On boundary behaviour of the Bergman projection on pseudoconvex domains

M. Jasiczak

Studia Mathematica (2005)

  • Volume: 166, Issue: 3, page 243-261
  • ISSN: 0039-3223

Abstract

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It is shown that on strongly pseudoconvex domains the Bergman projection maps a space L v 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 L v 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.

How to cite

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M. Jasiczak. "On boundary behaviour of the Bergman projection on pseudoconvex domains." Studia Mathematica 166.3 (2005): 243-261. <http://eudml.org/doc/284624>.

@article{M2005,
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.},
author = {M. Jasiczak},
journal = {Studia Mathematica},
keywords = {weighted-sup seminorms; asymptotic expansion},
language = {eng},
number = {3},
pages = {243-261},
title = {On boundary behaviour of the Bergman projection on pseudoconvex domains},
url = {http://eudml.org/doc/284624},
volume = {166},
year = {2005},
}

TY - JOUR
AU - M. Jasiczak
TI - On boundary behaviour of the Bergman projection on pseudoconvex domains
JO - Studia Mathematica
PY - 2005
VL - 166
IS - 3
SP - 243
EP - 261
AB - 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.
LA - eng
KW - weighted-sup seminorms; asymptotic expansion
UR - http://eudml.org/doc/284624
ER -

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