Fractional type operators on Hardy spaces associated with ball quasi-Banach function spaces

TL;DR

This paper studies fractional operators on Hardy spaces linked to ball quasi-Banach spaces, establishing boundedness results that extend classical theory to new spaces like Lorentz and Orlicz spaces.

Contribution

It introduces boundedness results for fractional operators on Hardy spaces associated with ball quasi-Banach spaces, including new applications to Lorentz and Orlicz spaces.

Findings

Fractional operators extend boundedly between Hardy and quasi-Banach spaces.

Results apply to weighted, variable Lebesgue, Lorentz, and Orlicz spaces.

New boundedness results for Lorentz and Orlicz spaces.

Abstract

For $0 \leq \alpha < n$ and $m \in \mathbb{N} \cap \left(1 - \frac{\alpha}{n}, +\infty \right)$, we consider certain fractional type operators $T_{\alpha, m}$ generated by $m$-orthogonal matrices and prove that, for $0 < \alpha < n$, $T_{\alpha, m}$ can be extended to a bounded operator $H_X \to Y$ and, for $\alpha = 0$, $T_{0, m}$ can be extended to a bounded operator $H_X \to X$, where $X$ and $Y$ are certain ball quasi-Banach spaces related to each other and $H_X$ is the Hardy space associated with $X$. In particular, our results apply to weighted Lebesgue spaces, variable Lebesgue spaces, Lorentz spaces and Orlicz spaces, the last two are new. Our proofs rely on the ssumption that $X$ is $\mathcal{O}(n)$-invariant, the theory of weighted Hardy spaces, the Rubio de Francia iteration algorithm and the finite atomic decomposition of $H_X$.