Herz spaces and summability of Fourier transforms

Abstract

AbstractA general summability method is considered for functions from Herz spaces Kαp,r (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 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,∞ (ℝd ). Moreover, σθ T f (x) converges to f (x) at each p ‐Lebesgue point of f ∈ K–d /p p,∞ (ℝd ) if and only if the Fourier transform of θ is in the Herz space Kd /p p ′,1 (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, Weinheim)