Discrete Triebel-Lizorkin spaces and expansive matrices

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.