Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces

[Opérateurs elliptiques du second ordre à coefficients complexes mesurables bornés dans les espaces $L^p$, de Sobolev et de Hardy]

Hofmann, Steve ; Mayboroda, Svitlana ; McIntosh, Alan

Annales scientifiques de l'École Normale Supérieure, Tome 44 (2011), p. 723-800 / Harvested from Numdam

Résumé

Soit $L$ un opérateur elliptique du second ordre de formes de divergence, à coefficients complexes bornés et mesurables. Les opérateurs associés à $L$ tels que le semi-groupe de la chaleur ou la transformée de Riesz ne sont en général pas de type Calderón-Zygmund et présentent des comportements différents de leurs analogues construits à partir du laplacien. Cet article a pour objectif de décrire de manière exhaustive les propriétés de ces opérateurs dans $L^p$, dans les espaces de Sobolev ainsi que dans certains nouveaux espaces de Hardy naturellement associés à $L$. Tout d’abord, nous montrons que les plages de valeurs connues pour lesquelles ces opérateurs sont bornés en norme $L^p$ sont strictes. En particulier, le semi-groupe de la chaleur et la transformée de Riesz ne sont pas obligatoirement bornés si $p\notin [2n/(n+2),2n/(n-2)]$. Nous fournissons ensuite une description complète de tous les espaces de Sobolev pour lesquels $L$ admet un calcul fonctionnel borné, en particulier, pour lesquels $e^{-tL}$ est borné. Puis, nous développons une théorie extensive des espaces de Hardy et de Lipschitz associés à $L$, pour les valeurs de $p$ hors de $[2n/(n+2),2n/(n-2)]$. Cette théorie comprend, en particulier, des caractérisations par la fonction maximale « dièse » et par la transformée de Riesz (pour certaines plages de $p$), ainsi que leur décomposition moléculaire, leur dualité et les théorèmes d’interpolation.

Abstract

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the known ranges of boundedness in $L^p$ for the heat semigroup and Riesz transform of $L$, are sharp. In particular, the heat semigroup $e^{-tL}$ need not be bounded in $L^p$ if $p\notin [2n/(n+2),2n/(n-2)]$. Then we provide a complete description of all Sobolev spaces in which $L$ admits a bounded functional calculus, in particular, where $e^{-tL}$ is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to $L$, that serves the range of $p$ beyond $[2n/(n+2),2n/(n-2)]$. It includes, in particular, characterizations by the sharp maximal function and the Riesz transform (for certain ranges of $p$), as well as the molecular decomposition and duality and interpolation theorems.

Détail

Publié le : 2011-01-01

MR 2931518 | Zbl 1243.47072

DOI : https://doi.org/10.24033/asens.2154
Classification:  42B30,  42B35,  42B25,  35J15
Mots clés: espaces de Hardy et de Lipschitz, opérateurs elliptiques, coefficients complexes, semi-groupe de la chaleur, transformation de Riesz

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