Fractional Hardy inequalities and visibility of the boundary

Abstract

Abstract. We prove fractional order Hardy inequalities on open sets under a combined fatness and visibility condition on the boundary. We demonstrate by counterexamples that fatness conditions alone are not sufficient for such Hardy inequalities to hold. In addition, we give a short exposition of various fatness conditions related to our main result, and apply fractional Hardy inequalities in connection to the boundedness of extension operators for fractional Sobolev spaces.

Local maximal operators on fractional Sobolev spaces

“…The described transference of the problem to the 2n-dimensional Euclidean space is a typical step when dealing with norm estimates for the spaces W s,p (G), we refer to [4,6,21] for other examples. We plan to adapt the transference method to norm estimates on intrinsically defined Triebel-Lizorkin and Besov function spaces on open sets, [20].…”

“…holds for all functions f ∈ C c (G). These inequalities have attracted some interest recently, we refer to [2,3,4,6,7,8] and the references therein. In Theorem 4.3 we answer a question from [2], i.e., we characterize those bounded open sets which admit an (s, p)-Hardy inequality.…”

A framework for fractional Hardy inequalities

“…The simple proof of our main result is a refinement of techniques in [7] where, e.g., (s, p, 0)-Hardy inequalities for bounded Lipschitz domains are found. There has been recent interest in (s, p, 0)-Hardy inequalities and the boundary regularity of an open set D, we refer to [10,16,17,18]. In another direction, the sharp constants for fractional Hardy-type inequalities on general domains are obtained in [27], where the distance is replaced with an averaged pseudo distance.…”

“…Unlike in the case of inequality (1.4) with β = 0 and s = 1, the (s, p)-uniform fatness of ∂D (let alone D c ) is not a sufficient condition for an open set D to admit an (s, p, 0)-Hardy inequality (at least) in the case of 0 < sp ≤ 1. This 'non-local obstruction' is recognised and addressed in [16]. It affects certain fractional Hardy inequalities that are treated in [10].…”

Muckenhoupt Ap-properties of Distance Functions and Applications to Hardy–Sobolev -type Inequalities

“…An easy application of the doubling property of µ and inequality rad(B 1 ) ≤ 2 rad(B 2 ) yields these two growth conditions. The requirement lim rad(B)→∞ ψ(B) = 0 that appears in [31, (23) The last fact that we need for (20) is that h = v t/(t−p) belongs to the "dyadic A ∞ -class" A dy ∞ (ψ −1 ). To this end, let us first observe that h dµ is a doubling measure, see inequality (8).…”