On comparability of integral forms

Abstract

We prove comparability of certain homogeneous anisotropic integral forms. As a consequence we obtain a Hardy type inequality generalising that for the fractional Laplacian. We give an application to anisotropic censored stable processes in Lipschitz domains.  2005 Elsevier Inc. All rights reserved.

Fractional Sobolev spaces with power weights

“…However, it is also very interesting and nontrivial property itself. Similar results were obtained before by Dyda [6] for Gagliardo-type seminorms with the additional homogeneous kernels (like indicators of cones), by Prats and Saksman [22] in a more general context of Triebel-Lizorkin spaces and generalized later by Rutkowski [23] for the kernels of the form |x − y| −d ϕ(|x − y|) −q , with ϕ satisfying certain technical assumptions. Some versions of the reduction of the integration theorems can also be found in [3], [4] and [15].…”

“…If w or v take the value zero on Ω, then we can artificially augment them by adding a positive, locally comparable to a constant weights w ′ , v ′ , which in addition satisfy (6). New weights w + w ′ and v + v ′ are also locally comparable to a constant, positive and satisfy (6). In this case w and v should be replaced by w + w ′ and v + v ′ in all the computations below.…”

The best constant in a fractional Hardy inequality

“…These functions may then be used to investigate transience and boundary behavior of the underlying Markov processes ( [1], [7], [17], [14]). In particular, we expect that the results of [16,17] may be used to obtain, for the anisotropic stable ( [10,9]) censored processes, the ruin probabilities generalizing [7,Theorem 5.10], and to develop the boundary potential theory on Lipschitz domains ( [7,26,24]) in analogy with those of the killed stable processes ( [8,13,6]). We also like to mention the connection of optimal Hardy inequalities with critical Schrödinger perturbations and the so-called ground state representation [21].…”

Improved fractional Poincaré type inequalities on John domains

“…Observe that the seminorm appearing on the right hand side of inequality (1.2) is weaker than that of the usual fractional Sobolev space W s,p (Ω). More precisely, if we consider W s,p (Ω) to be the subspace of L p (Ω) induced by the seminorm for fixed s, τ ∈(0, 1), then it is known that both spaces coincide when Ω is Lipschitz (see [12]), but there are examples of John domains Ω⊂R n for which the inclusion W s,p (Ω)⊂ W s,p (Ω) is strict (see [9] for this result and characterizations of both spaces as interpolation spaces). This has led to call inequality (1.2) an "improved" fractional inequality.…”