Dirchlet's problem on Green lines and some convergence theorems for Denjoy's domains.
Doubling conditions for harmonic measure in John domains
We introduce new classes of domains, i.e., semi-uniform domains and inner semi-uniform domains. Both of them are intermediate between the class of John domains and the class of uniform domains. Under the capacity density condition, we show that the harmonic measure of a John domain satisfies certain doubling conditions if and only if is a semi-uniform domain or an inner semi-uniform domain.