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

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.