Restricted summability of Fourier transforms and local Hardy spaces

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