Cesàro summability of two-parameter Walsh-Fourier series

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11 Citations (Scopus)

Abstract

A new atomic decomposition of the two-parameter dyadic martingale Hardy spaces Hp defined by the quadratic variation is given. We introduce Hp-quasi-local operators and prove that if a sublinear operator V is Hp-quasi-local and bounded from L2 to L2 then it is also bounded from Hp to Lp (0 < p ≤ 1). By an interpolation theorem we get that V is of weak type (H#1, L1) where the Hardy space H#1 is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded from Hp to Lp (4/5 < p ≤ ∞) and is of weak type (H#1, L1). So we obtain that the Cesàro means of a function f ∈ H#1 converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded from Hp to Lp for every 2/3 < p ≤ ∞.

Original languageEnglish
Pages (from-to)168-192
Number of pages25
JournalJournal of Approximation Theory
Volume88
Issue number2
DOIs
Publication statusPublished - Feb 1997

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics

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