On well-posedness and maximal regularity for parabolic Cauchy problems on weighted tent spaces

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem \(\partial _t u - {{\,\textrm{div}\,}}A \nabla u = f, u(0)=0\) ${\partial }_{t}u-\text{div}A\mathrm{\nabla }u=f,u(0)=0$ , 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 [8] to obtain estimates on solutions u, and also \(\nabla u\) $\mathrm{\nabla }u$ , \(\partial _t u\) ${\partial }_{t}u$ , and \({{\,\textrm{div}\,}}A \nabla u\) $\text{div}A\mathrm{\nabla }u$ 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.