Resumen de Hardy spaces associated to self-adjoint operators on general domains

Xuan Thinh Duong, Ji Li, Ming-Yi Lee, Chin-Cheng Lin

• Let be the space of homogeneous type and be a measurable subset of X which may not satisfy the doubling condition. Let L denote a nonnegative self-adjoint operator on which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space associated to L on which provides an appropriate setting to obtain boundedness for certain singular integrals with rough kernels. This then implies boundedness for the rough singular integrals, , from interpolation between the spaces and . As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from to and on for . 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.