Estimates and structure of α-harmonic functions

Abstract

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c ∪{∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.

Martin Kernels for Markov Processes with Jumps

“…We remark that in fact one can prove uniformity also in D, just as in [11], by appropriately modifying the final part of the proof. More formally,…”

“…We prove Theorem 2 by considering separately two types of boundary points, which are called accessible and inaccessible in [11]. First, however, we introduce some further notation and prove preliminary estimates.…”

“…The representation given in part (d) essentially follows now from the general theory of Martin boundary, together with some ideas developed in [11]. For simplicity, in the remaining part of the proof we simply write that a function is harmonic when we refer to harmonicity in D. (25)).…”

“…The proof of our main result, Theorem 2, relies on the boundary Harnack inequality for Markov Mateusz Kwaśnicki mateusz.kwasnicki@pwr.edu.pl Tomasz Juszczyszyn tomasz.juszczyszyn@pwr.edu.pl 1 Faculty of Pure and Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland processes with jumps, proved recently in [12], and the oscillation reduction argument, developed in [6] and [11]. As an application, we obtain Martin representation of harmonic functions in Theorem 3.…”

“…For this reason, unlike in [11], we refer to the general theory of Martin boundary. Our argument still requires extension of some elements of [11] for more general Markov processes, but the most involved part of the proof is avoided. For an excellent exposition of the general theory of Martin boundary, we refer to Chapter 14 of [16].…”