On boundary behaviour of the Bergman projection on pseudoconvex domains
Volume 166 / 2005
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 ${\mit \Omega }$ be a bounded, pseudoconvex set with $C^{3}$ boundary. We show that if the Bergman projection is continuous on a space $E \supset L^{\infty }({\mit \Omega })$ 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 $L^{\infty }$ 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.
