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 L²(Ω) which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space HL1(Ω) associated to L on Ω which provides an appropriate setting to obtain HL1(Ω)→L1(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 L²(Ω) and HL1(Ω) . As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from HL1(Ω) to L¹(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.