Multidimensional Paley–Zygmund theorems and sharp L p estimates for some elliptic operators
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2723-2809.

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.

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.

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Accepted:
Published online:
DOI: 10.5802/aif.3306
Classification: 60G50,  15B52,  46B09
Keywords: Paley–Zygmund theorems, elliptic operators, wave equation, Sobolev embeddings
Imekraz, Rafik 1

1 Institut de Mathématiques de Bordeaux, UMR 5251 Université de Bordeaux 351 cours de la Libération F33405 Talence Cedex (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Imekraz, Rafik. Multidimensional Paley–Zygmund theorems and sharp $L^p$ estimates for some elliptic operators. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2723-2809. doi : 10.5802/aif.3306. https://aif.centre-mersenne.org/articles/10.5802/aif.3306/

[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

[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 | DOI | MR | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | Zbl

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

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

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

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

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

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[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 | DOI | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Zbl

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

[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 | DOI | Zbl

[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 | DOI | Zbl

[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 | DOI | MR | Zbl

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

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