The bilinear Bochner-Riesz problem

Frederic Bernicot (1) , Loukas Grafakos (2) , Liang Song (3) , Lixin Yan (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 MU Mathematics - Department of Mathematics - University of Missouri
3 Department of Mathematics, Sun Yat-sen University

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.

Mots clés: Bilinear Bochner-Riesz problem; bilinear Fourier multipliers; bilinear restriction-extension operators

Domaines: Analyse classique [math.CA]

Frederic Bernicot, Loukas Grafakos, Liang Song, Lixin Yan. The bilinear Bochner-Riesz problem. Journal d'analyse mathématique, 2015, 127, pp.179-217. ⟨hal-00766114v2⟩