Estimates for Riesz potential on weighted variable Hardy spaces revisited

TL;DR

This paper revisits estimates for the Riesz potential on weighted variable Hardy spaces, providing simpler proofs that do not require a previously assumed technical condition, thereby broadening the applicability of these bounds.

Contribution

The paper offers a new proof for Riesz potential estimates on weighted variable Hardy spaces without relying on a specific hypothesis, simplifying the previous approach.

Findings

Established boundedness of Riesz potential without hypothesis A2

Simplified proof techniques for variable Hardy space estimates

Broader applicability of Riesz potential bounds in weighted settings

Abstract

In [Math. Ineq. \& appl., Vol 26 (2) (2023), 511-530] and [Period. Math. Hung., 89 (1) (2024), 116-128], the present author proved that the Riesz potential $I_{\alpha}$ extends to a bounded operator $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to L^{q(\cdot)}_{\omega}(\mathbb{R}^n)$ and $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to H^{q(\cdot)}_{\omega}(\mathbb{R}^n)$ respectively, under the following two assumptions: $A1)$ $\omega \in \mathcal{W}_{q(\cdot)}$ with $q(\cdot) \in \mathcal{P}^{\log}(\mathbb{R}^{n})$ and $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \frac{\alpha}{n}$; $A2)$ for every cube $Q \subset \mathbb{R}^{n}$, $\| \chi_Q \|_{L^{q(\cdot)}_{\omega}} \approx |Q|^{-\alpha/n} \| \chi_Q \|_{L^{p(\cdot)}_{\omega}}$. In this note, we re-establish such estimates for $I_{\alpha}$ without assuming the hypothesis $A2)$. These proofs are simpler than the previous ones.