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Sobolev norm estimates for a class of bilinear multipliers

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  • 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 $0$.
    Mathematics Subject Classification: 42B15.

    Citation:

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  • [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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