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arXiv:0707.0940v2 [math.DS] 12 Jul 2007SOBOLEV REGULARITY OF SOLUTIONSOF THE COHOMOLOGICAL EQUATIONGIOVANNI FORNICONTENTS1. 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 Teichmller ow 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 1171. INTRODUCTIONWe prove the sharpest results available on the loss of regularity for so-lutions of the cohomological equation for translation ows. For any giventranslation surface and for the directional ow in almost all directions theDate: February 1, 2008.1991 Mathematics Subject Classication. 37A20, 37C10, 37C15.Key words and phrases. Invariant distributions, cohomological equation, Teich mullerow, Kontsevich-Zorich cocycle, distributional cocycles. The author gratefully acknowledges support from National Science Foundation grantDMS-0244463.12 GIOVANNI FORNIsmallest loss Sobolev regularity available to the Fourier analysis methodsdeveloped in [For97] is essentially 3 + (for any >0). We remark thatthis is the best result available for the ow 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 equation for almost allinterval exchange transformations (IETs), do not apply to rational billiardsfor the well-known reason that IETs induced by rational billiard ows forma zero measure set in the space of all IETs.For almost all translation surfaces in every stratum of the moduli space,we prove the rened, optimal result that the loss of Sobolev regularity forthe directional ow in almost all directions is1 + (for any >0). Inthis case, in fact we prove that for any function of Sobolev orders>1,the solution and its derivatives up to orderk 1) is ex-actly equal to the genus of the surface. For such coboundaries the transferfunction is actually in L.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 with Diophantine conditions on 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 include Oseledec regu-larity (or rather a weaker coherence property) with respect to Kontsevich-Zorich renormalization cocycle. The Oseledecs theorem is invoked to en-sure that the set of regular (coherent) IETs 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 Oseledecs 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 as triviality of time-changes forSOBOLEV REGULARITY OF SOLUTIONS 3ows, asymptotic of ergodic averages and the smooth 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 ows 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 ows. This in an exampleof elliptic dynamics (no divergence of nearby orbits) which can be studiedto a great extent by the classical theory of Fourier series. It is characterizedby the small divisors appearing in the Fourier coefcients of solutions,which lead to a loss of regularity. It is not difcult to see that the optimalloss of Sobolev regularity for the full measure set of Roth ows is 1 + andthat for such ows any zero average function of Sobolev order s>1 is anL2coboundary. 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 of constant type seems to havebeen explicitly considered there.For systems with intermediate behavior, that is, for elliptic systems withsingularities or for parabolic systems (characterized by polynomial diver-genceofnearbyorbits) muchlessisknown. Theauthordiscoveredin[For97] that the cohomological equation for generic translation ows (orequivalently for generic IETs) has nitely smooth solutions for sufcientlysmooth data under nitely many distributional conditions. In other terms,on one hand the problem shares a typical feature of small divisors prob-lems, namely the nite loss of regularity of solutions with respect to thedata; on the other hand, a new phenomenon appears: the existence of in-nitely many independent distributional obstructions (of increasing 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 ows (on surfaces constantnegative curvature) [FF03], for generic nilows on quotients of the Heisen-berg group [FF06] and generic nilows on general nilmanifolds [FF07]. Inall cases the fundamental features of nite loss derivatives and of the exis-tence of innitely many independent distributional obstructions have beenestablished (although the structure of the space of invariant distributions is4 GIOVANNI FORNIsignicantly different for IETs, horocycle ows and nilows). For horo-cycle ows and for Heisenberg nilows it was possible to estimate that theloss of Sobolev regularity is 1 + (for any > 0) and to establish the con-jectural relation that the Sobolev order of the distributional obstructions berelated to the Lyapunov exponents of the distribution under the appropri-ate renormalization dynamics. In this paper we prove analogous results forgeneric translation ows. We should point out that for generic nilows ongeneral nilmanifolds the loss of regularity and the regularity of the distri-butional obstructions seem to depend on the depth and rank of the nilpotentgroup considered, although no lower bound was established in [FF07].Letq be a holomorphic orientable quadratic differential on a Riemannsurface M of genus g 1. The horizontal and vertical measured foliations(in the Thurstons sense) associated to a holomorphic quadratic differen-tial q onMare dened asFq=(q1/2)=0 (the horizontal foliation)andFq=(q1/2) =0. Such foliations are well-dened even in thecase that there is no globally dened square root of the quadratic differen-tial. The horizontal foliation is endowed with the transverse measure givenby [(q1/2)[, the vertical foliation is endowed with the transverse measuregiven by [(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 at metric Rq associatedwith any quadratic differential q on M. Such a metric has conical singular-ities as the nite set q= p M[q(p) = 0. If q is orientable there existsa (positively oriented) parallel orthonormal frameSq, Tq of the tangentbundleTM[M qsuch thatSq is tangent to the horizontal foliationFqandTq is tangent to the vertical foliationFq everywhere onM q. Inother terms, the at metric Rq has trivial holonomy. In another equivalentformulation, any orientable holomorphic quadratic differential determinesa translation structure onM, that is, an equivalence class of atlases withtransition functions given by translation of the euclidean plane (see for in-stance the excellent survey [MT05], 1.8. For a given orientable quadraticdifferential q on a Riemann surface M, we will consider the one-parameterfamily of vector elds on M q dened as(1.1) S:= cos Sq+sin Tq , S1.The vector eldS is a parallel normalized vector eld in the direction atangle S1with the horizontal. We remark that it is not dened as thesingular set qof the at metric. hence the ow it generates is dened(almost everywhere) on the complement of the union of all separatricesSOBOLEV REGULARITY OF SOLUTIONS 5of 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 than 2.In fact, since the quadratic differential is supposed to be orientable it haszeroes of even order and the orbit foliations of the vector elds (1.1) has mstable and m unstable separatrices at any zero of order 2m.Our goal is to investigate the loss of (Sobolev) regularity of solutions of thecohomological equationSu=ffor Lebesgue almost all S1. Theauthor proved in [For97] that if the function f is sufciently regular, satis-es a nite number of independent distributional conditions (which includeconditions on the jets at the singularities) then there exists a nitely smoothsolution (unique up to additive constants). The loss of regularity was esti-mated in that paper to be no more than 9 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 family Hsq(M)[s Rof weighted Sobolev spaces. Such spaces were introduced in [For97] forall s Z as follows. Let qbe the standard (degenerate) volume formonMof the at metric Rq. The space H0q(M) is the space L2(M, q) ofsquare-integrable functions. For kN, the space Hkq(M) is the subspaceof functions fH0q(M) such that the weak derivatives SiqTjqfH0q(M)[andTiqSjqf H0q(M)] for alli + j k and the spaceHkq(M) is thedual Hilbert spaceHkq (M). In 2 of this paper we introduce weightedSobolev spaces with arbitrary (real) exponents by 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 weighted Sobolev spacescombine standard Sobolev smoothness conditions on M q with restric-tions on the jet of the functions at the singular set q M.As discovered in [For97], for functions fC0(M q) the space of alldistributional obstructions to the existence of a solution uC0(M q)of the cohomological equationSu=fcoincides for almost allS1with the innite dimensional space of all S-invariant distributions:(1.2) Iq,(M q) := D D(M q) [ SD = 0 in D(M q) .For dataf Hsq(M) of nite Sobolev differentiability, a complete set ofobstructions is given for almost allS1by the nite dimensional sub-space of invariant distributions(1.3) Isq,(M) := D Hsq(M) [ SD = 0 in Hsq(M) .The goal of this paper is to prove optimal estimates on the Sobolev reg-ularity of solutions of the cohomological equationSu =fand on the6 GIOVANNI FORNIdimension of the spacesIsq,(M) of invariant distributions for all s>0. In5.1, Theorem 5.1, we proveTheorem A1. Let q be any orientable holomorphic quadratic differential.LetkN be any integer such that k3 and lets>k andr 0 such that the following holds. If f Hsq(M) is suchthatD(f)=0 for allDIsq,(M), the cohomological equation Su=fhas a solution u Hrq(M) satisfying the following estimate:(1.4) [u[r Cr,s() [f[s.The dimensions of the spaces of invariant distributions can be estimated asfollows (see 3.3, Corollary 3.20 and Theorem 3.21):Theorem A2. Let q be any orientable holomorphic quadratic differential.Let k N be any integer such that k 3 and let k< s k +1. For almostall S1(with respect to the Lebesgue measure),(1.5) 1 + 2(k 2)(g 1) dimIsq,(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 an arbitrary orientable quadratic differential.We prove much sharper results for almost all orientable quadratic differ-entials. The moduli space of orientable holomorphic quadratic differen-tialsq on some Riemann surfaceMqwith a given pattern of zeroes, thatis, withzeroesof(even)multiplicities =(k1, . . . , k)at anitesetq=p1, . . . , p Mqis astratumMof the moduli spaceMgofall holomorphic quadratic differential. Let M(1)M(1)gbe the subsetsof quadratic differential of total area equal to 1. It was proved by H. Ma-sur [Mas82] and W. Veech [Vee86] that each stratumM(1)carries an ab-solutely continuous probability measure (1), invariant under the action ofthe Teichmller geodesic ow, which is ergodic when restricted to eachconnected component ofM(1)(the connected components of strata of ori-entable quadratic differentials were classied in [KZ03]). In fact, there isnatural action of the group SL(2, R) on the moduli space M(1)gsuch that theTeichmller geodesic ow corresponds to the action of the diagonal sub-group of SL(2, R) and the measure (1)is SL(2, R) invariant.In [Kon97] M. Kontsevich introduced a renormalization cocycle for trans-lation ows, inspired to the Rauzy-Veech-Zorich cocycle for interval ex-changetransformations. TheKontsevich-Zorich cocycleisadynamicalsystem on an orbifold vector bundle over M(1)with ber the rst cohomol-ogy H1(Mq, R) of the Riemann surface carrying the orientable holomorphicSOBOLEV REGULARITY OF SOLUTIONS 7quadratic differential qM(1). The action of such a dynamical system is(by denition of a cocycle) linear on the bers and projects onto the Teich-mller geodesic ow on the base M(1). Since the cocycle is symplectic, forany probability measure on a stratumM(1), the Lyapunov spectrum of theKontsevich-Zorich cocycle takes the form:(1.6) 1 g 0 g+1= g 2g= 1 .In addition, it is not difcult to prove that 1=1. A probability measure on a stratum M(1), invariant under the Teichmller geodesic ow, will becalled(a)SO(2, R)-absolutely continuous if it induces absolutely contin-uous measures on every orbit of the circle groupSO(2, R) SL(2, R);(b) KZ-hyperbolic if all the Lyapunov exponents in (1.6) are non-zero. Itis immediate that all SL(2, R)-invariant measures are SO(2, R)-absolutelycontinuous. It wasrstproved in[For02]that the measure(1)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 all quadratic differentials with re-spect to any SO(2, R)-absolutely continuous, KZ-hyperbolic, Teichmllerinvariant, probability measure on any stratumM(1)of orientable quadraticdifferentials. Thesmoothnessinformationsonthesolutionsisstrongerthan just Sobolev L2regularity and it is naturally encoded by the followingspaces. For any k N, let Bkq(M) be the space of all functions u Hkq(M)such that SiqTjqu = TiqSjqu L(M) for all pairs of integers (i, j) such that0i + j k. The spaceBkq(M) is endowed with the norm dened asfollows: for any u Bkq(M),(1.7) [u[k,:=_ i+jk[SiqTjqu[2_1/2=_ i+jk[TiqSjqu[2_1/2.For s [k, k +1), let Bsq(M) := Bkq(M) Hsq(M) endowed with the normdened as follows: for any u Bsq(M),(1.8) [u[s, :=_[u[2k,+[u[2s_1/2.In 5.3, Theorem 5.19, we prove the following:TheoremB1. Let be any SO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)of orientable quadratic differen-tials. Lets>1 and letr 0 such that the following holds. If fHsq(M) is suchthatD(f)=0 for allDIsq,(M), the cohomological equation Su=fhas a solution u Brq(M) satisfying the following estimate:(1.9) [u[r, Cr,s() [f[s.The regularity of invariant distributions can be precisely determined as fol-lows (see 4.3, Corollary 4.33, and 5.3, Theorem 5.19). For any q M(1)and any distributionDD(M q), the weighted Sobolev order is thenumber(1.10) OHq(D) = infs R+[D Hsq(M) .Let Iq,(M):= Isq,(M)[s 0 denote the space of all S-invariant dis-tribution of nite Sobolev order and let Iq,(M) Iq,(M) be the subspaceof invariant distributions vanishing on constant functions. It follows imme-diately by the denitions that Iq,(M) = C Iq,(M).TheoremB2. Let be any SO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)of orientable quadratic differen-tials. For -almost all qM(1)and for almost all S1(with respect tothe Lebesgue measure), the space Iq,(M) has a basis Di,j() such that(1.11) OHq(Di,j()) = i(j +1) , i 2, . . . , 2g 1,j N0 .In addition, the basis Di,j() can be generated fromthe nite dimensionalsubsystem D2, . . . , D2g1 by the following differential relations:(1.12) Di,j() = TjDi,0() , i 2, . . . , 2g 1,j N 0 .The above Theorems B1 and B2 are proved by methods based on renor-malization, whichwereinspiredbytheworkofMarmi-Moussa-Yoccoz[MMY05]. However, our approach differs from theirs since we explicitlyassume that the Lyapunov exponents of the Kontsevich-Zorich cocycle areall non-zero. The main idea of the argument, as in [MMY05], is to 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 in H1(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 distributional cocycles over theTeichmller ow 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 the denition ofSOBOLEV REGULARITY OF SOLUTIONS 9distributional cocycles, and we have strengthened the estimates proved in[For02] under the slightly stronger (and correct) assumption that the func-tions considered belong to Hsq(M) for some s > 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 difculties, hence the lack of precisionof Theorem A1 and A2 for intermediate Sobolev regularity. In the genericcase 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 denition (see Denition 5.11). A nite systemD1, . . . , DJHq(M) of nite order distributions is called -regular(with respect to the familyHsq(M) of weighted Sobolev spaces) if forany (0, 1] there exists a dual systemu1(), . . . , uJ()Hq (M)(that is, the identities Di(uj()) = ij hold for all i, j 1, . . . , J and all (0, 1]) such that the following estimates hold. For all 0 r and all > 0, there exists a constant Cr () > 0 such that, for all i, j 1, . . . , J,(1.13) [uj()[r Cr () OH(Dj)r.A nite system D1, . . . , DJ Hsq(M) of nite order distributions willbe calledregular if it is-regular for any s. A nite dimensionalsubspace I Hsq(M) of nite order distributions will be called -regular[regular] if it admits a -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. Let be any SO(2, R)-absolutely continuous, KZ-hyperbolicprobability measure on any stratumM(1)of orientable quadratic differen-tials. For -almost all qM(1), for almost all S1and for all s>0,the space Isq,(M) Hsq(M) of S-invariant distributions is regular.2. FRACTIONALWEIGHTED SOBOLEVSPACESIn [For97] we have introduced a natural scale of weighted Sobolev spaceswith integer exponent associated with any orientable holomorphic quadraticdifferential q on a Riemann surface M (of genus g 2). In this section weextend the denition of weighted Sobolev spaces to arbitrary (real) expo-nents by methods of interpolation theory.2.1. WeightedSobolevspaces. Let q:=p1, . . . , p Mbetheset ofzerosoftheholomorphicquadraticdifferential q, ofevenorders(k1, . . . , k) respectively with k1 + + k= 4g 4. Let Rq:= [q[1/2be10 GIOVANNI FORNIthe at metric with cone singularities at q induced by the quadratic differ-ential q on M. With respect to a holomorphic local coordinate z= x + iy,the quadratic differential q has the form q=(z)dz2, where is a locallydened holomorphic function, and, consequently,(2.1) Rq= [(z)[1/2(dx2+ dy2)1/2, q= [(z)[ dx dy .The metric Rq is at, it is degenerate at the nite set q of zeroes of q and,ifqis orientable, it has trivial holonomy, henceqinduces a structure oftranslation surface on M.The weightedL2space is the standard spaceL2q(M) :=L2(M, q) withrespect to the area elementqof the metricRq. Hence the weightedL2norm [[0 are induced by the hermitian product , )q dened as follows:for all functions u,v L2q(M),(2.2) u, v)q:=_Mu v q.Let Fq be the horizontal foliation, Fq be the vertical foliation for the holo-morphic quadratic differential q on M. The foliations Fq and Fq are mea-sured foliations (in the Thurstons sense):Fq is the foliation given (locally)by the equation (q1/2) = 0 endowed with the invariant transverse measure[(q1/2)[,Fq is the foliation given (locally) by the equation (q1/2)=0endowed with the invariant transverse measure [(q1/2)[. If the quadraticdifferential q is orientable, since the metric Rq is at with trivial holonomy,there exist commuting vector elds Sq and Tq on M q such that(1)The frame Sq, Tq is a parallel orthonormal frame with respect tothe metricRqfor the restriction of the tangent bundleTMto thecomplement M q of the set of cone points;(2)the vector eld Sq is tangent to the horizontal foliation Fq, the vectoreld Tq is tangent to the vertical foliation Fq on M q [For97].In the following we will often drop the dependence of the vector elds Sq,Tq on the quadratic differential in order to simplify the notations. We have:(1) LSq=LTq= 0 on M q , that is, the area form q is invariantwith respect to the ows generated by S and T;(2) Sq= (q1/2) and Tq= (q1/2), hence the 1-forms S:= Sq,T:= Tq are smooth and closed on M and q= T S.It follows from the area-preserving property(1) that the vector eldS,Tare anti-symmetric as densely dened operators onL2q(M), that is, for allfunctions u, v C0(M q), (see [For97], (2.5)),(2.3) Su, v)q= u, Sv)q, respectively Tu, v)q= u, Tv)q.SOBOLEV REGULARITY OF SOLUTIONS 11In fact, by Nelsons criterion [Nel59], Lemma 3.10, the anti-symmetric op-erators S, T are essentially skew-adjoint on the Hilbert space L2q(M).The weighted Sobolev norms[[k, with integer exponentk>0, are theeuclidean norms, introduced in [For97], induced by the hermitian productdened as follows: for all functions u, v L2q(M),(2.4) u, v)k:=12

