Anisotropic Lizorkin--Triebel Spaces with Mixed Norms --- Traces on Smooth Boundaries

TL;DR

This paper studies trace operators on anisotropic Lizorkin--Triebel spaces with mixed norms on smooth boundary domains, extending theoretical foundations and applying results to the heat equation with compatibility conditions.

Contribution

It extends Lizorkin--Triebel space theory to mixed norms on manifolds and cylinders, and constructs bounded extension and trace operators with applications to PDEs.

Findings

Boundedness of Rychkov's extension operator in mixed-norm spaces

Explicit construction of a support-preserving right-inverse of the trace

Application to heat equation with derived compatibility conditions

Abstract

This article deals with trace operators on anisotropic Lizorkin--Triebel spaces with mixed norms over cylindrical domains with smooth boundary. As a preparation we include a rather self-contained exposition of Lizorkin--Triebel spaces on manifolds and extend these results to mixed-norm Lizorkin--Triebel spaces on cylinders in Euclidean space. In addition Rychkov's universal extension operator for a half space is shown to be bounded with respect to the mixed norms, and a support preserving right-inverse of the trace is given explicitly and proved to be continuous in the scale of mixed-norm Lizorkin--Triebel spaces. As an application, the heat equation is considered in these spaces, and the necessary compatibility conditions on the data are deduced.