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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.
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