Hölder functions in Bergman type spaces
Hölder functions in Bergman type spaces
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DOI：
10.4064/sm212-3-3
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发表时间：
2012
影响因子：
0.8
通讯作者：
Yingwei Chen;G. Ren
中科院分区：
文献类型：
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作者：
Yingwei Chen;G. Ren
由于一般情况下，Bergman空间中的函数不存在边值，因此将Hardy空间的边值理论推广到Bergman空间似乎是不可能的。在这篇文章中，我们通过证明Hardy - Littlewood在Hardy空间中的一个结果在Bergman空间中的推广，提供了一个新的思路来证明什么是Bergman空间的正确版本，该结果用其导数的增长来表征单位盘中Hardy空间中函数的Hölder类边值。为此，利用连续模在Bergman空间中引入了一类Hölder函数，并建立了其径向导数的表征。Hardy - littlewood在Hardy空间中的经典结果可以被认为是极限情况，与Hardy空间是Bergman空间的极限这一事实相匹配。
It seems impossible to extend the boundary value theory of Hardy spaces to Bergman spaces since there is no boundary value for a function in a Bergman space in general. In this article we provide a new idea to show what is the correct version of Bergman spaces by demonstrating the extension to Bergman spaces of a result of Hardy– Littlewood in Hardy spaces, which characterizes the Hölder class of boundary values for a function from Hardy spaces in the unit disc in terms of the growth of its derivative. To this end, a class of Hölder functions in Bergman spaces is introduced in terms of the modulus of continuity and we establish its characterization in terms of radial derivatives. The classical result of Hardy–Littlewood in the Hardy space can be thought of as the limit case, matching the fact that the Hardy space is a limit of Bergman spaces.