). Finally, weprove that, by construction, the distributionsΦ2, . . . , Φg ∈ H2

q (M) are hor-izontally invariant. In fact, for anys > 2 and anyv ∈ Hs+1

q (M), thefunction fv := Sqv ∈ [I2

q(M)]⊥ ∩ Hsq (M), hence, by Theorem 5.10, the

cohomological equationsSqu = fv andSquT = Tqfv have unique Greensolutionsu(fv) anduT (fv) ∈ L∞(M) respectively. By the ergodicity ofthe horizontal foliation, sincev andTqv ∈ L2

q(M) are also zero-averagesolutions, the identitiesu(fv) = v anduT (fv) = Tqv hold. It follows thatuT (fv) − Tqu(fv) = 0, hence by (5.74), for allj ∈ 2, . . . , g,

(5.85) Φj(Sqv) = Fj(Sqv) = 0 , for all v ∈ Hs+1q (M) ,

thusΦ2, . . .Φg ⊂ Isq(M) and the claim is completely proved.

For µ-almost allq ∈ M(1)κ and for all integersk ≥ 2, let Ik

q (M) ⊂ Ikq (M)

be the subspaces defined as follows:

(5.86) Ikq (M) :=

k−2⋃

h=0

LhTq

[

J2q(M) ⊕

g⊕

j=2

C · Φj

]

.

It follows from the above construction that the following holds: for anyintegerk ≥ 2, for anys > k and for any functionf ∈ Hs

q (M) ∩ [Ikq (M)]⊥,

the Green solutionUq(f) ∈ Bk−1q (M) and satisfies the required estimate

(5.71). In fact, fork = 2 the statement follows by the construction of thesystem of invariant distributionsΦ2, . . . , Φg ⊂ H−2

q (M). Fork ≥ 3, thestatement can be proved by induction. In fact, by the induction hypothesis,for anys > k and anyf ∈ Hs

q (M)∩[Ikq (M)]⊥, the cohomological equations

Squ = f has a unique solutionu ∈ Bk−2q (M) such that

(5.87) |u|k−2,∞ ≤ Ck−1,s−1κ (q) |f |s−1 .

In addition, the functionuT := Tqu ∈ Hk−3q (M) is the unique solution

of the cohomological equationSquT = Tqf . SinceTqf ∈ Hs−1q (M) ∩

[Ik−1q (M)]⊥, by the induction hypothesis, the following estimate holds:

(5.88) |Tqu|k−2,∞ ≤ Ck−1,s−1κ (q) |Tqf |s−1 ≤ Ck−1,s−1

κ (q) |f |s .

Finally, by the Sobolev embedding theorem, there exists a continuous func-tion Cκ : M

(1)κ → R+ such that the following estimate holds:

(5.89)

[k−1∑

i=1

|Siqu|

2∞

]1/2

=

[k−2∑

i=0

|Siqf |

2∞

]1/2

≤ Cκ(q) |f |s .

The required estimate follows from (5.87), (5.88) and (5.89).

Page 114: Sobolev Regularity

114 GIOVANNI FORNI†

It remains to be proven that, forµ-almostq ∈ M(1)κ and all integersk ≥ 2,

the spaceIkq (M) is coherent. From the above argument it follows that

(5.90) Iq(M) =

+∞⊕

k=2

Ikq(M) .

In fact, if f ∈ H∞q (M) is such thatf ∈ [Ik

q (M)]⊥ for all k ∈ N, there existsu ∈ H∞

q (M) such thatSqu = f , hencef ∈ Iq(M)⊥. The identity (5.90)immediately implies that

(5.91) Ikq (M) = Ik

q(M) , for all integersk ≥ 2 .

The identities (5.91) in turn imply that the systemΦ2, . . . , Φg constructedabove is linearly independent overJ2

q(M), in particular

(5.92) dimC I2q(M) = dimC I2

q(M) = 3g − 2 .

In fact, by the above construction the following identity holds:

(5.93) Iq(M) = I1q(M) +

g⊕

j=2

C · Φj + LTq[Iq(M)] .

