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In this article we give a molecular reconstruction theorem for$H_{\omega}^{p(\cdot)}(\mathbb{R}^{n})$. As an application of this result andthe atomic decomposition developed in [5] we show that classical singularintegrals 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 $\frac{1}{p(\cdot)} :=\frac{1}{q(\cdot)} + \frac{\alpha}{n}$.

A molecular reconstruction theorem for $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n})$