i+jkSiTju, SiTjv)q +TiSju, TiSjv)q .The weighted Sobolev norms with integer exponent k 0, let Nq() := card EV(Q) / , where eacheigenvalue EV(Q) is counted according to its multiplicity.Theorem 2.3. ([For97], Th. 2.5) There exists a constant C> 0 such that(2.8) lim+Nq()=vol(M, Rq).Let q:= Sq Tq be the Cauchy-Riemann operators induced by the holo-morphic orientable quadratic differential qonM, introduced in [For97],3. Let Mq L2q(M) be the subspaces of meromorphic, respectively anti-meromorphic functions (with poles at q). By the Riemann-Roch theorem,the subspacesMqhave the same complex dimension equal to the genusg 1 of the Riemann surface M. In addition, M+q Mq= C, hence(2.9) Hq:=_M+q__Mq_= u L2q(M) [_Mu q=0 .Let H1q:=Hq H1q(M). By Theorem 2.2, the restriction of the hermitianform toH1qis positive denite, hence it induces a norm. By the Poincarinequality (see [For97], Lemma 2.2 or [For02], Lemma 6.9), the Hilbertspace (H1q, Q) is isomorphic to the Hilbert space (H1q, , )1).Proposition 2.4. ([For97], Prop. 3.2) The Cauchy-Riemann operators qare closable operators on the common domain C0(M q) L2q(M) andtheir closures (denote by the same symbols) have the following properties:(1)the domains D(q) = H1q(M) and the kernels N(q) = C;(2)the ranges Rq:= Ran(q) =_Mq_are closed in L2q(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 theHilbert space L2q(M) of eigenfunctions of the Dirichlet form (2.7) and let : N R+ 0 be the corresponding sequence of eigenvalues:(2.10) n:=Q(en, en) , for eachn N.SOBOLEV REGULARITY OF SOLUTIONS 13The Friedrichs weighted Sobolev norm ||s of order sR+is the norminduced by the hermitian product dened as follows: for all u, v L2q(M),(2.11) (u, v)s :=

nN(1 +n)su, en)qen, v)q .The inner products (2.4) and (2.11) induce equivalent Sobolev norms onthe weighted Sobolev space Hkq(M), for all kZ+. In fact, the followingresult, a sharp version of Lemma 4.2 of [For97], holds:Lemma 2.5. For eachkZ+there exists a constantCk>1 such that,for any orientable holomorphic quadratic differentialq onMand for allfunctions u Hkq(M),(2.12) C1k[u[k|u|kCk[u[k.Proof. By Proposition 2.4, (3), and Lemma 2.1, for all u Hk+1q(M),k N, the following identity holds (see (4.4) in [For97]):(2.13) [u[2k+1 = [u[20+

i+jkQ(SiTju, SiTju) == [u[20+

i+jkSiTjqu, SiTjqu)q= [u[20+[qu[2k.Hence in particularuHk+1q(M), k N, impliesquHkq(M). Ifk 1, a second application of the identity (2.13) yields(2.14) [u[2k+1 = [u[21+[qqu[2k1.The statement then follows by induction on k N. For k= 0 it is immedi-ate and for k = 1, by the identity (2.13),(2.15) [u[21 = [u[20+Q(u, u) =

nN(1 +n) [u, en)q[2.For k>1, by induction hypothesis we can assume that the norms [[k1and ||k1 are equivalent, that is, there exists a constant Ck1>1 suchthat, for all u Hk1q(M),(2.16) C1k1[u[k1|u|k1Ck1[u[k1.Since by (2.3) for all u, vH1q(M), qu, v)q=u, qv)q, the adjointoperator (q)= qon H1q(M). By Proposition 2.4, (3), we have(2.17) |qqu|2k1 =

nN(1 +n)k1[qu, qen)q[2==

nN(1 +n)k1[Q(u, en)[2=

nN(1 +n)k12n[u, en)q[2.14 GIOVANNI FORNIThere exists a constant Ck+1> 1 such that, for all 0,(2.18) C2k+1 (1 +)k+1 1 + + C2k12(1 +)k1 1 + + C2k12(1 +)k1 C2k+1 (1 +)k+1.By (2.14), (2.15), (2.16), (2.17) and (2.18), the estimate(2.19) C1k+1[u[k+1|u|k+1Ck+1[u[k+1follows, thereby completing the induction step. 2.2. Fractional Sobolev norms. Let q be any orientable quadratic differ-ential on M. For all s 0, let(2.20)Hsq(M) := u L2q(M) /

nN(1 +n)s[u, en)q[2< +,endowed with the hermitian product given by (2.11) and, for any s > 0, letHsq(M) be the dual space of the Hilbert spaceHsq(M). The spacesHsq(M)will be called the Friedrichs (fractional) weighted Sobolev spaces.Let H1H2 be Hilbert spaces such that H1 embeds continuously into H2with dense image. For all [0, 1], let[H1, H2]be the (holomorphic)interpolation space ofH1H2 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 interpolationfamily Hsq(M)sR of Hilbert spaces: for all r, s R with r < s,(2.21)H(1)r+sq(M) [ Hrq(M),Hsq(M)].The family Hsq(M)sR of fractional weighted Sobolev spaces will be de-ned as follows. Let[s] N denote the integer part and s[0, 1) thefractional part of any real number s 0.Denition 2.7. (1)The fractional weighted Sobolev norm[[s of or-der s0 is the euclidean norm induced by the hermitian productdened as follows: for all functions u, v Hq(M),(2.22) u, v)s:=12

