A domain integral equation for the Bergman kernel (Q1290400)

This paper is devoted to these integral operators which have a reproducing property. The authors consider special representations of the Szegő, the Bergman and the Cauchy kernels. In generalization of Henrici's function-theoretic approach they obtained a boundary integral equation of the second-order for the Bergman kernel. For this reason it is necessary to construct an analogue to the Cauchy transform with a non-hermitean kernel. This transform is called \(\widehat{B}\)-transform. The construction is lined out for some important example: circle, oval of Cassini and ellipse. In some sense there now exists a similar result for the Bergman kernel to earlier statements for the Szegő kernel by E. Stein, N. Kerzman. The reader can find in this interesting paper a lot of further details on kernel functions of this type.