Riesz means of Fourier transforms and Fourier series on Hardy spaces

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Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from HP(ℝ) to LP(ℝ) (1/(α + 1) < p < ∞) and is of weak type (1,1), where HP(ℝ) is the classical Hardy space. As a consequence we deduce that the Riesz means of a function f ∈ L1(ℝ) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on HP(ℝ)) whenever 1/(α + 1) < p < ∞. Thus, in case f ∈ Hp(ℝ), the Riesz means converge to f in Hp(ℝ) norm (1/(α + 1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.

Original languageEnglish
Pages (from-to)253-270
Number of pages18
JournalStudia Mathematica
Issue number3
Publication statusPublished - Jan 1 1998



  • Atomic decomposition
  • Fourier transforms
  • Hardy spaces
  • Interpolation
  • P-atom
  • Riesz means

ASJC Scopus subject areas

  • Mathematics(all)

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