p Harmonic Measure in Simply Connected Domains

TL;DR

This paper extends Makarov-type results on the Hausdorff dimension of p-harmonic measure to all planar simply connected domains, using gradient estimates and conformal maps to analyze boundary behavior.

Contribution

It generalizes previous results to all simply connected domains and introduces a new gradient estimate approach based on boundary distance.

Findings

Hausdorff dimension bounds for p-harmonic measure in simply connected domains

Gradient estimates depending only on p and boundary distance

Construction of quasicurves via conformal maps

Abstract

We extend to all planar simply connected domains Makarov-type results about the Hausdorff dimension of $p$-harmonic measure pioneered by Lewis and Bennewitz in the context of quasidisks. The key to our analysis is a gradient estimate using the distance to the boundary and constants that only depend on $p$. This is achieved by studying the conformal map from the unit disk to the simply connected domain to construct good quasicurves from a point in the domain to the boundary.