The bilinear Bochner-Riesz problem

TL;DR

This paper investigates the boundedness of bilinear Bochner-Riesz multipliers, establishing optimal results for their behavior from L^2×L^2 to L^1 with minimal smoothness, and extending to general radial multipliers.

Contribution

It provides the first optimal boundedness results for bilinear Bochner-Riesz means with minimal smoothness and extends the analysis to general radial bilinear multipliers.

Findings

Boundedness from L^2×L^2 to L^1 for all δ>0

Estimates for other space pairs with larger δ

Techniques include Fourier series, orthogonality, and bilinear restriction theorems

Abstract

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|\xi|^2-|\eta|^2)^\delta_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $\delta>0$, and we obtain estimates for other pairs of spaces for larger values of $\delta$. Our study is broad enough to encompass general bilinear multipliers $m(\xi,\eta)$ radial in $\xi$ and $\eta$ with minimal smoothness, measured in Sobolev space norms. Our results are based on a variety of techniques, that include Fourier series expansions, orthogonality, and bilinear restriction and extension theorems.