Triebel-Lizorkin spaces and expansive matrices

TL;DR

This paper surveys classification theorems for expansive matrices that generate equivalent anisotropic Triebel-Lizorkin function and sequence spaces, highlighting the differences in their classification criteria.

Contribution

It provides a precise classification of matrices based on the equivalence of associated homogeneous quasi-norms and explains the relationship between function and sequence space classifications.

Findings

Function spaces are classified by matrices with equivalent homogeneous quasi-norms.

Sequence spaces are classified by a stricter condition than function spaces.

Two sequence spaces are retracts of each other when their corresponding function spaces are identical.

Abstract

We survey recent classification theorems for expansive matrices that generate the same anisotropic homogeneous Triebel-Lizorkin function space or sequence space. The function spaces are classified precisely by those matrices for which their associated homogeneous quasi-norms on Euclidean space are equivalent, whereas the sequence spaces are classified by a strictly stronger condition. We unravel this discrepancy between function spaces and sequence spaces by showing that two sequence spaces are retracts of each other whenever the corresponding function spaces are the same.