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