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STRONG UNIQUE CONTINUATION PROPERTIES OF GENERALIZEDBAOUENDI-GRUSHIN OPERATORS

NICOLA GAROFALO AND DIMITER VASSILEV

1. Introduction and statement of the results.

The uniqueness in the Cauchy problem and the closely connected unique continuationproperty (ucp) for subelliptic operators is a subject which is far from being understoodand to a large extent unexplored. On the negative side there exists a general coun-terexample of Bahouri [Ba] to the ucp for zero order perturbations of sub-LaplaciansL − V =

∑mj=1 XjXj − V , when, besides the finite rank condition on the Lie algebra,

some additional geometric conditions are fulfilled by the vector fields X1, ..., Xm (such addi-tional assumptions are not necessary in dimension three or four). What happens, however,if one considers the unperturbed operator corresponding to the case V = 0? In this situationBony [Bo] has proved uniqueness in the Cauchy problem if the vector fields are real analytic.A general satisfactory answer to this question in the C∞ or less regular case does not seemto be presently available. In this paper we study the strong unique continuation property(sucp) for a class of variable coefficient operators whose “constant coefficient” model at onepoint is the so called Baouendi-Grushin operator [B], [Gr1], [Gr2]. We recall that the latteris the following operator on RN = Rm × Rn, N = n + m,

(1.1) Lo =N∑

i=1

XiXiu,

where the vector fields (which are not in fact constant coefficient) are given by

(1.2) Xk =∂

∂xk, k = 1, . . . , n, Xn+j = |x|α ∂

∂yj, j = 1, . . . , m.

Here α > 0 is a fixed parameter, x = (x1, . . . , xn) ∈ Rn and y = (y1, . . . , ym) ∈ Rm. Whenα = 0, Lo is just the standard Laplacian in RN . For α > 0 the ellipticity of the operator Lo

becomes degenerate on the characteristic submanifold M = Rn×0 of RN . When α = 2k,with k ∈ N, then Lo is a sum of squares of C∞ vector fields satisfying Hormander finiterank condition on the Lie algebra rank Lie[X1, ..., Xm] ≡ N . We note that there exists afamily of anisotropic dilations

(1.3) δt(ξ) = δt(x, y) = (tx, t(α+1)y) , t > 0

Date: September 5, 2005.The first author was supported in part by NSF Grant No. DMS-0300477.

1

2 NICOLA GAROFALO AND DIMITER VASSILEV

naturally associated with the vector fields in (1.2). Consequently, in the analysis of Lo thenumber

(1.4) Q = n + (α + 1)m (> N = n + m),

plays the role of a dimension. We refer to Q as the homogeneous dimension relative tothe vector fields (1.2). Operators modeled on (1.1) have been intensively studied after thepioneering works of Franchi and Lanconelli, see [FL], [FS], and the references therein.

The analysis of the operator Lo is subtle and, at least in the case α = 1, it is closelyconnected to that of the real part of the Kohn sub-Laplacian on the Heisenberg group Hn,see [RS], [FL], [G2], [GS1], [GS2]. Since the latter operator is real-analytic hypoelliptic,harmonic functions in Hn cannot vanish to infinite order at one point unless they areidentically zero. However, to present date there exists no quantitative proof of such sucpin Hn (by this we mean a proof based on estimates and which does not directly hinge onthe real-analyticity of solutions). In particular, it would be important to know whetherthe generalized frequency in Hn introduced in [GLa] is increasing, but this remains at themoment a challenging open question. Such and related questions constitute some of themotivations of the present paper. Returning to the operator Lo, we mention that it wasproved in [G2] that the frequency attached to the horizontal energy is indeed increasing atpoints of the degeneracy manifold M , thus the sucp holds for Lo. In the same paper this is

also proved for the operator Lo− <→b , Du > −V with suitable assumptions on

→b and V .

To give an idea, for example

|V | ≤ C

ρψ and | <

→b , Du > | ≤ C|Xu|ψ1/2

is enough. Here Du is the gradient of u, |Xu| is the horizontal gradient (1.10) of u, and ρand ψ are defined correspondingly in (1.8) and (1.9). With a completely different method,based on a subtle two-weighted Carleman estimate, the sucp was established in [GS1] for zeroorder perturbations Lo−V , where the potential V is allowed to belong to some appropriateLp spaces.

In this paper we consider equations of the type

(1.5) Lu =N∑

i,j=1

Xj(aij(x, y)Xiu) = 0.

We assume that A =(aij(x, y)

), i, j = 1, ..., N, is a N × N matrix-valued function on RN

which, for simplicity, we take such that

(1.6) A(0) = Id.

Furthermore, we assume A is symmetric and uniformly elliptic matrix. Thus aij(g) = aji(g)and there exists λ > 0 such that for any η ∈ RN

(1.7) λ|η|2 ≤ < Aη, η > ≤ λ−1|η|2.

