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2005 | 166 | 3 | 243-261
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On boundary behaviour of the Bergman projection on pseudoconvex domains

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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.
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  • Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm166-3-3
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