Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus

Let X be a space of homogeneous type. Assume that an operator L has a bounded holomorphic functional calculus on L²(X) and the kernel of the heat semigroup {e-tL}t>0 satisfies the Davies-Gaffney estimates. Without the assumption that L is self-adjoint, we develop a theory of Hardy spaces HLp(X), 0<p≤1, which includes a molecular decomposition, an atomic decomposition, a square function characterization, duality of Hardy and Lipschitz spaces, and a Marcinkiewicz type interpolation theorem. As applications, we show that L has a bounded holomorphic functional calculus on HLp(X) for all p>0 and certain Riesz transforms associated to L are bounded from HLp(X) to L^p(X) for all 0<p≤2.