The classical kernel functions of potential theory

Young-Bok Chung, Purdue University

Abstract

Suppose that $\Omega$ is a bounded domain with $C\sp\infty$ smooth boundary in the plane and let $\zeta \in \Omega$ be given. Let $f\sb\zeta$ denote the Ahlfors mapping function associated to the pair $(\Omega, \zeta).$ We show that, given an integer $k \ge 2,$ the function which maximizes Re $h\sp{(k)}(\zeta)$ among all holomorphic functions h mapping $\Omega$ into the unit disc with $h(\zeta)$ = $h\sp\prime(\zeta)$ = $\cdots$ = $h\sp{(k-1)}(\zeta)$ = 0; $h(a\sb{j})$ = $h\sp\prime(a\sb{j})$ = $\cdots$ = $h\sp{(k-2)}(a\sb{j})$ = 0 where $\zeta , a\sb1, a\sb2, \cdots, a\sb{n-1}$ are distinct simple zeroes of the Ahlfors map $f\sb\zeta ,$ is the power $f\sb\zeta\sp{k}$ of the Ahlfors map. We also show that the exact Bergman kernel function is equal to a constant times the derivative of the Ahlfors mapping function plus a finite sum which is explicitly written in terms of the Szego kernel. As an application, we see an explicit formula between the derivative of the Ahlfors map and the Bergman kernel function. We find a relationship between the Robin function and the Szego kernel function and give a computation of the Robin function in the case of an annulus. We prove that if $\Omega$ is a bounded domain with $C\sp\infty$ smooth boundary in the plane whose associated Bergman kernel or Szego kernel is an algebraic function, then any proper holomorphic mapping of $\Omega$ onto the unit disc must be algebraic. We find many interesting relationships among the classical kernel functions associated to the domain $\Omega$.

Degree

Ph.D.

Advisors

Bell, Purdue University.

Subject Area

Mathematics

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