  We prove well-posedness in weighted tent spaces of weak solutions to the
Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the
source $f$ also lies in (different) weighted tent spaces, provided the complex
coefficient matrix $A$ is bounded, measurable, time-independent, and uniformly
elliptic. To achieve this, we extend the theory of singular integral operators
on tent spaces via off-diagonal estimates introduced by [arXiv:1112.4292] to
obtain estimates on solutions $u$, and also $\nabla u$, $\partial_t u$, and
$\mathrm{div} A \nabla u$ in weighted tent spaces, showing at the same time
maximal regularity. Uniqueness follows from a different strategy using interior
representation for weak solutions and boundary behavior.
