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

الأصل · EN

Let D be a pseudoconvex domain in ᵏₜ× and let ϕ be a plurisubharmonic function in D. For each t we consider the n-dimensional slice of D, Dₜ={z; (t,z)∈ D}, let ϕᵗ be the restriction of ϕ to Dₜ and denote by Kₜ(z,ζ) the Bergman kernel of Dₜ with the weight function ϕᵗ. Generalizing a recent result of Maitani and Yamaguchi (corresponding to n=1 and ϕ=0) we prove that Kₜ(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.