  We establish higher integrability estimates for constant-coefficient systems
of linear PDEs
  \[
  \mathcal{A} \mu = \sigma,
  \]
  where $\mu \in \mathcal{M}(\Omega;V)$ and $\sigma\in \mathcal{M}(\Omega;W)$
are vector measures and the polar $\frac{\mathrm{d} \mu}{\mathrm{d} |\mu|}$ is
uniformly close to a convex cone of $V$ intersecting the wave cone of
$\mathcal{A}$ only at the origin. More precisely, we prove local compensated
compactness estimates of the form
  \[
  \|\mu\|_{\mathrm{L}^p(\Omega')} \lesssim |\mu|(\Omega) + |\sigma|(\Omega),
\qquad \Omega' \Subset \Omega.
  \]
  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 $\mathcal{A}$. 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.
