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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-3,
author = {Yingwei Chen and Guangbin Ren},
title = {H\"older functions in Bergman type spaces},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {237-258},
zbl = {1275.26006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-3}
}
Yingwei Chen; Guangbin Ren. Hölder functions in Bergman type spaces. Studia Mathematica, Tome 209 (2012) pp. 237-258. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm212-3-3/