Skip to main content
Log in

An improvement of the Marcinkiewicz multiplier theorem

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript


We provide an improvement of the Marcinkiewicz multiplier theorem on ℝn by relaxing the integrability hypothesis on the multiplier. In particular, when r> 1 and min(s1, …, sn) > 1/r, we replace the product-type Sobolev norm \({\left\| {{{(I - \partial _1^2)}^{{s_1}/2}} \cdots {{(I - \partial _n^2)}^{{s_n}/2}}[\Phi \,g]} \right\|_{{L^r}}}\) by the larger Sobolev norm \({\left\| {{{(I - \partial _1^2)}^{{s_1}/2}} \cdots {{(I - \partial _n^2)}^{{s_n}/2}}[\Phi \,g]} \right\|_{{L^{q,1}}}}\) built upon the Lorentz space with first indices q = 1/min(s1,…,sn) and 1. Here si > 0 are distinct numbers and Φ is a function with compact support away from the origin.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others


  1. C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Vol. 129, Academic Press, Boston, MA, 1988.

    MATH  Google Scholar 

  2. A. Carbery, Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem, Université de Grenoble. Annales de l’Institut Fourier 38 (1988), 157–168.

    Article  MathSciNet  Google Scholar 

  3. A. Carbery and A. Seeger, Hp- and Lp-Variants of multiparameter Calderón-Zygmund theory, Transactions of the American Mathematical Society 334 (1992), 719–747.

    MathSciNet  MATH  Google Scholar 

  4. A. Carbery and A. Seeger, Homogeneous Fourier multipliers of Marcinkiewicz type, Arkiv for Matematik 33 (1995), 45–80.

    Article  MathSciNet  Google Scholar 

  5. A. Córboda and R. Fefferman, A geometric proof of the strong maximal theorem, Annals of Mathematics 102 (1975), 95–100.

    Article  MathSciNet  Google Scholar 

  6. C. Fefferman and E. M. Stein, Some maximal inequalities, American Journal of Mathematics 93 (1971), 107–115.

    Article  MathSciNet  Google Scholar 

  7. L. Grafakos, Classical Fourier Analysis, Graduate Texts in Mathematics, Vol. 249, Springer, New York, 2014.

    MATH  Google Scholar 

  8. L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics, Vol. 250, Springer, New York, 2014.

    MATH  Google Scholar 

  9. L. Grafakos, Some remarks on the Miklhin-Hörmander and Marcinkiewicz multiplier theorems: a short historical account and recent improvements, Journal of Geometric Analysis, to appear,

  10. L. Grafakos and H. V. Nguyen, The Hörmander Multiplier Theorem, III: The complete bilinear case via interpolation, Monatschfete für Mathematik 190 (2019), 735–753.

    Article  Google Scholar 

  11. L. Grafakos and S. Oh, The Kato-Ponce inequality, Communications in Partial Differential Equations 39 (2014), 1128–1157.

    Article  MathSciNet  Google Scholar 

  12. L. Grafakos and L. Slavíková, The Marcinkiewicz multiplier theorem revisited, Archiv der Mathematik 112 (2019), 191–203.

    Article  MathSciNet  Google Scholar 

  13. L. Grafakos and L. Slavíková, A sharp version of the Hörmander multiplier theorem, International Mathematics Research Notices 2019 (2019), 4764–4783.

    Article  Google Scholar 

  14. L. Hörmander, Estimates for translation invariant operators in Lpspaces, Acta Mathematica 104 (1960), 93–139.

    Article  MathSciNet  Google Scholar 

  15. T. Hytönen, Fourier embeddings and Mihlin-type multiplier theorems, Mathematische Nachrichten 274–275 (2004), 74–103.

    Article  MathSciNet  Google Scholar 

  16. B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fundamenta Mathematicae 25 (1935), 217–234.

    Article  Google Scholar 

  17. J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier, Studia Mathematica 8 (1939), 78–91.

    Article  MathSciNet  Google Scholar 

  18. S. G. Mikhlin, On the multipliers of Fourier integrals, Doklady Akademii Nauk SSSR 109 (1956), 701–703.

    MathSciNet  Google Scholar 

  19. L. Slavíková, On the failure of the Hörmander multiplier theorem in a limiting case, Revista Matemática Iberoamerican 36 (2020), 1013–1020.

    Article  MathSciNet  Google Scholar 

  20. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, Vol. 30, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

Download references


The author would like to thank Harsh Kumar for useful conversations related to this problem. He also thanks the referee for comments that led to the improvement of the exposition.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Loukas Grafakos.

Additional information

The author acknowledges the support of the Simons Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grafakos, L. An improvement of the Marcinkiewicz multiplier theorem. Isr. J. Math. 244, 163–184 (2021).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: