Spring 2010 - Problem 6

Let $\mu$ be a finite, positive, regular Borel measure supported on a compact subset of the complex plane $\C$ and define the Newtonian potential of $\mu$ to be

1. Prove that $U_\mu$ exists at Lebesgue-almost all $z \in \C$ and that

   $$\iint_K U_\mu\p{z} \,\diff{x} \,\diff{y} < \infty$$

   for every compact $K \subseteq \C$. Hint: Fubini.

2. Prove that for almost every horizontal or vertical line $L \subseteq \C$, $\mu\p{L} = 0$ and $\int_K U_\mu\p{z} \,\diff{s} < \infty$ for every compact subset $K \subseteq L$ where $\diff{s}$ denotes Lebesgue linear measure on $L$. Hint: Fubini and (1). (Here a.e. vertical line means the vertical lines through $\p{x, 0}$ for a.e. $x \in \R$. Likewise for horizontal lines.)

3. Define the Cauchy potential of $\mu$ to be

   $$S_\mu\p{z} = \int_\C \frac{1}{z - w} \,\diff\mu\p{w},$$

   which trivially exists whenever $U_\mu\p{z} < \infty$. Let $R$ be a rectangle in $\C$ whose four sides are contained in lines $L$ having the conclusion of (2). Prove that

   $$\frac{1}{2\pi i} \int_{\partial R} S_\mu\p{z} \,\diff{z} = \mu\p{R}.$$

   Hint: Fubini and Cauchy.