Cesàro summability of two-dimensional Walsh-Fourier series

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Abstract

We introduce p-quasi-local operators and the two-dimensional dyadic Hardy spaces Hp defined by the dyadic squares. It is proved that, if a sublinear operator T is p-quasi-local and bounded from L to L, then it is also bounded from Hp to Lp (0 < p ≤ 1). As an application it is shown that the maximal operator of the Cesaβro means of a martingale is bounded from Hp to Lp (1/2 < p ≤ ∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function f ∈ 6 L1 converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from Hp to Lp for every 0 < p ≤ ∞.

Original languageEnglish
Pages (from-to)2169-2181
Number of pages13
JournalTransactions of the American Mathematical Society
Volume348
Issue number6
Publication statusPublished - Dec 1 1996

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Keywords

  • Atomic decomposition
  • Cesàro summability
  • Interpolation
  • Martingale hardy spaces
  • Walsh functions
  • p-Atom
  • p-Quasi-local operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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