Vol. 19, No. 3, 1966

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K- and L-kernels on an arbitrary Riemann surface

Myron Goldstein

Vol. 19 (1966), No. 3, 449–459

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 Re 1∕z and Im 1∕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.

Mathematical Subject Classification
Primary: 30.45
Received: 10 April 1965
Published: 1 December 1966
Myron Goldstein