Discrete Triebel-Lizorkin spaces and expansive matrices

TL;DR

This paper characterizes when two expansive matrices define the same discrete anisotropic Triebel-Lizorkin spaces, extending Triebel's results from diagonal to arbitrary matrices and revealing a new classification of dilations.

Contribution

It extends Triebel's classification of dilations for Triebel-Lizorkin spaces from diagonal to general expansive matrices, providing necessary and sufficient conditions.

Findings

Spaces coincide if the set of matrix powers is finite.

Equality holds trivially when p=q and determinants satisfy a specific relation.

The classification differs from previous results for anisotropic Triebel-Lizorkin spaces.

Abstract

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices $A$ and $B$, it is shown that $\dot{\mathbf{f}}^{\alpha}_{p,q}(A) = \dot{\mathbf{f}}^{\alpha}_{p,q}(B)$ for all $\alpha \in \mathbb{R}$ and $p, q \in (0, \infty]$ if and only if the set $\{A^j B^{-j} : j \in \mathbb{Z}\}$ is finite, or in the trivial case when $p = q$ and $|\det(A)|^{\alpha + 1/2 - 1/p} = |\det(B)|^{\alpha + 1/2 - 1/p}$. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.