Sharp Vanishing order of solutions to Stationary Schrodinger equations on Carnot groups of arbitrary step
aa r X i v : . [ m a t h . A P ] M a y SHARP VANISHING ORDER OF SOLUTIONS TO STATIONARYSCHR ¨ODINGER EQUATIONS ON CARNOT GROUPS OF ARBITRARYSTEP
AGNID BANERJEE
Abstract.
Based on a variant of the frequency function approach of Almgren([Al]), underappropriate assumptions we establish an optimal upper bound on the vanishing order of solutionsto stationary Schr¨odinger equations associated to sub-Laplacian on Carnot groups of arbitrarystep. Such a bound provides a quantitative form of strong unique continuation and can bethought of as a subelliptic analogue of the recent results obtained by Bakri ([Bk]) and Zhu([Zhu]) for the standard Laplacian. Introduction
We say that the vanishing order of a function u is ℓ at x , if ℓ is the largest integer such that D α u = 0 for all | α | ≤ ℓ , where α is a multi-index. In the papers [DF1], [DF2], Donnelly andFefferman showed that if u is an eigenfunction with eigenvalue λ on a smooth, compact andconnected n -dimensional Riemannian manifold M , then the maximal vanishing order of u is lessthan C √ λ where C only depends on the manifold M . Using this estimate, they showed that H n − ( x : u λ ( x ) = 0) ≤ C √ λ where u λ is the eigenfunction corresponding to λ and thereforegave a complete answer to a famous conjecture of Yau ( [Yau]). We note that the zero set of u λ is referred to as the nodal set. This order of vanishing is sharp. If, in fact, we consider M = S n ⊂ R n +1 , and we take the spherical harmonic Y κ given by the restriction to S n of thefunction f ( x , ..., x n , x n +1 ) = ℜ ( x + ix ) κ , then one has ∆ S n Y κ = − λ κ Y κ , with λ κ = κ ( κ + n − Y κ at the North pole (0 , ..., ,
1) is precisely κ = C √ λ κ .In his work [Ku1] Kukavica considered the more general problem(1.1) ∆ u = V ( x ) u, where V ∈ W , ∞ , and showed that the maximal vanishing order of u is bounded above by C (1 + || V || W , ∞ ). He also conjectured that the rate of vanishing order of u is less than or equalto C (1 + || V || / L ∞ ), which agrees with the Donnelly-Fefferman result when V = − λ . EmployingCarleman estimates, Kenig in [K] showed that the rate of vanishing order of u is less than C (1 + || V || / L ∞ ), and that furthermore the exponent is sharp for complex potentials V basedon a counterexample of Meshov. (see [Me]).Recently, the rate of vanishing order of u has been shown to be less than C (1 + || V || / W , ∞ )independently by Bakri in [Bk] and Zhu in [Zhu]. Bakri’s approach is based on an extensionof the Carleman method in [DF1]. In this connection, we also quote the recent interestingpaper by R¨uland [Ru], where Carleman estimates are used to obtain related quantitative uniquecontinuation results for nonlocal Schr¨odinger operators such as ( − ∆) s/ + V . On the otherhand, Zhu’s approach is based on a variant of the frequency function approach employed byGarofalo and Lin in [GL1], [GL2]), in the context of strong unique continuation problems. Suchvariant consists in studying the growth properties of the following average of the Almgren’sheight function H ( r ) = Z B r ( x ) u ( r − | x − x | ) α dx, α > − , first introduced by Kukavica in [Ku] to study quantitative unique continuation and vortex degreeestimates for solutions of the Ginzburg-Landau equation.In [Bk] and [Zhu] it was assumed that u be a solution in B to(1.2) ∆ u = V u, with || V || W , ∞ ≤ M and || u || L ∞ ≤ C , and that furthermore sup B | u | ≥
1. Then, it was provedthat u satisfies the sharp growth estimate(1.3) || u || L ∞ ( B r ) ≥ Br C (1+ √ M ) , where B, C depend only on n and C . Such an estimate has been recently extended to stationarySchr¨odinger equations associated to generalized Baouendi Grushin operators in [BG1] and alsofor elliptic equations with Lipschitz principle part at the boundary of Dini domains in [BG2].Over here, we would like to refer to[Ba], [Gr1] and [Gr2] for a detailed account on Baouendi-Grushin operators and corresponding hypoellipticity results.Therefore given the current interest in quantitative forms of strong unique continuation andthe crucial role played by them in the past to get Hausdorff measure estimates on the nodalsets as in [DF1] and [DF2] has provided us with a natural motivation to study quantitativeuniqueness for elliptic equations on Carnot groups. More precisely, we analyze equations of theform(1.4) ∆ H u = V u, where ∆ H is the sub-Laplacian on a Carnot group G ( see (2.9) below ) and the discrepancy E u of the solution u ( see (2.22) for the definition) at the identity e satisfies the growth assumption(2.23). The growth assumption (2.23) can be thought of as the measure of a certain symmetrytype property of u and we have kept a brief discussion on this aspect in Section 2.The assumptions on the potential function V are specified in (2.19) in the next section. Theyrepresent the counterpart on G with respect to certain non-isotropic dilations of the followingEuclidean requirements(1.5) | V ( x ) | ≤ M, | < x, DV ( x ) > | ≤ M, for the classical Schr¨odinger equation ∆ u = V u in R n . Such non-isotropic dilations will bedescribed in Section 2.Now in the case of Carnot groups, unlike the Euclidean case, the reader should notice thatalthough we have an additional assumption (2.23) on the discrepancy E u of u , it is still notvery