We establish higher integrability estimates for constant-coefficient systems of linear PDEs
\[
A
μ
=
σ
,
\mathcal {A} \mu = \sigma ,
\]
where
μ
∈
M
(
Ω
;
V
)
\mu \in \mathcal {M}(\Omega ;V)
and
σ
∈
M
(
Ω
;
W
)
\sigma \in \mathcal {M}(\Omega ;W)
are vector measures and the polar
d
μ
d
|
μ
|
\frac {\mathrm {d}\mu }{\mathrm {d}|\mu |}
is uniformly close to a convex cone of
V
V
intersecting the wave cone of
A
\mathcal {A}
only at the origin. More precisely, we prove local compensated compactness estimates of the form
\[
‖
μ
‖
L
p
(
Ω
′
)
≲
|
μ
|
(
Ω
)
+
|
σ
|
(
Ω
)
,
Ω
′
⋐
Ω
.
\|\mu \|_{\mathrm {L}^p(\Omega ’)} \lesssim |\mu |(\Omega ) + |\sigma |(\Omega ), \qquad \Omega ’ \Subset \Omega .
\]
Here, the exponent
p
p
belongs to the (optimal) range
1
≤
p
>
d
/
(
d
−
k
)
1 \leq p > d/(d-k)
,
d
d
is the dimension of
Ω
\Omega
, and
k
k
is the order of
A
\mathcal {A}
. We also obtain the limiting case
p
=
d
/
(
d
−
k
)
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.