Maximal estimates for the bilinear spherical averages and the bilinear Bochner-Riesz operators

TL;DR

This paper establishes optimal $L^p imes L^q o L^r$ bounds for bilinear spherical maximal functions and improves estimates for bilinear Bochner-Riesz operators, resolving longstanding open problems.

Contribution

It provides the first optimal range estimates for the bilinear spherical maximal function and enhances known bounds for bilinear Bochner-Riesz operators, connecting maximal and square function estimates.

Findings

Optimal $L^p imes L^q o L^r$ estimates for bilinear spherical maximal function.

Improved bounds for bilinear Bochner-Riesz operators.

Resolution of previously open problems in bilinear harmonic analysis.

Abstract

We study the maximal estimates for the bilinear spherical average and the bilinear Bochner-Riesz operator. Firstly, we obtain $L^p\times L^q \to L^r$ estimates for the bilinear spherical maximal function on the optimal range. Thus, we settle the problem which was previously considered by Geba, Greenleaf, Iosevich, Palsson and Sawyer, later Barrionevo, Grafakos, D. He, Honz\'ik and Oliveira, and recently Heo, Hong and Yang. Secondly, we consider $L^p\times L^q \to L^r$ estimates for the maximal bilinear Bochner-Riesz operators and improve the previous known results. For the purpose we draw a connection between the maximal estimates and the square function estimates for the classical Bochner-Riesz operators.