i+j[s](SiTju, SiTjv){s} + (TiSju, TiSjv){s}.(2)The fractional weighted Sobolev norm[[s of orders 0 by denition of theSobolev spaces Hsq(M).The operatorss: Hs+1q(M)Hsq(M) do not extend to bounded oper-atorsHs+1q(M) Hsq(M) unlessMq is a at torus. In fact, every nitecombination fof eigenfunctions of the Dirichlet form belongs toHsq(M),for allsR, butqf ,H1q(M) in all cases because of the presence ofobstructions in the Taylor expansion of eigenfunctions at the singular setq,=. In fact, if+qenH1q(M) (orqenH1q(M)) for allnN,then +qenH1q(M) and qenH1q(M), for all nN, since the eigen-functions en can be chosen real. It follows that en H2q(M), for all n N,hence H2q(M)=H2q(M) is the domain of the Friedrichs extension Fqofthe Laplacian q of the metric Rq. Thus the Laplacian q is self-adjoint onthe domain H2q(M), hence the metric Rq has no singularities and M is thetorus. In fact, the space H2q(M) is the domain of the closure of the Lapla-cian q on the common invariant domain Hq(M). If H2q(M)=H2q(M),then q is essentially self-adjoint on Hq(M) and by [Nel59], Th. 5 or Cor.9.1, the action of the commutative Lie algebra spanned by S, T integratesto a Lie group action. Hence, the singularity set q= and M is the torus.16 GIOVANNI FORNIFinally, the Friedrichs extensionFq , dened onH2q(M), has a boundedrestrictions:Hs+2q(M) Hsq(M), for all s 0. In fact, we have(2.25) Fq u =

nNnu, en)q en, for all u H2q(M) ,henceFq uHsq(M) ifuHs+2q(M), by denition of the Friedrichsweighted Sobolev spacesHsq(M) (in terms of eigenfunction expansions forthe Dirichlet form). Lemma 2.10. The fractional weighted Sobolev norms satisfy the followinginterpolation inequalities. For any 0 r < s there exists a constant Cr,s>0 such that, for any [0, 1] and any function u Hsq(M),(2.26) [u[(1)r+sCr,s[u[1r[u[s .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 existskN such thatkr 0 such that(2.35)[u[k1C(3)r,s[u[1k1+1rr[u[k1rk1+1rk1+1;[u[k2C(3)r,s[u[sk2sk2+1k21[u[1sk2+1s.18 GIOVANNI FORNIThe estimates in (2.35) imply, by bootstrap-type estimates based on (2.34)for k=k1 + 1 and k=k2 1, that there exists a constant C(4)r,s>0 suchthat(2.36)[u[k1C(4)r,s[u[k2k1k2rr[u[k1rk2rk2;[u[k2C(4)r,s[u[sk2sk1k1[u[k2k1sk1s.By (2.36) and again by bootstrap, there exists a constant C(5)r,s> 0 such that(2.37)[u[k1C(5)r,s[u[sk1srr[u[k1rsrs;[u[k2C(5)r,s[u[sk2srr[u[k2rsrs,Let (r, s). We have proved the interpolation inequality for the subcases=k1 and =k2. Let us prove that the general case can be reduced tothese subcases. If (r, k1), by step (1) there exists Cr> 0 such that(2.38) [u[Cr[u[k1k1rr[u[rk1rk1.The interpolation inequality in this case follows immediately from (2.37)and (2.38). If (k2, s), the argument is similar. If (k1, k2), then bystep (1), there exists C[]> 0 such that(2.39) [u[C[][u[1{}[][u[{}[]+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 manifold M(dened with respect to a Riemannian metric). Thecomparison lemma below claries 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) Hsq(M) , for 0 s < 1;(2) Hs(M) Hsq(M) Hsq(M) , for s = 1;(3) Hsq(M) Hsq(M) Hs(M) , for s > 1.Fors[0, 1], the spaceHs(M) is dense inHsq(M) and, fors>1, theclosure of Hsq(M) inHsq(M) or Hs(M) has nite codimension.Proof. By denition H0(M)=L2(M) and H0q(M)=H0q(M)=L2q(M).Since the area form induced by any quadratic differential is smooth on M,which is a compact surface, it follows that L2(M) L2q(M). The em-beddingH1q(M) H1q(M)follows byLemma 2.5andtheembeddingSOBOLEV REGULARITY OF SOLUTIONS 19H1q(M) H1q(M) holds since the eigenfunctions of the Dirichlet formare in H1q(M). The isomorphism H1(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 constant Ak> 0 such that, forall functions u Hq(M), we have(2.40) |u|2s= |(I Fq )ku|2{s} A2k

i+j2k|SiTju|2{s}= C2k[u[s.If [s] = 2k +1 is odd, we argue as follows. The Cauchy-Riemann operatorsq: H1q(M) L2q(M) are bounded and extend by duality to boundedoperators0: L2q(M)H1q(M). Hence, by the fundamental theoremof interpolation (see [LM68], Chap. 1, 5.1), for all [0, 1] the Cauchy-Riemann operators have bounded restrictions(2.41) :Hq (M) H1q(M) .It follows that there exists a constantBk>0 such that, for all functionsu Hq(M), we have(2.42)|u|2s= |(I Fq )k+1u|2{s}1 B2k

i+j2k+1|SiTju|2{s} = C2k[u[s.Thus the embeddings Hsq(M) Hsq(M) hold for all s > 1.It was proved in [For02], 6.2, that Hkq(M) Hk(M), for all k Z+. Weprove below the stronger statement thatHsq(M) Hs(M), for all s R+.Let R be a smooth Riemannian metric on M conformally equivalent to thedegenerate metricRqand letHsR(M), s0, denote the Sobolev spacesof the Riemannian manifold(M, R) which are dened as the domains ofthe powers of the essentially self-adjoint Laplacian R of the metric. SinceM is compact, the Sobolev spaces HsR(M)Hs(M) are independent, astopological vector spaces, of the choice of the Riemannian metric R, for allsR. We claim thatH2kq(M)H2kR(M), for all kZ+. In fact, thereexists a smooth non-negative real-valued function W on M (vanishing onlyat q) such that WFqR. Let Wbe the unique function such thatthe area forms of the metrics are related by the identityq=WR. Ifu H2q(M), then Fq u L2q(M), so that(2.43) Ru = WFq u L2(M, R) .Let us assume thatH2k2q(M) H2k2R(M) and let u H2kq(M). We have(2.44) kRu = k1RWFq u = [k1R, W]Fq u + Wk1RFq u .Since the commutator [k1R, W] and k1Rare differential operators of or-der 2k 2 on M and Fq uH2k2R(M) by the induction hypothesis, the20 GIOVANNI FORNIfunction kRuL2(M, R). The claim is therefore proved. It follows byinterpolation thatHsq(M) HsR(M), for all s 1. Thus (3) is proved.Fors [0, 1], the spaceC0(Mq) Hsq(M)is dense inHsq(M).Fors >1, the subset C(M)Hsq(M) is dense inHsq(M), since theeigenfunctions of the Dirichlet form (hence all nite linear combinations)belong to C(M) and the space C(M) is dense in Hs(M). Finally, thesubspace C(M) Hkq(M)C(M) can be described, for any kN,as the kernel of a nite number of distributions of nite order supported onthe nite setq(see [For02], (7.9)), hence for anyk>s the closure ofHkq(M) Hsq(M) inHsq(M) or in Hs(M) has nite codimension. 2.3. Local analysis. For each pMand all kZ+, let Hkq(p),Hkq (p),andHk(p) the spaces of germs of fuctions atp which belong toHkq(M),Hkq(M) andHk(M), endowed with the respective direct limit topologies.More precisely, a germ of function f at p belongs to the space Hkq (p),Hkq (p)or Hk(p) iff it can be realized by a function Fon M which belongs to thespace Hkq(M),Hkq(M) or Hk(M) respectively and the open sets in Hkq (p),Hkq(p) or Hk(p) are dened as the images of open sets in Hkq (M),Hkq(M)or Hk(M) under the natural maps Hkq(M) Hkq(p),Hkq(M) Hkq(p) orHk(M) Hk(p).By Lemma 2.11 we have the inclusions(2.45)H0(p) H0q(p) H0q(p) ;H1q(p) =H1q(p) = H1(p) ;Hkq(p) Hkq (p) Hk(p) .If p,q, since there is an open neighbourhood Dp of p in M isomorphicto a at disk and the operator Fqis elliptic of order 2 on Dp (isomorphicto the at Laplacian), all the inclusions in (2.45) are identities. We willdescribe precisely the inclusions Hkq(p) Hkq(p) Hk(p) for k> 1.Let pq be a zero of (even) order 2m of the (orientable) quadratic dif-ferential q on M. There exists a unique canonical holomorphic coordinatez: DpC, dened on a neighbourhood Dp of p M, such that z(p) = 0and q(z)=z2mdz2. With respect to the canonical coordinate the Cauchy-Riemann operators qcan be written in the following form:(2.46) +q=2 zm zand q=2zmz.SOBOLEV REGULARITY OF SOLUTIONS 21Let C(p) be the space of germs at p Mof smooth complex-valuedfunctions on M and for any u C(p) let(2.47) u(z, z) =

i,jNaij(u, p)zi zj.be its (formal) Taylor series at p (with respect to the canonical coordinate).Lemma 2.12. Let pq be a zero of (even) order 2m of the (orientable)quadratic differential q on M. For any k N, a germu 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, N(m + 1) and i < j ;(3)i , N(m+ 1) , j N(m + 1) and i > j .Proof. Let u C(p). For any n N, there is a local Taylor expansionu(z, z) =

i+jnaij(u, p)zi zj+Run(z, z)where the remainderRunis a smooth function vanishing at ordern at p.Astraightforwardcalculation(basedonformulas(2.46)yieldsthat anysmooth function R vanishing at p at order n belongs to the space Hkq(p) Hkq(p) if n > (k 1)m. It follows that u Hk(p) iff its Taylor polynomialof any order n > (k 1)m does. The argument can therefore be reduced tothe case of polynomials.It follows from formulas (2.46) that, for all N and all (i, j)NN,there exists a complex constant cm,ijsuch that(2.49) q(zi zj) = cm,ijzi(m+1) zj(m+1).The area form of the quadratic differential q can be written as(2.50) q= [z[2mdx dywith respect to the canonical coordinate z:= x+y. Hence, straightforwardcomputations in polar coordinates yield that, if cm,ij,= 0,(2.51)q(zi zj) H0q(p) i + j 2(m+ 1) > (m+ 1) ;q(zi zj) H1q(p) i + j 2(m+ 1) > 0 .If cm,ij= 0, then either i N (m+1) and i < (m+1) or j N (m+1)and j< (m+ 1). It follows that, if i, j, N(m + 1), then(2.52) zi zjHkq(p) i + j (k 1)(m+ 1) > 0 .22 GIOVANNI FORNIIf iN(m + 1), i=h(m + 1), andj,N(m + 1), then conditions(2.51) apply for all h, hence (2.52) holds if k 2h, while if k> 2h,(2.53) zi zjHkq(p) hq(zi zj) H1q(p) j> i .Similarly, if j N (m+1), j= h(m+1), and i , N (m+1), then (2.52)holds if k 2h, while if k> 2h,(2.54) zi zjHkq(p) hq(zi zj) H1q(p) i > j .It follows immediately from (2.52), (2.53) and (2.54) that the conditionslisted in the statement of the lemma are sufcient. The necessity followsfrom the following argument. For any r1< r2, let D(r1, r2) M be the an-nulus (centered at p) dened by the inequalities r1< [z[< r2. The systemof Laurent monomials zi zj[i, j Z is orthogonal inHk(Dr1,r2). In fact, acomputation in polar coordinates shows that, for all (i, j) ,= (i, j) ZZ,(2.55)_D(r1,r2)q(zi zj)q ( zizj)q=0 ,hence(2.56) |

