We establish higher integrability estimates for constant-coefficient systems of linear PDEs [ mathcal A mu = sigma, ] where $mu in mathcal M(Omega;V)$ and $sigmain mathcal M(Omega;W)$ are vector measures and the polar $frac{dmu}{d |mu|}$ is uniformly close to a subspace $L$ of $V$ intersecting the wave cone of $mathcal A$ only at the origin. More precisely, we prove local compensated compactness estimates of the form [ |mu|_{L^p(Omega)} lesssim |mu|(Omega) + |sigma|(Omega), qquad Omega Subset Omega. ] Here, the exponent $p$ belongs to the (optimal) range $1 leq p < d/(d-k)$, $d$ is the dimension of $Omega$, and $k$ is the order of $mathcal A$. For canceling constant-rank operators we also obtain the limiting case $p = d/(d-k)$. We consider applications to compensated compactness as well as applications to the theory of functions of bounded variation and bounded deformation.
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