Scale-invariant Boundary Harnack Principle on Inner Uniform Domains in Fractal-type Spaces

Abstract

We prove a scale-invariant boundary Harnack principle for inner uniform domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that the volume doubling property and two-sided sub-Gaussian heat kernel bounds are satisfied. We make no assumptions on the pseudo-metric induced by the Dirichlet form, hence the underlying space can be a fractal space.2010 Mathematics Subject Classification: 31C25, 60J60, 60J45.

The Dirichlet Heat Kernel in Inner Uniform Domains in Fractal-Type Spaces

“…These are proved [24,Section 6]. Similarly, we can show that the Markovian transition functions P t map the space C 0 of continuous functions vanishing at infinity into itself, by repeating the proof given in [25,Proposition 3.2] under the present weaker assumptions and by applying the results of [24,Section 6].…”

“…Proof The strong Feller property is proved in [25,Proposition 3.2] under the stronger assumption that (X, d) is complete and geodesic, VD holds, and weak heat kernel estimates HKE( ) hold. However, the same proof goes through under the present weaker assumptions.…”

“…Proof The relation between the inner uniformity of a domain in Euclidean space and the boundary Harnack principle has been studied in [1,2]. The boundary Harnack principle is proved in [25,Theorem 5.2] under the hypotheses that (X, d) is geodesic, and that VD and weak heat kernel estimates w-HKE( ) hold globally on X. These global hypotheses are used to prove the Feller property of (E, F), the Poincaré inequality, resistance estimates, elliptic Harnack inequality and certain bounds for the Dirichlet heat kernel on balls.…”

“…Indeed, the Dirichlet heat kernel estimates of Theorem 3.3 imply that the Dirichlet heat kernel and the ground state φ share the same boundary decay rate. However, φ and h share the same boundary decay rate due to a version of the boundary Harnack principle: See [27, Proof of Theorem 6.12] which, thanks to [24,25], generalizes to the present setting of fractal spaces.…”

Martin Kernels for Markov Processes with Jumps

“…The boundary Harnack inequality was first proved independently by A. Ancona ([5]), B. Dahlberg ([17]) and J.-M. Wu ([35]) for Lipschitz domains, and then extended by numerous authors to a wider class of domains and elliptic operators. We refer to [1][2][3][4]31] for further discussion and references.…”