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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 themetric $d$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assumethat the semigroup of $L$ satisfies the Davies-Gaffney estimates. In thispaper, 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 andthe molecular characterizations of elements in $H^p_{L,w}(X)$. As applications,we obtain the boundedness on $\HL$ for the generalized Riesz transformsassociated to $L$ and for the spectral multipliers of $L$.

Weighted Hardy spaces associated to operators and boundedness of singular integrals