Racines carrées d’opérateurs elliptiques et espaces de Hardy

Russ, Emmanuel

Confluentes Mathematici, Tome 3 (2011) no. 1, pp. 1-119.

Il est bien connu que, pour tout 1 < p < +∞, il existe C_p > 0 tel que

$\sum _{1\le i\le n}{\Vert {\partial }_{i}f\Vert }_{L^{P\left({\mathbb{R}}^n\right)}}\sim {\Vert (-\mathrm{\Delta }{)}^{1/2}f\Vert }_{L^{P\left({\mathbb{R}}^n\right)}}$     (0.1)

pour toute fonction . Lorsque p = 1, (0.1) est fausse pour L¹(ℝⁿ) mais devient

$\sum _{1\le i\le n}{\Vert {\partial }_{i}f\Vert }_{H^1\left({\mathbb{R}}^n\right)}\sim {\Vert (-\mathrm{\Delta }{)}^{1/2}f\Vert }_{H^1\left({\mathbb{R}}^n\right)}$  (0.2)

où H¹(ℝⁿ) est l'espace de Hardy réel classique. Dans cette vue d'ensemble, nous rassemblons des résultats qui étendent (0.1) et (0.2) dans deux directions. D'une part, nous nous plaçons dans un ouvert fortement lipschitzien de ℝⁿ, ou dans un cadre géométrique non euclidien (variété riemannienne complète ou graphe). D'autre part, nous remplaçons Δ par un opérateur elliptique d'ordre 2 plus général (opérateur sous forme divergence dans un ouvert de ℝⁿ, laplacien de Hodge dans une variété riemannienne, laplacien discret sur un graphe). Dans le cas des domaines fortement lipschitziens de ℝⁿ, ces questions conduisent à introduire des espaces de Hardy–Sobolev et à en donner des propriétés analogues à celles des espaces de Sobolev usuels. Dans le cas des variétés riemanniennes, nous introduisons des espaces de Hardy de formes différentielles exactes. Ces espaces sont adaptés au laplacien de Hodge et possèdent des propriétés analogues à l'espace de Hardy H¹(ℝⁿ). Le laplacien de Hodge possède un calcul fonctionnel holomorphe sur ces espaces, et en particulier la transformée de Riesz est continue. Sous des hypothèses géométriques convenables, nous comparons ces espaces de Hardy aux espaces de Hardy usuels (dans le cas d'espaces de fonctions) ou aux espaces L^p. Enfin, sur un graphe vérifiant certaines propriétés géométriques, nous obtenons des versions discrètes de la plupart des résultats correspondants pour les transformées de Riesz et inégalités reliées obtenues dans des variétés riemanniennes. Nous donnons également des résultats concernant les puissances fractionnaires d'opérateurs elliptiques d'ordre 2. Il s'agit de propriétés d'algèbre pour des espaces de Bessel fractionnaires sur des groupes de Lie unimodulaires, généralisant les résultats euclidiens. Nous utilisons également des estimations de la norme L² de puissances fractionnaires de certains opérateurs pour montrer une inégalité de Poincaré fractionnaire pour certaines mesures de probabilité dans ℝⁿ ou des groupes de Lie à croissance polynomiale.

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