Vol. 22, No. 1, 1967

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On Evans’ kernel

Mitsuru Nakai

Vol. 22 (1967), No. 1, 125–137

In classical potential theory on the plane, two important kernels are considered: the hyperbolic kernel log (|1 ζz||z ζ|) on |z| < 1 and the logarithmic kernel log(1|z ζ|) on |z| < +. The former is extended to a general open Riemann surface of positive boundary as the Green’s kernel.

The object of this note is to generalize the latter to an arbitrary open Riemann surface of null boundary, which we shall call Evans’ kernel. The symmetry (Theorem 1) and the joint continuity (Theorem 2) of Evans’ kernel are the main assertions of this note. It is also shown that Evans’ kernel is obtained on every compact set in the product space as a uniform limit of Green’s kernels of specified subsurfaces less positive constants (Theorem 3).

The hyperbolic and logarithmic kernels are characteristic of hyperbolic and parabolic simply connected Riemann surfaces, respectively. The corresponding rôle is played by the elliptic kernel log (1[z,ζ]) for an elliptic simply connected Riemann surface, i.e., a sphere. The generalization of it, which we call Sario’s kernel, is shown to be obtained in a natural manner from the Evan’s kernel.

Mathematical Subject Classification
Primary: 30.45
Received: 26 July 1966
Published: 1 July 1967
Mitsuru Nakai