We extend our recent results on propagation of semiclassical resolvent
estimates through trapped sets when a priori polynomial resolvent bounds hold.
Previously we obtained non-trapping estimates in trapping situations when the
resolvent was sandwiched between cutoffs microlocally supported away from the
trapping, a microlocal version of a result of Burq and Cardoso-Vodev. We now
allow one of the two cutoffs to be supported at the trapped set, giving
estimates which are intermediate between the trapping and non-trapping ones.Comment: 5 page

Datchev, Kiril

Vasy, András

English

arXiv

Kiril Datchev

András Vasy

Crossref

Annales de l’institut Fourier

Semiclassical resolvent estimates at trapped sets

Annales de l’institut Fourier (AIF)

ANNALESDEL’INSTITUTFOURIERANNALESDEL’INSTITUT FOURIERKiril DATCHEV & András VASYSemiclassical resolvent estimates at trapped setsTome 62, no 6 (2012), p. 2379-2384.<http://aif.cedram.org/item?id=AIF_2012__62_6_2379_0>© Association des Annales de l’institut Fourier, 2012, tous droitsréservés.L’accès aux articles de la revue « Annales de l’institut Fourier »(http://aif.cedram.org/), implique l’accord avec les conditionsgénérales d’utilisation (http://aif.cedram.org/legal/). Toute re-production en tout ou partie de cet article sous quelque forme quece soit pour tout usage autre que l’utilisation à fin strictement per-sonnelle du copiste est constitutive d’une infraction pénale. Toutecopie ou impression de ce fichier doit contenir la présente mentionde copyright.cedramArticle mis en ligne dans le cadre duCentre de diffusion des revues académiques de mathématiqueshttp://www.cedram.org/Ann. Inst. Fourier, Grenoble62, 6 (2012) 2379-2384SEMICLASSICAL RESOLVENT ESTIMATESAT TRAPPED SETSby Kiril DATCHEV & András VASY (*)Abstract. — We extend our recent results on propagation of semiclassicalresolvent estimates through trapped sets when a priori polynomial resolvent boundshold. Previously we obtained non-trapping estimates in trapping situations whenthe resolvent was sandwiched between cutoffs χ microlocally supported away fromthe trapping: ‖χRh(E + i0)χ‖ = O(h−1), a microlocal version of a result of Burqand Cardoso-Vodev. We now allow one of the two cutoffs, χ̃, to be supported atthe trapped set, giving ‖χRh(E+ i0)χ̃‖ = O(√a(h)h−1) when the a priori boundis ‖χ̃Rh(E + i0)χ̃‖ = O(a(h)h−1).Résumé. — Nous étendons nos résultats récents sur la propagation d’esti-mations de résolvantes semi-classiques à travers des ensembles captifs sous desbornes a priori de type polynomial. Précédemment, nous obtenions des estima-tions non-captives dans des situations captives quand la résolvante est contrôléepar au dessus et en dessous par des fonctions cutoff χ dont le support micro-local est situé loin de l’ensemble captif : ‖χRh(E + i0)χ‖ = O(h−1) (versionmicrolocale d’un résultat de Burq et Cardoso-Vodev). Nous considérons mainte-nant le cas où l’une des deux fonctions cutoff, χ̃, est à support dans l’ensemblecaptif, obtenant ‖χRh(E + i0)χ̃‖ = O(√a(h)h−1) lorsque la borne a priori est‖χ̃Rh(E + i0)χ̃‖ = O(a(h)h−1).This short article is an addendum to the previous paper by K. Datchevand A. Vasy.Let (X, g) be a Riemannian manifold which is asymptotically conic orasymptotically hyperbolic in the sense of [6], let V ∈ C∞0 (X) be real valued,let P = h2∆g + V (x), where ∆g > 0, and fix E > 0.Keywords: Resolvent estimates, trapping, propagation of singularities.Math. classification: 58J47, 35L05.