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 \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open set, 1<\alpha<\beta\leq 2 and \kappa\in C^{1}(\overline{G}\times \overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\partial G)^{\alpha-1} order decay for harmonic functions near the boundary. For a C^{1,\beta-1} open set D\subset \overline{D}\subset G, 0<\alpha\leq (1\vee\alpha)<\beta\leq 2, we prove a boundary Harnack inequality giving dist(x,\partial D)^{\alpha/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.