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,q ^α(A) for α∈R and p,q∈(0,∞]. It is shown that F˙ _p,q ^α(A)=F˙ _p,q ^α(B) if and only if the homogeneous quasi-norms ρ _A,ρ _B associated to the matrices A, B are equivalent, except for the case F˙ _p,2 ⁰=L ^p with p∈(1,∞). The obtained results complement and extend the classification of anisotropic Hardy spaces H ^p(A)=F˙ _p,2 ⁰(A), p∈(0,1], in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).