i+jnaijzi zj|2k=

i+jn[aij[2|zi zj|2k .It follows that only Laurent monomialszi zjHk(p) can appear in theTaylor expansion of a function f C(p) Hk(p). Lemma 2.13. Let p q be a zero of order 2mof the quadratic differentialq on M. For any k N, a germu 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 all N and all i N,there exists a complex constant cm,isuch that(2.57) (+q)(q)zi zj= cm,i cm,jzi(m+1) zj(m+1).As in the proof of Lemma 2.12, it follows by a straightforward computationin polar coordinates that, if cm,i cm,j,= 0,(2.58) (+)()zi zj H0q(p) i +j (+)(m+1) > (m+1) .Since there existsN such thatcm,i=0 iffi N(m + 1), eitheri, j N(m + 1), in which casezi zjHkq(p), or there exists(, )such that + =k andcm,i cm,j,=0, in which casezi zjHkq (p) iffi + j> (k 1)(m+ 1). SOBOLEV REGULARITY OF SOLUTIONS 23Let p q be a zero of order 2mp of the orientable quadratic differential q.Let Tp N N be the set of (i, j) such that(2.59)i N(mp + 1) , j, N(mp + 1) and i < j ori , N(mp + 1) , j N(mp + 1) and i > j,For any (i, j) Tp, let ijpbe the linear functional (distribution) on C(M)dened as follows. Let u(z, z) =

aij(u, p)zi zjdenote the Taylor expan-sion ofuC(M) atpqwith respect to the canonical coordinatez: Dp C for the differential q at p. Let(2.60) ijp (u) := aij(u, p) .It is clear from the denition that ijp= jipfor all (i, j) Tp. A calculationshows that, for any h NN(mp+1) we have the following representationin terms of the Cauchy principal value: for any u C(p),(2.61)h0p(u) = 14h PV_Mquzhq ;0hp(u) = 14h PV_Mqu zhq .(The above formulas can be derived one from the other by conjugation). Inaddition, for any N, by formulas (2.49) there exist complex constantscmp,0,h,=0 and cmp,h,0,= 0 such that the following identities hold in the senseof distributions:(2.62)cmp,0,h(mp+1),(mp+1)+hp= q_0,hp_;cmp,h,0(mp+1)+h,(mp+1)p= q_h,0p_.Let Tkp Tp be the subset of (i, j) such that i + j (k 1)(mp + 1).Lemma 2.14. For each(i, j)Tkp, the functional ijphas a unique (non-trivial) continuous extension to the spaceHkq (p) and the following holds:(2.63) Hkq(p) = u Hkq(p) [ ijp (u) = 0 for all (i, j) Tkp .Proof. The functionszhand zhL2q(D) for all1hmp and theoperatorFq:H2q(p)L2q(p) is bounded. Hence the linear functionals0hpandh0pare continuous onH2q(p) for all1hmp. Similarly, thedistributions PV(zh) and PV( zh) H1(Dp) for all mp< h < 2(mp +1) and the operatorFq:H3q(p) H1(p) is bounded. Hence the linearfunctionals 0hpand h0pare continuous onH3q(p) for all mp< h < 2(mp +1). Since the space Hkq (p) is equal to the closure inHkq(p) of the subspaceC(p) Hkq(p), the statement for k= 2, k= 3 follows from Lemmas 2.12and 2.13.24 GIOVANNI FORNIWe complete the argument by induction on k N. The Friedrichs extensiondenes bouned operatorsFq:Hk+1q(p) Hk1q(p) and its dual Fq:Hk+1q(p) Hk1q(p). By the induction hypothesis, since all functionalsinTk1pextend (uniquely) to bounded functionals inHk+1q(p), it followsthat all functionals in Fq (Tk1p) extend (uniquely) to bounded functionalsinHk1q(p) and the following holds. For any u Hk+1q(p),(2.64) Fq u Hk1q(p) ijp (u) = 0 for all ijp Fq (Tk1p) .Let Ek+1qHk+1q(p) the closed nite-codimensional subspace dened as(2.65) Ek+1q:= u Hk+1q(p) [ ijp (u) = 0 for all ijp Fq (Tk1p) .By formulas (2.62), any distribution pTk+1p Fq (Tk1p) is of the formp=0hpor p=h0pwith 1hk(mp + 1). By formulas (2.61), suchdistributions have a (unique) continuous extension to the subspace Ek+1qHk+1q(p). In fact, the distributions PV(zh) and PV( zh)Hk+1q(Dp),for all1h0 such that,for any (i, j) N N and for all (0, 1]:(2.68)[Kp()(Zijp) Zijp [k Ck 1+i+jm+1k, fork< 1 +i + jm + 1 ;[Kp()(Zijp)[k Ck[ log [1/2, fork= 1 +i + jm + 1 ;[Kp()(Zijp)[k Ck [k(1+i+jm+1)], fork> 1 +i + jm + 1 .Proof. Let z:DpC be a canonical coordinate at pq dened on anopen neighbourhood Dp M such that Dpq= p. There exists r1> 0such that D(r1)z(Dp), where D(r1) is the euclidean disk centered atthe origin of radiusr1>0 . Let 0 ( + 1)(m+ 1), since the functionszi(a)(m+1) zj(b)(m+1) L2q(Dp) ,for all 0 a and0 b , andabis bounded, by change ofvariables we obtain that for (a, b),=(0, 0) there exists a constant Ca,b>0such that(2.73) [ab(z)(a+b)zi(a)(m+1) zj(b)(m+1)[0 Ca,b1+i+jm+1(+),and that similarly, for a = b = 0, there exists a constant C0> 0(2.74) [((z) 1) zi(m+1) zj(m+1)[0 C01+i+jm+1(+).If i +j< [( a) + ( b) 1](m+ 1), there exists a constant Ca,b> 0such that(2.75) [ab(z)(a+b)zi(a)(m+1) zj(b)(m+1)[0 Ca,b1+i+jm+1(+),since the function abis bounded and supported outside the euclidean diskof radius r11m+1centered at the origin.If i + j= [( a) + ( b) 1](m+ 1), a similar calculation yields(2.76)[ab(z)(a+b)zi(a)(m+1) zj(b)(m+1)[0 Ca,b1+i+jm+1(+)[ log [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() con-structed in Lemma 2.16 with respect to the fractional weighted Sobolevnorms. Let pq be any zero of order 2mp of the quadratic differentialSOBOLEV REGULARITY OF SOLUTIONS 27q onM. For each pair(i, j) NN, leteijp: R+[0, 1] denote thefunction dened 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 family Kp() [ (0, 1] of local smoothing oper-atorsKp() : C(M) Hq(M), dened in (2.69),has the followingproperties. For each sR+, there exists a constant Cs>0 such that, forany pair (i, j) N N and all (0, 1]:[Kp()(Zijp) Zijp [s Cs1+i+jmp+1s[ log [eijp(s)2, fors < 1 +i + jmp + 1 ;[Kp()(Zijp)[s Cs[ log [ [ log [eijp(s)2, fors = 1 +i + jmp + 1 ;[Kp()(Zijp)[s Cs[s(1+i+jmp+1)][ log [eijp(s)2, fors > 1 +i + jmp + 1 .Proof. For any k N, the function Zijp Hkq(M), if k< 1+(i+j)/(m+1),since zi zjHkq(p). By Lemma 2.16, there exists a constant Ck>0 suchthat, for all (0, 1],(2.78)[Kijp() Kijp(/2)[k [(Kijp() Zijp) (Kijp(/2) Zijp)[k ; [Kijp() Zijp [k +[Kijp(/2) Zijp [k Ck 1+i+jm+1k.If k 1 + (i + j)/(m+ 1),(2.79) [Kijp() Kijp(/2)[k [Kijp()[k+[Kijp(/2)[khence, by Lemma 2.16,(2.80)[Kijp() Kijp(/2)[k 2Ck[ log [1/2, if k= 1 +i + jm+ 1 ;[Kijp() Kijp(/2)[k 2Ck 1+i+jm+1k, if k> 1 +i + jm+ 1 .As a consequence, by the interpolation inequality (Lemma 2.10), for everys R+there exists a constant Cs> 0 such that, for all (0, 1],[Kijp() Kijp(/2)[s Cs1+i+jm+1s, if [s], [s] + 1 ,= 1 +i + jm + 1 ;[Kijp() Kijp(/2)[s Cs1+i+jm+1s[ log [1{s}2, if [s] = 1 +i + jm+ 1 ;[Kijp() Kijp(/2)[s Cs1+i+jm+1s[ log [{s}2, if [s] =i + jm + 1 .28 GIOVANNI FORNIIf s < 1 + (i + j)/(m+ 1) and [s] ,= (i + j)/(m+ 1), for all n N,(2.81) [Kijp(/2n) Kijp(/2n+1)[s Cs1+i+jm+1s2n(1+i+jm+1s),hence, for any xed (0, 1], the sequenceKijp(/2n)nN is Cauchy,and therefore convergent, in the Hilbert spaceHsq(M). By Lemma 2.16Kijp(/2n)nN converges to Zijpin H[s]q(M). Since H[s]q(M)Hsq(M),by uniqueness of the limit Zijp Hsq(M) and Kijp(/2n)nN converges toZijpin Hsq(M). The estimate (2.81) also implies that(2.82)[Kijp() Zijp [s

nN[Kijp(/2n) Kijp(/2n+1)[s Cs1+i+jm+1s.If s < 1+(i +j)/(m+1) and [s] = (i +j)/(m+1), by a similar argumentwe again get that Zijp Hsq(M) and(2.83) [Kijp() Zijp[s Cs1+i+jm+1s[ log [{s}2.If s 1 +(i +j)/(m+1) and [s], [s] +1 ,= 1 +(i +j)/(m+1) we argueas follows. For each 1/2, we have(2.84) [Kijp(2) Kijp()[s Cs (2)1+i+jm+1s,hence if 2n, for all 0 k < n,(2.85) [Kijp(2k+1) Kijp(2k)[s Cs 2(k+1)(1+i+jm+1s)1+i+jm+1s.It follows that, there exists a constant Cs> 0 such that(2.86) [Kijp(2n) Kijp()[s Cs 2n(1+i+jm+1s)1+i+jm+1s.For every (0, 1], let n() be the maximum nN such that 2n 1.By this denition it follows that 1/2 < 2n() 1. Sincesup1/21[Kijp()[s sup1/21[Kijp()[[s]+1 0 such that(2.87)[Kijp()[s Cs1+i+jm+1s, if s > 1 +i + jm + 1 ;[Kijp()[s Cs[ log [ , if s = 1 +i + jm + 1 .By a similar argument, for s > 1 + (i + j)/(m+ 1) we have(2.88)[Kijp()[s Cs1+i+jm+1s[ log [1{s}2, if [s] = 1 +i + jm+ 1 ;[Kijp()[s Cs1+i+jm+1s[ log [{s}2, if [s] =i + jm + 1 ,SOBOLEV REGULARITY OF SOLUTIONS 29while for s = 1 + (i + j)/(m+ 1) we have(2.89)[Kijp()[s Cs[ log [ [ log [1{s}2, if [s] = 1 +i + jm+ 1 ;[Kijp()[s Cs[ log [ [ log [{s}2, if [s] =i + jm + 1 .

