Probabilistic Sobolev embeddings, applications to eigenfunctions estimates

Transcription

1 Contemporary Mathematics Probabilistic Sobolev embeddings, applications to eigenfunctions estimates Nicolas Bur and Gilles Lebeau Abstract. The purpose of this article is to present some probabilistic versions of Sobolev embeddings and an application to the growth rate of the L p norms of Spherical harmonics on spheres. More precisely, we prove that for natural probability measures, almost every Hilbert base of L S d made of spherical harmonics has all its elements uniformly bounded in any L p space p < +. We also show that most of the analysis extends to the case of Riemanian manifolds with groups of isometries acting transitively.. Introduction Consider M, g a compact Riemanian manifold of dimension d and the Laplace operator on M, g. Since the wors by Hörmander [] and Sogge [4], it is nown that the eigenfunctions satisfy Theorem. For any p +, there exists C > such that for any eigenfunctions of the Laplace operator u, g u = λ u, one has. u L p M Cλ δp u L with. δp = { d d p d d+ if p d p if p d+ d c copyright holder

2 NICOLAS BURQ AND GILLES LEBEAU δp d d 4 d d+ Figure.. Eigenfunction estimates p These estimates are also nown to be optimal on the spheres endowed with their standard metric. In the first regime, the so called zonal spherical harmonics which concentrate on two opposite points on a diameter of the sphere are optimizing. while in the second regime, the highest weight spherical harmonics ux, x d+ = x +ix n which concentrate on the euator x 3,, x d+ = are saturating.. On the other hand, the case of the tori shows that there exists manifolds on which these estimates can be improved. Indeed, on tori, we have by simple number theory arguments.3 δp d + ɛ p for d 5, the ɛ can be dropped. On the other hand, it is standard and can easily be shown see e.g. Section 4. and the wors by Bourgain [, 3] and Bourgain-Rudnic [4] for many more results on eigenfunctions on tori that on tori.4 δp d d p, p d d and conseuently, as soon as d 3, even on the torus, there exists seuences of L - normalized eigenfunctions having unbounded L p norms. On the other hand, of course, the usual eigenbasis of eigenfunctions e n x = e in x, n Z d has all its L p norms bounded and conseuently a natural uestion is to as whether this phenomenon is uniue or there exists other manifolds exhibiting the same ind of behaviour existence of both seuences having unbounded L p norms and seuences having bounded L p norms. Our result is that it is true on all spheres S d endowed with their standard metric for any p < +, despite the optimality of.. We actually prove a stronger result. Let us recall that the eigenfunctions of the Laplace operator on the spheres S d with its standard metric are the spherical harmonics of degree restrictions to the sphere S d of the harmonic polynomials of degree and they satisfy For any N, the vector space of spherical harmonics of degree, E, has dimension + d.5 N = d + d d + e d d! e

3 PROBABILISTIC SOBOLEV 3 For any e E, S de = + d e. The vector space spanned by the spherical harmonics is dense in L S d As a conseuence, any family B = B N = b,l l=,...,n N of orthonormal bases of E N is a Hilbert base of L S d. Our result is the following with z = + t z e t dt the classical Gamma function. Theorem. For any d, consider the space of spherical harmonics of degree, E endowed with the L S d norm and S its unit sphere endowed with the uniform probability measure P. Consider the space of Hilbert bases of L S d having all elements harmonic polynomials, with real coefficients, endowed with its natural probability measure, ν see Section 3 for a precise definition. Then for any < +, we have N /.6 A, = E u L S d u S = u L S d dp = VolS d e e A, = VolS d and there exists M, > such that C.7 M, A, C + O N + O d d N N P u S ; u L M, > Λ if d+ d < + if d+ d + d e N d r if d+ e N + N N + d < + r if d+ and.9 ν{b = b,l N,l=, N B;, l; b,l L S d M, > r} Ce cr Furthermore, there exists C, c, c > such that d,. ν{b = b,l N,l=, N B;, l; b,l L S d > c + r log} Ce cr Remar.. VanderKam [5] see also Zelditch [6] proves an estimate for the L - norm of spherical harmonics on S which involves a logarithmic loss log, in the same context. Though the results are in spirit very close, the methods of proof appear to differ significantly: VanderKam s approach which do not appear to be flexible enough to tacle the case p < +, or to give lower bounds is based upon the precise descriptions of some particular spherical harmonics, implied by Ramanujan s conjectures proved by Deligne [7]. As explained above, our proof is very general see Section 4 and the only sphere-specific fact we use is the exact spectral clustering σ = { + d, N }.

