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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

References

Benea, C. and Muscalu, C., ‘Multiple vector valued inequalities via the helicoidal method’, Anal. PDE, to appear.Google Scholar
Bernicot, F., ‘ L p boundedness for nonsmooth bilinear Littlewood–Paley square functions’, Math. Ann. 2351(1) (2011), 1–49.CrossRefGoogle Scholar
Bernicot, F., Grafakos, L., Song, L. and Yan, L., ‘The bilinear Bochner–Riesz problem’, J. Anal. Math. 127 (2015), 179–217.Google Scholar
Bernicot, F. and Shrivastava, S., ‘Boundedness of smooth bilinear square functions and applications to some bilinear pseudo-differential operators’, Indiana Univ. Math. J. 60(1) (2011), 233–268.CrossRefGoogle Scholar
Carleson, L., ‘On convergence and growth of partial sums of Fourier series’, Acta Math. 116 (1966), 135–157.Google Scholar
Coifman, R. and Meyer, Y., Wavelets, Cambridge Studies in Advanced Mathematics, 48 (Cambridge University Press, Cambridge, 1997), Calderón-Zygmund and multilinear operators, translated from the 1990 and 1991 French originals by David Salinger.Google Scholar
Diestel, G., ‘Some remarks on bilinear Littlewood–Paley theory’, J. Math. Anal. Appl. 307(1) (2005), 102–119.Google Scholar
Diestel, G. and Grafakos, L., ‘Maximal bilinear singular integral operators associated with dilations of planar sets’, J. Math. Anal. Appl. 332(2) (2007), 1482–1494.CrossRefGoogle Scholar
de Francia, J. R., ‘A Littlewood–Paley inequality for arbitrary intervals’, Rev. Mat. Iberoam. 1(2) (1985), 891–921.Google Scholar
Grafakos, L. and Li, X., ‘The disc as a bilinear multiplier’, Amer. J. Math. 128(1) (2006), 91–119.CrossRefGoogle Scholar
Hunt, R. A., ‘On the convergence of Fourier series’, inOrthogonal Expansions and their Continuous Analogues (Proc. Conf., Edwardsville, IL, 1967) (Southern Illinois University Press, Carbondale, IL, 1968), 235–255.Google Scholar
Journé, J.-L., ‘Calderón–Zygmund operators on product spaces’, Rev. Mat. Iberoam. 1(3) (1985), 55–91.CrossRefGoogle Scholar
Lacey, M., ‘On bilinear Littlewood–Paley square functions’, Publ. Mat. 40(2) (1996), 387–396.Google Scholar
Lacey, M. and Thiele, C., ‘On Calderón’s conjecture’, Ann. of Math. (2) 149(2) (1999), 475–496.Google Scholar
Mohanty, P. and Shrivastava, S., ‘A note on the bilinear Littlewood–Paley square function’, Proc. Amer. Math. Soc. 138(6) (2010), 2095–2098.Google Scholar
Muscalu, C., PhD thesis. Brown University, Providence, USA, 2000.Google Scholar
Muscalu, C. and Schlag, W., Classical and Multilinear Harmonic Analysis (Cambridge University Press, 2013).Google Scholar
Muscalu, C., Tao, T. and Thiele, C., ‘Multi-linear operators given by singular multipliers’, J. Amer. Math. Soc. 15(2) (2002), 469–496.Google Scholar
Muscalu, C., Tao, T. and Thiele, C., ‘ L p estimates for the biest. II. The Fourier case’, Math. Ann. 329(3) (2004), 427–461.Google Scholar
Thiele, C., Wave Packet Analysis, CBMS Regional Conference Series in Mathematics, 105 (American Mathematical Society, Providence, RI, 2006), Published for the Conference Board of the Mathematical Sciences, Washington, DC.CrossRefGoogle Scholar