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

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 Ho (Tohoku Math J 69 (3), 383–413, 2017) 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 the 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 \(\frac{1}{p(\cdot )}:= \frac{1}{q(\cdot )} + \frac{\alpha }{n}\).