Theorem 2.17 implies in particular the following smoothness results.Corollary 2.18. Letz : DpCbe a canonical coordinate for an ori-entable quadratic differentialqat a zeropqof order2m. For each(i, j) N N, the function(2.90) zi zj Hsq(p), for all s < 1 +i + jmp + 1.Corollary 2.19. Let p q be a zero of order 2mp and let (i, j) Tp. Thedistribution ijphas the following regularity properties:(2.91)ijpHsq(p) for s > 1 +i + jmp + 1 ,ijp, Hsq(p) for s < 1 +i + jmp + 1 .Proof. By the formulas (2.61) and by Lemma 2.14, for anyhN N (mp + 1) and for any N, there exist constants Cmp,h0,= 0 and Cmp,0h,= 0such that the following identities hold in the dual Hilbert spaceHkq(p) forany integer k 1 +h/(mp + 1):(2.92)Cmp,h0h0p= +1q_z(mp+1)h z(mp+1)_,Cmp,0h0hp= +1q_z(mp+1) z(mp+1)h_.By Corollary 2.18, if (mp + 1) h > 0, for all s < 2 + 1 h/(mp + 1),z(mp+1)h z(mp+1)and z(mp+1) z(mp+1)h Hsq(p) Hsq(p) .Hence h0p, 0hpHsq(p) for all s > 1 +h/(mp +1). By formulas (2.62) itthen follows that ijpHsq(p) for all s > 1 +(i +j)/(mp +1) as claimed.Let (i, j) Tp and s < 1+(i+j)/(mp+1). By Corollary 2.18, the functionzi zj Hsq(p). Since by denition Hq(p) is dense in Hsq(p) for any s > 0,the functional ijp 0 on Hq(p) and ijp (zi zj) = 1, it follows that ijpdoesnot extend to a bounded functional on Hsq(p). 30 GIOVANNI FORNILet p q be a zero of (even) order 2mp the quadratic differential q on M.For every s R+, let Tsp Tp be the subset dened as(2.93) Tsp:= (i, j) Tp[ i + j< (s 1)(mp + 1) .Let DsqHsq(M) be the set of distributions dened as follows:(2.94) Dsq:= ijp[ p qand(i, j) Tsp .Corollary 2.20. The closure of the subspace Hsq(M) inHsq(M) is a subsetof the (closed) kernel of the system Dsq onHsq(M), that is,(2.95) Hsq(M) u Hsq(M) [ (u) = 0 , for all DsqThe reverse inclusion holds if the following sufcient condition is satised:(2.96) s , 1 + (i + j)/(mp + 1) [ p qand(i, j) Tp .Proof. Since Hq(M) is dense in Hsq(M) Hsq(M),Hq(M) u Hsq(M) [ (u) = 0 , for all Dsqand DsqHsq(M), it follows thatHsq(M) u Hsq(M) [ (u) = 0 , for all Dsq .Conversely, if condition (2.96) is satised, by Corollary 2.18 the subspaceu C(M) [ (u) = 0 , for all Dsq Hsq(M) .Since C(M) Hsq(M) is dense inHsq(M), the result follows. The regularity result proved in Corollary 2.18 extends to a certain subset ofall pairs (i, j) ZZ if the functions zi zjare interpreted as distributionsin the sense of the Cauchy principal value:(2.97) PV_zi zj_(v) :=PV_Mzi zjv q , 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: DpC be a canonical coordinate for an orientableholomorphic quadratic differentialqat a zeropqof order2m. Foreach (i, j, h) N N N, the functionzi zjlogh[z[ Hsq(p) , for all s < 1 +i + jmp + 1 .SOBOLEV REGULARITY OF SOLUTIONS 31Proof. Simple calculations show that log [z[ L2q(M) and that by formulas(2.46) the following identities hold on Dp p:(2.98) +log [z[ =1 zm+1and log [z[ =1zm+1.It follows that, for each (i, j, h) NNNand each (, ) NN, thereexists a nite sequence of non-zero constants C1, . . . , Ch, which depend on(i, j, h, , , m), such that the following identity holds on Dp p:(2.99) (+)()_zi zjlogh[z[_= zi(m+1) zj(m+1)h

=0Clog[z[ .For all (i, j, h) NNN, let Lijhp C(M p) be any function suchthat Lijhp(z) = zi zjlogh[z[ for all z Dp. By (2.99), the functionLijhp Hkq (M) , if k Nand k< 1 +i + jmp + 1Let Kp() [ (0, 1] be the family of local smoothing operators denedby formulas (2.69). By computations similar to those carried out in theproof of Lemma 2.16, based on formulas (2.99), it is possible to prove thatfor each k N, there exists a constant Ck> 0 such that for all (0, 1]:[Kp()(Lijhp) Lijhp[k Ck 1+i+jm+1k[ log [h, for k < 1 +i + jm+ 1;[Kp()(Lijhp)[k Ck[ log [1/2[ log [h, for k = 1 +i + jm+ 1;[Kp()(Lijhp)[k Ck [k(1+i+jm+1)][ log [h, for k > 1 +i + jm+ 1.Reasoning as in the proof of Theorem 2.17, we can derive similar estimatesfor fractional Sobolev norms. For each (i, j) NN, let eijp: R+[0, 1]be the function dened in formula (2.77). By the interpolation Lemma 2.10,for any s < 1 + (i + j)/(m+ 1) there exists a constant Cs> 0 such that[Kp()(Lijhp) Kp(/2)(Lijhp)[s Cs1+i+jm+1s[ log [h+eijp(s)2.It follows that the sequence Kp(/2n)(Lijhp)nN is Cauchy and thereforeconverges in Hsq(M). By uniqueness of the limitLijhp Hsq(M) , for all s < 1 +i + jmp + 1 .32 GIOVANNI FORNIIn addition, the following estimates hold. For eachsR+there exists aconstant Cs> 0 such that for all (0, 1]:[Kp()(Lijhp)[s Cs[ log [ [ log [h+eijp(s)2, for s = 1 +i + jmp + 1 ;[Kp()(Lijhp)[s Cs[s(1+i+jmp+1)][ log [h+eijp(s)2, for s > 1 +i + jmp + 1 .

Theorem 2.22. Let z : DpCbe a canonical coordinate for an ori-entableholomorphicquadratic differential qatazerop qoforder2mp. For each (i, j)ZZ such that (1) i j ,Z(mp + 1) or (2)i > (mp + 1) or (3)j> (mp + 1), the distribution(2.100) PV_zi zjlogh[z[_ Hsq(p), for all s < 1 +i + jmp + 1.Proof. For all (i, j)ZZ such that i j,Z(mp+ 1) and for anyh Z the following formulas hold for all functions v Hq(p):(2.101)(a) PV_M+(zi zjlogh[z[)v q= PV_Mzi zjlogh[z[ +v q ;(b) PV_M(zi zjlogh[z[)v q= PV_Mzi zjlogh[z[ v q .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 germs v C(p).By taking into account the formulas (2.46) for the Cauchy-Riemann opera-tors with respect to a canonical coordinate, it follows from formulas (2.101)by induction on h N that ifPV(zi zjlogh[z[) Hsq(p) , for all h N,then, if i j, Z(mp + 1) or i > (mp + 1) and j Z,PV(zi zj(mp+1)logh[z[) Hs1q(p) , for all h N,and, if i j, Z(mp + 1) or j> (mp + 1) and i Z,PV(zi(mp+1) zjlogh[z[) Hs1q(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 : DpC be a canonical coordinate for an ori-entable holomorphic quadratic differential q at a zero p q of order 2mp.SOBOLEV REGULARITY OF SOLUTIONS 33If (i, j) , Z(mp + 1) Z(mp + 1), the distribution(2.102) PV_zi zjlogh[z[_, Hsq(p), for s > 1 +i + jmp + 1 ,and, if i j Z(mp + 1)and bothi (m + 1)andj (m + 1),(2.103) PV_zi zjlogh[z[_, Hq(p).Proof. We argue by contradiction. Assume there exists (i, j) Z Z suchthat (i, j) ,Z (mp+1)Z (mp+1) and PV(zi zj) Hrq(p) forsome r>1 + (i + j)/(mp + 1). By taking Cauchy-Riemann derivativesif necessary, we can assume that i0 and j0. By Theorem 2.22, thedistributionPV_zi(m+1) zj(m+1)_ Hsq(p) , for all s < 1 i + jmp + 1.It follows that, for any positive smooth functionC0(Dp) identicallyequal to 1 on a disk Dp Dp centered at p q, the principal valuePV_M(z) zi zjzi(m+1) zj(m+1)qis nite, which can be proved to be false by a simple computation in (ge-odesic) polar coordinates. This contradiction proves the rst part of thestatement.If i j Z (mp +1) and both i (m+1) and j (m+1), we argueas follows. It is not restrictive to assume thati j, hence the function zijHq(M). However, by a computation in polar coordinates, sincei (mp + 1),PV_M(z) zi zjlog [z[ zijq= +.It follows that PV_zi zjlogh[z[_ ,Hq(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 each kN, there exists a family Sk() [ (0, 1]of bounded operatorsSk() : L2q(M)Hkq(M) such that the followingestimates hold. For anys, r [0, k] and for any >0, there existsaconstant Ckr,s() > 0 such that, for all u Hsq(M) and for all (0, 1]:(2.104)[Sk()(u) u[r Ckr,s()|u|ssr, if s > r ;[Sk()(u)[r Ckr,s()|u|ssr, if s r .34 GIOVANNI FORNIProof. For each p q, let z: DpC be a canonical coordinate denedon a disk Dp (centered at p) such that Dp q= p. For each (i, j) Tp,let Zijp C(M) be a (xed) smooth extension, as in (2.67), of the locallydened functionzi zjC(p). LetPkbe the linear operator dened asfollows:(2.105) Pk(f) := f


(i,j)Tkpijp (f) Zijp, for all fHkq(M) .The operatorPk:Hkq(M)Hkq (M) is well-dened and bounded. It iswell-dened by Lemma 2.14 and Theorem 2.15. It is bounded since, for allpq and all(i, j)Tp, the functions ZijpHq(M) by Lemma 2.12and, for all (i, j)Tp, the distributions ijpHkq(M) by Corollary 2.19.In fact, the condition (i, j) Tp implies i + j< (k 1)(mp + 1).For each pq, let Kp() [ (0, 1] be the family of local smoothingoperators constructed in Lemma 2.16. Let Sk[ (0, 1] be the one-parameter family of bounded linear operatorsSk:Hkq(M) Hkq(M)dened as follows. For all fHkq(M), we let(2.106) Sk(f) := Pk(f)+


(i,j)Tkpijp (f) Kp()_Zijp_.By denition the following identity holds for all fHkq (M):Sk(f) f=


(i,j)Tkpijp (f)_Kp()_Zijp_Zijp.Since for all p q the condition (i, j) Tp implies i+j, N (mp+1), byLemma 2.16 the following estimate holds. For each N such that k,there exists a constant Ck> 0 such that, for any fHkq(M),(2.107) |Sk(f) f| Ck


(i,j)Tkp1+i+jmp+1[ijp (f)[ .In fact, for each pq and each(i, j)Tp, since ZijpHq(M), whichimpliesKp()_Zijp_ ZijpHq(M), if 0, C> 0 such that|Kp()_Zijp_Zijp| C[Kp()_Zijp_Zijp [ C1+i+jmp+1,while, if > 1 + (i + j)/(mp + 1), since ZijpHq(M),|Kp()_Zijp_Zijp | |Kp()_Zijp_| +|Zijp | C1+i+jmp+1.The scale of Friedrichs Sobolev spaces admits a standard family of smooth-ing operators T[ >0 such that the operator T:L2q(M)Hq(M)SOBOLEV REGULARITY OF SOLUTIONS 35is dened, for each > 0, by the following truncation of Fourier series. Leten[ nN be an orthonormal basis of eigenfunctions of the FriedrichsLaplacian Fqand let : N R+ 0 be the corresponding sequence ofeigenvalues. ThenT(u) :=

