[35] (with S. Bergman) Kernel functions and conformal mapping

Abstract

In this well-known and widely cited paper, the authors return to their study of the Bergman reproducing kernel $K(z,\overline{\zeta })$ in the Hilbert space of square integrable analytic functions on a bounded plane domain B whose boundary C consists of finitely many real analytic closed curves. The kernel $K(z,\overline{\zeta })$ becomes strongly unbounded whenever z and ζ tend to the same boundary point in C, so the standard theory of integral equations does not apply. To circumvent this difficulty, the authors consider the regularized (Hermitian) kernel $K(z,\overline{\zeta }) - \Gamma (z,\overline{\zeta }),$ where

$$\displaystyle{\Gamma (z,\overline{\zeta }) ={ \frac{1} {\pi }^{2}} \iint _{\mathbb{C}\setminus B} \frac{du\,dv} {{(w - z)}^{2}{(\overline{w} -\overline{\zeta })}^{2}}\,,\qquad w = u + iv,}$$

which is shown to be regular in the closed region B ∪ C and thus amenable to study via the classical theory.