Representation of harmonic functions with respect to subordinate Brownian motion

Abstract

In this article we prove a representation formula for non-negative generalized harmonic functions with respect to a subordinate Brownian motion in a general open set D ⊂ R d . We also study oscillation properties of quotients of Poisson integrals and prove that oscillation can be uniformly tamed.

Semilinear equations for non-local operators: beyond the fractional Laplacian

“…is a singular harmonic function with respect to X, we have that M D µ ∈ C 2 (D), and also by[8, Remark 5.12], it is in L 1 (D). We note further thatM D µ ≡ ∞ in D ifand only if µ is an infinite measure, see [8, Corollary 5.13].…”

“…The function M D (x, z) is called the Martin kernel of D with respect to X. It is shown in [8,Proposition 5.11] (cf. [13] for the case of the fractional Laplacian) that u : D → [0, ∞) is harmonic with respect to X D if and only if there exists a nonnegative finite measure µ on ∂ M D such that…”

“…The operator W D is a boundary trace operator first introduced in [12] in the case of the fractional Laplacian, and extended to more general non-local operators in [8] -see Subsection 2.6 for the precise definition.…”

“…This is possible due to potential-theoretic and analytic properties of such operators developed in the last ten years. Here we single out the construction of the boundary trace operator for the operator L in the recent preprint [8]. The second main contribution is that we obtain some of the results from [1,2] (which deals with C 1,1 open sets) for regular open subsets of R d .…”