SHARP VANISHING ORDER OF SOLUTIONS TO STATIONARY SCHRODINGER EQUATIONS ON CARNOT GROUPS OF ARBITRARY STEP AGNIDBANERJEE 7 1 0 Abstract. Based on a variant of the frequency function approach of Almgren([Al]), we es- 2 tablish an optimal upper bound on the vanishing order of solutions to stationary Schrödinger equationsassociated tosub-Laplacian on aCarnot group of arbitrary step G. Suchboundpro- n vides a quantitative form of strong unique continuation and can be thought of as an analogue a J of the recent results of Bakri and Zhu for thestandard Laplacian. 2 2 ] P 1. Introduction A In this paper we study quantitative uniqueness for zero-order perturbations of sub-Laplacian . h on Carnot groups. Precisely, we analyze equations of the form t a (1.1) ∆ u= Vu, m H [ where ∆H is the sub-Laplacian associated to a Carnot group G ( see (2.8) below) and the discrepancy E ( see (2.20)) at the identity e of the solution u satisfies the growth assumption 2 u v (2.21). The growth assumption (2.21) is to be thought of as the measure of certain symmetry 0 type properties of u and we will have a brief discussion on that in Section 2. 8 0 The assumptions on the potential function V are specified in (2.19) below. They represent 4 the counterpart, with respect to the non-isotropic dilations associated with certain horizontal 0 vector fields X ,...,X on G which will be described later in Section 2, of the requirements . 1 N 1 0 (1.2) V(x) M, < x,DV(x) > M, 7 | | ≤ | | ≤ 1 for theclassical Schro¨dingerequation ∆u =Vuin Rn. To putthis paperin theproperhistorical : perspective we recall that for this operator, and under the hypothesis (1.2), quantitative unique v i continuation resultsakintoourhave beenrecently obtainedin[Bk], byCarlemanestimates, and X in[Zhu],bymeansofavariantofAlmgren’sfrequencyfunctionintroducedin[Ku]. Wealsorefer r a to [BG1]wherea similar result has beenestablished for generalized Baouendi-Grushin operators and [BG2] where sharp quantitative estimates has been established at the boundary of Dini domains for general elliptic equations with Lipschitz principal part. In these papers the authors established sharp estimates on the order of vanishing of solution to Schro¨dinger equations which generalized those in [DF1] and [DF2] for eigenvalues of the Laplacian on a compact manifold. Now in the case of Carnot groups, unlike the Euclidean case, the reader will notice that although we have an additional assumption on the discrepancy E of the solution u to (1.1) but u it is nevertheless not very restrictive in the sense that strong unique continuation property is in general not true for solutions to (1.1). This follows from some interesting work of Bahouri ([Bah]) where the author showed that unique continuation is not true for even smooth and compactly supported perturbations of the sub-Laplacian. Therefore, one cannot expect any quantitative estimates either without further assumptions. On the other hand, it turns out that with precisely this growth assumption on E that we have in (2.21) , strong uniquecontinuation u property (sucp) for (1.1) has been established in [GLa] for the case