Abstract: The boundary Harnack principle states that the ratio of any two functions, which are positive and harmonic on a domain, is bounded near some part of the boundary where both functions vanish. A given domain may or may not have this property, depending on the geometry of its boundary and the underlying metric measure space.
In this talk, we will consider a scale-invariant boundary Harnack principle on domains that are inner uniform. This has applications such as two-sided bounds on the Dirichlet heat kernel, or the identification of the Martin boundary and the topological boundary for bounded inner uniform domains.
The inner uniformity provides a large class of domains which may have very rough boundary as long as there are no cusps. Aikawa and Ancona proved the scale-invariant boundary Harnack principle on inner uniform domains in Euclidean space. Gyrya and Saloff-Coste gave a proof in the setting of non-fractal strictly local Dirichlet spaces that satisfy a parabolic Harnack inequality.
I will present a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces that satisfy a parabolic Harnack inequality. This result applies to fractal spaces.