Restricted summability of the multi-dimensional Cesàro means of Walsh–Kaczmarz–Fourier series

Károly Nagy, Mohamed Salim

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

The properties of the maximal operator of the (C, α)-means (α = (α1, . . ., αd)) of the multi-dimensional Walsh–Kaczmarz–Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from dyadic Hardy space Hp γ to Lebesgue space Lp for p0 < p (p0 = max{1/(1 + αk): k = 1, . . ., d}) and is of weak type (1, 1). As a corollary, we get a theorem of Simon on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case p = p0, we show that the maximal operator σL κ,α, is not bounded from the dyadic Hardy space Hp γ 0 to the Lebesgue space Lp0.

Original languageEnglish
Pages (from-to)381-394
Number of pages14
JournalPublicationes Mathematicae
Volume94
Issue number3-4
DOIs
Publication statusPublished - 2019
Externally publishedYes

Keywords

  • A.e. convergence
  • Cesàro means
  • Maximal operator
  • Multi-dimensional system
  • Restricted summability
  • Walsh–Kaczmarz system

ASJC Scopus subject areas

  • Mathematics(all)

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