A new estimate for Bochner-Riesz operators at the critical index on the weighted Hardy spaces

TL;DR

This paper establishes a new boundedness result for Bochner-Riesz operators at the critical index on weighted Hardy spaces, extending known results to the weighted setting and even the unweighted case.

Contribution

It introduces a novel estimate for Bochner-Riesz operators at the critical index on weighted Hardy spaces using atomic decomposition techniques.

Findings

Bochner-Riesz operators are bounded from weighted Hardy spaces to weak Hardy spaces at the critical index.

The result is new even in the unweighted setting.

The proof employs atomic decomposition of weighted Hardy spaces.

Abstract

Let $w$ be a Muckenhoupt weight and $H^p_w(\mathbb R^n)$ be the weighted Hardy spaces. In this paper, by using the atomic decomposition of $H^p_w(\mathbb R^n)$, we will show that the Bochner-Riesz operators $T^\delta_R$ are bounded from $H^p_w(\mathbb R^n)$ to the weighted weak Hardy spaces $WH^p_w(\mathbb R^n)$ when $0<p<1$ and $\delta=n/p-(n+1)/2$. This result is new even in the unweighted case.