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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

 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 $0$.
Citation: Frédéric Bernicot, Vjekoslav Kovač. Sobolev norm estimates for a class of bilinear multipliers. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1305-1315. doi: 10.3934/cpaa.2014.13.1305
 [1] Á. Bényi, A. R. Nahmod and R. H. Torres, Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators,, \emph{J. Geom. Anal.}, 16 (2006), 431.  doi: 10.1007/BF02922061.  Google Scholar [2] Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators,, \emph{Comm. Partial Differential Equations}, 28 (2003), 1161.  doi: 10.1081/PDE-120021190.  Google Scholar [3] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Springer-Verlag, 223 (1976).   Google Scholar [4] F. Bernicot, A bilinear pseudodifferential calculus,, \emph{J. Geom. Anal.}, 20 (2010), 39.  doi: 10.1007/s12220-009-9105-8.  Google Scholar [5] F. Bernicot, Fiber-wise Calderón-Zygmund decomposition and application to a bi-dimensional paraproduct,, \emph{Illinois J. Math.}, 56 (2012), 415.   Google Scholar [6] R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels,, Soc. Math. Fr., 57 (1978).   Google Scholar [7] R. Coifman and Y. Meyer, Commutateurs d'intégrales singuliéres et opèrateurs multilinéaires,, \emph{Ann. Inst. Fourier} (Grenoble), 28 (1978), 177.   Google Scholar [8] R. Coifman and Y. Meyer, Ondelettes et opérateurs. III. Opérateurs multilinéaires,, Hermann, (1991).   Google Scholar [9] C. Demeter and C. Thiele, On the two-dimensional bilinear Hilbert transform,, \emph{Amer. J. Math.}, 132 (2010), 201.  doi: 10.1353/ajm.0.0101.  Google Scholar [10] V. Kovač, Boundedness of the twisted paraproduct,, \emph{Rev. Mat. Iberoam.}, 28 (2012), 1143.  doi: 10.4171/RMI/707.  Google Scholar [11] M. Lacey and C. Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2\emph{Ann. of Math.}, 146 (1997), 693. doi: 10.2307/2952458. Google Scholar [12] M. Lacey and C. Thiele, On Calderón's conjecture,, \emph{Ann. of Math.}, 149 (1999), 475. doi: 10.2307/120971. Google Scholar [13] C. Muscalu, T. Tao and C. Thiele, Multi-linear operators given by singular multipliers,, \emph{J. Amer. Math. Soc.}, 15 (2002), 469. doi: 10.1090/S0894-0347-01-00379-4. Google Scholar show all references ##### References:  [1] Á. Bényi, A. R. Nahmod and R. H. Torres, Sobolev space estimates and symbolic calculus for bilinear pseudodifferential operators,, \emph{J. Geom. Anal.}, 16 (2006), 431. doi: 10.1007/BF02922061. Google Scholar [2] Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators,, \emph{Comm. Partial Differential Equations}, 28 (2003), 1161. doi: 10.1081/PDE-120021190. Google Scholar [3] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction,, Springer-Verlag, 223 (1976). Google Scholar [4] F. Bernicot, A bilinear pseudodifferential calculus,, \emph{J. Geom. Anal.}, 20 (2010), 39. doi: 10.1007/s12220-009-9105-8. Google Scholar [5] F. Bernicot, Fiber-wise Calderón-Zygmund decomposition and application to a bi-dimensional paraproduct,, \emph{Illinois J. Math.}, 56 (2012), 415. Google Scholar [6] R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels,, Soc. Math. Fr., 57 (1978). Google Scholar [7] R. Coifman and Y. Meyer, Commutateurs d'intégrales singuliéres et opèrateurs multilinéaires,, \emph{Ann. Inst. Fourier} (Grenoble), 28 (1978), 177. Google Scholar [8] R. Coifman and Y. Meyer, Ondelettes et opérateurs. III. Opérateurs multilinéaires,, Hermann, (1991). Google Scholar [9] C. Demeter and C. Thiele, On the two-dimensional bilinear Hilbert transform,, \emph{Amer. J. Math.}, 132 (2010), 201. doi: 10.1353/ajm.0.0101. Google Scholar [10] V. Kovač, Boundedness of the twisted paraproduct,, \emph{Rev. Mat. Iberoam.}, 28 (2012), 1143. doi: 10.4171/RMI/707. Google Scholar [11] M. Lacey and C. Thiele,$L^p$estimates on the bilinear Hilbert transform for$2\emph{Ann. of Math.}, 146 (1997), 693.  doi: 10.2307/2952458.  Google Scholar [12] M. Lacey and C. Thiele, On Calderón's conjecture,, \emph{Ann. of Math.}, 149 (1999), 475.  doi: 10.2307/120971.  Google Scholar [13] C. Muscalu, T. Tao and C. Thiele, Multi-linear operators given by singular multipliers,, \emph{J. Amer. Math. Soc.}, 15 (2002), 469.  doi: 10.1090/S0894-0347-01-00379-4.  Google Scholar
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