Estimates of Poisson Kernels for Symmetric Levy Processes and Their Applications

Abstract

In this paper, we obtain sharp two-sided estimates of Poisson kernels for pure jump symmetric Levy processes in C-1,C-1 open sets. In particular, by combining Green function estimates obtained by Kim and Mimica (Electron. J. Probab., 23(64), 1-45, 2018), our result covers subordinate Brownian motions with high intensity of small jumps such as conjugate geometric stable process whose Laplace exponent is lambda -> lambda/(log (1 + lambda(alpha/2))) for 0 < alpha < 2. As an application, we show the existence of tangential limits for harmonic functions in C-1,C-1 open sets.