Hardy spaces with variable exponents and applications in Fourier analysis

Abstract

We summarize some results about the variable Lebesgue and Hardy spaces $L_{p(\cdot)}(\mathbb{R})$ and $H_{p(\cdot)}(\mathbb{R})$ and about the Fej\'{e}r-summability of Walsh-Fourier series and Fourier transforms. We prove that the maximal operator of the Fej\'{e}r-means is bounded from $H_{p(\cdot)}(\mathbb{R})$ to $L_{p(\cdot)}(\mathbb{R})$. This implies some norm and almost everywhere convergence results of the Fej\'{e}r means.