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A Multidimensional Hardy–Littlewood Theorem

  • Elijah Liflyand
  • Ulrich Stadtmüller
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Two classical results by F. and M. Riesz on absolute continuity and by Hardy and Littlewood on the absolutely convergence of Fourier series for a function f with bounded variation, whose conjugate is also of bounded variation, are generalized. We improve earlier obtained one-dimensional non-periodic versions and present multidimensional extensions for Hardy’s variation.

Keywords

Hardy–Littlewood theorem Brothers Riesz theorem Absolute continuity Fourier transform Hilbert transform Bounded variation Hardy space Hardy’s variation 

2010 Mathematics Subject Classification

Primary 42B10; Secondary 42B20 42B30 42B35 42A50 42A20 42A38 26B30 

Notes

Acknowledgement

The authors thank D. Khavinson, M. Rudelson, and M. Sodin for stimulating discussions.

References

  1. 1.
    C.R. Adams, J.A. Clarkson, Properties of functions f(x, y) of bounded variation. Trans. Am. Math. Soc. 36, 711–730 (1934)MathSciNetzbMATHGoogle Scholar
  2. 2.
    N.K. Bary, A Treatise on Trigonometric Series, I and II (MacMillan, New York, 1964)zbMATHGoogle Scholar
  3. 3.
    E. Berkson, T.A. Gillespie, Absolutely continuous functions of two variables and well-bounded operators. J. Lond. Math. Soc. (2), 30, 305–324 (1984)MathSciNetCrossRefGoogle Scholar
  4. 4.
    S.-Y.A. Chang, R. Fefferman, Some recent developments in Fourier analysis and H p-theory on product domains. Bull. Am. Math. Soc. (N.S.) 12, 1–43 (1985)Google Scholar
  5. 5.
    J.A. Cima, A.L. Matheson, W.T. Ross, The Cauchy Transform. Mathematical Surveys and Monographs, vol. 125 (American Mathematical Society, Providence, 2006)Google Scholar
  6. 6.
    J.A. Clarkson, C.R. Adams, On definitions of bounded variation for functions of two variables. Trans. Am. Math. Soc. 35, 824–854 (1934)MathSciNetCrossRefGoogle Scholar
  7. 7.
    R. Fefferman, Some recent developments in Fourier analysis and H P theory and product domains. II, in Function Spaces and Applications, Proc. US-Swed. Semin., Lund/Swed. Lecture Notes in Mathematics, vol. 1302 (1988), pp. 44–51CrossRefGoogle Scholar
  8. 8.
    J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics (North-Holland, Amsterdam, 1985)zbMATHGoogle Scholar
  9. 9.
    J.B. Garnett, Bounded Analytic Functions (Springer, New York, 2007)zbMATHGoogle Scholar
  10. 10.
    D.V. Giang, F. Móricz, Lebesgue integrability of double Fourier transforms. Acta Sci. Math. (Szeged) 58, 299–328 (1993)MathSciNetzbMATHGoogle Scholar
  11. 11.
    D.V. Giang, F. Móricz, Hardy spaces on the plane and double Fourier transform. J. Fourier Anal. Appl. 2, 487–505 (1996)MathSciNetCrossRefGoogle Scholar
  12. 12.
    G.H. Hardy, On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters. Q. J. Math. 37, 53–79 (1906)zbMATHGoogle Scholar
  13. 13.
    G.H. Hardy, J.E. Littlewood, Some new properties of Fourier constants. Math. Ann. 97, 159–209 (1926)MathSciNetCrossRefGoogle Scholar
  14. 14.
    E. Hille, J.D. Tamarkin, On the absolute integrability of Fourier transforms. Fundam. Math. 25, 329–352 (1935)CrossRefGoogle Scholar
  15. 15.
    E.W. Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, vol. 1, 3rd edn. (University Press, Cambridge, 1927; Dover, New York, 1957)Google Scholar
  16. 16.
    B. Jawerth, A. Torchinsky, A note on real interpolation of Hardy spaces in the polydisk. Proc. Am. Math. Soc. 96, 227–232 (1986)MathSciNetCrossRefGoogle Scholar
  17. 17.
    F.W. King, Hilbert Transforms. Encyclopedia of Mathematics and Its Applications, vol. 1 (Cambridge University Press, Cambridge, 2009)Google Scholar
  18. 18.
    H. Kober, A note on Hilbert’s operator. Bull. Am. Math. Soc. 48, 421–426 (1942)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H. Kober, A note on Hilbert transforms. Q. J. Math. Oxford Ser. 14, 49–54 (1943)MathSciNetCrossRefGoogle Scholar
  20. 20.
    E. Liflyand, Variations on the theorems of F. and M. Riesz and of Hardy and Littlewood. Georgian Math. J. 21, 337–341 (2014)Google Scholar
  21. 21.
    E. Liflyand, Multiple Fourier transforms and trigonometric series in line with Hardy’s variation. Contemp. Math. Nonlinear Anal. Optim. 659, 135–155 (2016)MathSciNetzbMATHGoogle Scholar
  22. 22.
    E. Liflyand, U. Stadtmüller, On a Hardy-Littlewood theorem. Bull. Inst. Math. Acad. Sinica (New Series) 8, 481–489 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    E. Liflyand, U. Stadtmüller, R. Trigub, An interplay of multidimensional variations in Fourier analysis. J. Fourier Anal. Appl. 17, 226–239 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    V. Matsaev, M. Sodin, Distribution of Hilbert transforms of measures. GAFA Geom. Funct. Anal. 10, 160–184 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.N. Pandey, The Hilbert transform of Schwartz distributions. Proc. Am. Math. Soc. 89, 86–90 (1983)MathSciNetCrossRefGoogle Scholar
  26. 26.
    A. Poltoratski, A problem on completeness of exponentials. Ann. Math. (2) 178, 983–1016 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    A.A. Talalyan, G.G. Gevorkyan, Representation of absolutely continuous functions of several variables. Acta Sci. Math. (Szeged) 54, 277–283 (1990, in Russian)Google Scholar
  28. 28.
    R.M. Trigub, E.S. Belinsky, Fourier Analysis and Approximation of Functions (Kluwer, Dordrecht, 2004)CrossRefGoogle Scholar
  29. 29.
    A. Zygmund, Trigonometric Series, vols. I, II (Cambridge University Press, Cambridge, 1966)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Elijah Liflyand
    • 1
  • Ulrich Stadtmüller
    • 2
  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of MathematicsUniversity of UlmUlmGermany

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