Hardy spaces meet harmonic weights

Abstract

We investigate the Hardy space H L 1 H^1_L associated with a self-adjoint operator L L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L L -harmonic non-negative function h h such that the semigroup exp ⁡ ( − t L ) \exp (-tL) , after applying the Doob transform related to h h , satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H L 1 H^1_L in terms of a simple atomic decomposition associated with the L L -harmonic function h h . Our approach also yields a natural characterisation of the B M O BMO -type space corresponding to the operator L L and dual to H L 1 H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in R n {\mathbb {R}^n} , Schrödinger operators with certain potentials, and Bessel operators.