Classification of anisotropic 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 ${\dot{\mathbf{F}}}_{p,q}^{\alpha }(A)$ for $\alpha \in \mathbb{R}$ and $p,q\in (0,\infty ]$. It is shown that ${\dot{\mathbf{F}}}_{p,q}^{\alpha }(A)={\dot{\mathbf{F}}}_{p,q}^{\alpha }(B)$ if and only if the homogeneous quasi-norms ${\rho }_A,{\rho }_B$ associated to the matrices $A,B$ are equivalent, except for the case ${\dot{\mathbf{F}}}_{p,2}^0=L^p$ with $p\in (1,\infty )$. The obtained results complement and extend the classification of anisotropic Hardy spaces $H^p(A)={\dot{\mathbf{F}}}_{p,2}^0(A)$, $p\in (0,1]$, in [Mem. Am. Math. Soc. 781, 122 p. (2003)].