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