Discrete Triebel-Lizorkin spaces and expansive matrices

TL;DR

This paper classifies when two expansive dilation matrices yield identical discrete Triebel-Lizorkin sequence spaces. It proves a sharp dichotomy: equality for all parameters occurs iff the orbit $\{A^{j}B^{-j}: j\in\mathbb{Z}\}$ is finite (a strong pairing condition) or, in a trivial isotropic regime, a determinant-exponent balance holds for $p=q$. The authors extend Triebel’s diagonal-dilation result to arbitrary expansive matrices and provide a spectral characterization of isotropic sequence spaces, showing that $A/2$ must be periodic up to similarity. The findings reveal that the dilation-driven structure of discrete Triebel-Lizorkin spaces behaves differently from anisotropic spaces, with implications for transference arguments and isotropy analysis in harmonic analysis.

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 $\dot{\mathbf{f}}^α_{p,q}(A) = \dot{\mathbf{f}}^α_{p,q}(B)$ for all $α\in \mathbb{R}$ 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 $|\det(A)|^{α+ 1/2 - 1/p} = |\det(B)|^{α+ 1/2 - 1/p}$. 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.

Theorems & Definitions (26)

• Theorem 1.1: triebel2006theory
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: bownik2008duality
• Remark 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Theorem 3.2
• proof