The l-kernel which was
first considered by Schiffer for plane regions is extended to arbitrary open
Riemann surfaces for a number of significant subspaces of the space of square
integrable harmonic dffierentials Γh. The l-kernel for each of the subspaces
considered is expressed in terms of the principal functions. Thus if W is an
open Riemann surface and p and q the L1 principal functions of W with
singularities Re1∕z and Im1∕z respectively, then the following result is
proved.
Theorem. The differential dp − dq∗ is an l-kernel for the space Γh.
The l-kernel and another kernel function called the k-kernel by Schiffer are
applied to the solution of some well known extremal problems on open Riemann
surfaces.