STRONG UNIQUE CONTINUATION 3

Our main concern is whether, under suitable assumptions on the matrix A, the sucp contin-ues to hold for the operator L. To put our result in perspective we mention that when α = 0in (1.2), so that Lo is the standard Laplacian, a famous result due to Aronszaijn, Krzywickiand Szarski [AKS] states that if the matrix A has Lipschitz continuous coefficients, thenthe operator L possesses the sucp. Furthermore, it was shown in [M] that such assumptionis optimal. Our results, Theorems 1.2 and 1.3 can be seen as a generalization of that in[AKS], in the sense that, in the limit as α → 0 we recapture both the assumptions and theconclusion of the elliptic case, see Remark 1.3. The approach, however, is different fromthat in [AKS], which is based on Carleman inequalities along with results from Riemanniangeometry that do not seem to be adaptable to our context due to the lack of ellipticity. In-stead, we have borrowed the ideas developed in [GL1], [GL2], [G2], see also the subsequentsimplification in [K]. Our main result is Theorem 1.2, which gives a quantitative control ofthe order of zero of a weak solution to (1.5). Such result is proved under some hypothesison the matrix A which are listed as assumptions (H) below. The latter are tailored on thegeometry of the operator Lo and should be interpreted as a sort of Lipschitz continuity withrespect to a suitable pseudo-distance associated to the system of vector fields (1.2).

In order to state the main result we recall the definition of the gauge ρ associated to Lo

[G2]. With ξ = (x, y) ∈ RN we let

(1.8) ρ = ρ(ξ)def= (|x|2(α+1) + (α + 1)2|y|2) 1

2(α+1) .

We stress that ρ is homogeneous of degree one with respect to the anisotropic dilations (1.3).In the sequel we indicate with Br = ρ < r the pseudo-balls centered at the origin in RN

with radius r with respect to the gauge ρ . Since ρ ∈ C∞(RN \ 0), the outer unit normalon ∂Br is given by ν = |Dρ|−1Dρ. As we mentioned, if α = 2k, with k ∈ N, then the system(1.2) satisfies Hormander’s condition, and the ensuing Carnot-Caratheodory distance of ξfrom the origin can be shown to be comparable to ρ(ξ). We will also need the angle functionψ defined as follows [G2]

(1.9) ψ = ψ(ξ)def= |Xρ|2(ξ) =

|x|2α

ρ2α, ξ 6= 0.

Hereafter, given a function f , we denote by

(1.10) Xf = (X1f, ...,XNf)

the gradient along the system of vector fields in (1.2) ( called also horizontal gradient of f),and let |Xf |2 =

∑Nj=1(Xjf)2. The function ψ vanishes at every point of the characteristic

manifold M , and clearly satisfies 0 ≤ ψ ≤ 1.

Definition 1.1. A weak solution to Lu = 0 in an open set Ω is a function u ∈ L2loc(Ω)

such that the (distributional) horizontal gradient Xu ∈ L2loc(Ω), and for which the equation

Lu = 0 is satisfied in the variational sense in Ω, i.e.,∫

Ω< AXu, Xφ > dV = 0

for every φ ∈ C∞o (Ω).


We note that, under the hypothesis in the present paper, thanks to the basic results in [FL],[FS] a weak solution u is (after modification on a set of measure zero) Holder continuouswith respect to the Euclidean distance. We are ready to state our main result.

Theorem 1.2. Let A be a symmetric matrix satisfying (1.7) and the hypothesis (H) belowwith relative constant Λ. Suppose u is a weak solution of (1.5) in a neighborhood of theorigin Ω. Under these assumptions, there exist positive constants C = C(u, α, λ,Λ, N) andro = ro(u, α, λ,Λ, N), such that, for any 2r ≤ ro, we have

∫

B2r

u2ψ dV ≤ C

∫

Br

u2ψ dV.

The dependence of the constant C on u is quite explicit. It involves the L2 norm of |Xu|on B1, and the L2 norm of u on ∂B1 with respect to the weighted measure ψ dHN−1. Weremark that, although we have stated Theorem 1.2 when the point of consideration is theorigin, this result continues to be true for any other point with the appropriate modificationof the hypothesis (H).

We say that u ∈ L2loc(RN ) vanishes to infinite order at some zo ∈ RN if for every k > 0

one has

limr→0

1rk

∫

Br(zo)|u|2 dV = 0 .

A given partial differential operator L in RN is said to possess the strong unique contin-uation property (SUCP) if for every zo ∈ RN , and any weak solution u of Lu = 0, theassumption that u vanishes to infinite order at zo implies that u ≡ 0 in some neighborhoodof zo. In other words non-trivial solutions can have at most finite order of vanishing. As itis well known [GL1], Theorem 1.2 implies the following sucp.

Theorem 1.3. With the assumptions of Theorem 1.2, the operator L has the SUCP.

In order to state our main assumptions (H) on the matrix A it will be useful to representthe latter in the following block form

A =(

A11 A12

A21 A22

),

Here, the entries are respectively n×n, n×m, m×n and m×m matrices, and we assumethat At

12 = A21. We shall denote by B the matrix

B = A − IIIN×N

and thus

(1.11) B(0) = OOON×N ,

thanks to (1.6). The proof of Theorem 1.2 relies crucially on the following assumptions onthe matrix A. These will be our main hypothesis and, without further mention, will beassumed to hold throughout the paper.


HYPOTHESIS. There exists a positive constant Λ such that, for some ε > 0, one has inBε the following estimates

|bij | = |aij − δij | ≤

Λρ, for 1 ≤ i, j ≤ n

Λψ12+ 1

2α ρ = Λ |x|α+1

ρα , else

(H)

|Xkbij | = |Xkaij | ≤

Λ, for 1 ≤ k ≤ n, and 1 ≤ i, j ≤ n

Λψ12 = Λ |x|α

ρα , else

An interesting, typical example of a matrix satisfying the conditions (H) is

A =(

1 + ρf(x, y) |x|α+1g(x, y)|x|α+1g(x, y) 1 + |x|α+1h(x, y)

),

where f, g and h are functions which are Lipschitz continuous at the origin of R2 withrespect to the Euclidean metric. In this example m = n = 1.