restrictive in the sense that strong unique continuation property is in general not truefor solutions to (1.4). This follows from some interesting work of Bahouri ([Bah]) where theauthor showed that unique continuation is not true for even smooth and compactly supportedperturbations of the sub-Laplacian. Therefore, one cannot expect any quantitative estimates tohold either without further assumptions. Once we introduce the appropriate notion in Section 2,the reader will also clearly see that the discrepancy E u is identically zero in the Euclidean case,i.e. when we view G = R n as a Carnot group of step 1. On the other hand, it turns out that sofar, only with this growth assumption on E u that we have in (2.23) , strong unique continuationproperty (sucp) for (1.4) is known. This follows from the interesting work of Garofalo andLanconelli ( see [GLa] ) in the case when G = H n (Heisenberg group which is a Carnot groupof step 2). Such a result has been recently generalized to Carnot groups of arbitrary step byGarofalo and Rotz in [GR]. It is to be noted that the results in [GLa] and [GR] follow the circleof ideas in the fundamental works [GL1] and [GL2].The purpose of our work is to therefore derive sharp quantitative estimates for equations(1.4) in the setup of [GR] where sucp is known so far, i.e. with the growth assumption on thediscrepancy term E u as in (2.23). Our main result Theorem 2.1 should be seen as a subelliptic ANISHING ORDER ETC. 3 generalization of the above mentioned Euclidean results in [Bk] and [Zhu]. As the reader willrealize, such a generalization relies on the deep link existing between the growth properties of acertain generalized Almgren frequency and the sub-elliptic structure of G . It turns out that inthe end, they beautifully combine.In this paper , similar to [Zhu], and [BG1], we work with an appropriate weighted version ofthe Almgren’s Frequency which is somewhat different from the one introduced in [GR]. Havingsaid that, we do follow [GR] closely in parts. Since we are interested in the question of sharpvanishing order estimates, it is worth emphasizing that as opposed to Theorem 7.3 in [GR], werequire some kind of monotonicity of the generalized frequency that is introduced in Section 3and not just the boundedness of the frequency (see Theorem 3.1). Moreover, in order to recoverthe sharp vanishing order in our subelliptic situation, we also need to keep track of how theseveral constants that appear in our computations depend on the subelliptic C norm of V as in(2.19) and this entails some novel work. As the reader will notice in Section 3, it turns out thatwe have to substantially modify an argument used in the proof of Theorem 7.3 in [GR]. Thisconstitutes one of the delicate aspects of our work and makes our proof quite different from thatof the Laplacian as in [Zhu] and also from that of the Baouendi Grushin operators as in [BG1].The paper is organized as follows. In Section 2, we introduce some basic notations, gatherthe relevant preliminary results from [DG], [GR], [GLa] and [GV1] and state our main result.In Section 3, we establish a monotonicity theorem for the generalized weighted Almgren typefrequency that we introduce and we then subsequently prove our main result. Acknowledgment:
The author would like to thank his former PhD. advisor Prof. NicolaGarofalo for introducing him to the very interesting subject of unique continuation and whosefundamental work on this subject has been his constant inspiration. The author would also liketo thank him for clarifying several results obtained in [GR].2.
Preliminaries and Statement of main result
Preliminaries.
In this section, we state some preliminary results that is relevant to ourwork and is similar to the one as in Section 2 in [GR]. Henceforth in this paper we follow thenotations adopted in [GR] with a few exceptions. For most of the discussion in this section, onecan find a detailed account in the book [BLU]. We recall that a Carnot group of step h is a simplyconnected Lie group G whose lie algebra g admits a stratification g = V ⊕ ... ⊕ V h which is h nilpotent., i.e., [ V , V j ] = V j +1 for j = 1 , ...h − V j , V h ] = 0 for j = 1 , ...h . A trivial exampleis when G = R n and in which case g = V = R n . The simplest non-Abelian example of a Carnotgroup of step 2 is the Heisenberg group H n , i.e. in R n +1 , we let ( x, y, t ) = ( x , ...x n , y , ...y n , t )and the group operation is as follows( x, y, t ) ◦ ( x ′ , y ′ , t ′ ) = ( x + x ′ , y + y ′ , t + t ′ + 2( x ′ .y − x.y ′ ))In such a case, we have that V is spanned by X i = ∂ x i + 2 y i ∂ t , i = 1 , .., n (2.1) Y j = ∂ y j − x j ∂ t , j = 1 , .., n and V is spanned by ∂ t . We note that the following holds,[ X i , Y j ] = − δ ij ∂ t and therefore V generates the whole lie algebra. We would like to mention that over here, weidentify the lie algebra g with the left invariant vector fields.Now in a Carnot group G , by the above assumptions on the Lie algebra, we see that any basisof the horizontal layer V generates the whole g . We will respectively denote by(2.2) L g ( g ′ ) = gg ′ , R g ( g ′ ) = g ′ g AGNID BANERJEE the left and right translation by an element g ∈ G .The exponential mapping exp : g → G defines an analytic diffeomorphism onto G . We recallthe Baker-Campbell-Hausdorff formula, see for instance section 2.15 in [V],(2.3) exp ( c ) exp ( c ) = exp ( c + c + 12 [ c , c ] + 112 { [ c , [ c , c ]] − [ c , [ c , c ]] } + .. )where the dots indicate commutators of order four and higher. Each element of the layer V j isassigned a formal degree j . Accordingly, one defines dilations on g by the rule(2.4) ∆ λ c = λc + .....