On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Pozna/n, Poland

Studia Mathematica, Tome 166 (2005) no. 3, pp. 243-261

Résumé

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.

Détail

DOI : 10.4064/sm166-3-3

Keywords: shown strongly pseudoconvex domains bergman projection maps space functions growing near boundary power bergman distance fixed point space functions which estimated consecutive power bergman distance property has local character mit omega bounded pseudoconvex set boundary bergman projection continuous space supset infty mit omega defined weighted sup seminorms equipped topology given these seminorms contain spaces each natural result strongly pseudoconvex domains inductive limit sequence spaces smallest extension infty class spaces defined weighted sup seminorms which bergman projection continuous generalization results taskinen the unit disc previous research author concerning unit ball