Herz spaces and summability of Fourier transforms

A general summability method is considered for functions from Herz spaces K-p,r(alpha) (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 theta-means sigma(theta)(T) f is also bounded on the corresponding Herz spaces and sigma(0)(T) f -> f a.e. for all f is an element of K-p,infinity(-d/p) (R-d). Moreover, sigma(theta)(T) f(x) converges to f(x) at each p-Lebesgue point of f is an element of K-p,infinity(-d/p) (R-d) if and only if the Fourier transform of theta is in the Herz space K-p',1(d/p) (R-d). Norm convergence of the theta-means is also investigated in Herz spaces. As special cases some results are obtained for weighted L-p spaces. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim