Herz spaces and summability of Fourier transforms

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)