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

Abstract

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.

41 pages; typos corrected; introduction rewritten, text inside unchanged