Herz spaces and summability of Fourier transforms |
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Authors: | Hans G. Feichtinger Ferenc Weisz |
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Affiliation: | 1. Numerical Harmonic Analysis Group, Faculty of Mathematics, University of Vienna, Nordbergstra?e 15, A–1090 Vienna, Austria;2. Department of Numerical Analysis, E?tv?s L. University, Pázmány P. sétány 1/C, H–1117 Budapest, Hungary |
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Abstract: | A general summability method is considered for functions from Herz spaces Kαp,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 σθ T f is also bounded on the corresponding Herz spaces and σθ T f → f a.e. for all f ∈ K–d /p p,∞ (?d ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (?d ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (?d ). Norm convergence of the θ ‐means is also investigated in Herz spaces. As special cases some results are obtained for weighted Lp spaces. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
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Keywords: | Herz spaces weighted Lp spaces weighted Wiener amalgam spaces Hardy− Littlewood maximal function θ ‐summability of Fourier series Lebesgue points |
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