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

In this article we give a molecular reconstruction theorem for \(H_{\omega }^{p(\cdot )}(\mathbb {R}^{n})\) $H_{\omega }^{p(·)}({\mathbb{R}}^n)$ . 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 \(H_{\omega }^{p(\cdot )}(\mathbb {R}^{n})\) $H_{\omega }^{p(·)}({\mathbb{R}}^n)$ . We also prove, for certain exponents \(q(\cdot )\) $q(·)$ and certain weights \(\omega \) $\omega$ , that the Riesz potential \(I_{\alpha }\) $I_{\alpha }$ , with \(0< \alpha < n\) $0<\alpha <n$ , can be extended to a bounded operator from \(H^{p(\cdot )}_{\omega }(\mathbb {R}^{n})\) $H_{\omega }^{p(·)}({\mathbb{R}}^n)$ into \(H^{q(\cdot )}_{\omega }(\mathbb {R}^{n})\) $H_{\omega }^{q(·)}({\mathbb{R}}^n)$ , for \(\frac{1}{p(\cdot )}:= \frac{1}{q(\cdot )} + \frac{\alpha }{n}\) $\frac{1}{p(·)}:=\frac{1}{q(·)}+\frac{\alpha }{n}$ .