Multidimensional Paley–Zygmund theorems and sharp L p estimates for some elliptic operators  [ Théorèmes de Paley–Zygmund multidimensionnels et estimées L p optimales pour quelques opérateurs elliptiques ]
Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2723-2809.

Le but de cet article est double. Premièrement, nous étudions des conditions suffisantes de convergence pour des séries aléatoires de fonctions propres dans L . Les fonctions propres sont considérées par rapport à un opérateur elliptique de référence tel que l’opérateur de Laplace–Beltrami ou un opérateur de Schrödinger avec un potentiel confinant de l’espace euclidien. Cela constitue une généralisation d’un vieux résultat de Paley et Zygmund. Dans un deuxième temps, nous obtenons quelques estimées L p optimales de fonctions propres incluant une généralisation de l’inégalité de Bernstein. Nous montrons que ces deux thèmes sont intimement liés.

The goal of the paper is twofold. Firstly we study sufficient conditions of convergence for random series of eigenfunctions in L . The eigenfunctions are considered with respect to a reference elliptic operator like the Laplace–Beltrami operator or a Schrödinger operator with a growing potential on the Euclidean space. That is a generalization of an old result due to Paley and Zygmund. Secondly, we obtain a few optimal L p bounds of eigenfunctions including a generalization of the Bernstein inequality. We show that the previous two themes are intimately linked.

Reçu le : 2017-07-23
Révisé le : 2018-03-15
Accepté le : 2018-04-25
Publié le : 2019-10-29
DOI : https://doi.org/10.5802/aif.3306
Classification : 60G50,  15B52,  46B09
Mots clés: Théorèmes de Paley–Zygmund, opérateurs elliptiques, équations des ondes, injections de Sobolev
@article{AIF_2019__69_6_2723_0,
     author = {Imekraz, Rafik},
     title = {Multidimensional Paley--Zygmund theorems and sharp $L^p$ estimates for some elliptic operators},
     journal = {Annales de l'Institut Fourier},
     pages = {2723--2809},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {6},
     year = {2019},
     doi = {10.5802/aif.3306},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2019__69_6_2723_0/}
}
Imekraz, Rafik. Multidimensional Paley–Zygmund theorems and sharp $L^p$ estimates for some elliptic operators. Annales de l'Institut Fourier, Tome 69 (2019) no. 6, pp. 2723-2809. doi : 10.5802/aif.3306. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2723_0/

[1] Anantharaman, Nalini Topologie des hypersurfaces nodales de fonctions aléatoires gaussiennes, Séminaire Bourbaki. Volume 2015/2016 (Astérisque) Volume 390, Société Mathématique de France, 2017 | Zbl 1386.60134

[2] Angst, Jürgen; Pham, Viet-Hung; Poly, Guillaume Universality of the nodal length of bivariate random trigonometric polynomials, Trans. Am. Math. Soc., Volume 370 (2018) no. 12, pp. 8331-8357 | Article | MR 3864378 | Zbl 1407.42001

[3] Ayache, Antoine; Tzvetkov, Nikolay L p properties for Gaussian random series, Trans. Am. Math. Soc., Volume 360 (2008) no. 8, pp. 4425-4439 | Article | MR 2395179 | Zbl 1145.60019

[4] Berezin, Feliks A.; Shubin, Mikhail A. The Schrödinger equation, Mathematics and Its Applications. Soviet Series, Volume 66, Kluwer Academic Publishers, 1991 | Zbl 0749.35001

[5] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Volume 194, Springer, 1971, vii+251 pages | Zbl 0223.53034

[6] Bony, Jean-Michel On the Characterization of Pseudodifferential Operators (Old and New), Studies in Phase Space Analysis with Applications to PDEs (Progress in Nonlinear Differential Equations and their Applications) Volume 84, Springer, 2013, pp. 21-34 | Article | MR 3185888 | Zbl 1273.35331

[7] de Bouard, Anne Construction de solutions pour des edp sur-critiques à données initiales aléatoires, Bourbaki seminar. Volume 2013/2014 (Astérisque) Volume 367-368, Société Mathématique de France, 2015 no. 1074 | Zbl 1356.35006

[8] Bouclet, Jean-Marc Semi-classical calculus on manifolds with ends and weighted L p estimates, Ann. Inst. Fourier, Volume 61 (2011) no. 3, pp. 1181-1223 | Article | MR 2918727 | Zbl 1236.58033

[9] Brézis, Haïm; Nirenberg, Louis Degree theory and BMO. I: Compact manifolds without boundaries, Sel. Math., New Ser., Volume 1 (1995) no. 2, pp. 197-263 | Article | MR 1354598 | Zbl 0852.58010

