Vol. 66, No. 1, 1976

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Variation of Green’s potential

Jang-Mei Gloria Wu

Vol. 66 (1976), No. 1, 295–303
Abstract

Let u be the Green’s potential of a nonnegative mass distribution μ on unit disk D, defined by

      ∫
u(z) =   G(z,w)dμ (w ),
D
(1.1)

where G is the Green’s function on D. The following will be proved.

Theorem 1. Suppose 0 < ρ < 1 and

∫            --1---
D(1 − |w|)log1− |w| dμ(w ) < ∞.
(1.2)

Then for almost all 𝜃, 0 𝜃 < 2π, u has finite variation on the line segment joining ρei𝜃 and ei𝜃.

Theorem 2. Fix 𝜃, 0 𝜃 < 2π and suppose

∫  1− |w|
--i𝜃-----dμ(w) < ∞.
D|e − w |
(1.3)

If L is a circular arc in D centered at ei𝜃, then for all a on L except a set of capacity zero, u has finite variation on the line segment joining a and ei𝜃.

Mathematical Subject Classification 2000
Primary: 31A15
Milestones
Received: 10 November 1975
Revised: 1 June 1976
Published: 1 September 1976
Authors
Jang-Mei Gloria Wu