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](a320x.png) | (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 ) < ∞.](a321x.png) | (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 |](a322x.png) | (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.
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