Weak type estimates for Bochner--Riesz means on Hardy-type spaces associated with ball quasi-Banach function spaces

TL;DR

This paper establishes weak type estimates for Bochner--Riesz means on Hardy-type spaces associated with ball quasi-Banach function spaces, covering critical cases and various specific function spaces.

Contribution

It proves boundedness of Bochner--Riesz means on Hardy spaces related to ball quasi-Banach spaces, including critical cases and multiple examples.

Findings

Boundedness of Bochner--Riesz means from Hardy to weak Hardy spaces

Extension to critical case of Bochner--Riesz means

Applicability to weighted Lebesgue, Herz, Lorentz, variable Lebesgue, and Morrey spaces

Abstract

Let $X\left(\mathbb{R}^{n}\right)$ be a ball quasi-Banach function space on $\mathbb{R}^{n}$, $WX\left(\mathbb{R}^{n}\right)$ be the weak ball quasi-Banach function space on $\mathbb{R}^{n}$, $H_{X}\left(\mathbb{R}^{n}\right)$ be the Hardy space associated with $X\left(\mathbb{R}^{n}\right)$ and $WH_{X}\left(\mathbb{R}^{n}\right)$ be the weak Hardy space associated with $X\left(\mathbb{R}^{n}\right)$. In this paper, we obtain the boundedness of the Bochner--Riesz means and the maximal Bochner--Riesz means from $H_{X}\left(\mathbb{R}^{n}\right)$ to $WH_{X}\left(\mathbb{R}^{n}\right)$ or $WX\left(\mathbb{R}^{n}\right)$, which includes the critical case. Moreover, we apply these results to several examples of ball quasi-Banach function spaces, namely, weighted Lebesgue spaces, Herz spaces, Lorentz spaces, variable Lebesgue spaces and Morrey spaces. This shows that all the results obtained in this article are of wide applications, and more applications of these results are predictable.