Boundary Harnack inequalities for regional fractional Laplacian

Abstract

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking κ(x,y)/|x-y|^{n+α}dxdy, x,y\in G as the jumping measure. When G is a C^{1,β-1} open set, 1<α<β\leq 2 and κ\in C^{1}(\overline{G}\times \overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\partial G)^{α-1} order decay for harmonic functions near the boundary. For a C^{1,β-1} open set D\subset \overline{D}\subset G, 0<α\leq (1\veeα)<β\leq 2, we prove a boundary Harnack inequality giving dist(x,\partial D)^{α/2} order decay for harmonic functions near the boundary. These inequalities are generalizations of the known results for the homogeneous case on C^{1,1} open sets. We also prove the boundary Harnack inequality for regional fractional Laplacian on Lipschitz domain.