Riesz means of d-dimensional fourier transforms and fourier series

Research output: Contribution to journalArticle

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Abstract

It is shown that the maximal operator of the Riesz means with parameter α = (α1, …, αd) of a d-dimensional tempered distribution is bounded from the Hardy space Hp(Rd) to Lp(Rd) if max{d/(d 4- 1), l/(αk + 1), k = 1, …, d} < p < and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the d-dimensional Riesz means of a function f L1(Rd) converge a.e. to f. Moreover, we prove that the Riesz means are uniformly bounded on the spaces Hp(Rd) whenever max{d/(d+1), l/(αk + l), k = 1, …, d} < p <. Thus the Riesz means converge in Hp(Rd) norm to f E Hp(Rd). The same results are proved for the conjugate Riesz means and for d-dimensional Fourier series of distributions, too.

Original languageEnglish
Pages (from-to)121-136
Number of pages16
JournalAnalysis (Germany)
Volume20
Issue number2
DOIs
Publication statusPublished - Jan 1 2000

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Riesz Means
Fourier series
Mathematical operators
Cones
Fourier transform
Fourier transforms
Maximal Operator
Converge
Tempered Distribution
Positive Cone
Hardy Space
Supremum
Norm

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

Riesz means of d-dimensional fourier transforms and fourier series. / Weisz, F.

In: Analysis (Germany), Vol. 20, No. 2, 01.01.2000, p. 121-136.

Research output: Contribution to journalArticle

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