A boxing Inequality for the fractional perimeter

Abstract

We prove the Boxing inequality:is the Hausdorff content of U of dimension d − α and the constant C > 0 depends only on d. We then show how this estimate implies a trace inequality in the fractional Sobolev space W α,1 (R d ) that includes Sobolev's L d d−α embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as α tends to 0 and 1, recovering in particular the classical inequalities of first order. Their counterparts in the full range α ∈ (0, d) are also investigated.

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

“…On the one hand, thanks to [28,Theorem 1.2], the fractional α-perimeter P α enjoys the following fractional analogue of Gustin's Boxing Inequality (see [19] and [16, (1.14)…”

A decomposition by non-negative functions in the Sobolev Space W^{k,1}

“…One deduces from Proposition 2.1 and a straightforward adaptation of the proof of Theorem 2.1 in [13] the following:…”

“…The plan of the paper is as follows. In Section 2 we show how recent work of the authors [13] on the Boxing inequality of Gustin implies a nonhomogeneous form of this inequality that involves the Choquet integral with respect to the Hausdorff outer measures H d−α δ for 0 < α ≤ d and 0 < δ < ∞. In this range of α such an estimate is equivalent to the strong capacitary inequality in Theorem 1.2.…”

“…In contrast, our argument works for every order α > 0 and ultimately relies on the Hahn-Banach theorem through the strong form of the Boxing inequality (see e.g. the proof of Lemma 4.6 in [13]).…”

A noninequality for the fractional gradient

“…for all non-negative Radon measures µ such that µ(B(x, r)) ≤ C ′ r d−α then as in Lemma 4.6 in[15] it would implŷR d |u| dH d−α ∞ ≤ CˆR d |D α u|, which when combined with equation (1.16) would yield…”