Vector-Valued Kernels of Bergman Type

摘要

Bergman reproducing integral formulas can be obtained for holomorphic mappings f : B - C-n, B the open unit ball of C-n, by applying the well-known formulas for scalar-valued functions on B to each coordinate function of f, provided those coordinate functions each lie in an appropriate Bergman space. Here, we consider an alternative formulation whereby f is reproduced as the integral of the product of a fixed vector-valued kernel and the scalar expression f (z), z , z is an element of B, where ., . is the Hermitian inner product in C-n. We provide two different classes of vector-valued kernels that reproduce holomorphic mappings lying in spaces properly containing the weighted vector-valued Bergman spaces. An analysis of these larger spaces is given. The first set of kernels arises naturally from the scalar-valued Bergman kernels, while the second yields the orthogonal projection onto an isomorphic space of scalar-valued functions in the unweighted case.