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May  2014, 13(3): 1305-1315. doi: 10.3934/cpaa.2014.13.1305

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, Croatia

Received  July 2013 Revised  September 2013 Published  December 2013

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: bilinear multiplier, pseudodifferential operator, Bilinear Hilbert transform, Sobolev space., paraproduct.
Mathematics Subject Classification: 42B1.
Citation: Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305
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.

show all references

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.

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