Sparse bounds for maximal triangle and bilinear spherical averaging operators

TL;DR

This paper extends recent methods to establish sparse bounds for bilinear and multilinear maximal operators, including spherical and triangle averages, based on $L^p$-improving bounds and continuity estimates.

Contribution

It adapts the framework of Roncal, Shrivastava, and Shuin to a broader class of bilinear operators beyond product type, establishing sparse bounds from $L^p$-improving properties.

Findings

Sparse bounds are derived for bilinear spherical and triangle averaging operators.

The framework applies to bilinear convolutions with compactly supported measures.

The method extends to a general class of bilinear operators satisfying certain estimates.

Abstract

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that $L^p$-improving bounds in the interior of the boundedness region for the corresponding single-scale maximal operators imply sparse bounds for the correpsonding full maximal operators. More generally we show that the framework applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $L^p$-improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted to obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $L^p$ exponents.