Remark 1.4. It is important to observe that, thanks to (1.9), if we take formally α = 0in (H) we obtain a Lipschitz condition at the origin for the matrix A. Our results thusencompass those in the cited paper [AKS], see also [GL1].

For a vector field F we denote by FA the matrix with elements(Faij

). We will apply the

same notation to all matrices under consideration. Throughout the paper we will tacitlyassume that all vectors are column vectors. Also, we will use the same notation for firstorder partial differential operators and for the corresponding tangent vectors, with meaningdetermined by the context.

The plan of the paper is as follows. In section two we prove Theorem 1.2. The proofinvolves various technical estimates. For the reader’s convenience and ease of exposition wehave collected all the auxiliary material in section three.

Acknowledgment: We thank Bruno Franchi for his kind help in clarifying the results inthe papers [FL], [FS].

2. The frequency function.

The purpose of this section is to prove Theorem 1.2. The main step is to show themonotonicity of the frequency Theorem 2.2. We begin by introducing the relevant quantitiesthat will appear in the proof. Since our results are local in nature, from now on, we focusour attention on the pseudo-ball B2. The notation dHN−1 will indicate (N−1)-dimensionalHausdorff measure in RN . Let u be a weak solution u of (1.5) in B2.


Definition 2.1. For every 0 < r < 2 we let

H(r) =∫

∂Br

u2 < AXρ, Xρ >

|Dρ| dHN−1,

D(r) =∫

Br

< AXu, Xu > dV.

The generalized frequency of u on Br is defined by

N(r)def=

rD(r)H(r) , if H 6= 0

0, if H = 0.

We shall denote by S the matrix relating the gradient along the vector fields in (1.2) andthe standard gradient in RN , i.e., X = SD , where

(2.1) S =(

IIIn×n 000000 |x|αIIIm×m

).

Trivially, we have

(2.2) S = St and Lu = div(SASDu).

The following theorem constitutes the main result of this section.

Theorem 2.2. Let u be a nontrivial weak solution of Lu = 0 in the pseudo-ball B2, thenthere exist positive constants ro = ro(α, λ, Λ, N) and M = M(u, α, λ,Λ, N) such that

N(r) = exp(Mr)N(r)

is a continuous monotonically nondecreasing function for r ∈ (0, ro).

Proof. The proof of Theorem 2.2 rests on Lemmas 2.5 and 2.12 below. Let M = maxC1, C2,where C1 and C2 are the constants from Lemmas 2.5 and 2.12. Let Q be the homogeneousdimension in (2.3) associated with the non-isotropic dilations (2.4). With ro as defined inLemma 2.5 we have that, either u ≡ 0 in Bro , or H(r) > 0 for 0 < r < ro. In the formercase the frequency is identically zero on (0, ro), so let us consider the latter case, in whichH(r) > 0. The continuity of N(r) follows from the continuity of each of the functionsinvolved in its definition. Furthermore, for a.e. r ∈ (0, ro) we have

(ln

rD(r)H(r)

e2Mr)′ =

1r

+D′(r)D(r)

− H ′(r)H(r)

+ 2M

≥ 1r

+Q− 2

r+

2D(r)

∫

∂Br

< AXu, Xρ >2

< AXρ, Xρ >

dHN−1

|Dρ|

− Q− 1r

− 2D(r)H(r)

≥ 0 ,

where we have applied first Lemmas 2.5 and 2.12, and then Proposition 2.4 and the Cauchy-Schwarz inequality.

¤


With the help of the monotonicity it is easy to prove Theorem 1.2, see Section 3 of [GL1].We include the proof in the current setting for completeness.

Proof of Theorem 1.2. If the solution vanishes in some neighborhood of the origin then thedoubling for all sufficiently small balls is trivially satisfied. Let us consider next the caseof a non-trivial solution. Let ro be the number defined in Lemma 2.5 and 2r ≤ ro. By theco-area formula ∫ R

0

∫

∂Br

u2ψdHN−1

|Dρ| dr =∫

BR

u2ψ dV.

¿From the ellipticity of A in (1.7), we have∫ R

0H(r) dr ≈

∫

BR

u2ψ dV ,

with constant of proportionality depending only on λ > 0. This shows it is enough to provethe doubling property for the height function H. Now, we obtain from Lemma 2.5

lnH(2r)

2Q−1H(r)= ln

H(2r)2Q−1rQ−1

− lnH(r)rQ−1

=∫ 2r

r

H ′(t)H(t)

− Q− 1t

dt

≤∫ 2r

r

2D(t)H(t)

+ C1

dt ≤

∫ 2r

r2N(t)

e−2Mt

tdt + Mr

≤ 2N(ro)∫ 2r

r

1t

dt + M = 2N(ro) ln 2 + M ,

where in the last inequality we have used the monotonicity of the modified frequency ex-pressed by Theorem 2.2. We thus conclude

H(2r) ≤ 2Q−1 e2N(ro) ln 2 + M H(r) .

Integrating the latter inequality we obtain the doubling property in the conclusion of The-orem 1.2.

¤

Remark 2.3. We observe that for non-trivial solution we have the doubling property forall balls B2r ⊂ Ω and 2r ≤ 1, since for ”big” balls, i.e., 2r ≥ ro we have

∫B2r

u2ψ dV∫Br

u2ψ dV≤

∫B1

u2ψ dV∫Bro/2

u2ψ dV.