[10] Burq, Nicolas; Gérard, Patrick; Tzvetkov, Nikolay Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Am. J. Math., Volume 126 (2004) no. 3, pp. 569-605 | Article | Zbl 1067.58027

[11] Burq, Nicolas; Lebeau, Gilles Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér., Volume 46 (2013) no. 6, pp. 917-962 | Article | MR 3134684 | Zbl 1296.46031

[12] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations I: local theory, Invent. Math., Volume 173 (2008) no. 3, pp. 449-475 | Article | MR 2425133 | Zbl 1156.35062

[13] Burq, Nicolas; Tzvetkov, Nikolay Random data Cauchy theory for supercritical wave equations II: A global existence result, Invent. Math., Volume 173 (2008) no. 3, pp. 477-496 | Article | MR 2425134 | Zbl 1187.35233

[14] Burq, Nicolas; Tzvetkov, Nikolay Probabilistic well-posedness for the cubic wave equation, J. Eur. Math. Soc., Volume 16 (2014) no. 1, pp. 1-30 | Article | MR 3141727 | Zbl 1295.35387

[15] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, Volume 268, Cambridge University Press, 1999 | MR 1735654 | Zbl 0926.35002

[16] Filbir, Frank-Dieter; Mhaskar, Hrushikesh N. A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel, J. Fourier Anal. Appl., Volume 16 (2010) no. 5, pp. 629-657 | Article | MR 2673702 | Zbl 1204.41005

[17] Hanin, Boris; Zelditch, Steve; Zhou, Peng Nodal sets of random eigenfunctions for the isotropic harmonic oscillator, Int. Math. Res. Not., Volume 2015 (2015) no. 13, pp. 4813-4839 | Article | MR 3439093 | Zbl 1319.81042

[18] Helffer, Bernard; Robert, Didier Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques, Ann. Inst. Fourier, Volume 31 (1981) no. 3, pp. 169-223 | MR 638623 | Zbl 0451.35022

[19] Helffer, Bernard; Robert, Didier Proprietes asymptotiques du spectre d’operateurs pseudo-differentiels sur R n , Commun. Partial Differ. Equations, Volume 7 (1982) no. 7, pp. 795-882 | Article | Zbl 0501.35081

[20] Hofer, Helmut; Zehnder, Eduard Symplectic invariants and Hamiltonian dynamics, Modern Birkhäuser Classics, Birkhäuser, 2011 | Zbl 1223.37001

[21] Hörmander, Lars The spectral function of an elliptic operator, Acta Math., Volume 121 (1968) no. 1, pp. 193-218 | Article | MR 609014 | Zbl 0164.13201

[22] Hörmander, Lars The analysis of linear partial differential operators. III : Pseudo-differential operators, Grundlehren der Mathematischen Wissenschaften, Volume 274, Springer, 1985 | Zbl 0601.35001

[23] Imekraz, Rafik Concentration et randomisation universelle de sous-espaces propres, Anal. PDE, Volume 11 (2018) no. 2, pp. 263-350 | Article | MR 3724490 | Zbl 1378.15021

[24] Imekraz, Rafik; Robert, Didier; Thomann, Laurent On random Hermite series, Trans. Am. Math. Soc., Volume 368 (2016) no. 4, pp. 2763-2792 | Article | MR 3449257 | Zbl 1357.60047

[25] Jain, Naresh C.; Marcus, Michael B. Integrability of infinite sums of independent vector-valued random variables, Trans. Am. Math. Soc., Volume 212 (1975), pp. 1-36 | Article | MR 385995 | Zbl 0318.60036

[26] John, Fritz; Nirenberg, Louis On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961) no. 3, pp. 415-426 | Article | MR 131498 | Zbl 0102.04302

[27] Kahane, Jean-Pierre Some random series of functions, Heath Mathematical Monographs, D.C. Heath and Company, 1968 | Zbl 0192.53801

[28] Karadzhov, Georgi E. Riesz summability of multiple Hermite series in L p spaces, Math. Z., Volume 219 (1995) no. 1, pp. 107-118 | Article | MR 1340852 | Zbl 0824.42019

[29] Koch, Herbert; Tataru, Daniel L p eigenfunction bounds for the Hermite operator, Duke Math. J., Volume 128 (2005) no. 2, pp. 369-392 | Article | MR 2140267 | Zbl 1075.35020

[30] Koch, Herbert; Tataru, Daniel; Zworski, Maciej Semiclassical L p estimates, Ann. Henri Poincaré, Volume 8 (2007) no. 5, pp. 885-916 | Article | MR 2342881 | Zbl 1133.58025

