Marcinkiewicz-θ-summability of Fourier transforms

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Abstract

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function θ. Under some conditions on θ we show that the maximal operator of the Marcinkiewicz-θ-means of a tempered distribution is bounded from Hp(R2) to Lp(R2) for all p0 < p ≦ ∞ and, consequently, is of weak type (1, 1), where p0 < 1 depends only on θ. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-θ-means of a function f ∈ L1(R2) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-θ-means are uniformly bounded on the spaces Hp(R2) and so they converge in norm (p0 < p <). Some special cases of the Marcinkiewicz-θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de la Vallée-Poussin, Rogosinski and Riesz summations.

Original languageEnglish
Pages (from-to)135-146
Number of pages12
JournalActa Mathematica Hungarica
Volume96
Issue number1-2
Publication statusPublished - Jul 1 2002

Keywords

  • Fourier transforms
  • Hardy spaces
  • Interpolation
  • Marcinkiewicz-θ-summation
  • p-atom

ASJC Scopus subject areas

  • Mathematics(all)

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