POLARIZATION, CONFORMAL INVARIANTS, AND BROWNIAN MOTION
Dimitrios Betsakos · 1998
Abstract. The polarization P(A) of a closed or open set A ⊂ C is defined as follows: If z, z ̄ ∈ A then z, z ̄ ∈ P(A). If neither z, z ̄ ∈ A then neither z, z ̄ ∈ P(A). If exactly one of z = x+ iy, z ̄ belongs to A then x+ i|y | ∈ P(A) and x − i|y | / ∈ P(A). We prove theorems that describe the behaviour of har-monic measure, Green function, Robin function, Brownian motion and ex-tremal length under polarization. These theorems, combined with an approxi-mation technique due to Dubinin, lead to a new proof of symmetrization results of Baernstein. 1.