Hölder domains and the boundary Harnack principle
Rodrigo Bañuelos, Richard F. Bass, Krzysztof Burdzy · Duke Mathematical Journal · 1991
We will prove the following version of the boundary Harnack principle (see below for the definition of a "Hölder domain," "L-harmonic function," etc.).an open set and K is a compact subset of V .Then there exists c < ∞ such that whenever u and v are positive and L-harmonic in V ∩ D, u and v vanish continuously on the regular points of ∂D ∩ V and u and v are bounded in a neighborhood of ∂D ∩ V , then u(x) v(x) ≤ c u(y) v(y) for all x, y ∈ K ∩ D. The constant c depends on L only through the constant c L defined in (1) below.Of course, c also depends on D, V and K.The above result shows that the boundary Harnack principle holds in all Hölder domains.It was first proven for the range α ∈ (1/2, 1] by Bass and Burdzy (1990a), Theorems 3.5 and 4.5.It was subsequently extended by Bañuelos (unpublished) to the full range of α provided the domain also satisfies a uniform capacity condition on the boundary.In Bañuelos (1990) a class of domains called uniformly Hölder domains of order β was introduced.A boundary Harnack principle may be obtained for such domains (for all β ∈ (0, 1))by a variation of the proof of Theorem 1.In order to make this note compact, we refer