Weighted Hardy spaces associated to operators and boundedness of singular integrals

Let $(X, d, \mu)$ be a space of homogeneous type, i.e. the measure $\mu$ satisfies doubling (volume) property with respect to the balls defined by the metric $d$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup of $L$ satisfies the Davies-Gaffney estimates. In this paper, we study the weighted Hardy spaces $H^p_{L,w}(X)$, $0 < p \le 1$, associated to the operator $L$ on the space $X$. We establish the atomic and the molecular characterizations of elements in $H^p_{L,w}(X)$. As applications, we obtain the boundedness on $\HL$ for the generalized Riesz transforms associated to $L$ and for the spectral multipliers of $L$.