Hardy-Sobolev inequalities for vector fields and canceling linear differential operators Pierre BousquetJean Van Schaftingen 46E35 (26D10 42B20)Hardy inequalityHardy-Sobolev inequalityoverdetermined elliptic operatorhomogeneous differential operatorcanceling operatorcocanceling operatorexterior derivativesymmetric derivativeHodge-Hardy inequalityKorn-Hardy inequality 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5395 10.1512/iumj.2014.63.5395 en Indiana Univ. Math. J. 63 (2014) 1419 - 1445 state-of-the-art mathematics http://iumj.org/access/