On boundary behaviour of the Bergman projection on pseudoconvex domains

Authors:

• Michał Jasiczak

Abstract

It is shown that on strongly pseudoconvex domains the Bergman projection maps a space Lv_k of functions growing near the boundary like some power of the Bergman distance from a fixed point into a space of functions which can be estimated by the consecutive power of the Bergman distance. This property has a local character. Let Ω be a bounded, pseudoconvex set with C³ boundary. We show that if the Bergman projection is continuous on a space E ⊃ L^∞(Ω) defined by weighted-sup seminorms and equipped with the topology given by these seminorms, then E must contain the spaces Lv_k for each natural k. As a result, in the case of strongly pseudoconvex domains the inductive limit of this sequence of spaces is the smallest extension of L^∞ in the class of spaces defined by weighted-sup seminorms on which the Bergman projection is continuous. This is a generalization of the results of J. Taskinen in the case of the unit disc as well as of the previous research of the author concerning the unit ball.