Hardy spaces associated to self-adjoint operators on general domains

Let ( X , d , μ ) be the space of homogeneous type and Ω be a measurable subset of X which may not satisfy the doubling condition. Let L denote a nonnegative self-adjoint operator on L 2 ( Ω ) which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space H L 1 ( Ω ) associated to L on Ω which provides an appropriate setting to obtain H L 1 ( Ω ) → L 1 ( Ω ) boundedness for certain singular integrals with rough kernels. This then implies L p boundedness for the rough singular integrals, 1 < p ≤ 2 , from interpolation between the spaces L 2 ( Ω ) and H L 1 ( Ω ) . As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from H L 1 ( Ω ) to L 1 ( Ω ) and on L p ( Ω ) for 1 < p < ∞ . We also study the case of the domains with finite measure and the case of the Gaussian upper bound on the semigroup replaced by the weaker assumption of the Davies–Gaffney estimate.