Fractional Hardy-type inequalities in domains with uniformly fat complement

Abstract

We establish fractional Hardy-type inequalities in a bounded domain with plump complement. In particular our results apply in bounded C ∞ domains and Lipschitz domains. Date: November 20, 2018. 2010 Mathematics Subject Classification. 46E35 (26D10). Key words and phrases. fractional Hardy-type inequality, domain with plump complement, Lipschitz domain, C ∞ domain.A.

In between the inequalities of Sobolev and Hardy

“…The set E is porous if and only if dim A (E) < n. In particular, we have that (E) Let us outline the proof; the reader will find it straightforward to fill in the details. We fix a number 0 < R < 2 d(E) and assume that (6) holds whenever x ∈ E and 0 < r < R. It suffices to prove that there exists a constant C > 0 such that inequality (6) holds whenever x ∈ E and 0 < r < ∞; clearly, we may also assume that 0 < s < n. If 0 < r < R, inequality (6) holds by the assumption. If R ≤ r ≤ 2 d(E), we use the compactness of E to find points x 1 , .…”

“…a compact set in R n and let s > 0. Then s ∈ A(E) if and only if for every (or, equivalently, for some) 0 < R ≤ ∞ there exists a constant C > 0 such that inequality(6) holds whenever x ∈ E and 0 < r < R.Proof. (A) This is proven in[25].…”

Local maximal operators on fractional Sobolev spaces

“…Let us consider Q ∈ W 2 and x ∈ Q. Observe that 2 −1 Q ⊂ B(x,4 5 diam(Q)). Hence, by inequalities (2.4),…”

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

“…When X is a metric space, the weights in these inequalities are of the type δ −α E , α ∈ R, where δ E (x) = dist(x, E) denotes the distance from a point x ∈ X to a closed set E ⊂ X. We refer to [25] and [23] for recent results related to Hardy-Sobolev -inequalities in R n and Hardy inequalities in metric spaces, respectively, and to [9,19] and [8] for fractional counterparts of such inequalities, respectively. Now, two natural questions arise: (i) When does such a weight δ −α E belong to (some) Muckenhoupt A p -class?…”