Boundary Potential Theory for Schrödinger Operators Based on Fractional Laplacian

Abstract

Precise boundary estimates and explicit structure of harmonic functions are closely related to the so-called Boundary Harnack Principle (BHP). The proof of BHP for classical harmonic functions was given in 1977-78 by H. Dahlberg in [65], A. Ancona in [3] and J.-M. Wu in [153] (we also refer to [99] for a streamlined exposition and additional results). The results were obtained within the realm of the analytic potential theory. A probabilistic proof of BHP, one which employs only elementary properties of the Brownian motion, was given in [11]. The proof encouraged subsequent attempts to generalize BHP to other processes, in particular to the processes of jump type.

BHP asserts that the ratio u(x)/v(x) of nonnegative functions harmonic on a domain D which vanish outside the domain near a part of the domain’s boundary, ?D, is bounded inside the domain near this part of ?D. The result requires assumptions on the underlying Markov process and the domain. For Lipschitz domains and harmonic functions of the isotropic ?-stable Lévy process (0 < ? < 2), BHP was proved in [27]. Another proof, motivated by [11], was obtained in [31] and extensions beyond Lipschitz domains were obtained in [150] and [38]. In particular the results of [38] provide a conclusion of a part of the research in this subject, and offer techniques that may be used for other jump-type processes.