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

Abstract

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