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 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 {\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