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).