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