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.

35 pages. The accepted manuscript. (Final version appeared on 4 March 2016 in Mathematische Nachrichten.)

References in corpus (5)

• On the trace problem for Triebel--Lizorkin spaces with mixed norms
• Pointwise multiplication of Besov and Triebel--Lizorkin spaces
• A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms
• Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms
• Pointwise estimates of pseudo-differential operators

Cited by in corpus (2)

• Wavelet transforms for homogeneous mixed-norm Triebel--Lizorkin spaces
• On Seeley-type Universal Extension Operators for the Upper Half Space