Abstract
For any \(f \in H^p({\mathbb {D}})\) \((p>0)\), the Hardy space over the unit disk, we characterize completely the triple \((\alpha , p, q)\in (-1,\infty )\times (0,\infty )^2\) such that the Cesàro partial sum \(\sigma _n^\alpha f\) converges in norm in \(H^q({\mathbb {D}})\), and in the Bergman space \(L^q_a({\mathbb {D}})\), respectively. This extends recent results of McNeal–Xiong and of Park–Zhao–Zhu, and complements a classical theorem of Hardy-Littlewood in 1934.
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Acknowledgements
G. Cheng is supported by NSFC (11871482, 12171075). X. Fang is supported by MOST of Taiwan (108-2628-M-008-003-MY4). C. Liu is supported by NSFC (12101103). T. Yu is supported by NSFC (11971087).
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Chen, B., Cheng, G., Fang, X. et al. Sharp Cesàro Convergence in the Hardy Spaces Revisited. J Geom Anal 33, 290 (2023). https://doi.org/10.1007/s12220-023-01354-2
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DOI: https://doi.org/10.1007/s12220-023-01354-2