In this article we give a molecular reconstruction theorem for $H_{ \omega}^{ p(\cdot)}(\mathbb{ R}^{ n})$. As an application of this result and the atomic decomposition developed in [5] we show that classical singular integrals can be extended to bounded operators on $H_{ \omega}^{ p(\cdot)}(\mathbb{ R}^{ n})$. We also prove, for certain exponents $q(\cdot)$ and certain weights $\omega$, that Riesz potential $I_{ \alpha}$, with $0 < \alpha < n$, can be extended to a bounded operator from $H^{ p(\cdot)}_{ \omega}(\mathbb{ R}^{ n})$ into $H^{ q(\cdot)}_{ \omega}(\mathbb{ R}^{ n})$, for $\frac{ 1}{ p(\cdot)} := \frac{ 1}{ q(\cdot)} + \frac{ \alpha}{ n}$.
Categories: math.CA