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

TL;DR

This paper establishes boundedness results for convolution operators of type $(eta, N)$ on Hardy spaces associated with ball quasi-Banach function spaces, extending classical results to a broader functional framework.

Contribution

It introduces new boundedness theorems for convolution operators on Hardy spaces linked to ball quasi-Banach spaces, including fractional integrals and singular integrals.

Findings

Boundedness of $T_0$ from Hardy space to $X$ and itself.

Boundedness of $T_eta$ for $0<eta<n$ between Hardy spaces and other spaces.

Off-diagonal Fefferman-Stein inequality for fractional maximal operators.

Abstract

Let $0 \leq \alpha < n$, $N \in \mathbb{N}$, 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.