Maximal estimates for bilinear Bochner-Riesz means

TL;DR

This paper improves and sharpens $L^p$ estimates for maximal bilinear Bochner-Riesz means across all dimensions, extending prior results and introducing new decomposition techniques for the multiplier.

Contribution

It introduces a novel decomposition of the bilinear Bochner-Riesz multiplier and establishes sharp $L^p$ estimates, extending previous work by Jeong and Lee.

Findings

Established improved $L^p$ bounds for bilinear Bochner-Riesz means.

Extended the results to all dimensions $n \\geq 1$.

Developed a new decomposition method for the multiplier.

Abstract

We establish improved and sharp $L^p$ estimates for the maximal bilinear Bochner-Riesz means in all dimensions $n\geq 1$. This work extends the results proved by Jeong and Lee \cite{JL}. We also recover the known results for the bilinear Bochner-Riesz means. The method of proof involves a new decomposition of the bilinear Bochner-Riesz multiplier $(1-|\xi|^2-|\eta|^2)_+^{\alpha}$ and delicate analysis in proving $L^p$ estimates for frequency localized square functions.