Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions

TL;DR

This paper establishes sharp endpoint $L^p$ estimates for bilinear and multilinear spherical maximal functions, advancing understanding of their boundedness properties especially in low dimensions and for local variants.

Contribution

It provides new endpoint estimates for bilinear spherical maximal functions in dimensions 1 and 2, and improves bounds for local spherical maximal functions in all dimensions.

Findings

Borderline restricted weak type estimates for bilinear spherical maximal functions in dimensions 1 and 2.

Sharp endpoint estimates for multilinear spherical maximal functions.

Improved $L^p$ bounds for local spherical maximal functions in all dimensions.

Abstract

In this article, we address endpoint issues for the bilinear spherical maximal functions. We obtain borderline restricted weak type estimates for the well studied bilinear spherical maximal function $$\mathfrak{M}(f,g)(x):=\sup_{t>0}\left|\int_{\mathbb S^{2d-1}}f(x-ty_1)g(x-ty_2)\;d\sigma(y_1,y_2)\right|,$$ in dimensions $d=1,2$ and as an application, we deduce sharp endpoint estimates for the multilinear spherical maximal function. We also prove $L^p-$estimates for the local spherical maximal function in all dimensions $d\geq 2$, thus improving the boundedness left open in the work of Jeong and Lee (https://doi.org/10.1016/j.jfa.2020.108629). We further study necessary conditions for the bilinear maximal function, \[\mathcal M (f,g)(x)=\sup_{t>0}\left|\int_{\mathbb S^{1}}f(x-ty)g(x+ty)\;d\sigma(y)\right|\] to be bounded from $L^{p_1}(\mathbb R^2)\times L^{p_2}(\mathbb R^2)$ to $L^p(\mathbb R^2)$ and prove sharp results for a linearized version of $\mathcal M$.