Classification of anisotropic Triebel-Lizorkin spaces

TL;DR

This paper classifies when different expansive matrices generate the same anisotropic Triebel-Lizorkin spaces, extending previous work on Hardy spaces and providing a criterion based on the equivalence of associated quasi-norms.

Contribution

It establishes a classification theorem for anisotropic Triebel-Lizorkin spaces generated by expansive matrices, linking space equality to quasi-norm equivalence, thus extending prior Hardy space classifications.

Findings

Spaces coincide iff associated quasi-norms are equivalent

Identifies exceptions for the case b0F^0_{p,2} = L^p

Extends classification from Hardy spaces to Triebel-Lizorkin spaces

Abstract

This paper provides a classification theorem for expansive matrices $A \in \mathrm{GL}(d, \mathbb{R})$ generating the same anisotropic homogeneous Triebel-Lizorkin space $\dot{\mathbf{F}}^{\alpha}_{p, q}(A)$ for $\alpha \in \mathbb{R}$ and $p,q \in (0,\infty]$. It is shown that $\dot{\mathbf{F}}^{\alpha}_{p, q}(A) = \dot{\mathbf{F}}^{\alpha}_{p, q}(B)$ if and only if the homogeneous quasi-norms $\rho_A, \rho_B$ associated to the matrices $A, B$ are equivalent, except for the case $\dot{\mathbf{F}}^0_{p, 2} = L^p$ with $p \in (1,\infty)$. The obtained results complement and extend the classification of anisotropic Hardy spaces $H^p(A) = \dot{\mathbf{F}}^{0}_{p,2}(A)$, $p \in (0,1]$, in [Mem. Am. Math. Soc. 781, 122 p. (2003)].