Survey on bilinear spherical averages and associated maximal operators

TL;DR

This survey reviews recent advances in understanding $L^{p}$ bounds for bilinear spherical averages and related maximal functions, highlighting known results, open questions, and potential generalizations.

Contribution

It compiles and discusses the latest progress, open problems, and possible extensions in the study of bilinear spherical averages and their maximal operators.

Findings

Describes necessary conditions for $L^{p}$ bounds.

Summarizes known boundedness regions.

Highlights open questions and future directions.

Abstract

In this survey, we collect recent progress in the understanding of $L^{p}$ bounds for bilinear spherical averages and some associated maximal functions like the bilinear spherical maximal function and its lacunary counterpart. We describe necessary conditions satisfied by triples in the $L^{p}$ improving region of a bilinear spherical averaging operator and the localized bilinear spherical maximal function, as well as describe the best-known boundedness regions to date. We state some open questions along the way to motivate future research on this topic, and we exploit some possible generalizations.