A generalization for Fourier transforms of a theorem due to Marcinkiewicz

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Abstract

It is shown that the maximal operator of the Marcinkiewicz 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. As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkiewicz and Zhizhiashvili, more exactly, the Marcinkiewicz means of a function f∈L1(R2) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz means are uniformly bounded on the spaces Hp(R2) and so they converge in the norm (p0<p<∞). Similar results for the Riesz transforms are also given.

Original languageEnglish
Pages (from-to)675-695
Number of pages21
JournalJournal of Mathematical Analysis and Applications
Volume252
Issue number2
DOIs
Publication statusPublished - Dec 15 2000

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Keywords

  • Atomic decomposition
  • Hardy spaces
  • Interpolation
  • Marcinkiewicz means
  • p-atom

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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