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 CalderonZygmund 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.