2n1u, en)q en, if u =

nNu, en)q en.If uHsq(M), then T(u) u inHsq(M) (as 0+) and the followingestimates hold. For all r R+, there exists a constant Cr,s(q) > 0 such that(2.108)|T(u) u|r Cr,s(q) |u|ssr, if s r ;|T(u)|r Cr,s(q) |u|s(rs), if r s .If pq and (i, j)Tp, the distribution ijpHsijq(M) for any sij>0such that i+j< (sij1)(mp+1). Hence there exists a constant Cijp(q) > 0such that, by estimates (2.108), for all u Hsq(M),(2.109) [ijp(T(u)) [ Cijp(q) |u|smax1, ssij .If u Hsq(M), then pij(u) = 0, for all p q and all (i, j) Tsp. Hence, ifsij s and i + j< (sij 1)(mp + 1),(2.110) [ijp(T(u)) [ = [ijp(T(u) u) [ Cijp(q) |u|sssij.The following estimates hold. Let s R+and N. For any > 0, thereexists a constant C,s() > 0 such that, for all (0, 1] and all u Hsq(M),(2.111) |Sk T(u) T(u)| C,s() |u|ss.In fact, if p q and (i, j) Tkp,(2.112) [ijp(T(u)) [ Cijp(q) |u|sssij;for any sij>1 + (i + j)/(mp + 1)s, if (i, j)Tkp Tsp, and for any1+(i+j)/(mp+1) < sij s, if (i, j) Tsp. The claim (2.111) then followsfrom (2.107). By estimates (2.108) and (2.111), for any >0, there existsa constant C,s() > 0 such that, for all (0, 1] and for all u Hsq(M),(2.113) |Sk T(u) u| C,s() |u|ss.Let Sk() [ (0, 1]bethe familyofoperators Sk() : L2q(M) Hkq(M) dened as follows: for each (0, 1],(2.114) Sk() := Sk T .By estimate (2.113), for any > 0, there exists a constant C,s() > 0 suchthat, for all (0, 1] and for all u Hsq(M),(2.115) |Sk()(u) Sk(/2)(u)| C,s() |u|ss.36 GIOVANNI FORNISinceSk()(u) Hkq(M) for all (0, 1], by the interpolation inequalityproved in Lemma 2.10 it follows that, for any r[0, k] and for any >0there exists Cr,s() > 0 such that, for all (0, 1] and for all u Hsq(M),(2.116) [Sk()(u) Sk(/2)(u)[r Cr,s() |u|ssr.It follows that, for every n N and for every (0, 1],(2.117)[Sk(/2n)(u) Sk(/2n+1)(u)[r Cr,s() |u|s2n(sr)sr.If s > r and 0 < < s r, the sequence Sk(/2n)(u)nN is Cauchy andtherefore convergent to the function uHsq(M)Hrq(M) in the Hilbertspace Hrq(M). It follows that, for all (0, 1] and for all u Hsq(M),(2.118) [Sk()(u) u[r Cr,s() |u|ssr.If sr and s r0 0 such that(2.121) [Sk(2n)(u) Sk()(u)[r Cr,s() |u|s 2n(sr)sr,For every (0, 1], let n() be the maximumn Nsuch that 2n 1. Bythis denition it follows that 1/2 0 such that, for all u Hsq(M),[Sk(2n())(u)[r sup1/21|Sk()(u)|k0 such that, for all (0, 1] and forall u Hsq(M),(2.122) [Sk()(u)[r Cr,s() |u|ssr,

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 any0 0 such that, for all u Hsq(M), the following inequalities hold:C1r|u|r [u[r Cr,s|u|s.SOBOLEV REGULARITY OF SOLUTIONS 37Finally, we derive a crucial interpolation estimate for the dual weightedSobolev norms:Corollary 2.26. Let 0s1s1andlet Sk() :L2q(M) Hkq(M) be the family of smoothing operators constructed above.By Theorem 2.24, since 0 r s1< s s1 and any r s2< s s2 0,there exists a constant Ckr,s>0 such that the following holds: for anyu Hs1q(M) 0, any v Hsq(M) and for all (0, 1],(2.124)[u, v)[ [u[s1[v Sk()(v)[s1+[u[s2[Sk()(v)[s2 Ckr,srs1[u[s1 + rs2[u[s2 [v[s.The interpolation inequality (2.123) then follows by taking=_[u[s2[u[s1_ 1s2s1 (0, 1] .

3. THE COHOMOLOGICALEQUATION3.1. 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 regularity within the reachof the methods of [For97]. We were initially motivated by a question ofMarmi, Moussa and Yoccoz who found for almost all orientable quadraticdifferentials a loss of regularity of1+BV (they nd bounded solutionsfor absolutely continuous data with rst derivative of bounded variation un-der nitely many independent compatibility conditions and correspondingresults for higher smoothness) [MMY03], [MMY05] . The results of thissection, as those of [For97], hold for all orientable quadratic differentials.There is a natural action of the circle groupS1SO(2, R) on the spaceQ(M) of holomorphic quadratic differentials on a Riemann surface M:r(q) := eiq , for all (r, q) SO(2, R) Q(M) .Let qdenote the quadratic differentialr(q) and letS, T denote theframe (introduced in 2.1) associated to the quadratic differential q for any38 GIOVANNI FORNI S1. We have the following formulas:(3.1)S= cos_2_Sq + sin_2_Tq=ei22+q+ei22q;T= sin_2_Sq + cos_2_Tq=ei22i+qei22iq;Denition 3.1. Let q be an orientable quadratic differential. A distributionu Hrq(M) will be called a (distributional) solution of the cohomologicalequation Squ = f for a given function fHsq(M) ifu, Sqv) = f, v) , for all v Hr+1q(M) Hsq(M) .LetHsq(M) Hsq(M), Hsq(M) Hsq(M) (for any s R) be the subspacesorthogonal to constant functions, that is(3.2)Hsq(M) := fHsq(M) [ f, 1)s= 0 ,Hsq(M) := f Hsq(M) [ (f, 1)s= 0 .The spacesHsq(M) Hsq(M) andHsq(M) Hsq(M) coincide with thesubspaces of functions of zero average for s 0, and with the subspaces ofdistributions vanishing on constant functions for s < 0.Theorem 3.2. Letr>2 andp(0, 1) be such that andrp>2. Thereexists a bounded linear operatorU :H1q(M) Lp_S1,Hrq(M)_such that the following holds. For anyf H1q(M)there existsa fullmeasure subset Fr(f) S1such that u :=U(f)() Hrq(M) is a distri-butional solution of the cohomological equation Su = f for all Fr(f).Proof. We claim that for any r>2, any p(0, 1) such thatpr>2 andany fH1q(M), there exists a measurable function Aq:=Aq(p, r, f)Lp(S1, R+) such that the following estimates hold. Let S1be such thatAq() < +. For all v Hr+1q(M) we have(3.3) [f, v)[ Aq() |Sv|r.In addition, the following bound for the Lpnorm of the function Aq holds.There exists a constant Bq(p) > 0 such that(3.4) [Aq[p Bq(p) |f|1.Assuming the claim, we prove the statement of the theorem. In fact, by theestimate (3.3) the linear map given by(3.5) Sv f, v), for all v Hr+1q(M),SOBOLEV REGULARITY OF SOLUTIONS 39is well dened and extends by continuity to the closure of the rangeRr()of the linear operator S inHrq(M). Let U(f)() be the extension uniquelydened by the condition thatU(f)() vanishes on the orthogonal comple-ment ofRr() inHrq(M). By construction, for almost all S1the linearfunctional u:=U(f)()Hrq(M) yields a distributional solution of thecohomological equation Su = f whose norm satises the bound|U(f)()|r Aq() .By (3.4) theLpnorm of the measurable functionU(f): S1Hrq(M)satises the required estimate[U(f)[p :=__S1|U(f)()|pr d_1/pBq(p) |f|1.We turn now to the proof of the above claim. LetRq=_Mq_be the(closed) ranges of the Cauchy-Riemann operators q:H1q(M)L2q(M)(see Proposition 2.4). Following [For97], we introduce the linear operatorUq: RqR+qdened as(3.6) Uq:= q(+q)1.By Proposition 2.4, (3), the operatorUqis a partial isometry onL2q(M),hence by the standard theory of partial isometries on Hilbert spaces, it hasa family of unitary extensionsUJ: L2q(M) L2q(M) parametrized byisometriesJ : M+qMq(see formulas(3.10)-(3.12) in [For97]). Bydenition the following identities hold on H1q(M) (see formulas (3.13) in[For97]) :(3.7) S=ei22_UJ+ ei_+q=ei22_U1J+ ei_q.The proof of estimate (3.3) is going to be based on properties of 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-tangentialboundarybehaviouroftheresolvent ofaunitaryoperatoronaHilbertspace, applied to the operatorsUJ, U1JonL2q(M). The Fourier analysisof [For97], 2, also plays a relevant role through Lemma 4.2 in [For97] andthe Weyls asymptotic formula (Theorem 2.3).Following [For97], Prop. 4.6A, or [For02], Lemma 7.3, we prove that thereexists a constant Cq> 0 such that the following holds. For any distributionf H1q(M) there exist (weak) solutions FL2q(M) of the equationsqF= f such that(3.8) [F[0 Cq|f|1.40 GIOVANNI FORNIIn fact, the maps given by(3.9) qv f, v), for all v H1q(M) ,are bounded linear functionals on the (closed) ranges Rq L2q(M) (of theCauchy-Riemann operator q: H1q(M) L2q(M). In fact, the functionalsare well-dened since f vanishes on constant functions, that is, on the ker-nel of the Cauchy-Riemann operators, and it is bounded since by Poincarinequality (see [For97], Lemma 2.2 or [For02], Lemma 6.9) there exists aconstant Cq> 0 such that, for any v H1q(M) H1q(M),(3.10) [f, v)[ |f|1|v|1 Cq|f|1[qv[0.Let be the unique linear extension of the linear map (3.9) toL2q(M)which vanishes on the orthogonal complement of Rqin L2q(M). By (3.10),the functionals are bounded on L2q(M) with norm|| Cq|f|1.Bythe Rieszrepresentationtheorem, thereexist two(unique)functionsF L2q(M) such thatv, F)q=(v) , for all v L2q(M) .The functionsFare by construction (weak) solutions of the equationsqF= f satisfying the required bound (3.8).The identities (3.7) immediately imply that(3.11)qv, F)q= 2ei2RJ (z)Sv, F)q(z + ei)RJ (z)qv, F)q ,whereR+J (z) andRJ (z) denote the resolvents of the unitary operators UJand U1Jrespectively, which yield holomorphic families of bounded opera-tors on the unit disk D C.Let r>2 and let p(0, 1) be such that pr>2. LetE=ekkN be theorthonormal Fourier basis of the Hilbert space L2q(M) described in 2. ByCorollary 3.4 in [For97] all holomorphic functions(3.12) Rk (z) := RJ (z)ek, F)q, k N,belong to the Hardy space Hp(D), for any 02andp (0, 1) be such thatpr >2. By Theorem 3.2, for anyk N 0 there exists a functionwith distributional valuesuk:=U(ek)Lp_S1,Hrq(M)_ such that thefollowing holds. There exists a constant Cq:= Cq(p, r) > 0 such that(3.19)__S1|uk()|pr d_1/pCq|ek|1=Cq (1 +k)1/2.In addition, for any kN 0, there exists a full measure setFkS1such that, for all Fk, the distributionu :=uk() Hrq(M) is a(distributional) solution of the cohomological equation S u = ek.Any function fHr1q(M) such that_M f q= 0 has a Fourier decompo-sition in L2q(M):f=

