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Herz spaces and summability of Fourier transforms
Author(s) -
Feichtinger Hans G.,
Weisz Ferenc
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200510604
Subject(s) - mathematics , lp space , hardy space , bounded function , norm (philosophy) , maximal operator , banach space , pure mathematics , fourier transform , space (punctuation) , operator (biology) , mathematical analysis , chemistry , repressor , political science , law , transcription factor , gene , linguistics , philosophy , biochemistry
A general summability method is considered for functions from Herz spaces K α p,r (ℝ d ). 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 σ θT f is also bounded on the corresponding Herz spaces and σ θT f → f a.e. for all f ∈ K – d / pp ,∞ (ℝ d ). Moreover, σ θT f ( x ) converges to f ( x ) at each p ‐Lebesgue point of f ∈ K – d / pp ,∞ (ℝ d ) if and only if the Fourier transform of θ is in the Herz space K d / pp ′,1 (ℝ d ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted L p spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)