TY - JOUR

T1 - Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces

AU - Hofmann, Steve

AU - Mayboroda, Svitlana

AU - McIntosh, Alan

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 2011

Y1 - 2011

N2 - 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 Lp, Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in Lp for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e-tL need not be bounded in Lp 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.

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 Lp, Sobolev, and some new Hardy spaces naturally associated to L. First, we show that the known ranges of boundedness in Lp for the heat semigroup and Riesz transform of L, are sharp. In particular, the heat semigroup e-tL need not be bounded in Lp 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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U2 - 10.24033/asens.2154

DO - 10.24033/asens.2154

M3 - Article

AN - SCOPUS:84861480105

VL - 44

SP - 723

EP - 800

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

SN - 0012-9593

IS - 5

ER -