Abstract
Let \(p(\cdot ) \mathbb{T}\it ^n\rightarrow (0,\infty )\) be a variable exponent function satisfying the globally log-Hölder 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.
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This research was supported by the Hungarian National Research, Development and Innovation Office – NKFIH, K115804 and KH130426.
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Weisz, F. Summability of Fourier series in periodic Hardy spaces with variable exponent. Acta Math. Hungar. 162, 557–583 (2020). https://doi.org/10.1007/s10474-020-01056-z
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DOI: https://doi.org/10.1007/s10474-020-01056-z
Key words and phrases
- variable Hardy space
- variable Hardy–Lorentz space
- atomic decomposition
- \(\theta\)-summability
- maximal operator