Sparse bounds for the bilinear spherical maximal function

TL;DR

This paper establishes sparse bounds for the bilinear spherical maximal function across all dimensions, leading to sharp $L^p\times L^q \to L^r$ bounds and new weighted inequalities, via novel continuity estimates.

Contribution

It introduces new continuity $L^p$ improving estimates for the bilinear spherical averaging operator, enabling sparse bounds and weighted inequalities.

Findings

Sharp $L^p\times L^q \to L^r$ bounds derived

Quantitative weighted norm inequalities established

New continuity estimates for bilinear spherical averages

Abstract

We derive sparse bounds for the bilinear spherical maximal function in any dimension $d\geq 1$. When $d\geq 2$, this immediately recovers the sharp $L^p\times L^q\to L^r$ bound of the operator and implies quantitative weighted norm inequalities with respect to bilinear Muckenhoupt weights, which seems to be the first of their kind for the operator. The key innovation is a group of newly developed continuity $L^p$ improving estimates for the single scale bilinear spherical averaging operator.