The Riesz Decomposition Theorem

Abstract

In Section 15 we defined a function to be superharmonic if its negative is subharmonic. This is equivalent to saying that F is superharmonic on the open set Ω in ℝⁿ if for every x_o ∈ Ω ∃ r(x₀) > 0 with

$$F\left( {x_0 } \right) \geqslant \frac{1} {A}\int\limits_{\left| {x - x_0 } \right| = a} {F\left( x \right)} dS,$$

(17.1)

for all a < r(x₀), where A is the area of the sphere {x| |x-x₀| = a}, and

$$\frac{{lim}} {{x \to x_0 }}F\left( x \right) \geqslant F\left( {x_0 } \right), for all x_0 \in \Omega .$$

(17.2)

on any ball.