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

TL;DR

This paper thoroughly analyzes second order divergence form elliptic operators with complex coefficients, describing their boundedness properties in various function spaces, and developing Hardy and Lipschitz space theories beyond classical ranges.

Contribution

It provides sharp boundedness ranges for associated operators, characterizes all Sobolev spaces with bounded functional calculus, and develops Hardy and Lipschitz spaces linked to such operators.

Findings

Heat semigroup boundedness range is sharp and limited to p in [2n/(n+2), 2n/(n-2)]

Complete characterization of Sobolev spaces with bounded functional calculus for L

Development of Hardy and Lipschitz spaces with characterizations and duality

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 Calder\'on-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 $p\not\in [2n/(n+2),2n/(n-2)]$. Then we provide a complete description of {\it 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.