when G = Hn(Heisenberg group which is a Carnot group of step 2). This has been recently generalized to Carnot groups 1 2 AGNIDBANERJEE of arbitrary step in [GR]. It is to be noted that the results in [GLa] and [GR] follow the circle of ideas in the fundamental works [GL1] and [GL2]. Therefore, given the current interest in problems of this type, i.e. quantitative form of strong unique continuation, the purpose of our work is to establish sharp quantitative uniqueness for equations (1.1) in the setup of [GR] where sucp has been established, i.e. with the growth assumption on the discrepancy term E as in (2.21). Our results should be seen as a subelliptic u generalization of the above mentioned Euclidean results in [Bk] and [Zhu]. As the reader will realize, such generalization relies on the deep link existing between the growth properties of a certain generalized Almgren frequency, the sub-elliptic structure on G and the growth assumption (2.21)onE . Itturnsoutthatintheend,theybeautifully combine. Someofthesefacts are u based on the previous work of Garofalo and Rotz ( [GR]) where among several other interesting results, the notion of Almgren’s frequency and discrepancy were introduced on a stratified nilpotent Lie group G (also known as Carnot group). In this paper , similar to [Zhu], and [BG1], we work with an appropriate weighted version of the Almgren’s Frequency which is somewhat different from the one introduced in [GR]. Having said that, we do follow [GR] closely in parts. Since we are interested in the question of sharp vanishing order estimates, it is worth emphasizing that as opposed to Theorem 7.3 in [GR], we need some kind of monotonicity properties of the generalized frequency that is introduced in Section 3 and not just its boundedness (see Theorem 3.1). Moreover, we also need to keep track of the dependence of the several constants that appear in front of the integrals on K ( where K is as in (2.19) and can be thought of as the ”sub-elliptic” Lipschitz norm of the potential V). Therefore, we have to substantially modify an argument used in the proof of Theorem 7.3 in [GR] which constitutes one of the delicate aspects of this work. Thepaperis organized as follows. InSection 2, we introducethe basic notations, gather some crucial preliminary results from [DG], [GR], [GLa] and [GV1] and state our main result. In Section 3, we introduce a monotonicity theorem for the generalized frequency and subsequently prove our main result. Acknowledgment: The author would like to thank his former PhD. advisor Prof. Nicola Garofalo for introducing him to the very interesting subject of unique continuation and whose fundamental work on this subject has been his constant inspiration. The author would also like to thank him for clarifying several results obtained in [GR]. 2. Preliminaries and Statement of main result 2.1. Preliminaries. In this section, we first gather some preliminary results as in Section 2 in [GR]. Henceforth in this paper we follow the notations adopted in [GR] with a few exceptions. We recall that a Carnot group of step h is a simply connected Lie group G whose lie algebra ð admitsastratification ð = V ... V whichishnilpotent., i.e., [V ,V ] = V forj = 1,...h 1 1 h 1 j j+1 and[V ,V ] = 0forj = 1,...h.