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