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

Abstract

Let D be a pseudoconvex domain in ℂtk × ℂzn and let φ be a plurisubharmonic function in D. For each t we consider the n-diraensional 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 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 ℝn.