Higher integrability for measures satisfying a PDE constraint

We establish higher integrability estimates for constant-coefficient systems of linear PDEs \[\Acal \mu = \sigma,\] where $\mu \in \Mcal(\Omega;V)$ and $\sigma\in \Mcal(\Omega;W)$ are vector measures and the polar $\frac{\di \mu}{\di |\mu|}$ is uniformly close to a convex cone of $V$ intersecting the wave cone of $\Acal$ only at the origin. More precisely, we prove local compensated compactness estimates of the form \[\|\mu\|_{\Lrm^p(\Omega')} \lesssim |\mu|(\Omega) + |\sigma|(\Omega), \qquad \Omega' \Subset \Omega.\] 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 $\Acal$. We also obtain the limiting case $p = d/(d-k)$ for canceling constant-rank operators. We consider applications to compensated compactness and {applications to the theory of} functions of bounded variation and bounded deformation.