Herz spaces and summability of Fourier transforms

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. © 2008 Wiley-VCH Verlag GmbH & Co. KGaA.