On the bilinear Bochner-Riesz problem at critical index

TL;DR

This paper investigates the behavior of maximal and square functions related to bilinear Bochner-Riesz means at the critical index, establishing weighted estimates and identifying endpoint limitations.

Contribution

It proves weighted $L^p$ estimates for bilinear Bochner-Riesz operators at the critical index and shows failure of weak-type estimates at the endpoint.

Findings

Weighted estimates hold for $p_1,p_2>1$ with $A_{oldsymbol{P}}$ weights.

Operators fail to satisfy weak-type estimates at the endpoint $(1,1,1/2)$.

Results advance understanding of bilinear Bochner-Riesz operators at critical index.

Abstract

In this paper we study maximal and square functions associated with bilinear Bochner-Riesz means at the critical index. In particular, we prove that they satisfy weighted estimates from $L^{p_1}(w_1)\times L^{p_2}(w_2)\rightarrow L^p(v_w)$ for bilinear weights $(w_1,w_2)\in A_{\vec{P}}$ where $p_1,p_2>1$ and $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. Also, we show that both the operators fail to satisfy weak-type estimates at the end-point $(1,1,\frac{1}{2})$.