Lindelof Maximum Principle

Theorem 1. Suppose u is bounded and harmonic in a bounded open connected set W. Suppose lim sup_{z→p∈∂W} u(z) ≤ M except for a countable set {q_{j}}. Then u(z) ≤ M.

Proof. Choose ϵ_{j} > 0 so that ∑_{1}^{∞} ϵ_{j} < ∞. Let d be the diameter of W. Let a ∈ W and let c be the distance

from a to ∂W. Then

w(z) = ∑ ϵ_{j} log \frac{|z − q_{j}|}{d}

converges uniformly on |z − a| < d/2. Hence u(z) − w(z) is harmonic on W and we can apply our proof for the case of a finite number of exceptional points.

Remark 1. It’s not obvious how to modify this for a set of measure 0.