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

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

• Let (X,d,\mu ) be the space of homogeneous type and \Omega be a measurable subset of X which may not satisfy the doubling condition. Let L denote a nonnegative self-adjoint operator on L^2(\Omega ) which has a Gaussian upper bound on its heat kernel. The aim of this paper is to introduce a Hardy space H^1_L(\Omega ) associated to L on \Omega which provides an appropriate setting to obtain H^1_L(\Omega )\rightarrow L^1(\Omega ) boundedness for certain singular integrals with rough kernels. This then implies L^p boundedness for the rough singular integrals, 1 p \le 2 , from interpolation between the spaces L^2(\Omega ) and H^1_L(\Omega ). As applications, we show the boundedness for the holomorphic functional calculus and spectral multipliers of the operator L from H^1_L(\Omega ) to L^1(\Omega ) and on L^p(\Omega ) for 1< p < \infty. 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.