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 H p L (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 H p L (X) for all p >0 and certain Riesz transforms associated to L are bounded from H p L (X) to Lp(X) for all 0