Hölder functions in Bergman type spaces

Volume 212 / 2012

Yingwei Chen, Guangbin Ren Studia Mathematica 212 (2012), 237-258 MSC: Primary 26A16; Secondary 32A36. DOI: 10.4064/sm212-3-3

Abstract

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.

Authors

  • Yingwei ChenCollege of Mathematics and Statistics
    Hebei University of Economics
    and Business
    050061 Shijiazhuang, China
    e-mail
  • Guangbin RenSchool of Mathematical Sciences
    University of Science and Technology of China
    230026 Hefei, China
    e-mail

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