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Monatshefte für Mathematik

, Volume 190, Issue 4, pp 735–753 | Cite as

The Hörmander multiplier theorem, III: the complete bilinear case via interpolation

  • Loukas GrafakosEmail author
  • Hanh Van Nguyen
Article

Abstract

We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space \(L^r_s({\mathbb {R}}^{2n})\), \(2\le r<\infty \), \(rs>2n\), uniformly over all annuli. More precisely, given such a symbol with smoothness index s, we find the largest open set of indices \((1/p_1,1/p_2 )\) for which we have boundedness for the associated bilinear multiplier operator from \(L^{p_1}({\mathbb {R}}^{ n})\times L^{p_2} ({\mathbb {R}}^{ n})\) to \( L^p({\mathbb {R}}^{ n})\) when \(1/p=1/p_1+1/p_2\), \(1<p_1,p_2<\infty \).

Keywords

Multilinear operator Multiplier operator Interpolation 

Mathematics Subject Classification

42B15 42B30 

Notes

References

  1. 1.
    Calderón, A.P., Torchinsky, A.: Parabolic maximal functions associated with a distribution, II. Adv. Math. 24, 101–171 (1977)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Fujita, M., Tomita, N.: Weighted norm inequalities for multilinear Fourier multipliers. Trans. Am. Math. Soc. 364, 6335–6353 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Grafakos, L.: Classical Fourier Analysis, Graduate Texts in Mathematics, GTM 249, 3rd edn. Springer, New York (2014)Google Scholar
  4. 4.
    Grafakos, L.: Modern Fourier Analysis, Graduate Texts in Mathematics, GTM 250, 3rd edn. Springer, New York (2014)Google Scholar
  5. 5.
    Grafakos, L., He, D., Honzík, P., Nguyen, H.V.: The Hörmander multiplier theorem I: the linear case. Ill. J. Math. 61, 25–35 (2017)CrossRefGoogle Scholar
  6. 6.
    Grafakos, L., He, D., Honzík, P.: The Hörmander multiplier theorem II: the local \(L^2\) case. Math. Zeit. 289, 875–887 (2018)CrossRefGoogle Scholar
  7. 7.
    Grafakos, L., Miyachi, A., Nguyen, H.V., Tomita, N.: Multilinear Fourier multipliers with minimal Sobolev regularity, II. J. Math. Soc. Japan 69, 529–562 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Grafakos, L., Miyachi, A., Tomita, N.: On multilinear Fourier multipliers of limited smoothness. Can. J. Math. 65, 299–330 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Grafakos, L., Nguyen, H.V.: Multilinear Fourier multipliers with minimal Sobolev regularity, I. Colloq. Math. 144, 1–30 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Grafakos, L., Oh, S.: The Kato–Ponce inequality. Comm. PDE 39, 1128–1157 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Grafakos, L., Si, Z.: The Hörmander multiplier theorem for multilinear operators. J. Reine Angew. Math. 668, 133–147 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hirschman Jr., I.I.: On multiplier transformations. Duke Math. J. 26, 221–242 (1959)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–139 (1960)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Mikhlin, S.G.: On the multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR (N.S.) 109, 701–703 (1956)MathSciNetGoogle Scholar
  16. 16.
    Miyachi, A.: On some Fourier multipliers for \(H^p({\mathbb{R}}^{n})\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 157–179 (1980)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers. Rev. Mat. Iberoam. 29, 495–530 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Miyachi, A., Tomita, N.: Boundedness criterion for bilinear Fourier multiplier operators. Tohoku Math. J. 66, 55–76 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Slavíková, L.: On the failure of the Hörmander multiplier theorem in a limiting case. Rev. Mat. Iberoamer (to appear) Google Scholar
  20. 20.
    Stein, E.M., Weiss, G.: On the interpolation of analytic families of operators acting on \(H^{p}\)-spaces. Tohoku Math. J. 9, 318–339 (1957)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tomita, N.: A Hörmander type multiplier theorem for multilinear operators. J. Funct. Anal. 259, 2028–2044 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Wainger, S.: Special trigonometric series in k-dimensions. Mem. Am. Math. Soc. 59, 1–102 (1965)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsThe University of AlabamaTuscaloosaUSA

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