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.

32 pages