A Multidimensional Hardy–Littlewood Theorem

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


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.


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 



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


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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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