⊕At⊕rivialexampleiswhenG = Rn andinwhichcaseð= V = R−n. j h 1 The simplest non-Abelian example of a Carnot group of step 2 is Hn, i.e. in R2n+1, we let (x,y,t) = (x ,...x ,y ,...y ,t) and the group operation is as follows 1 n 1 n (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 1 (2.1) X = ∂ +2y ∂ i xi i t Y = ∂ 2x ∂ i yi − i t and V is spanned by ∂ . We note that the following holds, 2 t [X ,Y ]= 4δ ∂ i j ij t VANISHING ORDER ETC. 3 and therefore V generates the whole lie algebra, We would like to mention that over here, we 1 identify the lie algebra ð with the left invariant vector fields. Now in a Carnot group G, by the above assumptions on the Lie algebra, we see that any basis of the horizontal layer V generates the whole ð. We will respectively denote by 1 (2.2) L (g ) = gg , R (g ) = g g g ′ ′ g ′ ′ the left and right translation by an element g G. The exponential mapping exp : ð G defin∈es an analytic diffeomorphism onto G. We recall → the Baker-Campbell-Hausdorff formula, see, section 2.15 in [V], 1 1 (2.3) exp(c )exp(c ) = exp(c +c + [c ,c ]+ [c ,[c ,c ]] [c ,[c ,c ]] +..) 1 2 1 2 1 2 1 1 2 2 1 2 2 12{ − } where the dots indicate commutators of order four and higher. Each element of the layer V is j assigned a formal degree j. Accordingly, one defines dilations on ð by the rule (2.4) ∆ c= λc +.....λhc λ 1 h The anisotropic dilations δ on G are then defined as λ (2.5) δ (g) = exp ∆ exp 1g λ λ − ◦ ◦ Throughout the paper, we will indicate by dg the bi-invariant Haar measure on G obtained by lifting via the exponential map exp the Lebesgue measure on G. Let m = dimV . One can j j check that (2.6) (d δ )(g) = λQdg λ ◦ where Q = Σh jm . Q is referred to as the homogeneous dimension of G and is in general j=1 j different from the topological dimension of G which is Σh m . j=1 j Let e ,...,e be an orthonormal basis of the first layer V of the Lie algebra. We define the 1 m 1 { } corresponding left invariant smooth vector fields by the formula (2.7) X (g) = dL (e ), i = 1,...,m i g i where dL denote the differential of L . We assume that G is endowed with a left-invariant g g Riemannian metric such that X ,....,X are orthonormal. The corresponding subLaplacian 1 m { } is defined by the formula (2.8) ∆ u= Σm X2u H i=1 i We note that by Hormander’s theorem, ∆ is hypoelliptic. We with indicate with e the identity H element of G. Let Γ(g,g ) = Γ(g ,g) be the positive fundamental solution of ∆ . It turns out that Γ is ′ ′ H − left invariant, i.e., (2.9) Γ(g,g ) = Γ˜(g 1 g ) ′ − ′ ◦ For every r > 0, let 1 (2.10) B = g G Γ(g,e) > r { ∈ | rQ 2} − It was proved by Folland [F1] that Γ˜(g) is homogeneous of order 2 Q with respect to the − non-isotrophic dilations (2.5). Therefore, if we define −1 (2.11) ρ(r) = Γ(g)Q−2 4 AGNIDBANERJEE then ρ is homogenous of degree 1. One can immediately see that B can be equivalently char- r acterized as (2.12) B = ρ(g) < r r { } We let S = ∂B . We note that since Γ is homogeneous of degree 2 Q, we have r r − (2.13) ZΓ= (2 