Estimates for Riesz potential on weighted variable Hardy spaces revisited

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 $I_{\hat{I}}\pm$ extends to a bounded operator $H_{\ddot{I}}^{p(\cdot )}‰({\mathbb{R}}^n)\to L_{\ddot{I}}^{q(\cdot )}‰({\mathbb{R}}^n)$ and $H_{\ddot{I}}^{p(\cdot )}‰({\mathbb{R}}^n)\to H_{\ddot{I}}^{q(\cdot )}‰({\mathbb{R}}^n)$ respectively, under the following two assumptions: $A1)$ $\ddot{I}‰\in {\mathcal{W}}_{q(\cdot )}$ with $q(\cdot )\in {\mathcal{P}}^{\log }({\mathbb{R}}^n)$ and $\frac{1}{p(\cdot )}:=\frac{1}{q(\cdot )}+\frac{\hat{I}}{\pm }n$; $A2)$ for every cube $Q\subset {\mathbb{R}}^n$, $\Vert \ddot{I}{‡}_Q{\Vert }_{L_{\ddot{I}}^{q(\cdot )}‰}\approx |Q{|}^{-\hat{I}\pm /n}\Vert \ddot{I}{‡}_Q{\Vert }_{L_{\ddot{I}}^{p(\cdot )}‰}$. In this note, we re-establish such estimates for $I_{\hat{I}}\pm$ without assuming the hypothesis $A2)$. These proofs are simpler than the previous ones.

11 pages