Of course, in this case the constant C in the doubling property depends on N(1).

Finally, we establish Theorem 1.3.


Proof of Theorem 1.3. Suppose u is a solution which vanishes to infinite order at the origin.Let |Br| = ωor

Q. Fix a number κ > 0 such that Co2−Qκ = 1. For any r sufficiently smalland p ∈ N the doubling property applied p times gives∫

Br

u2ψ dV ≤ Cpo

∫

Br/2p

u2ψ dV

≤ ωκo Cp

o

rQκ

2Qpκ

1|Br/2p |κ

∫

Br/2p

u2ψ dV

≤ ωκo rQκ 1

|Br/2p |κ∫

Br/2p

u2ψ dV → 0

when p →∞ since 0 ≤ ψ ≤ 1. This ends the proof.¤

The remainder of this section is devoted to establishing Lemmas 2.5 and 2.12.

Proposition 2.4. For a.e. r ∈ (0, 2) the horizontal energy of u on Br can be expressed bythe surface integral

D(r) =∫

∂Br

u< AXu, Xρ >

|Dρ| dHN−1.

Proof. By the definition of weak solution we have u is continuous and Xu ∈ L2(B2), thusfor a.e. r ∈ (0, 2) one has Xu ∈ L2(∂Br). The outer unit normal on ∂Br is given byν = |Dρ|−1Dρ and thus

u< AXu, Xρ >

|Dρ| = u< AXu, SDρ >

|Dρ| =< uSAXu, ν > .

The divergence theorem, (2.2) and the fact that Lu = 0 imply

∫

∂Br

u< AXu, Xρ >

|Dρ| dHN−1 =∫

Br

div(uSAXu

)dV

=∫

Br

< AXu, Xu > dV +∫

Br

uLu dV

=∫

Br

< AXu, Xu > dV ,

as claimed in the proposition. ¤

We proceed with proving the main estimate for the generalized height function H(r).This is the first place where the assumptions (H) on the matrix A play a decisive role. Weobserve that r → H(r) is absolutely continuous, thus differentiable a.e. on (0, 2). In thesubsequent analysis the number


(2.3) Q = n + (α + 1)m (> N = n + m),

will play an important role. We note that Q is the homogeneos dimension relative to theanisotropic dilations

(2.4) δt(ξ) = δt(x, y) = (tx, t(α+1)y) , t > 0

naturally associated with the vector fields in (1.2). The infinitesimal generator of (2.4) is

(2.5) Z =∑

1≤i≤n

xi∂

∂xi+ (α + 1)

∑

1≤j≤m

yi∂

∂yi,

so that a function u is δt-homogeneous of degree k ∈ R if and only if Zu = ku. At thispoint it is worth observing that if u is homogeneous of degree k, and solves the ”constantcoefficient” equation Lou = 0 (i.e. u is a fundamental Lo-harmonic of degree k), then thecorresponding frequency is constant and equal to k. This justifies the name generalizedfrequency. To prove this fact one uses Proposition 2.4 with A ≡ I which gives

D(r) =∫

Br

< Xu, Xu > dV =∫

∂Br

u< Xu, Xρ >

|Dρ| dHN−1 .

A calculation, see (2.13) in [G2] or Proposition 3.1, shows ( X = SD !)

(2.6) Xρ =ψ

ρS−1Z ,

for any function u. When u is Lo-harmonic of degree k we have Zu = ku, and one infersfrom (2.6)

< Xu, Xρ > =ψ

ρZu .

Substitution of the latter identity in (2.6) gives

D(r) =k

r

∫

∂Br

u2 ψ

|Dρ|dHN−1 =k

rH(r) ,

which proves N(r) ≡ k.

Lemma 2.5. a) There exists a positive constant C1 = C1(α, λ, Λ, N) such that for a.e.r ∈ (0, 2) one has ∣∣∣H ′

(r)− Q− 1r

H(r)− 2D(r)∣∣∣ ≤ C1H(r).

b) There exists a positive number ro = ro(α, λ,Λ, N) ≤ 1 such that, either H(r) = 0 on(0, ro), or H(r) > 0 on (0, ro).


Proof. a) Using the definition (2.1) of S we have

< AXρ,Xρ >

|Dρ| = < SAXρ, ν > .

The divergence theorem gives

H(r) =∫

∂Br

u2 < SAXρ, ν > dHN−1 =∫

Br

div(u2SAXρ

)dV

=∫

Br

< AXρ,Xu2 > dV +∫

Br

u2Lρ dV

=∫

Br

2u < AXρ, Xu > dV +∫

Br

u2Lρ dV.

(2.7)

Since the gauge ρ is not smooth at the origin, to make rigorous the previous calculation onemust integrate on the set Br \Bε and then let ε → 0. We note that the last integral on thesecond line of the above chain of equalities is convergent since Lρ ∈ L1

loc(RN ). This can beseen from the remarkable formula

(2.8) Lo ρ =Q− 1

ρ|Xρ|2, in RN \ 0 ,

which is (2.18) in [G2]. Once (2.8) is available one easily obtains by a rescaling, using (2.4),that ρ−p ∈ L1

loc(RN ) if and only if p < Q. This shows, in particular, that Loρ ∈ L1loc(RN ).

We note explicitly that (2.8) expresses, in disguise, the fact that for a suitable constantC > 0 the function

(2.9) Γ = C ρ2−Q

is a fundamental solution of Lo with pole at 0.Returning to (2.7), after an application of the Federer’s co-area formula we differentiate

at a.e. r > 0, and use Proposition 2.4, obtaining

H′(r) = 2D(r) +

∫

∂Br

u2Lρ

|Dρ| dHN−1.