Q)Γ − Now bystrongmaximum principle(SinceΓ(g,e) isharmonicforg = e), wehave thatΓ(g,e) > 0 6 for all g = e. Now since ZΓ =< DΓ,Z >, where DΓ is the Riemannian gradient with respect to 6 theleftinvariant metric, weconcludefrom(2.13)thatDΓnever vanishes. Therefore,by implicit function theorem, we conclude that the level sets S are smooth hypersurfaces in G. r The position d(g,g ) defined by ′ (2.14) d(g,g ) = ρ(g 1 g ) ′ − ′ ◦ defines a pseudo-distance on G. We let Z to be the smooth vector field which corresponds to the infinitesimal generator of the non-isotrophic dilations (2.5). Note that Z is characterized by the following property d 1 (2.15) u(δ g) = Zu(δ g) r r dr r Therefore if u is homogenous of degree k with respect to (2.5), i.e., u(δ g) = rku(g), then we r have that Zu= ku. In what follows, we denote by (2.16) u = Σm X uX ∇H i=1 i i the horizontal gradient of u. We also let (2.17) u2 = Σ(X u)2 H i |∇ | 2.2. Statement of the main result. Let u be a solution to (2.18) ∆ u =Vu H We will assume apriori that u,X u,X X u exist classically and are continuous in B . We will i i j 1 assume that the potential V satisfies (2.19) V K ρ2 ZV K ρ2 H H | |≤ |∇ | | |≤ |∇ | for some K > 1. As in [GR], we define the discrepancy E at e by u Zu (2.20) E =< u, ρ > ρ2 u H H H ∇ ∇ − ρ |∇ | Similar to [GR], we will also assume that f(ρ) (2.21) E ρ2 u u H | | ≤ ρ |∇ | | | wheref :(0,1) (0, )isacontinuous increasingfunctionwhichsatisfiestheDiniintegrability → ∞ condition 1 f(t) (2.22) dt < K , f K Z t 0 | |≤ 1 0 We now list a few classes of examples from [GR] in which the assumption (2.21) holds. When u is radial, i.e., if u(g) = f(ρ(g)), then it follows from a straightforward calculation (See Proposition 9.6 in [GR]) that E 0. u ≡ Also, if we specialize to the case when G = Hn and when u is polyradial, i.e. with g = (w ,....,w ,t) where w = (x ,y ), we have that u(g) = φ(w ,..., w ,t), then E 0. ( see 1 n i i i 1 n u | | | | ≡ VANISHING ORDER ETC. 5 Proposition 9.11 in [GR]). Itis however not truethatfor general groups of Heisenberg type(see Section 9.1 in [GR] for the precise definition of groups of Heisenberg type), polyradial functions have zero discrepancy. ( see Section 9 in [GR] for a counterexample) Nevertheless, given these two classes of examples, we would like to think of the growth condition (2.21) on E as the u measure of certain symmetry type properties 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 which additionally satisfies 1 0 | |≤ (2.21). Let V satisfy (2.19). Then there exists a universal a (0,1/3), and constants C ,C 1 2 ∈ depending on Q,C and K ,K in (2.22) and also u2 ρ2dg such that for all 0 < r < a, 0 0 1 B1/3 |∇H | 3 one has R (2.23) ||u||L∞(Br) ≥ C1rC2√K Remark 2.2. We note that in the elliptic case as in [BG2], [Zhu], we have that ρ2 1 and H |∇ | ≡ in such a case, the constant K in (2.19) can be taken to be C( V W1,∞ +1) for some universal || || C. We thus see that in the elliptic case, (2.23) reduces to the Euclidean result in [Bk] and [Zhu]. Therefore, Theorem 2.1 can be thought of as a subelliptic analogue of the sharp quantitative uniqueness result for