$L^{p}-$estimates for uncentered spherical averages and lacunary maximal functions

TL;DR

This paper introduces bilinear analogues of uncentered spherical averages and maximal functions, establishing $L^p$-estimates in higher dimensions and revealing new linear case results, including $L^p$-improving properties and lacunary maximal function bounds.

Contribution

It develops the first bilinear uncentered spherical averages and maximal functions, providing $L^p$-estimates and extending linear results to this new bilinear setting.

Findings

Established $L^p$-estimates for bilinear maximal functions in dimensions $d\,\geq 2$

Proved $L^p$-improving properties for single scale averaging operators

Derived $L^p$-estimates for lacunary maximal functions in the linear case

Abstract

The primary goal of this paper is to introduce bilinear analogues of uncentered spherical averages, Nikodym averages associated with spheres and the associated bilinear maximal functions. We obtain $L^p$-estimates for uncentered bilinear maximal functions for dimensions $d\geq2$. Moreover, we also discuss the one-dimensional case. In the process of developing these results, we also establish new and interesting results in the linear case. In particular, we will prove $L^p$-improving properties for single scale averaging operators and $L^p$-estimates for lacunary maximal functions in this context.