Radial rearrangement, harmonic measures and extensions of Beurling's shove theorem (Q1426902)

This paper contains several nice results and interesting conjectures for harmonic measure. One such result is an extension of Beurling's shove theorem. Let \(I\) be a union of finitely many closed intervals in \([-1,0]\). Let \(I_o\) be a single interval of the form \([-1,-a]\) chosen to have the same logarithmic length as \(I\). Let \(\mathbb{D}\) be the unit disk. Beurling's theorem asserts that the harmonic measure of the unit circle at \(0\) in \(\mathbb{D}\setminus I\) is increased if \(I\) is replaced by \(I_o\). In the paper under review the author proves a higher dimensional analog of Beurling's result with the intervals replaced by polar rectangles centered on a coordinate axis and having fixed constant angular width. In the proof the author uses the Markus radial symmetrization and other symmetrization results due to A. Baernstein and K. Haliste. Let \(F\) be the collection of all continuous functions \(\Phi:[0,\infty)\to [-\infty,\infty)\) for which \(t\mapsto \Phi(e^{it})\) is convex. Given \(\Phi\in F\) and a domain \(D\), the author defines the functional \(\Gamma_\Phi(D)\) as the value at \(0\in D\) of the solution of the Dirichlet problem in \(D\) with boundary values \(\Phi(| z| )\). He then proves various theorems and formulates several conjectures for this functional. These results and conjectures involve circular and radial symmetrization, harmonic measures, Green functions, and the Chang-Marshall inequality. The author introduces a geometric transformation which he calls cutting procedure. He makes some natural and very interesting conjectures about the behavior of harmonic measure under this transformation. He explores the consequences of these conjectures, and he proves some partial results on the cutting procedure.