Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called theta-summability is considered for multi-dimensional Fourier transforms. Under some conditions on theta, it is proved that the maximal operator of the theta-means defined in a cone is bounded from the amalgam Hardy space W(h (p) , a"" (a)) to W(L (p) , a"" (a)). This implies the almost everywhere convergence of the theta-means in a cone for all f a W(L (1), a"" (a)) aS integral L (1).