Weighted reproducing kernels and the Bergman space

Reproducing kernels , Bergman kernels , Poisson kernels

Asymptotic Behaviour of Reproducing Kernels of Weighted Bergman Spaces

Let Ω be a domain in Cn, F a nonnegative and G a positive function on Ω such that 1/G is locally bounded, Aα the space of all holomorphic functions on Ω square-integrable with respect to the measure FαGdλ, where dλ is the 2n-dimensional Lebesgue measure, and Kα(x, y) the reproducing kernel for Aα. It has been known for a long time that in some special situations (such as on bounded symmetric do...

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...

WEIGHTED BERGMAN KERNELS AND QUANTIZATIONMiroslav

Weighted Bergman kernels on orbifolds

We describe a notion of ampleness for line bundles on orbifolds with cyclic quotient singularities that is related to embeddings in weighted projective space, and prove a global asymptotic expansion for a weighted Bergman kernel associated to such a line bundle.