Illinois Journal of Mathematics

The Hörmander multiplier theorem, I: The linear case revisited

Loukas Grafakos, Danqing He, Petr Honzik, and Hanh Van Nguyen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We discuss $L^{p}(\mathbb{R}^{n})$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance $\vert \frac{1}{p}-\frac{1}{2}\vert $ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $\vert \frac{1}{p}-\frac{1}{2}\vert <\frac{s}{n}$ and we discuss the endpoint case $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.

Article information

Illinois J. Math., Volume 61, Number 1-2 (2017), 25-35.

Received: 2 December 2016
Revised: 17 November 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B15: Multipliers 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces 42B99: None of the above, but in this section


Grafakos, Loukas; He, Danqing; Honzik, Petr; Van Nguyen, Hanh. The Hörmander multiplier theorem, I: The linear case revisited. Illinois J. Math. 61 (2017), no. 1-2, 25--35. doi:10.1215/ijm/1520046207.

Export citation


  • J. Bergh and J. Löfström, Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin–New York, 1976.
  • A. P. Calderón and A. Torchinsky, Parabolic maximal functions associated with a distribution, II, Adv. Math. 24 (1977), 101–171.
  • A. Carbery, G. Gasper and W. Trebels, On localized potential spaces, J. Approx. Theory 48 (1986), 251–261.
  • M. Christ, On almost everywhere convergence of Bochner–Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985), 16–20.
  • W. C. Connett and A. L. Schwartz, A remark about Calderón's upper s method of interpolation, Interpolation spaces and allied topics in analysis (Lund, 1983), Lecture Notes in Math., vol. 1070, Springer, Berlin, 1984, pp. 48–53.
  • L. Grafakos, Classical Fourier analysis, 3rd ed., GTM, vol. 249, Springer-Verlag, NY, 2014.
  • L. Grafakos and M. Mastyło, Analytic families of multilinear operators, Nonlinear Anal. 107 (2014), 47–62.
  • L. Grafakos and S. Oh, The Kato–Ponce inequality, Comm. Partial Differential Equations 39 (2014), 1128–1157.
  • I. I. \bsuffixJr. Hirschman, On multiplier transformations, Duke Math. J. 26 (1959), 221–242.
  • L. Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–139.
  • T. Kato and G. Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.
  • S. G. Mikhlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956), 701–703. (In Russian.)
  • A. Miyachi, On some Fourier multipliers for $H^{p}(\mathbb{R} ^{n})$, J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math. 27 (1980), 157–179.
  • A. Miyachi and N. Tomita, Minimal smoothness conditions for bilinear Fourier multipliers, Rev. Mat. Iberoam. 29 (2013), 495–530.
  • V. A. Olevskii, A note on Fourier multipliers and Sobolev spaces, Functions, series, operators (Budapest, 1999), Janos Bolyai Math. Soc., Budapest, 2002, pp. 321–325.
  • A. Seeger, A limit case of the Hörmander multiplier theorem, Monatsh. Math. 105 (1988), 151–160.
  • A. Seeger, Estimates near $L^{1}$ for Fourier multipliers and maximal functions, Arch. Math. (Basel) 53 (1989), 188–193.
  • A. Seeger, Remarks on singular convolution operators, Studia Math. 97 (1990), 91–114.
  • E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1971.
  • S. Wainger, Special trigonometric series in k-dimensions, Mem. Amer. Math. Soc. 59 (1965), 1–102.
  • T. H. Wolff, Lectures on harmonic analysis, (I. Łaba and C. Shubin, eds.), University Lecture Series, vol. 29, Amer. Math. Soc., Providence, RI, 2003. With a foreword by C. Fefferman and preface by I. Łaba.