Hardy-Sobolev inequalities for vector fields and canceling linear differential operators

Abstract

Given a homogeneous k-th order differential operator A(D) on R n between two finite dimensional spaces, we establish the Hardy inequalityand the Sobolev inequalitywhen A(D) is elliptic and satisfies a recently introduced cancellation property. We recover in particular a Hardy inequality due to V. Maz ′ ya, and a Sobolev inequality due to J. Bourgain and H. Brezis. We also study the necessity of these two conditions. α∈N n |α|=kHere, A α is a linear map in L(V ; E), for every α ∈ N n with |α| = k.

Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations

“…A positive answer to Open problem 1 or 2 would imply some limiting Sobolev-type inequalities into L ∞ which have been proved since [71,Proposition 3], [109, p. 911], [21].…”

“…An interesting consequence is the characterization of the operators such that a Gagliardo-Nirenberg-Sobolev inequality or a Hardy inequality holds [109, Theorem 1.3 and Proposition 6.1], [21]. Theorem 3.8.…”

“…The cancelation condition is not necessary [109, Remark 5.1], [21]. Indeed, the derivative D on R is not canceling, but the inequality u L ∞ ≤ u holds nevertheless.…”

“…The ellipticity assumption in Theorem 3.8 is necessary for the first-order Sobolev inequality of (i) [109, Theorem 1.3 and Corollary 5.2] and for HardySobolev inequalities [21]; it is not necessary for Hardy inequalities of the form (ii) [21] nor for higher-order Sobolev inequalities [109,Proposition 5.4].…”

A note on estimates for elliptic systems with L1 data

“…To this end let us recall what can already be said in light of the literature [1,2,4,7,10]. We first consider the case of the estimates (1.6).…”