Weighted Bergman Kernels and Quantization}

Abstract:

Let Ω be a bounded pseudoconvex domain in C ^N, φ, ψ two positive functions on Ω such that − log ψ, − log φ are plurisubharmonic, and z∈Ω a point at which − log φ is smooth and strictly plurisubharmonic. We show that as k→∞, the Bergman kernels with respect to the weights φ^{k ψ} have an asymptotic expansion

for x,y near z, where φ(x,y) is an almost-analytic extension of &\phi;(x)=φ(x,x) and similarly for ψ. Further, . If in addition Ω is of finite type, φ,ψ behave reasonably at the boundary, and − log φ, − log ψ are strictly plurisubharmonic on Ω, we obtain also an analogous asymptotic expansion for the Berezin transform and give applications to the Berezin quantization. Finally, for Ω smoothly bounded and strictly pseudoconvex and φ a smooth strictly plurisubharmonic defining function for Ω, we also obtain results on the Berezin–Toeplitz quantization.