(*) The first author is partially supported by a National Science Foundation postdoc-toral fellowship, and the second author is partially supported by the National ScienceFoundation under grants DMS-0801226 and DMS-1068742.2380 Kiril DATCHEV & András VASYTheorem 1. — [6, Theorem 1.2] Suppose that for any χ0 ∈ C∞0 (X)there exist C0, k, h0 > 0 such that for any ε > 0, h ∈ (0, h0] we have‖χ0(h2∆g + V − E − iε)−1χ0‖L2(X)→L2(X) 6 C0h−k. (1)Let KE ⊂ T ∗X be the set of trapped bicharacteristics at energy E, andsuppose that b ∈ C∞0 (T ∗X) is identically 1 near KE . Then there existC1, h1 > 0 such that for any ε > 0, h ∈ (0, h1] we have the followingnontrapping estimate:‖〈r〉−1/2−δ(1−Op(b))(h2∆g + V − E − iε)−1(1−Op(b))〈r〉−1/2−δ‖L2(X)→L2(X)6 C1h−1. (2)Here by bicharacteristics at energy E we mean integral curves in p−1(E)of the Hamiltonian vector field Hp of the Hamiltonian p = |ξ|2 +V (x), andthe trapped ones are those which remain in a compact set for all time. Weuse the notation r = r(z) = dg(z, z0), where dg is the distance function onX induced by g and z0 ∈ X is fixed but arbitrary.If KE = ∅ then (1) holds with k = 1. If KE 6= ∅ but the trapping issufficiently ‘mild’, then (1) holds for some k > 1: see [6] for details andexamples. The point is that the losses in (1) due to trapping are removedwhen the resolvent is cutoff away from KE . Theorem 1 is a more preciseand microlocal version of an earlier result of Burq [1] and Cardoso andVodev [3], but the assumption (1) is not needed in [1, 3]. See [6] for addi-tional background and references for semiclassical resolvent estimates andtrapping.In this paper we prove that an improvement over the a priori estimate(1) holds even when one of the factors of (1−Op(b)) is removed:Theorem 2. — Suppose that there exist k > 0 and a(h) 6 h−k suchthat for any χ0 ∈ C∞0 (X) there exists h0 > 0 such that for any ε > 0,h ∈ (0, h0] we have‖χ0(h2∆g + V − E − iε)−1χ0‖L2(X)→L2(X) 6 a(h)/h. (3)Suppose that b ∈ C∞0 (T ∗X) is identically 1 near KE . Then there existC1, h1 > 0 such that for any ε > 0, h ∈ (0, h1],‖〈r〉−1/2−δ(1−Op(b))(h2∆g + V − E − iε)−1〈r〉−1/2−δ‖L2(X)→L2(X)6 C1√a(h)/h. (4)Note that by taking adjoints, analogous estimates follow if 1 − Op(b) isplaced to the other side of (h2∆g + V − E − iε)−1.ANNALES DE L’INSTITUT FOURIERSEMICLASSICAL RESOLVENT ESTIMATES AT TRAPPED SETS 2381Such results were proved by Burq and Zworski [2, Theorem A] and Chris-tianson [4, (1.6)] when KE consists of a single hyperbolic orbit. Theorem 2implies an optimal semiclassical resolvent estimate for the example opera-tor of [6, §5.3]: it improves [6, (5.5)] to‖χ0(P − λ)−1χ0‖ 6 C log(1/h)/h.Further, this improved estimate can be used to extend polynomial resolventestimates from complex absorbing potentials to analogous estimates fordamped wave equations; this is a result of Christianson, Schenk, Wunschand the second author [5].Theorems 1 and 2 follow from microlocal propagation estimates in aneighborhood of KE , or more generally in a neighborhood of a suitablecompact invariant subset of a bicharacteristic flow.To state the general results, suppose X is a manifold, P ∈ Ψm,0(X) a selfadjoint, order m > 0, semiclassical pseudodifferential operator on X, withprincipal symbol p. For I ⊂ R compact and fixed, denote the characteristicset by Σ = p−1(I), and suppose that the projection to the base, π : Σ→ X,is proper (it is sufficient, for example, to have p classically elliptic). Supposethat Γ b T ∗X is