R\'esum\'e
We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic map** on the unit disk with an appropriate growth condition.
Résumé
Nous obtenons des versions non euclidiennes des théorèmes classiques dus à Hardy et Littlewood concernant la régularité de la fonction frontière d’une fonction analytique sur le disque unité avec une condition de croissance appropriée.
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References
S. Bloom and G.S. De Souza, Weighted Lipschitz Spaces and Their Analytic Characterizations, Constr. Approx. 10 (1994) 339–376.
P.L. Duren, Theory of HP spaces, Academic Press, New York and London, 1970.
G.H. Hardy and J.E. Littlewood, A convergence criterion for Fourier series, Math. Z. 28 (1928), 612–634.
St. G. Krantz, Complex Analysis: The Geometric Viewpoint, MAA, Washington, (2004)
M. Marković, Representations for the Bloch Type Semi-norm of Fréchet Differentiable Map**s, J. Geom. Anal. 31 (2021), 7947–7967.
K. Zhu, Distances and Banach spaces of holomorphic functions on complex domains, J. Lond. Math. Soc. 49 (1994), 163–182.
N. Nikolov and M. Trybula, Estimates of the Bergman distance on Dini–smooth bounded planar domains, Coll. Math. 67 (2016), 407–414.
M. Pavlović, Function classes on the unit disc, De Gruyter Studies in Mathematics, (2019)
Sh. Yamashita, Smoothness of the boundary values of functions bounded and holomorphic in the disk, Trans. Amer. Math. Soc. 272 (1982), 539–544.
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I would like to thank the referee of this work for carefully reading and pointing out several errors in previous versions of the manuscript.
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Marković, M. Hardy and Littlewood theorems and the Bergman distance. Ann. Math. Québec 48, 143–156 (2024). https://doi.org/10.1007/s40316-022-00205-w
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DOI: https://doi.org/10.1007/s40316-022-00205-w
Keywords
- Hardy spaces
- Lipschitz classes
- Mean Lipschitz classes
- The Bergman metric
- The hyperbolic metric
- The quasi-hyperbolic metric