Weighted variable Hardy spaces associated with operators satisfying Davies-Gaffney estimates

We introduce the weighted variable Hardy space $H^{p(\cdot)}_{L,w}(\mathbb{R}^n)$ associated with the operator $L$, which has a bounded holomorphic functional calculus and fulfills the Davies-Gaffney estimates. More precisely, we establish the molecular characterization of $H^{p(\cdot)}_{L,w}(\mathbb{R}^n)$ and we show that the new weighted variable bounded mean oscillation-type space $BMO^{p(\cdot),M}_{L^*,w}$ represents the dual space of $H^{p(\cdot)}_{L,w}(\mathbb{R}^n)$, where $L^*$ denotes the adjoint operator of $L$ on $L^2(\mathbb{R}^n)$.