Cesàro summability of multi-dimensional trigonometric-Fourier series

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Abstract

The d-dimensional classical Hardy spaces Hp(Td) are introduced and it is shown that the maximal operator of the Cesàro means of a distribution is bounded from Hp(Td) to Lp (Td) ((2d + 1)/(2d + 2) < p ≤ ° ∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain the summability result due to Marcinkievicz and Zygmund, more exactly, the Cesàro means of a function f ∈ L1(Td) converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Similar results for the (C, β) summability are also formulated.

Original languageEnglish
Pages (from-to)419-431
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume204
Issue number2
DOIs
Publication statusPublished - Dec 1 1996

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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