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

Berndtsson, Bo

Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1633-1662.

Soit $D$ un domaine pseudoconvexe en ${ℂ}_t^k\times {ℂ}_z^n$ et soit $\phi$ une fonction plurisousharmonique dans $D$. Pour $t$ fixé, soit $D_t=\{z;(t,z)\in D\}$ la tranche correspondante de $D$, ${\phi }^t$ la restriction de $\phi$ à $D_t$, et $K_t(z,\zeta )$ le noyau de Bergman pour le domaine $D_t$ et le poid ${\phi }^t$. En généralisant un résultat récent de Maitani et Yamaguchi (correspondant à $n=1$ et $\phi =0$), on montre que $log{K}_t(z,z)$ 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 ${ℝ}^n$.

Let $D$ be a pseudoconvex domain in ${ℂ}_t^k\times {ℂ}_z^n$ and let $\phi$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z;(t,z)\in D\}$, 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 ${ℝ}^n$.

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

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