kN\{0}f, ek)q ek.A (formal) solution of the cohomological equationSu=fis thereforegiven by the series(3.20) u:=

kN\{0}f, ek)q uk().By the triangular inequality inHrq(M) and by Hlder inequality, we have|u|r__

kN\{0}|uk()|2r(1 +k)r1__1/2|f|r1,hence by the triangular inequality for Lpspaces (with 0< p 1 the series in (3.21) is convergent, hence by Chebyshevinequality for the spaceLp(S1), there exists a full measure set BS1such that, for allB, formula (3.20) yields a well-dened distributionuHrq(M) and there exists a constant Cq() > 0 such that(3.22) |u|rCq() |f|r1.The set F = FkBhas full measure and for all F, for all k N0,the distribution uk()Hrq(M) is a solution of the equation Su=ek.It follows that uHrq(M) is a solution of the cohomological equationSu = f which satises the required bound (3.22). SOBOLEV REGULARITY OF SOLUTIONS 43We nally derive a result on distributional solutions of the cohomologicalequation for distributional data of arbitrary regularity:Corollary 3.4. For anysR there existsr>0 such that the followingholds. For almost all S1(with respect to the Lebesgue measure), thereexists a constant Cr,s()>0 such that, for all FHsq(M) orthogonal toconstant functions, the cohomological equationSU=Fhas a distribu-tional solution UHrq(M) satisfying the following estimate:|U|r Cr,s() |F|s.Proof. SinceF Hsq(M) is orthogonal to constant functions, for everykN there exists fkHs+2kq(M), orthogonal to constant functions, suchthat(I Fq )kfk=F. In fact, the family of Friedrichs Sobolev spacesHsq(M)[sR is dened in terms of the Friedrichs extension Fqof theLaplace operatorq of the at metric determined by the quadratic differ-ential. Let n N be the minimum integer such that :=s + 2n + 1>2.By Theorem 3.3, for almost all S1(with respect to the Lebesgue mea-sure) the cohomological equationSu=fnhas a distributional solutionuHq(M) satisfying the following estimate:(3.23) |u| C() |fn|1.Let U:=(I Fq )nuH2nq(M). It follows immediately from theestimate (3.23) and from the denitions that|U|2n= |un| C() |fn|1= C() |F|s.Finally UH2nq(M) is a distributional solution of the cohomologicalequation SU= F, for almost all S1. In fact, for any v H+2n+1q(M),the function Sv H+2nq(M), hence(I q)nSv= S(I q)nvand,since the distributionuHq(M) is a solution of the cohomologicalequation Su = fn, for almost all S1, and (I Fq )nv H+1q(M),U, Sv) = (I Fq )nu, Sv) = u, S(I q)nv)= fn, (I Fq )nv) = (I Fq )nfn, v) = F, v) ,as required by the denition of distributional solution of the cohomologicalequation (Denition 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 of invariant distri-butions. We then recall the structure theorem proved in that paper on thespace of invariant distributions (see [For02], Th. 7.7).44 GIOVANNI FORNIInvariant distributions for the horizontal [respectively vertical] vector eldof an orientable quadratic differential q are closely related to basic currents(of dimension and degree equal to 1) for the horizontal [vertical] foliation Fq[ Fq]. 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 some of the basicconstructions and results on basic currents and invariant distributions whichcarry over without modications to the more general context of 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 M be a nite subset. The space D(M) will denote the standardspace of de Rham currents on the open manifold M , that is the dual ofthe Frchet space c(M ) of differential forms with compact support inM . A homogenous current of dimensiondN (and degree2 d)on M is a continuous linear functional on the subspace dc(M ) ofdiferential forms of degreed. The subspace of homogeneous currents ofdimension d on M will be denote by Dd(M ).Let q be an orientable quadratic differential on a Riemann surface M. Letq be the (nite) set of its zeroes. In [For02] we have introduced the fol-lowing space q(M) of smooth test forms on M.Denition 3.5. For any p M of (even) order k= 2m N (m = 0 if p ,q), let z:UpC be a canonical complex coordinate on a neighbourhoodUp of p M, that is a complex coordinate such that z(p) = 0 and q= zkdz2on Up. Let p:Up C be the (local) covering map dened byp(z) :=zm+1m+ 1 , z C.The spaceq(M) is dened as the space of smooth forms onMsuchthat the following holds: for all pM, there exists a smooth form p ona neighbourhood of 0C such that =p(p) onUpUp. The spaceq(M) is the direct sum of the subspaces dq(M) of homogeneous forms ofdegree d 0, 1, 2. The spaces dq(M), for any d 0, 1, 2, and q(M)can be endowed with a natural Frchet topology modeled on the smoothtopology in every coordinate neighbourhood.Lemma 3.6. For any orientable quadratic differential q Q(M), the spaceof functions 0q(M) is dense in the space Hq(M) endowed with the inverselimit Frchet topology induced by the family of weighted Sobolev norms.Proof. By denition, the MacLaurin series of any f q(M) with respectto a canonical complex coordinate z for q at every pq (of order 2mp)SOBOLEV REGULARITY OF SOLUTIONS 45has the following form:(3.24) f(z) =

h,kNfhkzh(mp+1) zk(mp+1).By Lemmas 2.12 and 2.13, f Hq(M). Thus q(M) Hq(M).Let F Hq(M). By Lemma 2.11 the functionF C(M) and byLemmas 2.12 and 2.13 its MacLaurin series has the form (3.24) at everypq (of order2mp). By Borels thereorem and by a partition of unityargument, there exists a function fq(M) such that F fC(M)vanishes at innite order atq. LetUbe the open neighbourhood of qwhich is the union of a nite number of disjoint geodesic disksD(p) ofradius (0, 0), each centered at a point p q. Let : M [0, 1] bea smooth function such that (a) C0(M q), (b) 1 on M Uand (c)for each (i, j)NN there exists a constant Cij>0 such that,for all (0, 0),maxxM[SiTj(x)[ Ciji+j.If can be proved that, since Ff vanishes at innite order at q,f+ (Ff) F in Hq(M) , as 0+,which implies, since by constructionf+(Ff) 0q(M), that Fbelongs to the closure of 0q(M) in Hq(M). Denition 3.7. The spaceSq(M) D(M q) ofq-tempered currents(introduced in [For02], 6.1) is the dual space of the Frchet space q(M).A homogeneous q-tempered current of dimension d (and degree 2 d) is acontinuous functional on the subspace dq(M)q(M) of homogeneousforms of degree d0, 1, 2. The space of homogeneous currents of di-mension d (and degree 2 d) will be denoted by Sdq(M).For any quadratic differential q on M, there is a natural operator, whichmaps the spaceD0(M q) of currents of dimension0 and degree2 onthe non-compact manifoldM q(which is naturally identied with thespace of distributions on M q) bijectively onto the space D2(M q) ofcurrents of dimension 2 and degree 0 on M q. The operator:D0(M q) D2(M q)is dened as follows. Let q be the smooth area form associated with the(orientable) quadratic differential q on M. It is a standard fact in the theoryof currents that any distribution U on the 2-dimensional surface M q canbe written as U= Uq for a unique current Uof dimension 2 and degree0. Since q 2q(M), the mapextends to a bijective map:S0q(M) S2q(M) .46 GIOVANNI FORNIDenition 3.8. A distribution D D2(M q) is horizontally [vertically]quasi-invariant if SD=0[TD=0] inD2(Mq). AdistributionDS2q(M) is horizontally [vertically] invariant if SD=0 [TD=0] inS2q(M). The space of horizontally [vertically] quasi-invariant distributionswill be denoted by Iq(Mq) [Iq(Mq)] and the subspace of horizontally[vertically] invariant distributions will be denoted by Iq(M) [Iq(M)].Denition 3.9. For any s R+, let(3.25)Isq(M q) :=Iq(M q) Hsq(M) ;Isq(M) :=Iq(M) Hsq(M) .The subspacesIsq(M)Hsq(M) of horizontally [vertically] invariantdistributions can also be dened as follows:(3.26)Isq(M) := D Hsq(M) [ SD = 0 in Hs1q(M) ;[Isq(M) := D Hsq(M) [ TD = 0 in Hs1q(M)] .The subspaces of horizontally [vertically] invariant distributions which canbe extended to bounded functionals on Friedrichs weighted Sobolev spaceswill be denoted by(3.27)Isq(M q) :=Iq(M q) Hsq(M) ;Isq(M) :=Iq(M) Hsq(M) .Let Vq(M) be the space of vector elds X on M q such that the contrac-tion X and the Lie derivative LX q(M) for all q(M).Denition 3.10. A current CD1(M q) is horizontally [vertically]quasi-basic, that is basic forFq [Fq] in the standard sense on M q, ifthe identities(3.28) XC=LXC=0hold in D(M q) for all smooth vector elds X tangent to Fq [Fq] withcompact support on M q. A q-tempered current C S1q(M) is horizon-tally [vertically] basic if the identities (3.28) holds inSq(M) for all vectoreldsXVq(M), tangent toFq[Fq] onMq. The vector spacesof horizontally [vertically] quasi-basic (real) currents will be denoted byBq(M q) [Bq(M q)] and the subspace of horizontally [vertically]basic (real) currents will be denoted by Bq(M) [Bq(M)].Denition 3.11. For any sR, the Friedrichs weighted Sobolev space of1-currentsWsq(M) Sq(M) and the weighted Sobolev space of 1-currentsWsq(M) Sq(M) are dened as follows:(3.29)Wsq(M) := Sq(M) [ (S, T) Hsq(M) Hsq(M) ;Wsq(M) := Sq(M) [ (S, T) Hsq(M) Hsq(M) .SOBOLEV REGULARITY OF SOLUTIONS 47Denition 3.12. For any 1-current CD(M q), the weighted SobolevorderOWq(C) and the Friedrichs weighted Sobolev orderOWq(C) are thereal numbers dened as follows:(3.30)OWq(C) :=infs R[ D Wsq(M) ;OWq(C) :=infs R[ CWsq(M) .Denition 3.13. For any s R, let(3.31)Bsq(M q) :=Bq(M q) Wsq (M) ;Bsq(M) :=Bq(M) Wsq(M) .The subspaces Bsq(M) Wsq(M) of horizontally [vertically] basic cur-rents can also be dened as follows:(3.32)SC= 0 in Hsq(M) and LSC= 0 in Ws1q(M) ;[TC= 0 in Hsq(M) and LTC= 0 in Ws1q(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)Bsq(M q) :=Bq(M q) Wsq(M) ;Bsq(M) :=Bq(M) Wsq(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 currentCBsq(M q) [CBsq(M)] if and only ifthe distributionC (q1/2) Isq(M q) [C (q1/2) Isq(M)]. Acurrent CBsq(M q) [CBsq(M)] if and only if the distributionC (q1/2) Isq(M q) [C (q1/2) Isq(M)]. In addition, the map(3.34)Dq: CC (q1/2);[Dq: CC (q1/2)];is a bijection from the spaceBsq(M q) [Bsq(M q)] onto the spaceIsq(M q) [Isq(M q)], which maps the subspaceBsq(M) [Bsq(M)]onto the subspaceIsq(M) [Isq(M)]. The map (3.34) also maps the spaceBsq(M) [Bsq(M)] onto Isq(M) [Isq(M)].3.3. Basic cohomology. Let Z(M ) D1(M ) denote the subspaceof all (real) closed currents, that is, the space of all (real) de Rham currentsC D1(M ) such that the exterior derivative dC= 0 in D(M ). LetZq(M)S1q(M) be the subspace of all (real) closed q-tempered currents,that is, the space of allq-tempered (real) currentsC such thatdC=0 in48 GIOVANNI FORNISq(M). It was proved in [For02], Lemma 6.2, that the natural de Rhamcohomology map(3.35) jq:Z(M q) H1(M q, R)has the property that the subspace of closed q-tempered currents is mappedonto the absolute real cohomology of the surface, that is,(3.36) jq:Zq(M) H1(M, R) H1(M q, 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 inclusions hold:(3.37)Bq(M q) Z(M q) ,Bq(M) Zq(M) .The images of the restrictions of the natural cohomology map to the variousspaces of basic currents are called the horizontal [vertical] basic cohomolo-gies , namely the spaces(3.38)H1q(M q, R) := jq (Bq(M q)) H1(M q, R) ;H1,sq(M q, R) := jq_Bsq(M q)_H1q(M q, R) ;H1q(M, R) := jq (Bq(M)) H1(M, R) ;H1,sq(M, R) := jq_Bsq(M)_H1q(M, R) .Following [For02], Theorem 7.1, we give below a description of the hori-zontal [vertical] basic cohomologies for the orientable quadratic differentialq, for any orientable holomorphic quadratic differential q on M and for al-most all S1. The result we obtain below is stronger than Theorem 7.1of [For02] since it requires weaker Sobolev regularity assumptions.(Absolute) real cohomology classes onMcan be represented in terms ofmeromorphic (or anti-meromorphic) functions in L2q(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 isomorphismbetweenthe space Hol+(M) [Hol(M)] of holomorphic [anti-holomorphic] differ-entials and the space of square-integrable meromorphic [anti-meromorphic]functions. Let M+q[Mq ] be the space of meromorphic [anti-meromorphic]functions onMwhich 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 atq=q=0 of ordersbounded in terms of the multiplicity of the points pq as zeroes of thequadratic differential q. In fact, if p q is a zero of q of order 2m, that pis a pole of order at most m for any m Mq .SOBOLEV REGULARITY OF SOLUTIONS 49Let q1/2beaholomorphic square rootof qonM. Holomorphic [anti-holomorphic] differentials h+[h] on M can be written in terms of mero-morphic [anti-meromorphic] functions in L2q(M) as follows:(3.39)h+:= m+q1/2, m+ M+q;h:= m q1/2, m Mq.The following representations of real cohomology classes therefore hold:(3.40)c H1(M, R) c = [(m+q1/2)] , m+ M+q;c H1(M, R) c = [(m q1/2)] , m Mq.The mapscq: MqH1(M, R) given by the representations (3.40) arebijective and it is in fact isometric if the spacesMqare endowed with theeuclidean structure induced by L2q(M) and H1(M, R) with the Hodge prod-uct relative to the complex structure of the Riemann surface M. In fact, theHodge norm |c|2H of a cohomology class cH1(M, R) is dened as fol-lows:(3.41) |c|2H:=2_Mh hif c = [(h)] , h Hol(M) .We remark that the Hodge norm is dened in terms of the complex structureof the Riemann surfaceM(carrying a holomorphic quadratic differentialq Q(M)) but does not depend on the quadratic differential. If q Q(M)is any orientable quadratic differential on M, by the representation (3.40),we can also write:(3.42)|c+q (m+)|2H:=_M[m+[2q , for all m+ M+q;|cq (m)|2H:=_M[m[2q , for all m Mq.The representation (3.39)-(3.40) can be extended to the punctured cohomol-ogy H1(M q, R) as follows. For any nite set M, let Hol+(M )[Hol(M )] be the space of meromorphic [anti-meromorphic] differen-tials with at most simple poles at . By Riemann surface theory, any realcohomology classc H1(M , R) can be represented as the real (orimaginary) part of a differential h+ Hol+(M ) or h Hol(M ).Let Mbe a nite set and let M+() [M+()] be the space of allmeromorphic [anti-meromorphic] functions which are holomorphic [anti-holomorphic] on M . The spacesM() can be identied with a sub-space of the distributional spaceD2(M ). In fact, if q is any orientableholomorphic quadratic differential on M, the spaces M(q) identify withsubspaces of the space S2q(M) of q-tempered distributions. The distributiondetermined by a function M+(q) or M(q) is dened by integration50 GIOVANNI FORNI(in the standard way) as a linear functional on C0(M q), which can beextended to the space 0q(M) as follows:(v) := PV_M vq , for all v 0q(M) .The Sobolev regularity of a distributions M(q) depend on the orderof its poles. In fact, by Theorem 2.22 we have the following:Lemma 3.15. Let M+(q) [ M+(q) ] be a meromorphic [anti-meromorphic] function with poles atq. For anysR, the associateddistribution Hsq(M) if at every pqof order2mp the function has a pole of order < (mp + 1)(s + 1).We introduce the following notation: for all s > 0,(3.43) Ms (q) :=M(q) Hsq(M) .There exist natural maps q: Hol(Mq) M(q) dened as follows:for all h Hol(M q),+q (h+) = h+/q1/2[q (h) = h/ q1/2] .By Lemma 3.15 the range of the mapsqis contained in the weightedSobolev space Hsq(M) for all s > 0, hence there are well-dened maps(3.44) q: Hol(M q) Ms (q) for all s > 0 .The maps (3.44) are clearly injective and by Corollary 2.23 there existssq> 0 such that, for any s (0, sq), they are also surjective. Let(3.45) M (q) =

