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

TL;DR

This paper establishes a molecular reconstruction theorem for variable exponent Hardy spaces with weights, enabling the extension of classical singular integrals and Riesz potentials as bounded operators on these spaces.

Contribution

It introduces a molecular reconstruction theorem for weighted variable exponent Hardy spaces and demonstrates the boundedness of classical singular integrals and Riesz potentials on these spaces.

Findings

Classical singular integrals extend to bounded operators on $H_{ ext{omega}}^{p( ext{cdot})}( ext{R}^n)$.

Riesz potential $I_ ext{ extalpha}$ is bounded from $H^{p( extalpha)}_{ ext{omega}}( ext{R}^n)$ to $H^{q( extalpha)}_{ ext{omega}}( ext{R}^n)$.

The paper develops atomic decomposition techniques for these variable exponent spaces.

Abstract

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 $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$.