[31] Ledoux, Michel; Talagrand, Michel Probability in Banach Spaces. Isoperimetry and processes, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 23, Springer, 1991 | Zbl 0748.60004

[32] Lerner, Nicolas Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Applications, Volume 3, Birkhäuser, 2010 | MR 2599384 | Zbl 1186.47001

[33] Li, Daniel; Queffélec, Hervé Introduction à l’étude des espaces de Banach, Cours Spécialisés (Paris), Volume 12, Société Mathématique de France, 2004 | Zbl 1078.46001

[34] Marcus, Michael B.; Pisier, Gilles Random Fourier series with applications to harmonic analysis, Annals of Mathematics Studies, Volume 101, Princeton University Press, 1981 | MR 630532 | Zbl 0747.43004

[35] Maurey, Bernard; Pisier, Gilles Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach, Stud. Math., Volume 58 (1976), pp. 45-90 | Article | Zbl 0344.47014

[36] Muckenhoupt, Benjamin Mean convergence of Hermite and Laguerre series. II, Trans. Am. Math. Soc., Volume 147 (1970) no. 2, pp. 433-460 | Article | MR 256051

[37] Paley, Raymond E. A. C.; Zygmund, Antoni On some series of functions. I, Proc. Camb. Philos. Soc., Volume 26 (1930), pp. 337-357 | Article | Zbl 56.0254.01

[38] Poiret, Aurélien; Robert, Didier; Thomann, Laurent Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator, Anal. PDE, Volume 7 (2014) no. 4, pp. 997-1026 | Article | MR 3254351 | Zbl 1322.35190

[39] Poiret, Aurélien; Robert, Didier; Thomann, Laurent Random weighted Sobolev inequalities on d and applications to Hermite functions, Ann. Henri Poincaré, Volume 16 (2015) no. 2, pp. 651-689 | Article | MR 3302609 | Zbl 1310.81060

[40] Robbiano, Luc; Zuily, Claude Remark on the Kato smoothing effect for Schrödinger equation with superquadratic potentials, Commun. Partial Differ. Equations, Volume 33 (2008) no. 4, pp. 718-727 | Article | Zbl 1147.35359

[41] Robert, Didier; Thomann, Laurent Random weighted Sobolev inequalities and application to quantum ergodicity, Commun. Math. Phys., Volume 335 (2015) no. 3, pp. 1181-1209 | Article | MR 3320309 | Zbl 1341.60074

[42] Sledd, William T Random series which are BMO or Bloch, Mich. Math. J., Volume 28 (1981) no. 3, pp. 259-266 | MR 629359 | Zbl 0461.42010

[43] de Suzzoni, Anne-Sophie Consequences of the choice of a particular basis of L 2 (S 3 ) for the cubic wave equation on the sphere and the Euclidian space, Commun. Pure Appl. Anal., Volume 13 (2014) no. 3, pp. 991-1015 | MR 3177685 | Zbl 1284.35287

[44] Tao, Terence Topics in random matrix theory, Graduate Studies in Mathematics, Volume 132, American Mathematical Society, 2012 | MR 2906465 | Zbl 1256.15020

[45] Tao, Terence; Vu, Van Random matrices: The distribution of the smallest singular values, Geom. Funct. Anal., Volume 20 (2010) no. 1, pp. 260-297 | MR 2647142 | Zbl 1210.60014

[46] Triebel, Hans Theory of function spaces, Monographs in Mathematics, Volume 78, Birkhäuser, 1983 | MR 781540 | Zbl 0546.46027

[47] Tzvetkov, Nikolay Riemannian analogue of a Paley–Zygmund theorem, Sémin. Équ. Dériv. Partielles, Volume 2008-2009 (2010), XV, 14 pages | MR 2668635 | Zbl 1236.43007

[48] Yajima, Kenji; Zhang, Guoping Smoothing property for Schrödinger equations with potential superquadratic at infinity, Commun. Math. Phys., Volume 221 (2001) no. 3, pp. 573-590 | Article | Zbl 1102.35320

[49] Yajima, Kenji; Zhang, Guoping Local smoothing property and Strichartz inequality for Schrödinger operator with potentials superquadratic at infinity, J. Differ. Equations, Volume 202 (2004) no. 1, pp. 81-110 | Article | Zbl 1060.35121

[50] Zelditch, Steve Real and complex zeros of Riemannian random waves, Spectral analysis in geometry and number theory (Contemporary Mathematics) Volume 484, American Mathematical Society, 2009, pp. 321-342 | Article | MR 1500155 | Zbl 1176.58021

[51] Zygmund, Antoni Trigonometric series. II, Cambridge University Press, 2002 | Zbl 1084.42003