Relationship Between the Bergman and Cauchy-Szeg¨o Ker- nels in the Domains + (n -1) and ℜn I V

Abstract

The selection of classes biholomorphically equivalent domains has great importance in multidimensional analysis and its applications. It is well known that all simply connected proper open subsets of the plane C are conformally equivalent (Rieman mapping theorem). The situation is completely different in the multidimensional case. For instance, an open unit ball and an open unit polydisc are not biholomorphically equivalent. In fact, there does not exist any holomorphic mapping from one to the other. Therefore, it is very important to have stocks of domains that are biholomorphically equivalent to each other. Finding the kernels of representations of holomorphic functions in domains C and in the matrix domains from C [m×m] is a rather difficult task (see [1–4]). Usually, in classical theory, kernels of such kind are constructed in bounded symmetric domains (see [5]). One of such domain is the matrix ball. One considers the following problems for it (see [4, 6]): finding the transitive group of automorphisms of a matrix ball; computing the Bergman and Cauchy-Szegö kernels for this domain; finding Carleman’s formula, recovering values of a holomorphic function in a matrix ball by its values on some boundary (uniqueness) sets (see [7–9]). By writing down explicitly the transitive group of automorphisms of the matrix ball, by direct calculation, we can find the Bergman and Cauchy-Szegö kernels for this domain. And then (using the properties of the Poisson kernel) we can find Carleman’s formula, which recovers values of a holomorphic function in whole domain by its values on some boundary set of uniqueness