This implies


H′(r)− Q− 1

rH(r)− 2D(r) =

∫

∂Br

u2Lρ

|Dρ| dHN−1 − Q− 1r

H(r)

=∫

∂Br

u2 div(SBXρ)|Dρ| dHN−1 +

∫

∂Br

u2 L0ρ

|Dρ|dHN−1

− Q− 1r

∫

∂Br

u2 |Xρ|2|Dρ| dHN−1

− Q− 1r

∫

∂Br

u2 < BXρ,Xρ >

|Dρ| dHN−1.

We recall that (bij) = B = A− Id. Now, thanks to (2.8) the two middle terms in the lastequality above are equal. The last term is easily estimated as follows on ∂Br

< BXρ, Xρ >

|Dρ| ≤ C r< AXρ,Xρ >

|Dρ| ,

for some positive constant C = C(α, λ, Λ, N). This is recognized observing that by (H) wehave ||B||L∞(∂Br) ≤ Cr, and using also (1.7). Finally, we estimate the first term in theright-hand side. Writing the divergence term as

div(SBXρ) =N∑

i,j=1

Xi(bijXjρ) =N∑

i,j=1

XibijXjρ + bijXiXjρ,

and taking into account the assumptions (H), Proposition 3.1 and Proposition 3.3 we find,by splitting the terms into the four groups that appear in the block form of A ( and henceof B ), the following inequalities

N∑

i,j=1

|XibijXjρ| ≤ C(ψ1+ 1

2α + ψ12 ψ1+ 1

2α + ψ12 ψ

12 + ψ

12 ψ

12) ≤ Cψ,

N∑

i,j=1

|bijXiXjρ| ≤ C(ρψ

ρ+ ρψ

12+ 1

2α ρψ12− 1

2α + ρψ12+ 1

2αψ

32+ 1

2α

ρ+ ρψ

12+ 1

2αψ

ρ

)

≤ Cψ.

This completes the proof of part a).

b) From part a) we have

H ′(r) ≥ (Q− 1r

− C1

)H(r) + 2D(r).


Let r1 = min1, Q−12C1

so that H ′(r) ≥ C1H(r)+2D(r) ≥ 0 on the interval (0, r1). Thereforethere exists an 0 < ro ≤ r1 with the required properties. ¤

Our next objective is to obtain estimates of the first variation D′(r) of the horizontal

energy. Let

(2.10) µdef= < AXρ, Xρ > .

Consider the vector field F defined as follows

(2.11) F = ρN∑

i,j=1

aijXjρ

µXi, x 6= 0,

i.e.Fu =

ρ

µ< AXρ,Xu >=

ρ

µ< SAXρ, Du >,

for any smooth function u. We now see that the assumptions on the matrix A guaranteethat F can be continuously extended to all of RN . Furthermore, near the characteristicmanifold, such extension gives a small perturbation of the Euler vector field Z in (2.5). Toprove this latter claim, we recall (2.6), and let

(2.12) σdef= < BXρ, Xρ > = µ− ψ .

Thus, F can be re-written as

(2.13) F =ψ

µZ +

ρ

µSBXρ = Z − σ

µZ + ρ

N∑

i,j=1

bijXjρ

µXi.

From (H), the coercivity of A, and from Lemma 3.1 we find easily

∣∣∣σµ

Z∣∣∣ ≤ C

ρψ1+ 12α ψ1+ 1

2α + ρψ12+ 1

2α ψ12 ψ1+ 1

2α

ψ|Z|

≤ Λρψ1+ 1α |Z| ≤ Λ|x||Z|,

(2.14)

and ∣∣∣bijXjρ

µXi

∣∣∣ ≤ C ≤ Cρψ12α ≤ C|x| .

Substituting the two estimates (2.14) in (2.13), we obtain the above claim.Our next goal is establishing a basic Rellich-type identity involving the vector field F ,

Lemma 2.11, which we shall use to prove the main estimate on the derivative of the hori-zontal energy, see Lemma 2.12. The proof of such Rellich-type identity relies on some basicestimates on the divergence and the commutators of F which are collected in the subse-quent Lemmas 2.6, 2.7, 2.8, 2.9 and 2.10. We mention that, in turn, the proofs of thesefive lemmas rely on some auxiliary technical estimates which, in order to keep the flow of


this section, we have collected separately in the next section. Hereafter, the summationconvention over repeated indices will be adopted.

Lemma 2.6. There exists a constant C = C(α, λ,Λ, N) > 0 such that for 1 ≤ i ≤ N wehave: ∣∣∣[Xi,

ρ

µSBXρ]u

∣∣∣ ≤ Cρ|Xu|.

Proof. By a direct calculation

[Xi,ρ

µSBXρ]u = Xi <

ρ

µBXρ, Xu > − <

ρ

µBXρ, XXiu >=

= Xi

(ρ

µ

)bkjXjρXku +

ρ

µXi(bkjXjρ)Xku +

ρ

µbkjXjρ[Xi, Xk]u.

Now, Lemma 3.8, Lemma 3.9 and Remark 3.6 give the desired bound for the first and thesecond sum in the last line. To estimate the last sum we use that

|[Xi, Xk]u| ≤ α

|x| |Xu|and Lemma 3.9. ¤

Lemma 2.7. There exists a constant C = C(α, λ,Λ, N) > 0 such that for 1 ≤ i ≤ N wehave: ∣∣∣[Xi,−σ

µZ]u

∣∣∣ ≤ Cρ|Xu|.

