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

Steve Hofmann; Svitlana Mayboroda; Alan McIntosh

Annales scientifiques de l'École Normale Supérieure (2011)

- Volume: 44, Issue: 5, page 723-800
- ISSN: 0012-9593

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topHofmann, Steve, Mayboroda, Svitlana, and McIntosh, Alan. "Second order elliptic operators with complex bounded measurable coefficients in $L^p$, Sobolev and Hardy spaces." Annales scientifiques de l'École Normale Supérieure 44.5 (2011): 723-800. <http://eudml.org/doc/272188>.

@article{Hofmann2011,

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 ofallSobolev 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.},

author = {Hofmann, Steve, Mayboroda, Svitlana, McIntosh, Alan},

journal = {Annales scientifiques de l'École Normale Supérieure},

keywords = {Hardy and Lipschitz spaces; elliptic operators; complex coefficients; heat semigroup; Riesz transform},

language = {eng},

number = {5},

pages = {723-800},

publisher = {Société mathématique de France},

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

url = {http://eudml.org/doc/272188},

volume = {44},

year = {2011},

}

TY - JOUR

AU - Hofmann, Steve

AU - Mayboroda, Svitlana

AU - McIntosh, Alan

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

JO - Annales scientifiques de l'École Normale Supérieure

PY - 2011

PB - Société mathématique de France

VL - 44

IS - 5

SP - 723

EP - 800

AB - 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 ofallSobolev 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.

LA - eng

KW - Hardy and Lipschitz spaces; elliptic operators; complex coefficients; heat semigroup; Riesz transform

UR - http://eudml.org/doc/272188

ER -

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