Traces of anisotropic Besov--Lizorkin--Triebel spaces---a complete treatment of the borderline cases

TL;DR

This paper provides a comprehensive analysis of the trace spaces of anisotropic Besov--Lizorkin--Triebel spaces, including previously unstudied borderline cases, revealing new approximation spaces and extending interpolation and embedding results.

Contribution

It fully characterizes the trace spaces for anisotropic Besov--Lizorkin--Triebel spaces, including borderline cases, and introduces new approximation spaces with extended interpolation and embedding results.

Findings

Trace spaces are approximation spaces in all cases.

Borderline cases yield new approximation spaces.

Extended interpolation and Sobolev embedding results for anisotropic scales.

Abstract

Including the previously untreated borderline cases, the trace spaces in the distributional sense of the Besov--Lizorkin--Triebel spaces are determined for the anisotropic (or quasi-homogeneous) version of these classes. The ranges of the trace are in all cases shown to be approximation spaces, and these are shown to be different from the usual spaces precisely in the previously untreated cases. To analyse the new spaces, we carry over some real interpolation results as well as the refined Sobolev embeddings of J.~Franke and B.~Jawerth to the anisotropic scales.