Proof. ¿From Proposition 3.1 we have

Zu = =ρ

ψ< Xρ, Xu > .

Thus

[Xi,σ

µZ]u = Xi

(σ

µ

ρ

ψ< Xρ, Xu >

)− σ

µ

ρ

ψ< Xρ,XXiu >

= Xi

(σ

ψ

ρ

µXkρ

)Xku +

σρ

µψXkρ[Xi, Xk]u

=ρ

µXi

(σ

ψ

)XkρXku +

σ

ψXi

(ρ

µ

)XkρXku +

σρ

µψXiXkρXku +

σρ

µψXkρ[Xi, Xk]u.

Using Lemmas 3.7, 3.8, 3.5 and Propositin 3.3 together with

|[Xi, Xk]u| ≤ α

|x| |Xu|

we can bound each of the terms above and finish the proof. ¤


Lemma 2.8. There exists a constant C = C(α, λ, Λ, N) > 0 such that∣∣div

(ρ

µSBXρ

)∣∣ ≤ Cρ.

Proof. We have

div(ρ

µSBXρ

)=< BXρ,X(

ρ

µ) > +

ρ

µdiv

(SBXρ

)

=σ

µ− ρ

µ2

(< BXρ, Xσ > + < BXρ,Xψ >

)+

ρ

µXk(bkjXj).

Invoking Lemmas 3.5, 3.9, Proposition 3.2 and Remark 3.6, we end the proof.¤

Lemma 2.9. There exists a constant C = C(α, λ, Λ, N) > 0 such that∣∣div

(σ

µZ

)∣∣ ≤ Cρ.

Proof. The proof is straightforward after we make use of the fact that ψ is homogeneous oforder 0, i.e., Zψ = 0. Recall also that divZ = Q, and that µ = ψ + σ.

div(σ

µZ

)= Z

(σ

µ

)+ Q

σ

µ= Z

(µ− ψ

µ

)+ Q

σ

µ

= −Z(ψ

µ

)+ Q

σ

µ= −ψZ

( 1µ

)+ Q

σ

µ=

ψ

µ2Zσ + Q

σ

µ.

Clearly|σµ| ≤ Cρψ

while|Zσ| ≤ ρ

ψ|Xρ||Xσ| ≤ Cρψ

by Lemma 3.5.¤

Lemma 2.10. | < FAXu, Xu > | ≤ C ρ |Xu|2 .

Proof. It is enough to show that|Fars| ≤ C ρ ,

i.e.,ρ

ψ| < AXρ,Xars > | ≤ C ρ ,

which is the same as| < AXρ,Xars > | ≤ Cψ for all (r, s).


The assumption (H) implies

|aijXiρXjars| ≤ Cψ12 ψ

12 ≤ Cψ, n + 1 ≤ j ≤ N,

|aijXiρXjars| ≤ C(ψ1+ 12α + ψ

12+ 1

2α ρψ12 ) ≤ Cψ1+ 1

2α ≤ Cψ, 1 ≤ j ≤ n.

¤

We can now prove the above mentioned Rellich-type indentity.

Lemma 2.11. Let X1, . . . , XN and F be the above considered vector fields in RN . We havethe following identity

∫

∂Br

< AXu,Xu >< F, ν > dHN−1 =

= 2∫

∂Br

ajkXju < Xk, ν > Fu dHN−1 − 2∫

Br

(divXk) ajkXjuFudV

− 2∫

Br

ajk Xju [Xk, F ]u dV +∫

Br

(divF ) < AXu, Xu > dV

+∫

Br

< (FA)Xu,Xu > dV − 2∫

Br

FuLu dV,

where FA is the matrix with ellements Faij. Here, ν denotes the outer unit normal to Br .

Proof. The proof of the above integral identity is based on the divergence theorem andcan be carried similarly to its classical couterpart, see Ch.5 in [Ne]. Since the vectorfields and the matrix A are not smooth, one has to justify the use of such result by astandard approximation argument which can be carried using the following key estimatesfrom Lemmas 2.6 - 2.10. Specifically, Lemmas 2.8, 2.9 give

|Q − divF | ≤ C ρ ,

whereas Lemmas 2.6, 2.7 imply

|[X,F ]u−Xu| ≤ C ρ |Xu| .

Finally, Lemma 2.10 gives‖FA‖∞ ≤ C ρ .

¤

Lemma 2.12. There exists a constant C2 = C2(α, λ, Λ, N) > 0 such that

D′(r) ≥ 2

∫

∂Br

1µ

< AXu, Xρ >2

|Dρ| dV +Q− 2

rD(r) − C2 D(r) ,

where µ is defined in (2.10).


Proof. By the co-area formula D(r) =∫ r0

∫∂Bs

<AXu,Xu>|Dρ| dHN−1ds. Hence,

D′(r) =

∫

∂Br

< AXu, Xu >

|Dρ| dHN−1 =1r

∫

∂Br

< AXu, Xu >< F, ν > dHN−1 ,

taking into account that on ∂Br one has < F, ν > = r|Dρ| . The latter follows from the

following calculation

< F, ν > =Fρ

|Dρ| = ρ< SAXρ, Dρ >

µ|Dρ| = ρ< AXρ, Xρ >

µ|Dρ| =r

|Dρ| .