λ h c h The anisotropic dilations δ λ on G are then defined as(2.5) δ λ ( g ) = exp ◦ ∆ λ ◦ exp − g Throughout the paper, we will indicate by dg the bi-invariant Haar measure on G obtained bylifting via the exponential map exp the Lebesgue measure on g . Let m j = dimV j . One cancheck that(2.6) ( d ◦ δ λ )( g ) = λ Q dg where Q = P hj =1 jm j . Q is referred to as the homogeneous dimension of G and is in generaldifferent from the topological dimension of G which is P hj =1 m j .We let Z to be the smooth vector field which corresponds to the infinitesimal generator ofthe non-isotrophic dilations (2.5). Note that Z is characterized by the following property(2.7) ddr u ( δ r g ) = 1 r Zu ( δ r g )Therefore if u is homogeneous of degree k with respect to (2.5), i.e., u ( δ r g ) = r k u ( g ), then wehave that Zu = ku .Let { e , ..., e m } be an orthonormal basis of the first layer V of the Lie algebra. We define thecorresponding left invariant smooth vector fields by the formula(2.8) X i ( g ) = dL g ( e i ) , i = 1 , ..., m where dL g denote the differential of L g . We assume that G is endowed with a left-invariantRiemannian metric such that { X , ...., X m } are orthonormal. We note that in this case, thebi-invariant Haar measure dg agrees with the Riemannian volume element ( see for instance[BLU]). The corresponding subLaplacian is defined by the formula(2.9) ∆ H u = m X i =1 X i u We note that by Hormander’s theorem, ∆ H is hypoelliptic. We with indicate with e the identityelement of G .Let Γ( g, g ′ ) = Γ( g ′ , g ) be the positive fundamental solution of − ∆ H . It turns out that Γ isleft invariant, i.e.,(2.10) Γ( g, g ′ ) = ˜Γ( g − ◦ g ′ )For every r >
0, let(2.11) B r = { g ∈ G | Γ( g, e ) > r Q − } ANISHING ORDER ETC. 5
It was proved by Folland [F1] that ˜Γ( g ) is homogeneous of order 2 − Q with respect to thenon-isotrophic dilations (2.5). Therefore, if we define(2.12) ρ ( g ) = Γ( g ) − Q − then ρ is homogenous of degree 1. One can immediately see that B r can be equivalently char-acterized as(2.13) B r = { g : ρ ( g ) < r } We let S r = ∂B r . We note that since Γ is homogeneous of degree 2 − Q , therefore(2.14) Z Γ = (2 − Q )ΓNow by the strong maximum principle (Since Γ( g, e ) is harmonic for g = e ), we have thatΓ( g, e ) > g = e . Now since Z Γ = < D Γ , Z > , where D Γ is the Riemannian gradient withrespect to the left invariant metric, we conclude from (2.14) that D Γ never vanishes. Therefore,by implicit function theorem, we conclude that the level sets S r are smooth hypersurfaces in G .The position d ( g, g ′ ) defined by(2.15) d ( g, g ′ ) = ρ ( g − ◦ g ′ )defines a pseudo-distance on G . In what follows, we denote by(2.16) ∇ H u = m X i =1 X i uX i the horizontal gradient of u . We also let(2.17) |∇ H u | = m X i =1 ( X i u ) Statement of the main result.
In order to state our main result, we first describe ourframework. We will assume that u is a solution to(2.18) ∆ H u = V u in B . Since the regularity issues are not our main concern, we will assume apriori that u, X i u, X i X j u, Zu are in L ( B ) with respect to the Haar measure dg . Concerning the potential V , we assume that it satisfies(2.19) | V | ≤ K |∇ H ρ | | ZV | ≤ K |∇ H ρ | for some K >
1. An example of a smooth V which satisfies (2.19) is given by V ( g ) =˜ V ( g ) f ( ρ |∇ H ρ | ) where ˜ V is a smooth function defined on G and f : R → R is a smoothcompactly supported function which vanishes in a neighborhood of 0. The first condition in(2.19) is easy to see and for the second condition we note that(2.20) | ZV | ≤ | ( Z ˜ V ) f ( ρ |∇ H ρ | ) | + | ˜ V f ′ ( ρ |∇ H ρ | ) Zρ |∇ H ρ | | + | ˜ V f ′ ( ρ |∇ H ρ | ) ρZ ( |∇ H ρ | ) | Now since ρ has homogeneity 1, Zρ = ρ and since |∇ H ρ | has homogeneity 0 being the derivativeof 1 homogenous function, we have that Z ( |∇ H ρ | ) = 0 for g = e . We note that the derivative in(2.20) is only computed for g = e since f vanishes in a neighborhood of 0. Therefore we obtain(2.21) | ZV | ≤ | ( Z ˜ V ) f ( ρ |∇ H ρ | ) | + | ˜ V f ′ ( ρ |∇ H ρ | ) ρ |∇ H ρ | | From (2.21), it is easy to see that the second condition in (2.19) is satisfied for this choice of V .As in [GR], we define the discrepancy E u at e by(2.22) E u = < ∇ H u, ∇ H ρ > − Zuρ |∇ H ρ | Like in [GR], we will also assume that(2.23) | E u | ≤ f ( ρ ) ρ |∇ H ρ | | u | where f : (0 , → (0 , ∞ ) is a continuous increasing function which satisfies the Dini integrabilitycondition(2.24) Z f ( t ) t dt < K , | f | ≤ K We now list a few classes of examples from [GR] in which the assumption (2.23) holds.In the case when G = R n , we have that Z = Σ ni =1 x i ∂ i and ∇ H ρ = x | x | . Therefore, we clearlysee that E u ≡ G , when u is radial, i.e., if u ( g ) = f ( ρ ( g )), then it follows from astraightforward calculation (See Proposition 9.6 in [GR]) that E u ≡ G = H n and assume u to be polyradial, i.e. with g = ( w , ...., w n , t ) where w i = ( x i , y i ), we have that u ( g ) = φ ( | w | , ..., | w n | , t ), then E u ≡ E u as the measureof a certain symmetry type property of u .We now state our main result. Theorem 2.1.
Let u be a solution to (2.18) in B such that | u | ≤ C and the discrepancy E u of u at e satisfies (2.23) . Let V satisfy (2.19) . Then there exists a universal a ∈ (0 , / , andconstants C , C depending on Q, C and K , K in (2.24) and also R B / u |∇ H ρ | dg such thatfor all < r < a , one has (2.25) || u || L ∞ ( B r ) ≥ C r C √ K Remark 2.2.
We note that in the elliptic case as in [BG2] , [Zhu] , we have that |∇ H ρ | ≡ andin such a case, the constant K in (2.19) can be taken to be C ( || V || W , ∞ + 1) for some universal C . We thus see that in the elliptic case, (2.25) reduces to the Euclidean result as in [Bk] and [Zhu] since E u ≡ in the Euclidean case. Therefore our estimate (2.25) gives sharp bounds onthe vanishing order of u at the identity e in terms of a certain ”subelliptic” C norm of V withbounds as in (2.19) . Hence, our result can be thought of as a subelliptic analogue of the sharpquantitative uniqueness result in the Euclidean case. Remark 2.3.
It is worth mentioning the case when G = H n and E u ≡ . Now from thedefinition of the sublaplacian and the explicit representation of the horizontal vector fields for H n as in (2.1) , we have that u solves the following equation (2.26) ∆ z u + | z | ∂ tt u + ∂ t θ u = V u where θ u = Σ( x j ∂ y j u − y j ∂ x j u ) and E u = ρ tθ u ( see for instance Lemma 9.8 in [GR] ). Nowwhen E u ≡ , we get that θ u ≡ and hence u solves the stationary Schr¨odinger equationcorresponding to the Baouendi-Grushin operator (2.27) ∆ z u + | z | ∂ tt u = V u for which the sharp quantitative estimate (2.25) follows from Theorem 1.1 in [BG1] . ANISHING ORDER ETC. 7
However if we only assume that E u satisfies (2.23) , then from (2.23) and the fact that E u = ρ tθ u we can only assert that u solves (2.26) where θ u satisfies the following growth condition, (2.28) | θ u | ≤ f ( ρ ) ρ |∇ H ρ | | u | t and in this case, our result is not implied by [BG1] . Therefore our result is new even for G = H n . Remark 2.4.
It remains to be seen when the potential V only satisfies (2.29) | V | ≤ K |∇ H ρ | instead of (2.19) , then if it can be shown that the vanishing order of the solution u is boundedfrom above by CK / . This would constitute the subelliptic analogue of the result in [K] to whichwe would like to come back in a future study. Proof of Theorem 2.1
Monotonicity of a generalized frequency.
