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We investigate the Hardy space $H^1_L$ associated with a self-adjointoperator $L$ defined in a general setting in [S. Hofmann, et. al., Hardy spacesassociated to non-negative self-adjoint operators satisfying Davies-Gaffneyestimates, Mem. Amer. Math. Soc. 214 (2011), no. 1007, vi+78.]. We assume thatthere exists an $L$-harmonic non-negative function $h$ such that the semigroup$\exp(-tL)$, after applying the Doob transform related to $h$, satisfies theupper and lower Gaussian estimates. Under this assumption we describe anilluminating characterisation of the Hardy space $H^1_L$ in terms of a simpleatomic decomposition associated with the $L$-harmonic function $h$. Ourapproach also yields a natural characterisation of the $BMO$-type spacecorresponding to the operator $L$ and dual to $H^1_L$ in the samecircumstances.  The applications include surprisingly wide range of operators, such as:Laplace operators with Dirichlet boundary conditions on some domains in$\mathbb{R}^d$, Schr\"odinger operators with certain potentials, and Besseloperators.

Hardy spaces meet harmonic weights