A marriage between geometric measure theory and advanced parabolic theory

We prove that linear, non-autonomous parabolic equations 
\[
\frac {\partial u }{\partial t} - \mathcal A(\cdot) u = \rho 
\]
with measure-valued right hand side $\rho$ admit
a unique solution in the space of maximal parabolic regularity 
\[
L^r(J;W^{1 ,q}_D(\Omega) ) \cap W^{1,r}(J;W^{-1,q}_D(\Omega)), \quad q , r \sim 2
\]
if the measure $\rho$ is of the form 
\[
C_0(J \times \Omega) \ni f \mapsto \int_J \int_\Omega f(t,x) \, d\rho_t(x) \, dt
\]
and the measures $\rho_t$ live on upper l-sets $M_t \subset \Omega$ in the sense of Jonsson/Wallin.\\
These sets $M_t$ may be thought, in particular cases, as 'curves' or 'surfaces' within $\Omega$ and are 
allowed to 'wander in time'.\\
The regularity theorem rests, on one hand, on the insight that such measures may be identified with elements of
$W^{-1,q}_D(\Omega))$ and, on the other, on a non-trivial regularity result for non-autonomous
equations in the spaces $W^{-1,q}_D(\Omega))$ .