Hardy spaces meet harmonic weights

We investigate the Hardy space H_L¹ associated with a self-adjoint operator 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-harmonic non-negative function h such that the semigroup exp(−tL), after applying the Doob transform related to h, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H_L¹ in terms of a simple atomic decomposition associated with the L-harmonic function h. Our approach also yields a natural characterisation of the BMO-type space corresponding to the operator L and dual to H_L¹ in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in Rⁿ, Schrödinger operators with certain potentials, and Bessel operators.