Estimates for Riesz potential on weighted variable Hardy spaces revisited

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 $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)\to L_{\omega }^{q(\cdot )}({\mathbb{R}}^n)$ and $H_{\omega }^{p(\cdot )}({\mathbb{R}}^n)\to H_{\omega }^{q(\cdot )}({\mathbb{R}}^n)$ 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 $Q\subset {\mathbb{R}}^n$, $\mathrm{\parallel }{\chi }_Q{\mathrm{\parallel }}_{L_{\omega }^{q(\cdot )}}\approx \mathrm{\mid }Q{\mathrm{\mid }}^{-\alpha \mathrm{/}n}\mathrm{\parallel }{\chi }_Q{\mathrm{\parallel }}_{L_{\omega }^{p(\cdot )}}$. In this note, we re-establish such estimates for $I_{\alpha }$ without assuming the hypothesis $A2)$. These proofs are simpler than the previous ones.