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

TL;DR

This paper establishes well-posedness and maximal regularity for parabolic Cauchy problems in weighted tent spaces, extending singular integral operator theory to handle solutions and their derivatives under minimal assumptions on the coefficients.

Contribution

It extends the theory of singular integral operators on weighted tent spaces to prove well-posedness and maximal regularity for parabolic equations with measurable, time-independent coefficients.

Findings

Proves well-posedness of weak solutions in weighted tent spaces.

Establishes maximal regularity for solutions and their derivatives.

Extends singular integral operator theory to 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.