article

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

Authors: Pierre Bousquet and Jean Van Schaftingen

Issue: Volume 63 (2014), Issue 5, 1419-1445

Abstract:

$Given a homogeneous $k$-th order differential operator $A(\mathrm{D})$ on $\mathbb{R}^n$ between two finite dimensional spaces, we establish the Hardy inequality \[ \int_{\mathbb{R}^n}\frac{|\mathrm{D}^{k-1}u(x)|}{|x|}\,\mathrm{d}x\leq C\int_{\mathbb{R}^n}|A(\mathrm{D})u| \] and the Sobolev inequality \[ \|\mathrm{D}^{k-n}u\|_{L^{\infty}(\mathbb{R}^n)}\leq C\int_{\mathbb{R}^n}|A(\mathrm{D})u| \] when $A(\mathrm{D})$ is elliptic and satisfies a recently introduced cancellation property. We recover in particular a Hardy inequality due to V.\ Maz\cprime{}ya, and a Sobolev inequality due to J.\ Bourgain and H.\ Brezis. We also study the necessity of these two conditions.$