The Boundary Harnack Principle for Non-Divergence form Elliptic Operators

If L is a uniformly elliptic operator in non-divergence form,the boundary Harnack principle for the ratio of positive L-harmonicfunctions holds in Hölder domains of order if > . A counterexample shows that is sharp. For Hölder domainsof order with (0,1], the boundary Harnack principle holds providedthe domain also satisfies a strong uniform regularity condition.