Classification of anisotropic local Hardy spaces and inhomogeneous Triebel–Lizorkin spaces

Abstract This paper provides a characterization of when two expansive matrices yield the same anisotropic local Hardy and inhomogeneous Triebel–Lizorkin spaces. The characterization is in terms of the coarse equivalence of certain quasi-norms associated to the matrices. For nondiagonal matrices, these conditions are strictly weaker than those classifying the coincidence of the corresponding homogeneous function spaces. The obtained results complete the classification of anisotropic Besov and Triebel–Lizorkin spaces associated to general expansive matrices.