¿From Lemma 2.11 we obtain

D′(r) = 2

∫

∂Br

1µ

< AXu,Xρ >2

|Dρ| dV +1r

∫

Br

(divF ) < AXu,Xu > dV

− 2r

∫

Br

ajk Xju [Xk, F ]u dV +1r

∫

Br

< (FA)Xu, Xu > dV.

In view of (2.13), the fact that divZ = Q, and of the identities [Xi, Z] = Xi, i = 1, . . . , N ,we can rewrite the above formula in the following form

D′(r) = 2

∫

∂Br

1µ

< AXu, Xρ >2

|Dρ| dHN−1 +Q− 2

rD(r)

+1r

∫

Br

div(−σ

µZ +

ρ

µSBXρ) < AXu, Xu > dV

− 2r

∫

Br

ajk Xju [Xk,−σ

µZ +

ρ

µSBXρ]u dV

+1r

∫

Br

< (FA)Xu,Xu > dV.

At this point we are left with showing that the assumption (H) implies the correct estimatesfor the last three integrals. The absolute value of the integral involving the divergence isestimated by Lemmas 2.8 and 2.9. The integral involving the commutators is estimated byLemmas 2.6 and 2.7, using also the ellipticity of A, cf. (1.7). Finally, the absolute valueof the last integral is estimated by Lemma 2.10 and by (1.7). This finishes the proof ofLemma 2.12. ¤


3. Auxiliary results

In this section we collect some basic estimates that have been used in section two. Recallthat the matrix S was defined in (2.1)

Proposition 3.1. i) The following formula holds true

Z =ρ

ψSXρ

ii) The horizontal gradient of the gauge satisfies

|Xkρ| ≤ ψ1+ 12α for 1 ≤ k ≤ n ,

|Xn+kρ| ≤ (α + 1)ψ12 for 1 ≤ k ≤ m .

Proof. By definition

Xkρ = ψxk

ρfor 1 ≤ k ≤ n ,

Xn+kρ = (α + 1)ψ12

yk

ρα+1for 1 ≤ k ≤ m .

In other words, we have

Xρ =(ψ

ρx, (α + 1)

ψ1/2

ρα+1y)

=ψ

ρ

(x, (α + 1)|x|−α y

)=

ψ

ρS−1Z,

having in mind the definition of the radial vector field Z, see (2.5). From|x|ρ

= ψ12α and

|y| ≤ ρα+1 we obtain that the estimates in ii).¤

In the next proposition we compute the horizontal gradient of the angle function

Proposition 3.2. The angle function ψ satisfies the estimates

|Xkψ| ≤ Cαψ

|x| , if 1 ≤ k ≤ n

|Xn+kψ| ≤ Cαψ

ρ, if 1 ≤ k ≤ m .

Proof. Since ψ =|x|2α

ρ2αwe have


Xψ =2α|x|2α−1X|x|

ρ2α− 2α|x|2α

ρ2α+1Xρ

=2α|x|2α−1

ρ2α

( x|x|0

)− 2α|x|3α

ρ2(2α+1)

( |x|αx(α + 1)y

)

= 2αψ

( x|x|20

)− 2α

ψ2

ρ2

(x

(α + 1)|x|−αy

).

This shows that

Xiψ =

2αψ xi|x|2 − 2αψ2 xi

ρ2 if 1 ≤ i ≤ n,

−2α(α + 1)ψ yi−n|x|αρ2α+2 if n + 1 ≤ i ≤ N .

Now, |x| ≤ ρ and |y| ≤ ρα+1 lead to the desired estimates.¤

In the proof of Theorem 1.2 the following estimates on the horizontal Hessian of ρ play animportant role.

Proposition 3.3.

|XiXjρ| ≤ Cψ

ρfor 1 ≤ i, j ≤ n or n + 1 ≤ i, j ≤ N,

|XiXn+jρ| ≤ Cψ

12

|x| = Cρψ12− 1

2α for 1 ≤ i ≤ n, 1 ≤ j ≤ m,

|Xn+jXiρ| ≤ Cψ

32 |x|ρ2

= Cψ

32+ 1

2α

ρfor 1 ≤ i ≤ n, 1 ≤ j ≤ m.

Proof. We need to compute the second derivatives of ρ and this is done easily for exampleby using the product rule and the formulas from Propositions 3.1 and 3.2. We shall writeonly the expressions for the second derivatives.If 1 ≤ i, j ≤ n, we have:

XiXjρ = −(2α + 1)ψ2

ρ3xixj + 2α

ψ

ρ|x|2 xixj +ψ

ρδij .

If 1 ≤ i ≤ n and 1 ≤ j ≤ m we have:

XiXn+jρ = −(2α + 1)ψ2

ρ3(α + 1)|x|−αxiyj

+ 2αψ

ρ|x|2 (α + 1)|x|−αxiyj − ψ

ρα(α + 1)|x|−α−2xiyj .


If 1 ≤ i ≤ n and 1 ≤ j ≤ m we have:

Xn+jXiρ = −(2α + 1)ψ2

ρ3(α + 1)|x|−α−2xiyj .

If 1 ≤ i, j ≤ m, we have:

Xn+iXn+jρ = −(2α + 1)ψ2

ρ3(α + 1)|x|−α−2xjyi + (α + 1)

ψ

ρδij .

At this point the estimates follow in an obvious way using |x| ≤ ρ and |y| ≤ ρα+1. ¤

Definition 3.4. Let:µ

def= < AXρ,Xρ >,

and alsoB

def= A− Id, σ

def= < BXρ, Xρ > .

One more notation we will use is: (bij) = B.