invariant under the bicharacteristic flow in Σ. Define theforward, resp. backward flowout Γ+, resp. Γ−, of Γ as the set of pointsρ ∈ Σ, from which the backward, resp. forward bicharacteristic segmentstend to Γ, i.e. for any neighborhood O of Γ there exists T > 0 such that−t > T , resp. t > T , implies γ(t) ∈ O, where γ is the bicharacteristic withγ(0) = ρ. Here we think of Γ as the trapped set or as part of the trappedset, hence points in Γ−, resp. Γ+ are backward, resp. forward, trapped.Suppose V , W are neighborhoods of Γ with V ⊂W , W compact. Supposealso thatIf ρ ∈W \ Γ+, resp. ρ ∈W \ Γ−,then the backward, resp. forward bicharacteristic from ρ intersects W \V .(5)This means that all bicharacterstics in V which stay in V for all time tendto Γ.The main result of [6], from which the other results in the paper follow,is the following:Theorem 3. — [6, Theorem 1.3] Suppose that ‖u‖H−Nh6 h−N for someN ∈ N and (P − λ)u = f , Reλ ∈ I and Imλ > −O(h∞). Suppose f isO(1) on W , WFh(f) ∩ V = ∅, and u is O(h−1) on W ∩ Γ− \ V . Then u isO(h−1) on W ∩ Γ+ \ Γ.TOME 62 (2012), FASCICULE 62382 Kiril DATCHEV & András VASYHere we say that u is O(a(h)) at ρ ∈ T ∗X if there exists B ∈ Ψ0,0(X)elliptic at ρ with ‖Bu‖L2 = O(a(h)). We say u is O(a(h)) on a set E ⊂ T ∗Xif it is O(a(h))) at each ρ ∈ E.Note that there is no conclusion on u at Γ; typically it will be merelyO(h−N ) there. However, to obtain O(h−1) bounds for u on Γ+ we onlyneeded to assume O(h−1) bounds for u on Γ− and nowhere else. Note alsothat by the propagation of singularities, if u is O(h−1) at one point on anybicharacteristic, then it is such on the whole forward bicharacteristic. If| Imλ| = O(h∞) then the same is true for backward bicharacteristics.In this paper we show that a (lesser) improvement on the a priori boundholds even when f is not assumed to vanish microlocally near Γ:Theorem 4. — Suppose that ‖u‖H−Nh6 h−N for some N ∈ N and(P −λ)u = f , Reλ ∈ I and Imλ > −O(h∞). Suppose f is O(1) on W , u isO(a(h)h−1) onW , and u is O(h−1) onW ∩Γ−\V . Then u is O(√a(h)h−1)on W ∩ Γ+ \ Γ.In [6] Theorem 1 is deduced from Theorem 3. Theorem 2 follows fromTheorem 4 by the same argument.Proof of Theorem 4. — The argument is a simple modification of theargument of [6, End of Section 4, Proof of Theorem 1.3]; we follow thenotation of this proof. Recall first from [6, Lemma 4.1] that if U− is aneighborhood of (Γ− \ Γ) ∩ (W \ V ) then there is a neighborhood U ⊂ Vof Γ such that if α ∈ U \ Γ+ then the backward bicharacteristic from αenters U−. Thus, if one assumes that u is O(h−1) on Γ− and f is O(1) onV , it follows that that u is O(h−1) on U \Γ+, provided U− is chosen smallenough that u is O(h−1) on U−. Note also that, because U ⊂ V , f is O(1)on U . We will show that u is O(√a(h)h−1) on U ∩ Γ+ \ Γ: the conclusionon the larger set W ∩ Γ+ \ Γ follows by propagation of singularities.Next, [6, Lemma 4.3] states that if U1 and U0 are open sets with Γ ⊂U1 b U0 b U then there exists a nonnegative function q ∈ C∞0 (U) suchthatq = 1 near Γ, Hpq 6 0 near Γ+, Hpq < 0 on ΓU0+ \ U1.Moreover, we can take q such that both √q and√−Hpq are smoothnear Γ+.Remark. — The last paragraph in the proof of [6, Lemma 4.3] shouldbe replaced by the following: To make√−Hpq̃ smooth, let ψ(s) = 0 fors 6 0, ψ(s) = e−1/s for s > 0, and assume as we may that Uρ ∩ Sρ isa ball with respect to a Euclidean metric (in local coordinates near ρ) ofANNALES DE L’INSTITUT FOURIERSEMICLASSICAL RESOLVENT ESTIMATES AT TRAPPED SETS 2383radius rρ > 0 around ρ. We then choose ϕρ to behave like ψ(r′ρ2 − |.