Classification of anisotropic Triebel-Lizorkin spaces

Abstract

This paper provides a characterization of expansive matrices $$A \in \textrm{GL}(d, {\mathbb {R}})$$ A ∈ GL ( d , R ) generating the same anisotropic homogeneous Triebel–Lizorkin space $$\dot{\textbf{F}}^{\alpha }_{p,q}(A)$$ F ˙ p , q α ( A ) for $$\alpha \in {\mathbb {R}}$$ α ∈ R and $$p,q \in (0,\infty ]$$ p , q ∈ ( 0 , ∞ ] . It is shown that $$\dot{\textbf{F}}^{\alpha }_{p,q}(A) = \dot{\textbf{F}}^{\alpha }_{p,q}(B)$$ F ˙ p , q α ( A ) = F ˙ p , q α ( B ) if and only if the homogeneous quasi-norms $$\rho _A, \rho _B$$ ρ A , ρ B associated to the matrices A, B are equivalent, except for the case $$\dot{\textbf{F}}^0_{p, 2} = L^p$$ F ˙ p , 2 0 = L p with $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) . The obtained results complement and extend the classification of anisotropic Hardy spaces $$H^p(A) = \dot{\textbf{F}}^{0}_{p,2}(A)$$ H p ( A ) = F ˙ p , 2 0 ( A ) , $$p \in (0,1]$$ p ∈ ( 0 , 1 ] , in Bownik (Mem Am Math Soc 164(781):vi+122, 2003).