Classification of anisotropic Triebel-Lizorkin spaces

This paper provides a characterization of expansive matrices A ∈ GL(d, R) generating the same anisotropic homogeneous Triebel-Lizorkin space F˙p,qα(A) for α ∈ R and p, q ∈ (0, ∞]. It is shown that F˙p,qα(A) = F˙p,qα(B) if and only if the homogeneous quasi-norms ρA, ρB associated to the matrices A, B are equivalent, except for the case F˙p,20 = Lp with p ∈ (1, ∞). The obtained results complement and extend the classification of anisotropic Hardy spaces Hp(A) = F˙p,20(A), p ∈ (0, 1], in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).