s>0Ms (q) =Ms (q) , for any s (0, sq) .The representation (3.40) of the absolute real cohomology generalizes tothe punctured real cohomology as follows.(3.46)c H1(M q, R) c = [(m+q1/2)] , m+ M+ (q) ;c H1(M q, R) c = [(m q1/2)] , m M (q) .The following lemma, proved in [For02], Lemma 7.6, for weighted Sobolevspaces with integer exponent, holds:Lemma 3.16. LetsR+. LetCWsq(M) be any real current of di-mension (and degree) equal to 1, closed in the space D(Mq) of currentson M q. There exists a distribution U Hs+1q(M) and a meromorphicdifferential h+ Hol+(M q) such that(3.47) dU=(h+)C inWsq(M) .SOBOLEV REGULARITY OF SOLUTIONS 51If C is closed in the space Sq(M) of q-tempered currents there exists a dis-tribution UHs+1q(M) and a holomorphic differential h+ 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 order s R+. In fact, it follows from thedistributional identity (3.47) inSq(M) that the current UHs+1q(M) ifand only if the current C Wsq(M), for any s 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 surface M. Let m+ M+s (q) be a meromorphic function withpolesat qM. AdistributionU Hs+1q(M)isa(distributional)solution in D(M q) of the cohomological equation(3.48) SU= (m+)[TU= (m+)] in D(M q) ,if and only if the current C Wsq(M) uniquely determined by the identity(3.49) dU= (m+q1/2)+Cishorizontally [vertically]quasi-basic. If (m+q1/2) Hol+(M), thedistribution UHs+1q(M) is a solution of the cohomological equation(3.48) in the spaceSq(M) of q-tempered currentsif and only if the cur-rent CWsq(M) uniquely determined by formula (3.49) is horizontally[vertically] basic.Proof. If formula (3.49) holds inD(M q), then C is closed inD(Mq). If the differential (m+q1/2) is holomorphic and formula (3.49) holdsinSq(M), thenCis closed inSq(M). The standard formula for the Liederivative of a current,(3.50) LXC= XdC+dXC=0 ,holds in D(M q) for any vector eld X with compact support containedin M q and it holds in Sq(M) for any vector eld X Vq(M) It followsthat a current CD(M q) is horizontally [vertically] quasi-basic ifand only if it is closed andSC=0 [TC=0] inD(Mq) and itis horizontally [vertically] basic if and only ifCSq(M) is closed andSC=0 [TC=0] inSq(M). The distribution UHsq(M) in formula(3.49) is a solution of the cohomological equation (3.48) inD(Mq)or Sq(M) if and only ifSC=0 [TC=0] inD(Mq) or Sq(M)respectively. As a consequence, the lemma is proved. 52 GIOVANNI FORNILet q be an orientable holomorphic quadratic differential on a Riemann sur-face M(of genus g1). Let q(M q, R)H1(M q, R) be thecodimension 1 subspaces dened as follows:(3.51)1q(M q, R) := c H1(M q, R) [ c [(q1/2)] = 0 ;1q(M q, R) := c H1(M q, R) [ c [(q1/2)] = 0 .Since the absolute cohomology can be regarded as a subspace of the punc-tured cohomology it also is possible to dene the subspaces(3.52)1q(M, R) := 1q(M q, R) H1(M, R) ;1q(M, R) := 1q(M q, R) H1(M, R) .Theorem 3.18. For any s > 3 there exists a full measure set Fs S1suchthat the following holds. For any Fs, the following inclusions hold(3.53)1q(M q, R) H1,sq(M q, R) ,1q(M, R) H1,sq(M, R) .Proof. Let m+M+1(q) be any meromorphic function such that theinduced distribution PV(m+)H1q(M). A computation shows that, forall S1,(3.54) PV_M(m+) q= 0 [(m+q1/2)] 1q(M q, R) .Under the zero-average condition (3.54), by Theorem 3.2 for anyr >2there exists a full measure set Fr(m+) S1such that, for all S1,the cohomological equationSU =(m+) has a distributional solutionU(m+) Hrq(M). Let U(m+) be the current of dimension2 corre-sponding to the distributionU(m+). LetC(m+) Wr1q(M) be the1-dimensional current determined by the identity(3.55) dU(m+) =(m+q1/2)C(m+) .By Lemma 3.17, we have thus proved that, for all meromorphic functionsm+M+1(q) and for all Fr(m+), there exists a quasi-basic currentC(m+) Br+1q(M q) such that[C(m+)] = [(m+q1/2)] 1q(M q, R) ;in addition, whenever m+M+q , the current C(m+) Br+1q(M) is basicand has a cohomology class[C(m+)] = [(m+q1/2)] 1q(M, R) .Let N be the cardinality of the set q M and letm+1 , . . . , m+2g1, . . . , m+2g+1SOBOLEV REGULARITY OF SOLUTIONS 53be a basis (overR) of the real subspace ofM+ (q) dened by the zero-average condition (3.54). For any s > 3, letF+s:=2g+1

i=1Fs1(m+i) .Clearly he set F+shas full Lebesgue measure. We claim that for all F+sthe following inclusions hold:(3.56)1q(M q, R) H1,sq(M q, R) ,1q(M, R) H1,sq(M R) .The claim is proved as follows. For any cH1(M q, R) there exists aunique meromorphic function m+M+ (q) such that c=[(m+q1/2)].The functionm+M+qfor anycH1(M, R). Ifc1q(M, R), thedistribution (m+) vanishes on constant functions as in (3.54), hence forallF+s , there exists a solution UHs+1q(M) of the cohomologicalequation SU= (m+). The current C Bsq(Mq) such that [C] = c 1q(Mq, R) is then given by the identity (3.55). By the above discussionthe current C Bsq(M) for all c 1q(M, R).By a similar argument it is possible to construct a full measure set Fssuchthat, for all Fs , the following inclusions hold:(3.57)1q(M q, R) H1,sq(M q, R) ,1q(M, R) H1,sq(M, R) .Thus the set Fs:=F+sFshas 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 anys >3 there exists a full measure set FsS1such that, for all Fs, the spaces Isq(M) Hsq(M) of horizontally orvertically quasi-invariant distributions have dimension at least 2g + 1and the spaces Isq(M)Hsq(M) of horizontally or vertically invariantdistributions have dimension at least 2g 1.Corollary 3.20. For any s > 3 and for almost all S1,(3.58) H1,sq(M, R) = 1q(M, R) .For any s > 4 and for almost all S1,(3.59) H1,sq(M q, R) = H1(M q, R) .54 GIOVANNI FORNIProof. The inclusionsH1,sq(M, R) 1q(M, R) hold for an

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