Lemma 3.5. If (H) holds then:

|σ| ≤ Cρψ32+ 1

2α ,

|Xkσ| ≤ Cψ32 1 ≤ k ≤ N.

Proof. We have σ = bijXiρXjρ. Thus Proposition 3.3 and (H) give:

|σ| ≤ C(ρψ1+ 12α ψ1+ 1

2α + ρψ12+ 1

2α ψ1+ 12α ψ

12 + ρψ

12+ 1

2α ψ12 ψ

12 )

≤ C(ρψ2+ 1α + ρψ2+ 1

α + ρψ32+ 1

2α ) ≤ Cρψ32+ 1

2α .

The derivatives are given by Xkσ = bijXkXiρXjρ + XkbijXiρXjρ and we can use Proposi-tions 3.1 and 3.3 to obtain the desired estimates.

For 1 ≤ k ≤ n we have

|Xkσ| ≤ C(ρψ

ρψ1+ 1

2α + ρψ12+ 1

2αψ

ρψ

12

+ ρψ12+ 1

2α ρψ12− 1

2α ψ1+ 12α + ρψ

12+ 1

2α ρψ12− 1

2α ψ12 )

+ C(ψ1+ 12α ψ1+ 1

2α + ψ12 ψ1+ 1

2α ψ12 + ψ

12 ψ

12 ψ

12 )

≤ C(ψ2+ 12α + ψ2+ 1

α + ψ32 ) ≤ Cψ

32 .

For n + 1 ≤ k ≤ N we find


|Xkσ| ≤ C(ρψ

32+ 1

2α

ρψ1+ 1

2α + ρψ12+ 1

2αψ

32+ 1

2α

ρψ

12

+ ρψ12+ 1

2αψ

ρψ1+ 1

2α + ρψ12+ 1

2αψ

ρψ

12

)

+ Cψ12 (ψ1+ 1

2α ψ1+ 12α + ψ1+ 1

2α ψ12 + ψ

12 ψ

12 )

≤ C(ψ

52+ 1

α + ψ2+ 12α + ψ

32

)≤ Cψ

32 .

¤

Remark 3.6. Notice that a careful examination of the second part of the above proof showsthat we also proved: ∣∣XkbijXiρ

∣∣ ≤ Cψ.

Lemma 3.7. If (H) holds then:∣∣∣Xk

(ψ

µ

)∣∣∣ ≤ Cψ12 for 1 ≤ k ≤ N.

Proof. It is enough to estimate the reciprocal µψ since

Xk

(ψ

µ

)= −ψ2

µ2Xk

(µ

ψ

)and 0 < λ ≤ µ

ψ≤ λ−1.

From Xk

( µψ

)= Xk

(σψ

), using Lemma 3.5 and Proposition 3.2 we obtain:

∣∣∣Xk

(σ

ψ

)∣∣∣ =∣∣∣Xkσ

ψ− σ

ψ2Xkψ

∣∣∣ ≤ C(ψ

12 +

ρψ32+ 1

2α

ψ2ρψ1− 1

2α

)= Cψ

12 .

The proof is complete. ¤

Lemma 3.8. If (H) holds then:∣∣∣Xk

(ρ

µ

)∣∣∣ ≤ Cψ−1− 12α for 1 ≤ k ≤ N.


Proof.

Xk

(ρ

µ

)= Xk

(ψ

µ

ρ

ψ

)= Xk

(ψ

µ

) ρ

ψ+

Xkρ

µ− ρXkψ

ψµ.

Now Lemmma 3.7 and Propositions 3.1 and 3.2 give:∣∣∣Xk

(ρ

µ

)∣∣∣ ≤ C(ψ

12 ρ

ψ+

ψ12

ψ+

ρ

ψ2

ψ

|x|) ≤ Cψ−1− 1

2α ,

recalling also that 0 < λ ≤ µψ ≤ λ−1.

¤

Lemma 3.9. If (H) holds then:

|bkjXjρ| ≤ Cρψ1+ 12α

Proof. If 1 ≤ j ≤ n we have |Xjρ| ≤ Cψ1+ 12α and bkj ≤ Cρ. If n + 1 ≤ j ≤ N we have

|Xjρ| ≤ Cψ12 and bkj ≤ Cρψ

12+ 1

2α . ¤

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[FS] B. Franchi, R. Serapioni, Pointwise estimates for a class of strongly degenerate elliptic operators:Ageometrical approach, Ann. Sc. Norm. Sup. Pisa, 14 (1987), 527-568.

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[G1] N. Garofalo, Lecture notes on unique continuation, Summer School on Elliptic Equations, Cortona,July 1991.

[G2] N. Garofalo, Unique continuation for a class of ellipic operators which degenerate on a manifold ofarbitrary codimension, J. Diff. Eq., 104-1 (1993), 117-146.

[GLa] N. Garofalo, E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principleand unique continuation, Ann. Inst. Fourier, Grenoble, 44-2 (1990), 313-356.

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[Gr2] V. Grushin, On a class of hypoelliptic pseudodifferential operators degenerate on a submanifold, Math.USSR Sbornik, 13-2 (1971), 155-186.

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[Ne] J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Masson, Paris, 1967.[RS] L. P. Rothschild, E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math.,37 (1976), 247-320.

(Nicola Garofalo) Department of Mathematics, Purdue University, West Lafayette IN 47907-1935

E-mail address: [email protected]

(Dimiter Vassilev) University of California, Riverside, Riverside, CA 92521E-mail address: [email protected]

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