"Hardy spaces with variable exponents in Fourier analysis"Weisz, FerencLet $p(\cdot):\ \mathbb{R}\to(0,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder condition. We introduce the variable Hardy spaces $H_{p(\cdot)}(\mathbb{T})$ and $H_{p(\cdot)}[0,1)$ and give their atomic decompositions. It is proved that the maximal operator of the Fejér means of the Fourier series and Walsh-Fourier series is bounded on these spaces. This implies some norm and almost everywhere convergence results for the Fejér-means, amongst others the generalization of the well known Lebesgue's theorem. |
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