Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains
[Sous-harmonicité du noyau de Bergman et d’autres fonctions associées à des domaines pseudoconvexes]
Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1633-1662.

Soit D un domaine pseudoconvexe en t k × z n et soit φ une fonction plurisousharmonique dans D. Pour t fixé, soit D t ={z;(t,z)D} la tranche correspondante de D, φ t la restriction de φ à D t , et K t (z,ζ) le noyau de Bergman pour le domaine D t et le poid φ t . En généralisant un résultat récent de Maitani et Yamaguchi (correspondant à n=1 et φ=0), on montre que logK t (z,z) est plurisousharmonique en D. On donne aussi une généralisation d’un résultat de Yamaguchi concernant la fonction de Robin et on discute des résultats du même style pour  n .

Let D be a pseudoconvex domain in t k × 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 logK 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  n .

DOI : 10.5802/aif.2223
Classification : 32A25
Keywords: Bergman spaces, plurisubharmonic function, $\bar{\partial }$-equation, Lelong number
Mot clés : espace de Bergman, fonction plurisousharmonique, équation $\bar{\partial }$, nombre de Lelong
Berndtsson, Bo 1

1 Chalmers University of Technology and the University of Göteborg Department of Mathematics 412 96 Göteborg (Sweden)
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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, Tome 56 (2006) no. 6, pp. 1633-1662. doi : 10.5802/aif.2223. http://www.numdam.org/articles/10.5802/aif.2223/

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