the standard Laplacian. Remark 2.3. It is also worth mentioning as well that when G = Hn and E 0, then from u ≡ Proposition 9.8 in [GR], it follows that u solves the Schrodinger equation corresponding to the Baouendi-Grushin operator z 2 (2.24) ∆ u+ | | ∆ u= Vu z t 4t for which the sharp quantitative estimate (2.23) follows from Theorem 1.1 in [BG1]. The Baouendi Grushin operator was first introduced in [Ba]. We would also like to refer to [Gr1], [Gr2] where hypoelliptic results for generalized Baouendi-Grushin operators have been proved. 3. Proof of Theorem 2.1 3.1. Monotonicity of a generalized frequency. Following [Zhu] and [BG1], for α> 0 to be decided later, we let (3.1) H(r)= u2 ρ2(r2 ρ2)αdg Z |∇H | − Br We would like to mention that in the context of Almgren’s frequency, in the Euclidean case such weights first appeared in [Ku]. For notational convenience, we let ρ2 = ψ. Therefore with H |∇ | this new notation, we have that (3.2) H(r)= u2(r2 ρ2)αψ Z − Br By differentiating with respect to r, we get that (3.3) H (r)= 2αr u2(r2 ρ2)α 1ψ ′ − Z − Using the identity 1 ρ2 (3.4) (r2 ρ2)α 1 = (r2 ρ2)α+ (r2 ρ2)α 1 − − − r2 − r2 − the latter equation can be rewritten as 2α 2α (3.5) H (r) = H(r)+ u2(r2 ρ2)α 1ρ2ψ ′ − r r Z − 6 AGNIDBANERJEE Now since ρ has homogeneity 1, we have that Zρ= ρ. Hence (r2 ρ2)α 1ρ2 can be rewritten as − − 1 (3.6) (r2 ρ2)α 1ρ2 = Z(r2 ρ2)α − − −2α − Therefore we get that 2α 1 (3.7) H (r)= u2Z(r2 ρ2)αψ ′ r − r Z − Now we note that since ρ is homogeneous of degree 1, hence ρ2 = ψ is homogeneous of H |∇ | degree 0. Therefore we have that (3.8) Zψ(g) = 0 for g = e. We also note that 6 (3.9) divGZ = Q (see for instance [DG]) Note that over here, divG denotes the Riemmanian divergence on G. Now by using the Divergence theorem on G with respect to its Riemmanian structure and also by using (3.8), (3.9), we get that 2α+Q 2 (3.10) H (r)= H(r)+ uZu(r2 ρ2)αψ ′ r r Z − Over here, wecrucially usethefactthat since ρ2 hashomogeneity 0, thereforeitis bounded H |∇ | and hence the integration by parts can be justified by an approximation type argument. Now by using (2.21), we get that 2α+Q 2 (3.11) H (r) = H(r)+ uρ < u, ρ > (r2 ρ2)α+K(r) ′ r r Z ∇H ∇H − where f(r) (3.12) K(r) H(r) | | ≤ r (3.11) can be rewritten as 2α+Q 1 (3.13) H (r)= H(r)+ I(r)+K(r) ′ r (α+1)r where (3.14) I(r) = 2(α+1) u < u, ρ > (r2 ρ2)αρ= u < u, (r2 ρ2)α+1 > Z ∇H ∇H − −Z ∇H ∇H − Now we note that the following identity holds( see for instance [GV1]) (3.15) divGXi = 0 Therefore, by applying integrating by parts to (3.14) and by using the equation (2.18) and the identity (3.15) we get that, (3.16) I(r) = u2(r2 ρ2)α+1+Vu2(r2 ρ2)α+1 Z |∇H | − − We now define the generalized frequency of u as I(r) (3.17) N(r) = H(r) The central result of this section which implies (2.23) in Theorem 2.1 is the following monotonicity result of N(r). VANISHING ORDER ETC. 7 Theorem 3.1. For α = √K, we have that there exists universal C ,C depending on Q,K 1 2 0 such that (3.18) r → eC1R0r f(tt)(N(r)+C2K(r2+Z r f(tt)dt)) 0 is monotone increasing on (0,1). Proof. The proof will be divided into several steps. We first calculate I (r). By differentiating ′ the expression in (3.16) with respect to r, we get that (3.19) I (r) = 2(α+1)r u2(r2 ρ2)α+2(α+1)r Vu2(r2 ρ2)α ′ Z |∇H | − Z − This can be rewritten as (3.20) 2(α+1) 2(α+1) I (r) = u2(r2 ρ2)α+1+ u2(r2 ρ2)αρ2+2(α+1)r Vu2(r2 ρ2)α ′ r Z |∇H | − r Z |∇H | − Z − Using the fact that Zρ = ρ, the second term on the right hand side of above expression can be rewritten as 2(α+1) 1 (3.21) u2(r2 ρ2)αρ2 = u2Z(r2 ρ2)α+1 r Z |∇H | − −r Z |∇H | − At this point, we need the following Rellich type identity established in corollary 3.3 in [GV1] which can bethought of as the sub-elliptic analogue of Rellich typeidentity established in [PW]. For a C1 vector field F, we have that (3.22) 2Z Fu < ∇Hu,NH > dHn−1+Z divGF|∇Hu|2dg ∂Br Br −2Z Xiu[Xi,F]udg−2Z Fu∆Hudg Br Br = u2 dHn 1 Z |∇H | − ∂Br We now apply the identity (3.22) to the vector field F = (r2 ρ2)α+1Z. We note that the − boundary terms don’t appear due to the presence of the weight (r2 ρ2)α+1. Therefore we get, − 1 1 Q (3.23) − r Z |∇Hu|2Z(r2−ρ2)α+1 = −r Z |∇Hu|2divG(F)+ r Z |∇Hu|2(r2−ρ2)α+1 where we used the fact that divGZ = Q. Now by applying (3.22), we get that (3.24) 1 2 2 Q u2Z(r2 ρ2)α+1 = X u[X ,F]udg Fu∆ udg+ u2(r2 ρ2)α+1 −r Z |∇H | − −r Z i i −r Z H r Z |∇H | − Now we note that the following holds (3.25) [X ,Z]= X i i See for instance [DG]. Therefore by using (3.25), we have that (3.26) [X ,F]u = X (r2 ρ2)α+1Z+(r2 ρ2)α+1X = 2(α+1)ρ(r2 ρ2)αX ρZ+(r2 ρ2)α+1X i i i i i − − − − − By using (3.26) in (3.24) we get that, 1 4(α+1) (3.27) u2Z(r2 ρ2)α+1 = < u, ρ > ρZu(r2 ρ2)α − r Z |∇H | − r Z ∇H ∇H − 2 Q 2 VuZu(r2 ρ2)α+1+ − u2(r2 ρ2)α+1 − r Z − r Z |∇H | − 8 AGNIDBANERJEE Now by using the assumption (2.21) we get that (3.28) 1 4(α+1) Q 2 u2Z(r2 ρ2)α+1 = (Zu)2(r2 ρ2)αψ+ − u2(r2 ρ2)α+1 − r Z |∇H | − r Z − r Z |∇H | − 2 VuZu(r2 ρ2)α+1+K (r) − r Z − 1 where 4(α+1)f(r) (3.29) K (r) (r2 ρ2)α u Zuψ | 1 | ≤ r Z − | || | Therefore by substituting the above expression in (3.20) we get that, 2α+Q 4(α+1) (3.30) I (r) = u2(r2 ρ2)α+1+ (Zu)2(r2 ρ2)αψ ′ r Z |∇H | − r Z − 2 +2(α+1)r Vu2(r2 ρ2)α VuZu(r2 ρ2)α+1+K (r) Z − − r Z − 1 Recalling the definition of I(r), we can write the above as 2α+Q 2α+Q 4(α+1) (3.31) I (r) = I(r) Vu2(r2 ρ2)α+1 + (Zu)2(r2 ρ2)αψ ′ r − r Z − r Z − 2 +2(α+1)r Vu2(r2 ρ2)α VuZu(r2 ρ2)α+1+K (r) Z − − r Z − 1 Now by integrating by parts and using the fact that divGZ = Q, we get that 2 1 (3.32) VuZu(r2 ρ2)α+1 = Z(u2)V(r2 ρ2)α+1 − r Z − −r Z − 1 Q (3.33) = u2divG((r2 ρ2)α+1VZ)= Vu2(r2 ρ2)α+1 r Z − r Z − 1 2(α+1) + u2ZV(r2 ρ2)α+1 Vu2ρ2(r2 ρ2)α r Z − − r Z − Now we note that by (2.19) Q (3.34) Vu2(r2 ρ2)α+1 CKrH(r) |r Z − |≤ and also 1 (3.35) u2ZV(r2 ρ2)α+1 CKrH(r) |r Z − |≤ some universal C. We now write the expression 2α+Q Vu2(r2 ρ2)α+1 as r − R 2α+Q 2α+Q (3.36) Vu2(r2 ρ2)α+1 = (2α+Q)r Vu2(r2 ρ2)α Vu2(r2 ρ2)αρ2 r Z − Z − − r Z − Therefore, by using (3.32), (3.34), (3.35) and (3.36) in (3.31) and also (2.19) we get that 2α+Q 4(α+1) (3.37) I (r)= I(r)+ (Zu)2(r2 ρ2)αψ+O(1)KrH(r)+K (r) ′ r r Z − 1 Now from (2.21) it follows that (3.38) uZu(r2 ρ2)αψ = u < u, ρ > ρ(r2 ρ2)α+O(f(r)H(r)) Z − Z ∇H ∇H − VANISHING ORDER ETC. 9 Therefore from