Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Bo Berndtsson[1]

  • [1] Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 6, page 1633-1662
  • ISSN: 0373-0956

Abstract

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Let D be a pseudoconvex domain in ℂ t k × ℂ z n and let φ be a plurisubharmonic function in D . For each t we consider the n -dimensional slice of D , D t = { z ; ( t , z ) ∈ D } , let φ t be the restriction of φ to D t and denote by K t ( z , ζ ) the Bergman kernel of D t with the weight function φ t . Generalizing a recent result of Maitani and Yamaguchi (corresponding to n = 1 and φ = 0 ) we prove that log K t ( z , z ) is a plurisubharmonic function in D . We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of  ℝ n .

How to cite

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Berndtsson, Bo. "Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains." Annales de l’institut Fourier 56.6 (2006): 1633-1662. <http://eudml.org/doc/10187>.

@article{Berndtsson2006,
abstract = {Let $D$ be a pseudoconvex domain in $\mathbb\{C\}^k_t\times \mathbb\{C\}^n_z$ and let $\phi $ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\lbrace z; (t,z)\in D\rbrace $, let $\phi ^t$ be the restriction of $\phi $ to $D_t$ and denote by $K_t(z,\zeta )$ the Bergman kernel of $D_t$ with the weight function $\phi ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi =0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of $\mathbb\{R\}^n$.},
affiliation = {Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden)},
author = {Berndtsson, Bo},
journal = {Annales de l’institut Fourier},
keywords = {Bergman spaces; plurisubharmonic function; $\bar\{\partial \}$-equation; Lelong number},
language = {eng},
number = {6},
pages = {1633-1662},
publisher = {Association des Annales de l’institut Fourier},
title = {Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains},
url = {http://eudml.org/doc/10187},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Berndtsson, Bo
TI - Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 6
SP - 1633
EP - 1662
AB - Let $D$ be a pseudoconvex domain in $\mathbb{C}^k_t\times \mathbb{C}^n_z$ and let $\phi $ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\lbrace z; (t,z)\in D\rbrace $, let $\phi ^t$ be the restriction of $\phi $ to $D_t$ and denote by $K_t(z,\zeta )$ the Bergman kernel of $D_t$ with the weight function $\phi ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi =0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of $\mathbb{R}^n$.
LA - eng
KW - Bergman spaces; plurisubharmonic function; $\bar{\partial }$-equation; Lelong number
UR - http://eudml.org/doc/10187
ER -

References

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