Skip to main content
SpringerLink
Log in

Examples of Fourier Multipliers of the Sobolev Space W1,1(ℝd)

  • Published:
Analysis Mathematica Aims and scope Submit manuscript

Abstract

Fourier multipliers of the space W1,1(ℝd) are bounded functions m such that the convolution by F−1m extends into a bounded operator on W1,1(ℝd). Poornima exhibited in the eighties a family of such Fourier multipliers among which some are not Fourier transforms of bounded measures for d > 1. Her counterexamples are based on celebrated non-inequalities of Ornstein. We will give new examples of Fourier multipliers, which are not Fourier transforms of measures and do not belong to her family. The examples are constructed on the d-dimensional torus and then transferred to the Euclidean space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Bonami, Fourier multipliers of the homogeneous Sobolev space Ẇ 11 , in: Harmonic Analysis, Partial Differential Equations, Complex Analysis, Banach Spaces, and Operator Theory, Vol. 1, Assoc. Women Math. Ser., 4, Springer (Cham, 2016), pp. 135–142.

    Google Scholar 

  2. A. Bonami and S. Poornima, Nonmultipliers of the Sobolev spaces W k,1(ℝn), J. Funct. Anal., 71 (1987), 175–181.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Conti, D. Faraco and F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal., 175 (2005), 287–300.

    Google Scholar 

  4. M. Jodeit, Restrictions and extensions of Fourier multipliers, Studia Math., 34 (1970) 215–226.

    MATH  Google Scholar 

  5. K. Kazaniecki and M. Wojciechowski, On the continuity of Fourier multipliers on the homogeneous Sobolev spaces Ẇ 11 (ℝd), Ann. Inst. Fourier (Grenoble), 66 (2016), 1247–1260.

    Article  MathSciNet  MATH  Google Scholar 

  6. K. Kazaniecki, D. Stolyarov and M. Wojciechowski, Anisotropic Ornstein noninequalities, Anal. PDE, 10 (2017), 351–366.

    Article  Google Scholar 

  7. D. Ornstein, A non-inequality for differential operators in the L1 norm, Arch. Rational Mech. Anal., 11 (1962), 40–49.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Poornima, Multipliers of the Sobolev spaces, J. Funct. Anal., 45 (1982), 1–27.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Schikorra, D. Spector and J. Van Schaftingen, An L1-type estimate for Riesz potentials, Rev. Mat. Iberoam., 33 (2017), 291–303.

    Google Scholar 

  10. S. Wainger, Special trigonometric series in k-dimensions, Mem. Amer. Math. Soc., 59 (1965), 102 pp.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Denis Poisson, CNRS-UMR 2013, Université d’Orléans, France

    A. Bonami

  2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, India

    P. Mohanty

Authors
  1. A. Bonami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Mohanty
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Bonami.

Additional information

Dedicated to the memory of Jean-Pierre Kahane

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonami, A., Mohanty, P. Examples of Fourier Multipliers of the Sobolev Space W1,1(ℝd). Anal Math 44, 325–334 (2018). https://doi.org/10.1007/s10476-018-0504-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10476-018-0504-6

Key words and phrases

Mathematics Subject Classification

Search

Navigation