The boundedness of Bochner-Riesz operators on the weighted weak Hardy spaces

TL;DR

This paper proves the boundedness of Bochner-Riesz operators on weighted weak Hardy spaces, extending known results to new weighted and unweighted cases using atomic decomposition techniques.

Contribution

It establishes the boundedness of maximal and standard Bochner-Riesz operators on weighted weak Hardy spaces for the first time, including unweighted scenarios.

Findings

Boundedness of maximal Bochner-Riesz operators from $WH^p_w$ to $WL^p_w$.

Boundedness of Bochner-Riesz operators on $WH^p_w$.

Results are new even without weights.

Abstract

Let $w$ be a Muckenhoupt weight and $WH^p_w(\mathbb R^n)$ be the weighted weak Hardy spaces. In this paper, by using the atomic decomposition of $WH^p_w(\mathbb R^n)$, we will show that the maximal Bochner-Riesz operators $T^\delta_*$ are bounded from $WH^p_w(\mathbb R^n)$ to $WL^p_w(\mathbb R^n)$ when $0<p\le1$ and $\delta>n/p-{(n+1)}/2$. Moreover, we will also prove that the Bochner-Riesz operators $T^\delta_R$ are bounded on $WH^p_w(\mathbb R^n)$ for $0<p\le1$ and $\delta>n/p-{(n+1)}/2$. Our results are new even in the unweighted case.