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

Abstract

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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 - t L need not be bounded in  L p if p ∉ [ 2 n / ( n + 2 ) , 2 n / ( n - 2 ) ] . Then we provide a complete description ofallSobolev spaces in which L admits a bounded functional calculus, in particular, where e - t L is bounded. Secondly, we develop a comprehensive theory of Hardy and Lipschitz spaces associated to  L , that serves the range of  p beyond [ 2 n / ( n + 2 ) , 2 n / ( 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.

How to cite

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Hofmann, 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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