The bilinear Bochner-Riesz problem

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.