Variable Hardy and Hardy-Lorentz spaces and applications in Fourier analysis

Abstract

We summarize some results about the variable Hardy and Hardy-Lorentz spaces Hp()(Rd) and Hp();q(Rd) and about the -summability of multi-dimensional Fourier transforms. We prove that the maximal operator of the -means is bounded from Hp()(Rd) to Lp()(Rd) and from Hp();q(Rd) to Lp();q(Rd). This implies some norm and almost everywhere convergence results for the Riesz, Bochner-Riesz, Weierstrass, Picard and Bessel summations.