Summability of Fourier series in periodic Hardy spaces with variable exponent

Abstract

Let $$p(\cdot ) \mathbb{T}\it ^n\rightarrow (0,\infty )$$ be a variable exponent function satisfying the globally log-Holder condition and $$0<q \le \infty $$ . We introduce the periodic variable Hardy and Hardy–Lorentz spaces $$H_{p(\cdot )}(\mathbb{T}\it ^d)$$ and $$H_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ and prove their atomic decompositions. A general summability method, the so called $$\theta $$ -summability is considered for multi-dimensional Fourier series. Under some conditions on $$\theta $$ , it is proved that the maximal operator of the $$\theta $$ -means is bounded from $$H_{p(\cdot )}(\mathbb{T}\it ^d)$$ to $$L_{p(\cdot )}(\mathbb{T}\it ^d)$$ and from $$H_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ to $$L_{p(\cdot ),q}(\mathbb{T}\it ^d)$$ . This implies some norm and almost everywhere convergence results for the summability means. The Riesz, Bochner–Riesz, Weierstrass, Picard and Bessel summations are investigated as special cases.