Weighted Bergman Kernels and Quantization

Let Ω be a bounded pseudoconvex domain in C N , φ, ψ two positive functions on Ω such that − logψ,− log φ are plurisubharmonic, z ∈ Ω a point at which − log φ is smooth and strictly plurisubharmonic, and M a nonnegative integer. We show that as k → ∞, the Bergman kernels with respect to the weights φkψM have an asymptotic expansion KφkψM (x, y) = kN πNφ(x, y)kψ(x, y)M ∞ ∑ j=0 bj(x, y) k −j , b0(x, x) = det [ − ∂ 2 log φ(x) ∂zj∂zk ] , for x, y near z, where φ(x, y) is an almost-analytic extension of φ(x) = φ(x, x) and similarly for ψ. If in addition Ω is of finite type, φ, ψ behave reasonably at the boundary and − log φ,− logψ 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 about the Berezin-Toeplitz quantization on Ω. Let Ω be a domain in C , ρ a positive continuous function on Ω, and Kρ the reproducing kernel of the weighted Bergman space A(Ω, ρ) of all holomorphic functions on Ω squareintegrable with respect to the measure ρ(z) dz, dz being the Euclidean volume element in C ; we call Kρ the weighted Bergman kernel corresponding to ρ, and for ρ ≡ 1 we will speak simply of the Bergman kernel KΩ of Ω. The Berezin transform Bρ is the integral operator defined by Bρf(y) = ∫ Ω f(x) |Kρ(x, y)|2 Kρ(y, y) ρ(x) dx (1) for all y for which Kρ(y, y) 6= 0. In terms of the operator Mf of multiplication by f on the space L(Ω, ρ dz) this can be rewritten as Bρf(y) = 〈MfKρ(·, y),Kρ(·, y)〉 ‖Kρ(·, y)‖2 , from which it is immediate that the integral (1) converges, for instance, for any bounded measurable function f . The Berezin transform was first introduced by F.A. Berezin [Ber] in the context of quantization of Kähler manifolds. More specifically, let φ be a positive function on Ω such that − log φ is strictly plurisubharmonic, and set gjk = ∂ (− log φ)/∂zj∂zk (2) 2000 Mathematics Subject Classification. Primary 32A25, 53D55; Secondary 32A07, 32W05, 47B35.