Lipschitz continuity of the Green function in Denjoy domains

Abstract

In this paper a Wiener-type characterization is presented of those boundary points of a Denjoy domain where the Green function is Lipschitz continuous. This property is linked with the splitting of a Euclidean boundary point into two minimal Martin boundary points.

Polynomial approximation on a compact subset of the real line

“…For the geometry of E satisfying (1.2), we refer the reader to [9,14,10,3] and the many references therein.…”

Polynomial approximation of piecewise analytic functions on a compact subset of the real line

“…We refer the reader to [9,17,10,1] Next, we derive some estimates of E n from above. According to [17,Theorem 10.5] the only nontrivial unknown case is where E consists of an infinite number of components.…”

Positive Harmonic Functions that Vanish on a Subset of a Cylindrical Surface

“…Using ideas from [10], we will show that condition (ii) in Theorem 1.2 is equivalent to the corresponding condition in Theorem 1.1. For any compact set K ⊂ R N , we denote by v K the capacitary potential of K, and by μ K the associated Riesz measure.…”

“…Our aim is to obtain precise conditions on the set E for the existence of a positive harmonic function u on that vanishes on E and satisfies u ≥ h + on , where h + is assigned the value 0 outside U. We will do this firstly in terms of harmonic measure, by adapting ideas from Benedicks [7], Gardiner [15] and Cranston and Salisbury [12], and then more explicitly in terms of capacity, using methods from Carleson and Totik [9] and Carroll and Gardiner [10]. We have to develop additional techniques to cope with the absence of the symmetry and reflection arguments that are available when ∂ is contained in a hyperplane or union of lines, and with the fundamentally differing potential theoretic natures of U and R N \U.…”