A framework for fractional Hardy inequalities

Abstract

Abstract. We provide a general framework for fractional Hardy inequalities. Our framework covers, for instance, fractional inequalities related to the Dirichlet forms of some Lévy processes, and weighted fractional inequalities on irregular open sets.

Interpolation Theory for Sobolev Functions with Partially Vanishing Trace on Irregular Open Sets

“…The next lemma shows that E and R defined in (14) and (15) are well-behaved with respect to the Dirichlet conditions defined in (18) and (19). Proof.…”

“…Proof. We apply Proposition 3.5 with the pair (E, R) defined in (14) and (15). Owing to the mapping properties derived in Lemmas 4.10 and 4.12, we get equal sets…”

“…For the converse inclusion we argue as in [3,Lem. 3.2], using the localization formalism of Section 4.4 and in particular the maps E and R defined in (14) and (15). We let σ be the extension of functions from (−1, 0) × (−1, 1) d−1 to (−1, 1) d−1 by even reflection and consider the operator T : f → R((Ef ) 0 , (σ(Ef ) i ) i∈I ).…”

Geometric inequalities for fractional Laplace operators and applications

“…For fractional Hardy inequalities we refer to [11][12][13]19,20,22], and in particular we refer to [19] for the best constant in the case of the entire space, and to [20] for the best constant in general domains.…”

“…Concerning Hardy inequalities of fractional order and their generalizations to weights, we recall the following recent works [11][12][13].…”