*preprint*

**Inserted:** 5 jun 2023

**Year:** 2021

**Abstract:**

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.