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The bilinear Bochner-Riesz problem

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Journal d'Analyse Mathématique Aims and scope

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.

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Authors and Affiliations

  1. Laboratoire Jean Leray, Cnrs - Université De Nantes, 44322, Nantes Cedex 3, France

    Frédéric Bernicot

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

  3. Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275, P. R. China

    Liang Song & Lixin Yan

Authors
  1. Frédéric Bernicot
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  2. Loukas Grafakos
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  3. Liang Song
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  4. Lixin Yan
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Corresponding author

Correspondence to Frédéric Bernicot.

Additional information

F. Bernicot is supported by the ANR under the project AFoMEN no. 2011-JS01-001-01.

L. Grafakos is supported by the National Science Foundation (USA) under grant number 0900946.

L. Song is supported by the Fundamental Research Funds for the Central Universities.

L. Yan is supported by NNSF of China (Grants No. 10925106 and 11371378), Guangdong Province Key Laboratory of Computational Science, and Grant for Senior Scholars from the Association of Colleges and Universities of Guangdong.

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Bernicot, F., Grafakos, L., Song, L. et al. The bilinear Bochner-Riesz problem. JAMA 127, 179–217 (2015). https://doi.org/10.1007/s11854-015-0028-y

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  • DOI: https://doi.org/10.1007/s11854-015-0028-y

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