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arXiv · 1212.4018

The bilinear Bochner-Riesz problem

Frederic Bernicot·Loukas Grafakos·Liang Song·Lixin Yan
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Abstract

Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ 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 $δ>0$, 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. Our results are based on a variety of techniques, that include Fourier series expansions, orthogonality, and bilinear restriction and extension theorems.

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BibTeXRIS

Frederic Bernicot, Loukas Grafakos, Liang Song, Lixin Yan. 2013-04-03. The bilinear Bochner-Riesz problem. https://arxiv.org/abs/1212.4018

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