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

Bousquet, Pierre Van Schaftingen, Jean

Given a homogeneous $k$-th order differential operator A (D) on \R^n between two finite dimensional spaces, we establish the Hardy inequality $$ \int_{\R^n} \frac{\abs{D^{k-1}u}}{\abs{x}} \dif x \leq C \int_{\R^n} \abs{A(D)u} $$ and the Sobolev inequality $$ \norm{D^{k-n} u}_{L^{\infty}(\R^n)}\leq C \int_{\R^n} \abs{A(D)u} $$ when $A(D)$ is elliptic and satisfies a recently introduced cancellation property. We also study the necessity of these two conditions.