Estimates for Riesz potential on weighted variable Hardy spaces revisited
Pablo Rocha
TL;DR
This work addresses the boundedness of the Riesz potential $I_{\alpha}$ on weighted variable Hardy spaces byRemoving the previous $A2$ hypothesis. It develops two vector-valued inequalities in the weighted variable setting and employs the maximal characterization of $H^{p(\cdot)}_{\omega}$ to obtain $I_{\alpha}$-boundedness directly from atomic decompositions, avoiding the $A2$-based framework. The main contributions prove that $I_{\alpha}$ extends to a bounded operator $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n}) \to L^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$ and $H^{p(\cdot)}_{\omega}(\mathbb{R}^{n}) \to H^{q(\cdot)}_{\omega}(\mathbb{R}^{n})$ under $0<\alpha<n$, $q(\cdot) \in \mathcal{P}^{\log}(\mathbb{R}^{n})$, and $\omega \in \mathcal{W}_{q(\cdot)}$, with the relation $\frac{1}{p(\cdot)}=\frac{1}{q(\cdot)}+\frac{\alpha}{n}$. The approach yields simpler, self-contained proofs and broadens the applicability of Riesz potential estimates in the weighted variable-exponent setting, pertinent to harmonic analysis and PDEs with non-constant growth.
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 $I_α$ extends to a bounded operator $H^{p(\cdot)}_ω(\mathbb{R}^n) \to L^{q(\cdot)}_ω(\mathbb{R}^n)$ and $H^{p(\cdot)}_ω(\mathbb{R}^n) \to H^{q(\cdot)}_ω(\mathbb{R}^n)$ respectively, under the following two assumptions: $A1)$ $ω\in \mathcal{W}_{q(\cdot)}$ with $q(\cdot) \in \mathcal{P}^{\log}(\mathbb{R}^{n})$ and $\frac{1}{p(\cdot)} := \frac{1}{q(\cdot)} + \fracα{n}$; $A2)$ for every cube $Q \subset \mathbb{R}^{n}$, $\| χ_Q \|_{L^{q(\cdot)}_ω} \approx |Q|^{-α/n} \| χ_Q \|_{L^{p(\cdot)}_ω}$. In this note, we re-establish such estimates for $I_α$ without assuming the hypothesis $A2)$. These proofs are simpler than the previous ones.