Discrete Triebel-Lizorkin spaces and expansive matrices

2024/09/03 by van Velthoven, Jordy Timo, Voigtlaender, Felix
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2409.01849

Abstract

We provide a characterization of two expansive dilation matrices yielding equal discrete anisotropic Triebel-Lizorkin spaces. For two such matrices A and B, it is shown that f^α_p,q(A) = f^α_p,q(B) for all α∈ ℝ and p, q ∈ (0, ∞] if and only if the set \A^j B^-j : j ∈ ℤ\ is finite, or in the trivial case when p = q and |det(A)|^{α+ 1/2 - 1/p} = |det(B)|^{α+ 1/2 - 1/p}. This provides an extension of a result by Triebel for diagonal dilations to arbitrary expansive matrices. The obtained classification of dilations is different from corresponding results for anisotropic Triebel-Lizorkin function spaces.

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