The maximal (C, α, β) operator of two-parameter Walsh-Fourier series

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Abstract

The two-parameter dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α + 1), 1/(β + 1) < p < ∞) and is of weak type (H1 #, L1), where the Hardy space H1# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H1# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α + 1), 1/(β + 1) < p < ∞. Thus, in case f ∈ Hp, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too.

Original languageEnglish
Pages (from-to)389-401
Number of pages13
JournalJournal of Fourier Analysis and Applications
Volume6
Issue number4
DOIs
Publication statusPublished - Jan 1 2000

Keywords

  • (C, α, β) summability
  • Atomic decomposition
  • Interpolation
  • Martingale Hardy spaces
  • P-quasi-local operator
  • Rectangle p-atom
  • Walsh functions

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Applied Mathematics

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