4 4 NICOLAS BURQ AND GILLES LEBEAU Remar.. We actually prove a stronger result. Indeed, when proving the first part of Theorem euations.6,.7,.8, modulo a small change in the estimates, we only use the fact that the group of isometries on the manifold M acts transitively, whereas for.9 and., we use only the fact that the dimension of the eigenspaces is bounded from below by a positive power of the eigenvalue see Theorem 3 and Remar 3. Let us end this introduction by mentioning that most of the wor presented here is extracted from more general results on probabilistic Sobolev embeddings and applications to the study of the behaviour of solutions to Partial Differential Euations [5].. Probabilistic estimates.. The spectral function. In this section, we consider a compact Riemanian manifold M, g. We assume that the group of isometries of M acts transitively. Denote by E λ = Ker g λ Id, λ σ, an eigenspace of the Laplace operator of dimension. Let e j,λ j= be an orthonormal basis of the space E λ, and denote by F λ = e j,λ j=, and f λ x = e j,λ x /. j= We can identify a point U = u j j= S on the unit sphere in R with the function. ux = u j e j,λ x = U F λ x S, j= the sphere of functions in E λ having L M-norm eual to. Lemma.. Under the assumptions above, we have Proof. We notice that fλ x = e j,λ x = j= VolS d. K λ x, y = e j,λ xe j,λ y j= is the ernel of the orthogonal projector on the space E λ which is invariant by the action of any isometry R. As a conseuence, we deduce K λ x, R y = K λ Rx, y, which implies that fλ x is invariant by the isometries of the manifold M, hence constant on M, with integral. Lemma.. Under the same assumptions as in Lemma., there exists C > such that for any λ σ, for any u E λ we have Nλ. u L M u VolM L M

5 PROBABILISTIC SOBOLEV 5 Proof. Let us denote by T the orthogonal projector on E. By the usual T T argument the norm of T seen as an operator from L M to L M is eual to the suare root of the norm of T T = T seen as an operator from L M to L M. This norm is eual to the L M M-norm of its ernel Kx, y. By Cauchy-Schwartz ineuality, we have.3 K λ x, y K λ x, xk λ y, y / = VolM K λ L M M = f λ x L M = VolM which proves.. Notice that actually the proof also gives that there exists a non trivial function u E λ for which Nλ u L M = u VolM L M... Almost sure L estimates. In this section, we prove the main estimate on the L norms. The proof is very much inspired from Shiffman-Zelditch [3]. The main step in our proof is the proof of the estimates.8. According to Theorem, with F u = u L S d,.4 F u F v = u L v L u v L with We deduce γ = { d d CN γ if d+ d < if d+ d u v L CN γ distu, v, F Lips CN γ and conseuently, according to the concentration of measure property see Proposition 5. applied to the function F u = u L on the N - dimensional sphere S we get γ cn.5 P F u M, > r e r where M, is the median value of the function F on S. This implies.8. To get.7, we need to estimate from below and above M,. According to., 5. and Lemma. and the rotation invariance of the uniform measure on the sphere S, we have N P ux > λ = N λ θ N /, VolS d / sin N ϕdϕ, where the value of θ in the expression above is fied by the relation λ = N / cosθ. VolS / Using E g = + λ P g > λdλ

6 6 NICOLAS BURQ AND GILLES LEBEAU we obtain.6 A, = E u L = = We deduce N N = S N VolS d N N ux dxdp = λ P ux > λdλ S d S d π / θ cosθ sinθ sin N ϕdϕdθdx N VolS d.7 N N A, = VolS d + N + S d /VolS d π sin N ϕ cosϕ dϕ = e + = VolS d = N VolS d N + N +. e e + O N e e VolS d + O + o + N We also have.8 A, M, = F L S M, L S + F M, L S = λ P F M, > λdλ We deduce.9 + A, M, λ e c N γ λ dλ = C N γ which, according to.7, implies.7. We can now deduce an estimation of the L norm N γ c / /. Theorem. Denote by M, the median value of the function F u = u L on the unit sphere S endowed with its uniform probability measure. Then there exists c, c > for any N we have. M, [c log, c log] and conseuently according to.4 and Proposition 5., P u S ; u L > c log + Λ e cλ,. P u S ; u L < c log Λ e cλ

