Summability of Fourier transforms in variable Hardy and Hardy-Lorentz spaces

Let p(.) : R-n -> (0, infinity) be a variable exponent function satisfying the globally log- Holder condition and 0 < q <= infinity. We introduce the variable Hardy and Hardy-Lorentz spaces H-p(.)(R-d) and H-p(.),H-q(R-d). 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.-means is bounded from H-p(.)(R-d) to L-p(.)(R-d) and from H-p(.),H-q(R-d) to L-p(.),L-q(R-d). This implies some norm and almost everywhere convergence results for the theta-means, amongst others the generalization of the well known Lebesgue's theorem. Some special cases of the theta-summation are considered, such as the Riesz, Bochner-Riesz, Weierstrass, Picard and Bessel summations.