the definition of N(r) as in (3.17), from (3.13), (3.37) and also by using (3.38) we get that I (r) H (r) ′ ′ (3.39) N (r)= N(r) ′ H(r) − H(r) f(r) C N(r) C Kr 1 2 ≥− r − 4(α+1) (Zu)2(r2 ρ2)αψ 4(α+1)( (r2 ρ2)αuZuψ)2 +( − − ) R rH(r) − RrH(r)2 8(α+1)f(r)( u Zu(r2 ρ2)αψ) | || | − − RrH(r) where C ,C are universal. At this point, we need an adapted form of the argument used in 1 2 Section 7 in [GR]. Before proceeding further, we make the following remark. Remark3.2. Inthesubsequentexpressions, alltheconstants C ’s, C˜’sthatappear areuniversal i i and only depends on C ,Q and K in (2.22). 0 0 We first note that I(r) (3.40) uZu(r2 ρ2)αψ = +H (r) Z − 2(α+1) 1 where (3.41) H (r) f(r)H(r) 1 | | ≤ Now from the representation of I(r) as in (3.16) and from the assumption on V as in (2.19), we note that (3.42) I(r)+Kr2H(r) 0 ≥ Therefore, since α = √K, we get that I(r) (3.43) +√Kr2H(r) 0 2(α+1) ≥ By substituting (3.43) in (3.40), we get that (3.44) uZu(r2 ρ2)αψ+√Kr2H(r) f(r)H(r) Z − ≥ − which can be rewritten as (3.45) uZu(r2 ρ2)αψ+√Kr2H(r)+f(r)H(r) 0 Z − ≥ We now distinguish 2 cases. Either we have that (3.46) ( u2(r2 ρ2)αψ)1/2( (Zu)2(r2 ρ2)αψ)1/2 √2( uZu(r2 ρ2)αψ+8√Kr2H(r)+f(r)H(r)) Z − Z − ≤ Z − or (3.47) ( u2(r2 ρ2)αψ)1/2( (Zu)2(r2 ρ2)αψ)1/2 > √2( uZu(r2 ρ2)αψ+8√Kr2H(r)+f(r)H(r)) Z − Z − Z − If (3.46) occurs, then by applying Cauchy-Schwartz inequality to the expression 4(α+1) (Zu)2(r2 ρ2)αψ 4(α+1)( (r2 ρ2)αuZuψ)2 ( − − ) R rH(r) − RrH(r)2 10 AGNIDBANERJEE in (3.39), we see that the expression is non-negative. 8(α+1)f(r)(R u Zu(r2 ρ2)αψ) Now we estimate the term | || | − in (3.39) by Cauchy-Schwartz and the − rH(r) estimate (3.46) and thereby obtain 8(α+1)f(r)( u Zu(r2 ρ2)αψ) (3.48) | || | − | RrH(r) | 8(α+1)f(r)( uZu(r2 ρ2)αψ+8√Kr2H(r)+f(r)H(r)) − ≤ R rH(r) Now from (3.38) and the definition of I(r) we get that f(r) (3.49) N (r) C N(r) C Kr ′ 1 2 ≥ − r − f2(r) f(r) C˜K C˜ K 1 − r − r In (3.49), we crucially used the fact that α= √K K. Now since f K we therefore obtain 1 ≤ | |≤ that f(r) f(r) (3.50) N (r) C N(r) C Kr C K ′ 1 2 6 ≥ − r − − r If instead (3.47) occurs, then there are 2 sub-cases. Either (3.51) uZu(r2 ρ2)αψ 0 Z − ≥ or (3.52) uZu(r2 ρ2)αψ 0 Z − ≤ If (3.51) occurs, then by using the inequality (a+b)2 a2 when a,b 0 , we get that ≥ ≥ (3.53) ( u2(r2 ρ2)αψ)( (Zu)2(r2 ρ2)αψ) 2( uZu(r2 ρ2)αψ)2 Z − Z − ≥ Z − From (3.53), it follows that 4(α+1) (Zu)2(r2 ρ2)αψ 4(α+1)( (r2 ρ2)αuZuψ)2 (3.54) − − R rH(r) − RrH(r)2 2(α+1) (Zu)2(r2 ρ2)αψ (3.55) − ≥ R rH(r) By using the above inequality in (3.39), we get that f(r) (3.56) N (r) C N(r) C Kr ′ 1 2 ≥ − r − 2(α+1) (Zu)2(r2 ρ2)αψ 8(α+1)f(r)( u Zu(r2 ρ2)αψ) + − | || | − R rH(r) − RrH(r) Now by applying Cauchy-Schwartz inequality with ε, i.e. the inequality b2 (3.57) ab εa2 + ≤ ε 8(α+1)f(r)(R u Zu(r2 ρ2)αψ) to the term | || | − in (3.56) for small enough ε, we get that rH(r) f(r) f(r) (3.58) N (r) C N(r) C Kr C K ′ 1 2 8 ≥ − r − − r

Sharp Vanishing order of solutions to Stationary Schrodinger equations on Carnot groups of arbitrary step

Similar Sharp Vanishing order of solutions to Stationary Schrodinger equations on Carnot groups of arbitrary step