Tangential limits of harmonic functions for subordinate Brownian motions

Abstract

In this thesis, we study the integral kernel and boundary behavior of harmonic functions for certain non-local operators. First, using elementary calculus only, we give a simple proof that Green function estimates imply the sharp two-sided Poisson kernel estimates for pure-jump subordinate Brownian motions including geometric stable processes. The infinitesimal generators of pure-jump subordinate Brownian motions are non-local operators. Second, we show the existence of tangential limits of regular harmonic functions with respect to such non-local operators in $C^{1,1}$ open sets when the exterior functions are local $L_p$-H\"older continuous functions of order $\beta$.