7 PROBABILISTIC SOBOLEV 7 Proof. According to Sobolev embeddings, there exists C > such that for any, any >, and any u E u u L C d/ u and choosing = aε log, we obtain with C = C e d/aε independent of. E u E u L C E u On the other hand for any [, ], by Proposition 5. we have E u M,h u M,h dp = P u M,h > λdλ.3 e c λ = π/c Since ε logn,.7 imply M,h. We deduce conseuently from.3 E u, and also according to., E u L, hence, again according to.3 M,. Let us now remar that in the proof above, we used only Lemma. and the L p estimates on eigenfunctions given by Theorem. However, in the case of a riemanian manifold with a group of isometries acting transitively, we can use Lemma. and interpolation to obtain that there exists C > such that for any λ σ and any u E λ, u L p M CN p λ u L M Plugging this estimate instead of Theorem in the proof above gives Theorem 3. Consider M, g a riemanian manifold on which the group of isometries act transitively. for any λ σ, denote by E λ the eigenspace of the Laplace operator of dimension >, endowed with the L M norm and S λ its unit sphere endowed with the uniform probability measure P λ. Then for any < +, we have N / λ.4 A,λ = E u L S d u S = u L S d dp λ = λ VolS d A,λ = VolS d and there exists M,λ > such that.5 + O + O e M,λ A,λ C N / λ.6 P λ u S λ ; u L M,λ > Λ e N., λ r + Nλ Nλ +,

8 8 NICOLAS BURQ AND GILLES LEBEAU 3. Eigenbases We come bac to the case of the sphere and spherical harmonic. We can identify the space of orthonormal basis of E endowed with the L -norm with the orthogonal group, ON and endow this space with its Haar measure, ν. Let us recall that 3. N E is dense in L S d. The set of Hilbert bases of L compatible with the decomposition 3. is B = N ON and is endowed with the product probability measure ν = ν. Now, we can proceed to prove.9 and.. According to.8, we get Proposition 3.. There exists C, c > such that for any N, 3. ν {B = b l N l= SUN ; l N ; b l L M, > Λ c e c d γ Λ d Indeed, the map B = b l N l= SUN b S sends the measure ν to the measure P and conseuently, according to.8 for any l N ν{b = bl N l= SUN ; b l L M, > Λ c e c N γ Λ. We deduce 3.3 ν {B = b l N l= SUN ; l {,..., N }; b l L M, > Λ which implies.9. Let us define now F,r = {B = b SUN ; l ; b,l L M, r} and c e c d γ Λ d, F r = F r, = {B = b,l ; N, l =, N ;, l ; b,l L M, r} We have 3.4 νf r c ν F,r c c e c d γ r d ce c d γ r Ce c r. This ends the proof of.9. To prove., it sufficies to follow the same strategy and use Theorem instead of.8. Remar 3.. In the more general case of a riemanian manifold having a group of isometries acting transitively, the results above on Hilbert bases extend straightforwardly under the additional assumption that lim inf λ σ log >.

9 PROBABILISTIC SOBOLEV 9 4. Other manifolds 4.. Tori. A natural class of examples for which the group of isometries acts transitively are the tori. As a conseuence, as already mentionned, we deduce that Theorem 3 holds. Finally, Theorem holds also with log replaced by logn, with the notable difference that the multiplicity N do not tend to + if d 4. On the other hand, the proofs of.9 and. use the fact that the multiplicity tends to infinity at least polynomially, and it conseuently holds on rational tori only if d 5 see Grosswald [8, Chapter ]. For dimensions d 4, we only obtain the weaer 4. A >, ν{b = b,l N,l=, N B;, l; b,l L S d A} =. Finally, to complete our probabilistic upper bounds, we can also get lower bounds on tori: For n N Let us denote by E n,d the eigenspace of the Laplace operator associated with the eigenvalue λ and r λ the number of presentations of the integer n as a sum of d suares of integers: r d n = #E λ,d, E λ,d = {n,..., n d Z d ; According to Weyl s formula, r d m = Cn n <m n d d + On d n j = λ}. From which we deduce that for any d there exists a seuence λ p + such that r d λ p cλ d p and by considering the following eigenfunction on T d having L -norm eual to, u p x = r d λ p we have according to Sobolev embeddings Hence which is.4. cλ d 4 p j= N=n,...,n d E λp,d e in x u p L Cλ r p u p L r cλ d 4 r p u p L r 4.. Other manifolds. We can extend our results on arbitrary manifolds, to the price of considering only approximate eigenfunctions. Here the Weyl asymptotics are playing a crucial role. Let us consider < a h < < b h c two functions defined for, h such that 4. lim h b h = = lim h a h We assume that 4.3 b h a h Mh

