On boundary behaviour of the Bergman projection on pseudoconvex domains

Tom 166 / 2005

M. Jasiczak Studia Mathematica 166 (2005), 243-261 MSC: 32A25, 32A36, 32A40, 32A70, 32T15, 32T40, 32F45. DOI: 10.4064/sm166-3-3


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.


  • M. JasiczakFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    61-614 Pozna/n, Poland

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