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A BILINEAR RUBIO DE FRANCIA INEQUALITY FOR ARBITRARY SQUARES

Part of: Rings and algebras with additional structure Modules, bimodules and ideals

Published online by Cambridge University Press:  19 September 2016

CRISTINA BENEA  and
FRÉDÉRIC BERNICOT
CRISTINA BENEA
Affiliation:
CNRS — Université de Nantes, Laboratoire Jean Leray, Nantes 44322, France; cristina.benea@univ-nantes.fr, frederic.bernicot@univ-nantes.fr
FRÉDÉRIC BERNICOT
Affiliation:
CNRS — Université de Nantes, Laboratoire Jean Leray, Nantes 44322, France; cristina.benea@univ-nantes.fr, frederic.bernicot@univ-nantes.fr

Abstract

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We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane

$$\begin{eqnarray}(f,g)\mapsto \biggl(\mathop{\sum }_{\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}}\biggl|\int _{\mathbb{R}^{2}}\hat{f}(\unicode[STIX]{x1D709}){\hat{g}}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D714}}(\unicode[STIX]{x1D709},\unicode[STIX]{x1D702})e^{2\unicode[STIX]{x1D70B}ix(\unicode[STIX]{x1D709}+\unicode[STIX]{x1D702})}\,d\unicode[STIX]{x1D709}\,d\unicode[STIX]{x1D702}\biggr|^{r}\biggr)^{1/r},\end{eqnarray}$$
provided $r>2$. More exactly, we show that the above operator maps $L^{p}\times L^{q}\rightarrow L^{s}$ whenever $p,q,s^{\prime }$ are in the ‘local $L^{r^{\prime }}$’ range, that is,
$$\begin{eqnarray}\frac{1}{p}+\frac{1}{q}+\frac{1}{s^{\prime }}=1,\quad 0\leqslant \frac{1}{p},\frac{1}{q}<\frac{1}{r^{\prime }},\quad \text{and}\quad \frac{1}{s^{\prime }}<\frac{1}{r^{\prime }}.\end{eqnarray}$$
 Note that we allow for negative values of $s^{\prime }$, which correspond to quasi-Banach spaces $L^{s}$.

MSC classification

Primary: 16W10: Rings with involution; Lie, Jordan and other nonassociative structures
Secondary: 16D50: Injective modules, self-injective rings
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e26
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

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