A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces

TL;DR

This paper develops an optimal regularity theory for second-order parabolic PDEs in weighted mixed norm Sobolev-Zygmund spaces, extending classical estimates to less regular coefficients and initial data.

Contribution

It introduces a new regularity framework that handles measurable coefficients in time and critical regularity cases, with sharp trace theorems for initial data.

Findings

Extended Schauder estimates to measurable coefficients

Established optimal regularity in weighted mixed norm spaces

Proved sharp trace theorems for initial data

Abstract

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.