Herz spaces and summability of Fourier transforms

Hans G. Feichtinger, F. Weisz

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

A general summability method is considered for functions from Herz spaces Kp,rα (ℝd). The boundedness of the Hardy-Littlewood maximal operator on Herz spaces is proved in some critical cases. This implies that the maximal operator of the θ-means σTθ f is also bounded on the corresponding Herz spaces and σTθ f → f a.e. for all f ∈ Kp,∞-d/p (ℝd). Moreover, σTθf(x) converges to f(x) at each p-Lebesgue point of f ∈ Kp,∞-d/p (ℝd) if and only if the Fourier transform of θ is in the Herz space K p′,1d/p (ℝd). Norm convergence of the θ-means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces.

Original languageEnglish
Pages (from-to)309-324
Number of pages16
JournalMathematische Nachrichten
Volume281
Issue number3
DOIs
Publication statusPublished - Mar 2008

Fingerprint

Herz Space
Summability
Fourier transform
Maximal Operator
Lebesgue Point
Critical Case
Lp Spaces
Weighted Spaces
Boundedness
If and only if
Converge
Norm
Imply

Keywords

  • θ-summability of Fourier series
  • Hardy-Littlewood maximal function
  • Herz spaces
  • Lebesgue points
  • Weighted L spaces
  • Weighted Wiener amalgam spaces

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Herz spaces and summability of Fourier transforms. / Feichtinger, Hans G.; Weisz, F.

In: Mathematische Nachrichten, Vol. 281, No. 3, 03.2008, p. 309-324.

Research output: Contribution to journalArticle

Feichtinger, Hans G. ; Weisz, F. / Herz spaces and summability of Fourier transforms. In: Mathematische Nachrichten. 2008 ; Vol. 281, No. 3. pp. 309-324.
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