Estimates for convolution operators on Hardy spaces associated with ball quasi-Banach function spaces

Let $0\le \alpha <n$, $N\in \mathbb{N}$, and let $X$ and $Y$ be ball quasi-Banach function spaces on ${\mathbb{R}}^n$. We consider operators $T_{\alpha }$ defined by convolution with kernels of type $(\alpha ,N)$. Assuming that the powered Hardy-Littlewood maximal operator satisfies some Fefferman-Stein vector-valued maximal inequality on $X$ and is bounded on the associated space, we prove that $T_0$, $\alpha =0$, extends to a bounded operator $H_X({\mathbb{R}}^n)\to X$ and $H_X({\mathbb{R}}^n)\to H_X({\mathbb{R}}^n)$; and, under certain additional assumptions on $X$ and $Y$, $T_{\alpha }$, $0<\alpha <n$, extends to a bounded operator $H_X({\mathbb{R}}^n)\to Y$ and $H_X({\mathbb{R}}^n)\to H_Y({\mathbb{R}}^n)$. In particular, from these results, it follows that singular integrals and the Riesz potential satisfy such estimates, respectively. We also provide an off-diagonal Fefferman-Stein vector-valued inequality for the fractional maximal operator on the $p$-convexification of ball quasi-Banach function spaces.