Given a homogeneous k-th order differential operator A(D) on R n between two finite dimensional spaces, we establish the Hardy inequalityand the Sobolev inequalitywhen A(D) is elliptic and satisfies a recently introduced cancellation property. We recover in particular a Hardy inequality due to V. Maz ′ ya, and a Sobolev inequality due to J. Bourgain and H. Brezis. We also study the necessity of these two conditions. α∈N n |α|=kHere, A α is a linear map in L(V ; E), for every α ∈ N n with |α| = k.
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