A Forelli–Rudin Construction and Asymptotics of Weighted Bergman Kernels

Abstract

Let Ω be a pseudoconvex domain in CN with smooth boundary, −φ, −ψ two smooth defining functions for Ω={φ>0} such that −logψ, −logφ are plurisubharmonic, z∈Ω a point at which −logφ is strictly plurisubharmonic, and M⩾0 an integer. We show that as k→∞, the Bergman kernels with respect to the weights φkψM have the asymptotic expansionKφkψM(z, z)=kNπNφ(z)kψ(z)M∑j=0∞bj(z)k−j,b0=det−∂2logφ∂zj∂zk.For Ω strongly pseudoconvex with real-analytic boundary, φ, ψ real analytic and −logφ, −logψ strictly plurisubharmonic on Ω, we obtain also the analogous result for KφkψM(x, y) for (x, y) near the diagonal and discuss consequences for the asymptotics of the Berezin transform and for the Berezin quantization. The proofs rely on Fefferman's expansion for the Bergman kernel in a certain Forelli–Rudin type domain over Ω; as another application, they also yield a generalization of the cited Fefferman's expansion to a class of weighted Bergman kernels.