|2)with r′ρ < rρ for |.| close to r′ρ, bounded away from 0 for smaller values of|.|, and choose −χ′ρ to vanish like ψ at the boundary of its support. Thatsums of products of such functions have smooth square roots follows from[7, Lemma 24.4.8].The proof of Theorem 4 proceeds by induction: we show that if u is O(hk)on a sufficiently large compact subset of U ∩ Γ+ \ Γ, then u is O(hk+1/2)on ΓU0+ \ U1, provided√a(h)h−1 6 Chk+1/2.Now let U− be an open neighborhood of Γ+∩ supp q which is sufficientlysmall that Hpq 6 0 on U− and that√−Hpq is smooth on U−. Let U+ bean open neighborhood of supp q \U− whose closure is disjoint from Γ+ andfrom T ∗X \ U . Define φ± ∈ C∞(U+ ∪ U−) with suppφ± ⊂ U± and withφ2+ + φ2− = 1 near supp q.Putbdef= φ−√−Hpq2, edef= φ2+Hpq2.Let Q,B,E ∈ Ψ−∞,0(X) have principal symbols q, b, e, and microsupportssupp q, supp b, supp e, so thatih[P,Q∗Q] = −B∗B + E + hF,with F ∈ Ψ−∞,0(X) such that WF′h F ⊂ supp dq ⊂ U \ Γ. Butih〈[P,Q∗Q]u, u〉 = 2hIm〈Q∗Q(P − λ)u, u〉+ 2h〈Q∗Q Imλu, u〉> −2h−1‖Q(P − λ)u‖ ‖Qu‖ − O(h∞)‖u‖2 > −Ch−2a(h)−O(h∞),where we used Imλ > −O(h∞) and that on supp q, (P − λ)u is O(1). So‖Bu‖2 6 〈Eu, u〉+ h〈Fu, u〉+ Ch−2a(h) +O(h∞).But |〈Eu, u〉| 6 Ch−2 because WF′hE ∩ Γ+ = ∅ gives that u is O(h−1) onWF′hE by the first paragraph of the proof. Meanwhile |〈Fu, u〉| 6 C(h−2 +h2k) because all points of WF′h F are either in U\Γ+, where we know u isO(h−1) from the first paragraph of the proof, or on a single compact subsetof U∩Γ+\Γ, where we know that u is O(hk) by inductive hypothesis. Sinceb =√−Hpq2 > 0 on ΓU0+ \ U1, we can use microlocal elliptic regularity toconclude that u is O(hk+1/2) on ΓU0+ \ U1, as desired. TOME 62 (2012), FASCICULE 62384 Kiril DATCHEV & András VASYBIBLIOGRAPHY[1] N. Burq, “Lower bounds for shape resonances widths of long range Schrödingeroperators”, Amer. J. Math. 124 (2002), no. 4, p. 677-755.[2] N. Burq & M. Zworski, “Geometric control in the presence of a black box”, J. Amer.Math. Soc. 17:2 (2004), no. 4, p. 443-471.[3] F. Cardoso & G. Vodev, “Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II”, Ann. Henri Poincaré3 (2002), no. 4, p. 673-691.[4] H. Christianson, “Semiclassical non-concentration near hyperbolic orbits”, J.Funct. Anal. 246 (2007), no. 2, p. 145-195, Corrigendum, J. Funct. Anal., 258 (2010),no. 3 p. 1060-1065.[5] H. Christianson, E. Schenck, A. Vasy & J. Wunsch, “From resolvent estimates todamped waves”, To appear in J. Anal. Math. Preprint available at arXiv:1206.1565,2012.[6] K. Datchev & A. Vasy, “Propagation through trapped sets and semiclassical resol-vent estimates”, Annales de l’Institut Fourier 62.6 (2012), p. 2345-2375.[7] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators, Springer Verlag, 1994.Manuscrit reçu le 12 juin 2012,accepté le 10 janvier 2013.Kiril DATCHEVDepartment of Mathematics,Massachusetts Institute of Technology,Cambridge, MA 02139-4397,U.S.A.datchev@math.mit.eduAndrás VASYDepartment of Mathematics,Stanford University,Stanford, CA 94305-2125,U.S.A.andras@math.stanford.eduANNALES DE L’INSTITUT FOURIER

Semiclassical resolvent estimates at trapped sets

András Vasy

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Semiclassical Resolvent Estimates at Trapped Sets

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Semiclassical resolvent estimates at trapped sets

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