10 NICOLAS BURQ AND GILLES LEBEAU for a constant M to be precised later Let E h be the subspace of L M 4.4 E h = {u = I h z e x, z C}, I h = { N; hω a h, b h } Ẽ h = {u = I h z e x, z R}, Let N h = dime h. Let us recall that Weyl formula with precise remainder reads see Hörmander [] 4.5 N h = πh d VolMVolS d ρ d dρ + Oh d+ and conseuently, we have Vol M 4.6 C > ; λ > { N; ω λ} c d π d λ d λ d and 4.7 Which implies 4.8 a,b Vol M {; ω I h } c d π d h b h d h a h d Ch d+ M > ; b h a h M h β > α > ; αh d b h a h N h = {; ω I h } βh d b h a h Let e h,j N h j= be an orthonormal basis of the space E h, and f h x = F h x l with the function F h x = e h,j x N h j=, = N h j= e h,j x /. We can identify a point U = uj N h j= S h 4.9 ux = u j e h,j x = U F h x. N h j= We have the following other conseuence of Weyl s formula or rather a conseuence of the proof by Hörmander [] Lemma 4.. Let a h, b h as above. We assume that b h a h M h so that 4.8 is satisfied. We also assume if a = b that their common value is positive.then there exists C > such that for any x M and any h ], ] we have 4. e x,h C h d a h b h By decomposing L M as a direct sum of spaces L M = E h where E h correspond to the choice b h a h = Mh, we can prove the analogs of Theorems,. The only essential difference is that of course the elements of E h are no more exact eigenfunctions, but rather approximate eigenfunctions: e = h e + Oh L M

11 PROBABILISTIC SOBOLEV On Zoll manifolds where all geodesics are periodic, the clustering of the eigenvalues see [] allows to improve to u E h u = h u + O L M 5. Probability calculus on spheres Proposition 5.. Let us denote by p N the uniform probability measure on the unit sphere SN of R N. We have see [5, Appendix A] θ 5. p N x > cosθ = t [,[ D N sin N ϕdϕ with D N = c c N c N Notice that where c is the -dimensional volume of the sphere S. π/ 5. = D N sin N ϕdϕ = D N t N 3 = D N where β is the beta function of Weierstrass, βx, y = xy x + y, u N 3 u D N du = βn, which implies N D N = N. Finally let us recall the standard concentration of measure property see e.g. Ledoux [] Proposition 5.. Let us consider a Lipshitz function F on the sphere S d = Sd + endowed with its geodesi distance and with the uniform probability measure, µ d. Let us define its median value MF by Then for any r >, we have µf MF, µf MF. 5.3 µ F MF > r e d r F Lips References [] Besse A.. Manifolds all of whose geodesics are closed. Springer-Verlag, 978. Berlin-New Yor. [] Bourgain, J.. Eigenfunction bounds for the Laplacian on the n-torus. Internat. Math. Res. Notices n 3, 993, pp [3] Bourgain J. Moment ineualities for trigonometric polynomials with spectrum in curved hypersurfaces preprint arxiv.org/abs/7.9 [4] Bourgain J. and Rudnic Z. Restriction of toral eigenfunctions to hypersurfaces and nodal sets. Geom. Funct. Anal., no. 4, [5] Bur N. et Lebeau G.. Injections de Sobolev probabilistes et applications. Preprint,. [6] Bur N. and Tzvetov N.. Probabilistic Well-posedness for the cubic wave euation. Preprint, To appear Journal of the Eur. Math. Soc..

12 NICOLAS BURQ AND GILLES LEBEAU [7] Deligne, P. La conjecture de Weil, I Inst. Hautes Études Sci. Publ. Math , [8] Grosswald E.. Representations of integers as sums of suares. Springer-Verlag, New Yor, 985. xi+5 pp. [9] Guillemin V.. Some classical theorems in spectral theory revisited. In : Seminar on Singularities of Solutions of Linear Partial Differential Euations Inst. Adv. Study, Princeton, N.J., 977/78, pp Princeton, N.J., Princeton Univ. Press, 979. [] Hörmander L.. The spectral function of an elliptic operator. Acta. Math., vol., 968, pp [] Hörmander L.. The Analysis of Linear Partial Differential Operators IV. Springer Verlag, 985, Grundlehren der mathematischen Wissenschaften, volume 75. [] Ledoux M.. The concentration of measure phenomenon. Providence, RI, Amer. Math. Soc.,, Math Surveys and Mono. [3] Shiffman B. et Zelditch S.. Random polynomials of high degree and Levy concentration of measure. Asian J. Math., vol. 7, n 4, 3, pp [4] Sogge C.. Concerning the L p norm of spectral clusters for second order elliptic operators on compact manifolds. Jour. of Funct. Anal., vol. 77, 988, pp [5] VanderKam J. M.. L norms and uantum ergodicity on the sphere. Internat. Math. Res. Notices, no7, 997, pp [6] Zelditch S. A random matrix model for uantum mixing. Internat. Math. Res. Notices no. 3, 996, pp Laboratoire de Mathématiue d Orsay, Université Paris-Sud, UMR 868 du CNRS, 945 Orsay Cedex, France address: nicolas.bur@u-psud.fr Département de Mathématiues, Université de Nice Sophia-Antipolis Parc Valrose 68 Nice Cedex, France et Institut Universitaire de France address: lebeau@unice.fr