Let Dq : Bq(M) → Iq(M) the isomorphism between the spaceBq(M) ofhorizontally basic currents and the space of horizontally invariant distribu-tions defined in (3.34). Letδq : Bq(M) → Bq(M) be the differential mapon the space of basic currents introduced in (3.61). From (5.93) it followsimmediately that

(5.94) Bq(M) = B1q(M) +

g⊕

j=2

C · D−1q (Φj) + δq[Bq(M)] .

By the structure theorem for (real) basic currents (Theorem3.21), the coho-mology mapjq : Bq(M) → H1(M, R) vanishes on the spaceδq[Bq(M)].On the other hand, by Corollary 3.20, forµ-almost allq ∈ M

(1)κ , the co-

homology map onBq(M) has rank of codimension1 in the homologyH1(M, R), hence of dimension2g − 1. SinceB1

q(M) has dimensiongand the mapDq : Bq(M) → Iq(M) is an isomorphism, it follows that thesystemΦ2, . . . , Φg is linearly independent overJ2

q(M) and (5.92) holds.

We claim that, forµ-almost allq ∈ M(1)κ , the spaceI2

q(M) = I2q(M) is

coherent. For any Lyapunov exponentl < 0 of the cocycleΦ2t |I

2q(M),

let E2q (l) ⊂ I2

q(M) denote the corresponding Oseledec subspace and letF 2

q (l) ⊂ E2q (l) be the subspace of coherent distributions. Let

l1 < · · · < ld

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SOBOLEV REGULARITY OF SOLUTIONS 115

be the distinct Lyapunov exponents of the cocycleΦ2t |I

2q(M) on the the

Oseledec complement of the subspaceJ2q(M). SinceJ2

q(M) is coherentand

I2q(M) = I2

q(M) = J2q(M) ⊕

d⊕

i=1

E2q (li) ,

it is sufficient to prove the identities:

(5.95) F 2q (li) = E2

q (li) , for all i ∈ 1, . . . , d .

By Theorem 4.30 and Lemma 4.32, the Lyapunov spectrum ofΦ2t |J

2q(M)

is the ordered set

0 > λµ2 − 1 ≥ · · · ≥ λµ

g − 1 ≥ λµ2 − 2 ≥ · · · ≥ λµ

g − 2 .

Hence, by the description of the Lyapunov spectrum ofΦ2t |I

2q(M) given

in Corollary 4.33, the set

l1, . . . , ld = −1 − λµ2 , . . . ,−1 − λµ

g .

For anys ≥ 0, letIq(s) ⊂ E2q (l1)⊕· · ·⊕E2

q (ld) be the subset of horizontallyinvariant distributions of Sobolev order less or equal tos ≥ 0. It followsfrom Lemma 4.26 that the following inclusions hold:

(5.96) Iq(|li|) ⊂⊕

j≤i

E2q (lj) , for all i ∈ 1, . . . , d .

By the estimate (5.76) on Sobolev orders of the distribuitons in the systemΦ2, . . . , Φg, the following lower bounds hold:

(5.97) dimCIq(|li|) ≥i∑

j=1

dimCE2q (lj) , for all i ∈ 1, . . . , d .

It follows from (5.96) and (5.97) that the inclusions in (5.96) are in factidentities, for alli ∈ 1, . . . , d, and by Lemma 4.26 the claim (5.95) holds.

We have thus proved that the spaceI2q(M) = I2

q(M) is coherent. It follows

from definition (5.86) and Lemma 4.32 that the spaceIkq (M) is coherent for

any integerk ≥ 3. Since by (5.91) the identityIkq(M) = Ik

q(M) holds forall k ≥ 2, the spaceIk

q (M) is coherent and the proof is complete.

Theorem 5.19.Letµ be aSO(2, R)-absolutely continuous, KZ-hyperbolicmeasure on a stratumM(1)

κ ⊂ M(1)g of orientable quadratic differentials.

