Hofmann, Steve; Mayboroda, Svitlana; McIntosh, Alan
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 ]
Annales scientifiques de l'École Normale Supérieure, Série 4 : Tome 44 (2011) no. 5 , p. 723-800
Zbl 1243.47072 | MR 2931518
doi : 10.24033/asens.2154
URL stable : http://www.numdam.org/item?id=ASENS_2011_4_44_5_723_0

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
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[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.
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[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.

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