Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications (Q1923682)

An extensive literature on the boundary Harnack principle focusses on the following result: Let \(\Omega\) be a domain in \(\mathbb{R}^n\) with certain geometric property of \(\partial \Omega\) and let \(u,v\) be positive harmonic functions on \(\Omega\) vanishing on \(V\cap \partial\Omega\) \((V\) is an open set). Then there exists a constant \(C=C(\Omega,V,K)\) such that \[ {u(x)/v(x) \over u(y)/v(y)} \leq C, \quad \text{for } x,y\in K\cap \Omega, \] where \(K\) is a compact in \(V\). The authors extend investigations of the previous result to a wider class, namely to the class of John domains.