For µ-almost allq ∈ M(1)κ and for anys ∈ R+, the spaceIs

q(M) is coherent,hence regular (with respect to the familyHs

q (M)|s ≥ 0) and, for any

0 < r < s − 1 there exists a measurable functionCr,sκ : M

(1)κ → R+ such

that the following holds. For any functionf ∈ [Isq(M)]⊥ ∩ Hs

q (M) the

Page 116: Sobolev Regularity

116 GIOVANNI FORNI†

Green solutionUq(f) of the cohomological equationSqu = f belongs tothe spaceHr

q (M) and satisfies the estimates:

(5.98) |Uq(f)|r ≤ Cr,sκ (q) |f |s

Proof. By Theorem 5.18, the subspaceIkq (M) is coherent forµ-almost all

q ∈ M(1)κ and for anyk ∈ N. Since for anys < k the sub-bundleIs

κ,+(M) ⊂

Ikκ,+(M) is Φk

t -invariant, it follows thatIsq(M) is coherent, hence regular

by Lemma 5.13, forµ-almost allq ∈ M(1)κ .

By Theorem 5.16, forµ-almost all q ∈ M(1)κ there exists a smoothing

projectionP σ(τ)|τ ∈ (0, 1] of any given degreeσ > 0 relative to thesubspaceIσ

q (M). Let s > 1 and letσ ∈ R+ and k ∈ N be such thatσ > k + 1 ≥ s ≥ σ − 1 > k. Let f ∈ [Is

q(M)]⊥ ∩ Hsq (M). By Defini-

tion 5.15 of a smoothing projection, for anyf ∈ [Isq(M)]⊥ ∩ Hs

q (M), thefunctionP σ(τ)(f) ∈ [Iσ

q (M)]⊥ ∩Hσq (M) and satisfies the Sobolev bounds

(5.63). By Theorem 5.18 the cohomological equationSqu = P σ(τ)(f) hasa (unique) Green solutionu(τ) ∈ Hk−1

q (M) and there exists a measurable

functionCκ : M(1)κ → R+ such that , for allτ ∈ (0, 1],

(5.99)|u(τ) − u(τ/2)|k ≤ Cκ(q) |P

σ(τ)(f) − P σ(τ/2)(f)|σ ,

|u(τ) − u(τ/2)|k−1 ≤ Cκ(q) |Pσ(τ)(f) − P σ(τ/2)(f)|σ−1 .

By the interpolation inequality proved in Lemma 2.10, for any r ∈ [k−1, k]there existsCk,r > 0 such that, for allτ ∈ (0, 1],

(5.100) |u(τ)−u(τ/2)|r ≤ Ck,r |u(τ)−u(τ/2)|k−rk−1 |u(τ)−u(τ/2)|r−k+1

k .

By the bounds (5.63), it follows from (5.99) and (5.99) that,for anyǫ > 0there existsCσ

r,s(ǫ) > 0 such that

(5.101) |u(τ) − u(τ/2)|r ≤ Cσr,s(ǫ) |f |s τ (k−r)(s−σ+1−ǫ)τ (r−k+1)(s−σ−ǫ) .

Sincer < s − 1, it is possible to chooseσ ∈ R+, ǫ > 0 andk ∈ N so that

α = (k − r)(s − σ + 1 − ǫ) + (r − k + 1)(s − σ − ǫ) > 0 .

It follows then from the bound (5.101), that the sequenceu(2−n) |n ∈ Nis Cauchy inHr

q (M), hence it converges to a functionu ∈ Hrq (M) of zero

average. SinceP σ(τ)(f) → f in Hsq (M) asτ → 0+, it follows that the

function u ∈ Hrq (M) is the unique zero-average (Green) solution of the

cohomological equationSqu = f . The required Sobolev bound (5.98) alsofollows from (5.101). In fact, by the interpolation inequality (Lemma 2.10),by Theorem 5.18 and by the bounds (5.63) for the smoothing projection,there exists a measurable functionC ′

κ : M(1)κ → R+ such that,

(5.102) |u(1)|r ≤ C ′κ(q) |f |s .

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SOBOLEV REGULARITY OF SOLUTIONS 117

The bound (5.98) can then be derived from (5.101) and (5.102).

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