Following [Zhu] and [BG1], for α > H ( r ) = Z B r u |∇ H ρ | ( r − ρ ) α dg For notational convenience, we will let |∇ H ρ | = ψ . Therefore with this new notation, we havethat(3.2) H ( r ) = Z B r u ( r − ρ ) α ψ By differentiating with respect to r , we get that(3.3) H ′ ( r ) = 2 αr Z u ( r − ρ ) α − ψ Using the identity(3.4) ( r − ρ ) α − = 1 r ( r − ρ ) α + ρ r ( r − ρ ) α − the latter equation can be rewritten as(3.5) H ′ ( r ) = 2 αr H ( r ) + 2 αr Z u ( r − ρ ) α − ρ ψ Now by using the fact that Zρ = ρ , we see that ( r − ρ ) α − ρ can be rewritten as(3.6) ( r − ρ ) α − ρ = − α Z ( r − ρ ) α Therefore we get that(3.7) H ′ ( r ) = 2 αr − r Z u Z ( r − ρ ) α ψ Now we note that the following two identity holds(3.8) Z ( |∇ H ρ | ) = 0 , g = e and(3.9) div G Z = Q AGNID BANERJEE
For (3.9), for instance the reader can refer to [DG]. Note that over here, div G denotes theRiemmanian divergence on G . Now by using the Divergence theorem on G with respect to itsRiemmanian structure and also by using (3.8), (3.9), we get that(3.10) H ′ ( r ) = 2 α + Qr H ( r ) + 2 r Z uZu ( r − ρ ) α ψ Over here, we crucially use the fact that since |∇ H ρ | has homogeneity 0, therefore it is boundedand hence the integration by parts can be justified by an approximation type argument. Nowby using (2.23), we get that(3.11) H ′ ( r ) = 2 α + Qr H ( r ) + 2 r Z uρ < ∇ H u, ∇ H ρ > ( r − ρ ) α + K ( r )where(3.12) | K ( r ) | ≤ f ( r ) r H ( r )(3.11) can hence be rewritten as(3.13) H ′ ( r ) = 2 α + Qr H ( r ) + 1( α + 1) r I ( r ) + K ( r )where(3.14) I ( r ) = 2( α + 1) Z u < ∇ H u, ∇ H ρ > ( r − ρ ) α ρ = − Z u < ∇ H u, ∇ H ( r − ρ ) α +1 > Now we note that the following identity holds( see for instance [GV1])(3.15) div G X i = 0Therefore, by applying integrating by parts to (3.14) and by using the equation (2.18) andthe identity (3.15) we get that,(3.16) I ( r ) = Z |∇ H u | ( r − ρ ) α +1 + V u ( r − ρ ) α +1 We now define the generalized frequency of u as(3.17) N ( r ) = I ( r ) H ( r )The central result of this section which implies our main estimate (2.25) in Theorem 2.1 is thefollowing monotonicity result of N ( r ). Theorem 3.1.
For α = √ K , we have that there exists universal C depending on Q, K , K such that (3.18) r → e C R r f ( t ) t ( N ( r ) + CK ( r + Z r f ( t ) t dt )) is monotone increasing on (0 , .Proof. The proof will be divided into several steps. We first calculate I ′ ( r ). By differentiatingthe expression in (3.16) with respect to r , we get that(3.19) I ′ ( r ) = 2( α + 1) r Z |∇ H u | ( r − ρ ) α + 2( α + 1) r Z V u ( r − ρ ) α This can be rewritten as(3.20) I ′ ( r ) = 2( α + 1) r Z |∇ H u | ( r − ρ ) α +1 + 2( α + 1) r Z |∇ H u | ( r − ρ ) α ρ +2( α +1) r Z V u ( r − ρ ) α ANISHING ORDER ETC. 9
Using the fact that Zρ = ρ , the second term on the right hand side of above expression can berewritten as(3.21) 2( α + 1) r Z |∇ H u | ( r − ρ ) α ρ = − r Z |∇ H u | Z ( r − ρ ) α +1 At this point, we need the following Rellich type identity which corresponds to Corollary 3.3 in[GV1]. This can can be thought of as the sub-elliptic analogue of Rellich type identity establishedin [PW]. For a C vector field F , we have that2 Z ∂B r F u < ∇ H u, N H > dH n − + Z B r div G F |∇ H u | dg (3.22) − Z B r X i u [ X i , F ] udg − Z B r F u ∆ H udg = Z ∂B r |∇ H u | < F, ν > dH n − We now apply the identity (3.22) to the vector field F = ( r − ρ ) α +1 Z . We note that theboundary terms don’t appear due to the presence of the weight ( r − ρ ) α +1 . Therefore we get,(3.23) − r Z |∇ H u | Z ( r − ρ ) α +1 = − r Z |∇ H u | div G ( F ) + Qr Z |∇ H u | ( r − ρ ) α +1 where we used the fact that div G Z = Q . Now by applying (3.22), we get that(3.24) − r Z |∇ H u | Z ( r − ρ ) α +1 = − r Z X i u [ X i , F ] udg − r Z F u ∆ H udg + Qr Z |∇ H u | ( r − ρ ) α +1 At this point, we note that the following identity holds ( See for instance [DG])(3.25) [ X i , Z ] = X i Therefore by using (3.25), we have(3.26) [ X i , F ] u = X i ( r − ρ ) α +1 Z +( r − ρ ) α +1 X i = − α +1) ρ ( r − ρ ) α X i ρZ +( r − ρ ) α +1 X i By using (3.26) in (3.24) we get that, − r Z |∇ H u | Z ( r − ρ ) α +1 = 4( α + 1) r Z < ∇ H u, ∇ H ρ > ρZu ( r − ρ ) α (3.27) − r Z V uZu ( r − ρ ) α +1 + Q − r Z |∇ H u | ( r − ρ ) α +1 Now by using the growth assumption (2.23) on the discrepancy E u we get that − r Z |∇ H u | Z ( r − ρ ) α +1 = 4( α + 1) r Z ( Zu ) ( r − ρ ) α ψ + Q − r Z |∇ H u | ( r − ρ ) α +1 (3.28) − r Z V uZu ( r − ρ ) α +1 + K ( r )where(3.29) | K ( r ) | ≤ α + 1) f ( r ) r Z ( r − ρ ) α | u || Zu | ψ Therefore by substituting the above expression in (3.20) we get that, I ′ ( r ) = 2 α + Qr Z |∇ H u | ( r − ρ ) α +1 + 4( α + 1) r Z ( Zu ) ( r − ρ ) α ψ (3.30) + 2( α + 1) r Z V u ( r − ρ ) α − r Z V uZu ( r − ρ ) α +1 + K ( r )Recalling the definition of I ( r ), we can rewrite I ′ as I ′ ( r ) = 2 α + Qr I ( r ) − α + Qr Z V u ( r − ρ ) α +1 + 4( α + 1) r Z ( Zu ) ( r − ρ ) α ψ (3.31) + 2( α + 1) r Z V u ( r − ρ ) α − r Z V uZu ( r − ρ ) α +1 + K ( r )Now by integrating by parts and again by using the fact that div G Z = Q , we get that − r Z V uZu ( r − ρ ) α +1 = − r Z Z ( u ) V ( r − ρ ) α +1 (3.32) = 1 r Z u div G (( r − ρ ) α +1 V Z ) = Qr Z V u ( r − ρ ) α +1 + 1 r Z u ZV ( r − ρ ) α +1 − α + 1) r Z V u ρ ( r − ρ ) α At this point, we note from (2.19) that the following estimate holds(3.33) | Qr Z V u ( r − ρ ) α +1 | ≤ CKrH ( r )and also(3.34) | r Z u ZV ( r − ρ ) α +1 | ≤ CKrH ( r )for some universal C . We now write the expression α + Qr R V u ( r − ρ ) α +1 as(3.35) 2 α + Qr Z V u ( r − ρ ) α +1 = (2 α + Q ) r Z V u ( r − ρ ) α − α + Qr Z V u ( r − ρ ) α ρ Therefore, by using (3.32), (3.33), (3.34) and (3.35) in (3.31) and also by using (2.19) we getthat(3.36) I ′ ( r ) = 2 α + Qr I ( r ) + 4( α + 1) r Z ( Zu ) ( r − ρ ) α ψ + O (1) KrH ( r ) + K ( r )Finally from the definition of N ( r ) as in (3.17) and from (3.10) and (3.36) we get that thefollowing inequality holds N ′ ( r ) = I ′ ( r ) H ( r ) − H ′ ( r ) H ( r ) N ( r )(3.37) ≥ − C Kr + ( 4( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1)( R ( r − ρ ) α uZuψ )( R ( r − ρ ) α u < ∇ H ρ, ∇ H u > ρ ) rH ( r ) − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )where C is universal. Now by using (2.23), we get that(3.38) 4( α + 1) Z ( r − ρ ) α u < ∇ H ρ, ∇ H u > ρ = 4( α + 1) Z uZu ( r − ρ ) α ψ + K ( r ) ANISHING ORDER ETC. 11 where(3.39) | K ( r ) | ≤ α + 1) f ( r ) H ( r )Therefore by using (3.38) and (3.39) in (3.37) we get that N ′ ( r ) = I ′ ( r ) H ( r ) − H ′ ( r ) H ( r ) N ( r )(3.40) ≥ − C Kr + ( 4( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1)( R ( r − ρ ) α uZuψ ) rH ( r ) − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )At this point, we need a modified form of an argument used in the proof of Theorem 7.3 in[GR]. Before proceeding further, we make the following remark. Remark 3.2.
In the subsequent expressions, all the constants C i ’s, ˜ C i ’s that will appear are alluniversal and only depends on C , Q and K , K as in (2.24) . Note that from the definition of E u and the growth condition (2.22) that the following holds(3.41) Z uZu ( r − ρ ) α ψ = I ( r )2( α + 1) + H ( r )where(3.42) | H ( r ) | ≤ f ( r ) H ( r )Now from the expression of I ( r ) as in (3.16) and also from the assumption on V as in (2.19),we have that(3.43) I ( r ) + Kr H ( r ) ≥ α = √ K , by taking into account (3.43) we get that(3.44) I ( r )2( α + 1) + √ Kr H ( r ) ≥ Z uZu ( r − ρ ) α ψ − H ( r ) + √ Kr H ( r ) ≥ Z uZu ( r − ρ ) α ψ + √ Kr H ( r ) + f ( r ) H ( r ) ≥ Case 1 :(3.47)( Z u ( r − ρ ) α ψ ) / ( Z ( Zu ) ( r − ρ ) α ψ ) / ≤ √ Z uZu ( r − ρ ) α ψ +8 √ Kr H ( r )+ f ( r ) H ( r ))or Case 2 :(3.48)( Z u ( r − ρ ) α ψ ) / ( Z ( Zu ) ( r − ρ ) α ψ ) / > √ Z uZu ( r − ρ ) α ψ +8 √ Kr H ( r )+ f ( r ) H ( r )) If Case α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1)( R ( r − ρ ) α uZuψ ) rH ( r ) )in (3.40), we see that the above expression is non-negative.Now we estimate the term − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )in (3.40) by using Cauchy-Schwartz and also by using the estimate (3.47) and consequentlyobtain | α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r ) | (3.49) ≤ α + 1) f ( r )( R uZu ( r − ρ ) α ψ + 8 √ Kr H ( r ) + f ( r ) H ( r )) rH ( r )Now from (3.41) we get that(3.50) | α + 1) f ( r ) R uZu ( r − ρ ) α ψrH ( r ) | ≤ f ( r ) r N ( r ) + ˜ CK f ( r ) r Therefore by using (3.50) in (3.49) we have that the following holds N ′ ( r ) ≥ − C f ( r ) r N ( r ) − C Kr (3.51) − ˜ C K f ( r ) r − ˜ C K f ( r ) r In (3.51), we crucially used the fact that α = √ K ≤ K . Now since | f | ≤ K we obtain from(3.51) that the following holds(3.52) N ′ ( r ) ≥ − C f ( r ) r N ( r ) − C Kr − C K f ( r ) r If instead
