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

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