Herz spaces and summability of Fourier transforms

✍ Scribed by Hans G. Feichtinger; Ferenc Weisz

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 218 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200510604

✦ Synopsis

Abstract

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^θ^ ~T~ f is also bounded on the corresponding Herz spaces and σ^θ^ ~T~ f → f a.e. for all f ∈ K^–d /p^ ~p,∞~ (ℝ^d^ ). Moreover, σ^θ^ ~T~ f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K^–d /p^ ~p,∞~ (ℝ^d^ ) if and only if the Fourier transform of θ is in the Herz space K^d /p^ ~p ′,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)