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

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

[1]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[2]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[3]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[4]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009

[5]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190

[6]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363

[7]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[8]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[9]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[10]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017

[11]

Saima Rashid, Fahd Jarad, Zakia Hammouch. Some new bounds analogous to generalized proportional fractional integral operator with respect to another function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021020

[12]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[13]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[14]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (134)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]