Case subcase 1 (3.53) Z uZu ( r − ρ ) α ψ ≥ subcase 2 (3.54) Z uZu ( r − ρ ) α ψ ≤ subcase a + b ) ≥ a when a, b ≥ Z u ( r − ρ ) α ψ )( Z ( Zu ) ( r − ρ ) α ψ ) ≥ Z uZu ( r − ρ ) α ψ ) From (3.55), it follows that4( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1)( R ( r − ρ ) α uZuψ ) rH ( r ) (3.56) ≥ α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r )(3.57) ANISHING ORDER ETC. 13
By using the above inequality in (3.40), we get that N ′ ( r ) ≥ − C Kr (3.58) + 2( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )Now by applying Cauchy-Schwartz inequality with ε , i.e. the inequality(3.59) 2 ab ≤ εa + b ε to the term 8( α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )in (3.58) for small enough ε , we get that N ′ ( r ) ≥ − C Kr − C K f ( r ) r (3.60)If instead subcase
2( i.e. (3.54) ) occurs, then we first note that (3.48) trivially implies that(3.61)( Z u ( r − ρ ) α ψ ) / ( Z ( Zu ) ( r − ρ ) α ψ ) / > √ Z uZu ( r − ρ ) α ψ + √ Kr H ( r ) + f ( r ) H ( r ))Now by squaring the above inequality in (3.61)( where we taking into account that the right handside in the above inequality is non-negative due to (3.46)) and then by using ( a + b ) ≥ a + 2 ab for b ≥ a = Z uZu ( r − ρ ) α ψ (3.62) b = √ Kr H ( r ) + f ( r ) H ( r )we get that H ( r )( Z ( Zu ) ( r − ρ ) α ψ ) ≥ Z uZu ( r − ρ ) α ψ ) (3.63) + 4 √ Kr ( Z uZu ( r − ρ ) α ψ ) H ( r )+ 4 f ( r )( Z uZu ( r − ρ ) α ψ ) H ( r )Now we note that (3.46) and (3.54) together imply that(3.64) − √ Kr H ( r ) − f ( r ) H ( r ) ≤ Z uZu ( r − ρ ) α ψ ≤ N ′ ( r ) ≥ − C Kr (3.65) + 2( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − K / r − C K f ( r ) r − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )where we used the fact that | f | ≤ K and α = √ K . In order to get to (3.65), we also used thefact that ( α + 1) ≤ α since α > Now if we consider the term 16 K / r in (3.65), we note that it appears with a negative signon the right hand side and the exponent of K in that term is which is more than 1. Thiswould not let us conclude the desired monotonicity result in (3.18). Therefore, we have to getrid of this term in the final expression of N ′ . In order to do so, we first note that since (3.46)holds, therefore we get that the following inequality holds(3.66) Z uZu ( r − ρ ) α ψ + 8 √ Kr H ( r ) + f ( r ) H ( r ) ≥ √ Kr H ( r )Now because we are in Case
2, (3.48) and (3.66) together imply that(3.67) ( Z u ( r − ρ ) α ψ ) / ( Z ( Zu ) ( r − ρ ) α ψ ) / ≥ √ √ Kr H ( r )By squaring the above inequality and by cancelling off H ( r ) from both sides, we get that(3.68) ( Z ( Zu ) ( r − ρ ) α ψ ) ≥ Kr H ( r )By dividing both sides by rH ( r ) we get(3.69) ( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) ≥ K / r Therefore by writing2( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) = ( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) + ( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r )and by using (3.69) in (3.65), we get that N ′ ( r ) ≥ − C Kr − C K f ( r ) r + 94 K / r (3.70) + ( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − K / r − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )Therefore we see that the estimate (3.69) allows us to get rid of the undesirable term 16 K / r and now from (3.70) one can easily infer that the following inequality holds N ′ ( r ) ≥ − C Kr − C K f ( r ) r (3.71) + ( α + 1) R ( Zu ) ( r − ρ ) α ψrH ( r ) − α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )Again by applying Cauchy-Schwartz inequality with ε to the term8( α + 1) f ( r )( R | u || Zu | ( r − ρ ) α ψ ) rH ( r )with an appropriate choice of ε , we get that the following estimate holds(3.72) N ′ ( r ) ≥ − C Kr − C K f ( r ) r ANISHING ORDER ETC. 15
Therefore in conclusion, we have that in all the cases, either the estimate (3.52), (3.60) or(3.72) holds. Each of these estimates implies that for some universal constant ˜ C , the followinginequality holds(3.73) N ′ ( r ) ≥ − ˜ C ( f ( r ) r N ( r ) + Kr + K f ( r ) r )(3.18) now follows from (3.73) in a standard way. (cid:3) Proof of estimate (2.25) in Theorem 2.1.
We note that although (3.19) in the mono-tonicity Theorem 3.1 is different from its counterpart Theorem 3.1 in [BG1], nevertheless it stillimplies that the following inequality holds(3.74) N ( r ) ≤ ˜ C ( N ( s ) + ˜ C K ) , for 0 < r < s < . Using (3.74), we can argue in the same way as in Section 4 in [BG1] to conclude that our desiredestimate (2.25) in Theorem 2.1 holds.
