The Maximal (C, α, β) Operator of Two-Parameter Walsh-Fourier Series

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 (H# 1, L1), where the Hardy space H# 1 is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f ∈ H# 1 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 Knot Theory and its Ramifications
Volume6
Issue number4
Publication statusPublished - 2000

Fingerprint

Fourier series
Two Parameters
Operator
Hardy Space
Converge
Maximal Function
Maximal Operator
Martingale
Norm

Keywords

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

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

The Maximal (C, α, β) Operator of Two-Parameter Walsh-Fourier Series. / Weisz, F.

In: Journal of Knot Theory and its Ramifications, Vol. 6, No. 4, 2000, p. 389-401.

Research output: Contribution to journalArticle

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