Anisotropic Lizorkin-Triebel spaces with mixed norms - traces on smooth boundaries

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.