References [Al] F. Almgren,
Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integralcurrents. , Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), pp. 16,North-Holland, Amsterdam-New York, 1979.[Ba] M. Baouendi,
Sur une classe d’oprateurs elliptiques degnrs , Bull. Soc. Math. France (1967), 45-87.[Bah] H. Bahouri, Non prolongement unique des solutions d’oprateurs ”somme de carrs” (French) [Failure ofunique continuation for ”sum of squares” operators] , Ann. Inst. Fourier (Grenoble) (1986) 137-155.[Bk] L. Bakri, Quantitative uniqueness for Schr¨odinger operator , Indiana Univ. Math. J., (2012), no. 4, 1565-1580.[BG1] Agnid Banerjee & Nicola Garofalo, Quantitative Uniqueness for zero-order perturbations of general-ized baouendi-grushin operators , arXiv:1604.06000, to appear in Rendiconti dell’Istituto di Matematicadell’Universit di Trieste[BG2] Agnid Banerjee & Nicola Garofalo,
Quantitative uniqueness for elliptic equations at the boundary of C ,Dini domains , J. Differential Equations (2016), no. 12, 6718-6757. arXiv:1605.02363[BLU] A. Bonfiglioli, E. Lanconelli & Uguzzoni, Stratified Lie groups and potential theory for their sub-Laplacians ,Springer Monographs in Mathematics. Springer, Berlin, 2007. xxvi+800 pp. ISBN: 978-3-540-71896-3; 3-540-71896-6[DF1] H. Donnelly & C. Fefferman,
Nodal sets of eigenfunctions on Riemannian manifolds , Invent. Math, (1988), 161-183.[DF2] H. Donnelly & C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary
Analysis,Et Cetera, Academic Press, Boston, MA, 1(990,) 251-262.[DG] D. Danielli & N. Garofalo,
Geometric properties of solutions to subelliptic equations in nilpotent Lie groups,Reaction Diffusion systems (1985), 89-105, Lecture notes in Pure and Appl. Math, 194, Dekker, New York,1998.[GLa] N. Garofalo & E. Lanconelli,
Frequency functions on the Heisenberg group, the uncertainty principle andunique continuation , Ann. Inst. Fourier(Grenoble), -(1990), 313-356.[GL1] N. Garofalo & F. Lin, Monotonicity properties of variational integrals, A p weights and unique continuation ,Indiana Univ. Math. J. (1986), 245-268.[GL2] , Unique continuation for elliptic operators: a geometric-variational approach , Comm. Pure Appl.Math. (1987), 347-366.[GR] N. Garofalo & Kevin Rotz, Properties of a frequency of Almgren type for harmonic functions in Carnotgroups , Calc. Var. Partial Differential Equations (2015), no. 2, 2197-2238.[GV] N. Garofalo & D. Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin opera-tors. , Comm. Partial Differential Equations (2007), no. 4-6, 643-663.[GV1] N. Garofalo & D. Vassilev, Regularity near the characteristic set in the non-linear Dirichlet problem andconformal geometry of sub-Laplacians on Carnot groups , Math. Ann. (2000), 453-516.[Gr1] V. V. Gruˇ s in, A certain class of hypoelliptic operators , (Russian) Mat. Sb. (N.S.) (125) (1970), 456-473.[Gr2] , A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold , (Rus-sian) Mat. Sb. (N.S.) (126) (1971), 163-195.[F1] G. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups , Ark. Mat. (1975), 161-207. [K] C. Kenig, Some recent applications of unique continuation , Recent Developments in Nonlinear Partial Dif-ferential Equations, Contemporary Mathematics, (2007)[Ku1] I. Kukavica,
Quantitative uniqueness for second order elliptic operators , Duke Math. J., (1998), 225-240.[Ku] I. Kukavica, Quantitative, uniqueness, and vortex degree estimates for solutions of the Ginzburg-Landauequation , Electron. J. Differential Equations , No. 61, 15 pp. (electronic).[Me] V. Meshov,
On the possible rate of decrease at infinity of the solutions of second-order partial differentialequations , Math. USSR-Sb. 72
2, 343-361.[N] J. Necas,
Direct methods in the theory of elliptic equations , Translated from the 1967 French original byGerard Tronel and Alois Kufner. Editorial coordination and preface by S´arka Necasov´a and a contributionby Christian G. Simader. Springer Monographs in Mathematics. Springer, Heidelberg, 2012. xvi+372 pp.[PW] L. E. Payne & H. F. Weinberger,
New bounds for solutions of second order elliptic partial differentialequations , Pacific J. Math. (1958), 551-573.[Ru] A. R¨uland, On quantitative unique continuation properties of fractional Schr¨odinger equations: Doubling,vanishing order and nodal domain estimates , (accepted Trans. AMS), arXiv:1407.0817[V] V. S. Varadarajan,
Lie groups, Lie Algebras, and their representations. Springer, New York (1974)[Yau] S. T. Yau,
Seminar on differential geometry , , Princeton University Press, 1982.[Zhu] J. Zhu, Quantitative uniqueness for elliptic equations , Amer. J. Math. (2016), 733-762.
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