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 $H^{p(\cdot)}_{\omega}(\mathbb{R}^n) \to L^{q(\cdot)}_{\omega}(\mathbb{R}^n)$ 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.