Homogeneous Subsets of a Lipschitz Graph and the Corona Theorem

Abstract

This paper proves the Corona Theorem to be affirmative for domains in the complex plane bounded by thick subsets of a Lipschitz graph. Specifically, the boundary of these domains $E_0$ has a Carleson lower density:

$$ \Lambda\left(B(z,r) \cap E_0\right) > \epsilon_0 r \quad\text{for all } z\in E_0, \quad \text{and all } r>0. $$