Hölder functions in Bergman type spaces

Yingwei Chen¹ ; Guangbin Ren²

¹ College of Mathematics and Statistics Hebei University of Economics and Business 050061 Shijiazhuang, China
² School of Mathematical Sciences University of Science and Technology of China 230026 Hefei, China

Studia Mathematica, Tome 212 (2012) no. 3, pp. 237-258

Résumé

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.

Détail

DOI : 10.4064/sm212-3-3

Mots-clés : seems impossible extend boundary value theory hardy spaces bergman spaces since there boundary value function bergman space general article provide idea what correct version bergman spaces demonstrating extension bergman spaces result hardy littlewood hardy spaces which characterizes lder class boundary values function hardy spaces unit disc terms growth its derivative end class lder functions bergman spaces introduced terms modulus continuity establish its characterization terms radial derivatives classical result hardy littlewood hardy space thought limit matching the hardy space limit bergman spaces