Sobolev norm estimates for a class of bilinear multipliers

Frédéric Bernicot; Vjekoslav Kovač

doi:10.3934/cpaa.2014.13.1305

Article Contents

2014, Volume 13, Issue 3: 1305-1315. Doi: 10.3934/cpaa.2014.13.1305

This issue Previous Article On the Cauchy problem for a generalized Camassa-Holm equation with both quadratic and cubic nonlinearity Next Article Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime

Sobolev norm estimates for a class of bilinear multipliers

Frédéric Bernicot^1, and
Vjekoslav Kovač^2,

1.
Laboratoire Paul Painlevé - CNRS, Université Lille 1, 59655 Villeneuve d’Ascq Cedex
2.
University of Zagreb, Department of Mathematics, Bijenička cesta 30, 10000 Zagreb

Received: June 30, 2013

Revised: August 31, 2013

Published: April 2015

Abstract Related Papers Cited by

Abstract

We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing $\mathrm{L}^p$ estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodifferential operators of order $0$.

Keywords:

Mathematics Subject Classification: 42B15.

Citation:

Related Papers

Cited by

References

[1]	Á. Bényi, A. R. Nahmod and R. H. Torres, Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators, J. Geom. Anal., 16 (2006), 431-453.doi: 10.1007/BF02922061.
[2]	Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators, Comm. Partial Differential Equations, 28 (2003), 1161-1181.doi: 10.1081/PDE-120021190.
[3]	J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der mathematischen Wissenschaften, 223.
[4]	F. Bernicot, A bilinear pseudodifferential calculus, J. Geom. Anal., 20 (2010), 39-62.doi: 10.1007/s12220-009-9105-8.
[5]	F. Bernicot, Fiber-wise Calderón-Zygmund decomposition and application to a bi-dimensional paraproduct, Illinois J. Math., 56 (2012), 415-422.
[6]	R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Soc. Math. Fr., Paris, 1978. Astérisque, 57.
[7]	R. Coifman and Y. Meyer, Commutateurs d'intégrales singuliéres et opèrateurs multilinéaires, Ann. Inst. Fourier (Grenoble), 28 (1978), 177-202.
[8]	R. Coifman and Y. Meyer, Ondelettes et opérateurs. III. Opérateurs multilinéaires, Hermann, Paris, 1991.
[9]	C. Demeter and C. Thiele, On the two-dimensional bilinear Hilbert transform, Amer. J. Math., 132 (2010), 201-256.doi: 10.1353/ajm.0.0101.
[10]	V. Kovač, Boundedness of the twisted paraproduct, Rev. Mat. Iberoam., 28 (2012), 1143-1164.doi: 10.4171/RMI/707.
[11]	M. Lacey and C. Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2Ann. of Math., 146 (1997), 693-724.doi: 10.2307/2952458.
[12]	M. Lacey and C. Thiele, On Calderón's conjecture, Ann. of Math., 149 (1999), 475-496.doi: 10.2307/120971.
[13]	C. Muscalu, T. Tao and C. Thiele, Multi-linear operators given by singular multipliers, J. Amer. Math. Soc., 15 (2002), 469-496.doi: 10.1090/S0894-0347-01-00379-4.