### Minimal thinness with respect to symmetric Lévy processes

Panki Kim, Renming Song, Zoran Vondraček
2016 Transactions of the American Mathematical Society
Minimal thinness is a notion that describes the smallness of a set at a boundary point. In this paper, we provide tests for minimal thinness at finite and infinite minimal Martin boundary points for a large class of purely discontinuous symmetric Lévy processes. F D = {u ∈ F : u = 0 on D c except for a set of zero capacity}. The Hardy inequality is one of the main ingredients in Aikawa's construction of a measure comparable to the capacity, which is fundamental in proving quasi-additivity of
more » ... acity (see [1, 2] ). We introduce a local Hardy inequality for the Dirichlet form (E, F D ) in the next definition and show in Section 5 that it holds under natural conditions on the open set D. Definition 1.1 We say that (E, F D ) satisfies the local Hardy inequality at z ∈ ∂D (with a localization constant r 0 ) if there exist c > 0 and r 0 > 0 such that We recall now the definition of κ-fat open set and introduce the necessary notation. r). We say that an open set is κ-fat with localization radius Without loss of generality, we will assume that R ≤ 1/2 and κ ≤ 1/4. The first main result of this paper is the following Aikawa's version of the Wiener-type criterion for minimal thinness. For any open set D ⊂ R d , we use G D to denote the Green function of X D . See Definition 6.1 for the definition of minimal thinness in D with respect to X. (2) Conversely, if E is the union of a subfamily of Whitney cubes of D and is not minimally thin in D at z with respect to X, then ∫ When D is a half space, or when D is a C 1,1 open set and X is a purely discontinuous unimodal Lévy processes, we have an explicit form of the integral test. We first recall the definition of a C 1,1 open set. Definition 1.4 An open set D in R d is said to be a (uniform) C 1,1 open set if there exist a localization radius R > 0 and a constant Λ > 0 such that for every z ∈ ∂D, there exist a C 1,1 -function ψ = ψ z : R d−1 → R satisfying ψ(0) = 0, ∇ψ(0) = (0, . . . , 0), ∥∇ψ∥ ∞ ≤ Λ, |∇ψ(x) − ∇ψ(w)| ≤ Λ|x − w|, and an orthonormal coordinate system CS z with its origin at z such that The pair (R, Λ) is called the characteristics of the C 1,1 open set D. A C 1,1 open set D with characteristics (R, Λ) can be unbounded and disconnected; the distance between two distinct components of D is at least R. Recall that an open set D is said to satisfy the interior and exterior balls conditions with radius R 1 if for every z ∈ ∂D, there exist x ∈ D and y ∈ D c such that dist(x, It is known, see [3, Definition 2.1 and Lemma 2.2], that an open set D is a C 1,1 open set if and only if it satisfies the interior and exterior ball conditions. By taking R smaller if necessary, we will always assume a C 1,1 open set with characteristics (R, Λ) satisfies the interior and exterior balls conditions with radius R. Corollary 1.5 Suppose that either (i) D is a half space; or (ii) D ⊂ R d is a C 1,1 open set and γ = 1 in (1.2). Assume that E is a Borel subset of D. (1) If E is minimally thin in D at z ∈ ∂D with respect to X, then ∫ (2) Conversely, if E is the union of a subfamily of Whitney cubes of D and is not minimally thin in D at z ∈ ∂D with respect to X, then ∫ We sometimes write a point z = (z 1 , . . . , An open set D is said to be half-space-like if, after isometry, there exist two real numbers b 1 ≤ b 2 such that H b2 ⊂ D ⊂ H b1 . Without loss of generality, whenever we deal with a half-space-like open set D, we will always assume that H 1 ⊂ D ⊂ H. Now we state our results on minimal thinness at infinity. In Section 7 we will first extend the main result of [32] to purely discontinuous unimodal Lévy processes so that, for a large class of unbounded open sets including half-space-like open sets, the infinite part of the (minimal) Martin boundary consists of a single point. We call such a point infinity and denote it by ∞.