Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II

Abstract

This paper is a continuation of [arXiv:2104.14361]. It concerns maximal characterizations of anisotropic Triebel-Lizorkin spaces Ḟα p,q for the endpoint case of p = ∞ and the full scale of parameters α ∈ R and q ∈ (0, ∞]. In particular, a Peetre-type characterization of the anisotropic Besov space Ḃα ∞,∞ = Ḟα ∞,∞ is obtained. As a consequence, it is shown that there exist dual molecules of frames and Riesz sequences in Ḟα ∞,q .

Classification of anisotropic Triebel-Lizorkin spaces

“…In the proof of Theorem 1.2, the criterion (1.1) is used to control the overlap of the Fourier supports of the A-dilates and B-dilates of the analyzing vectors ϕ and ψ, respectively, that are used to define the spaces Ḟα p,q (A) and Ḟα p,q (B). Combined with our maximal characterizations of Triebel-Lizorkin spaces obtained in [18,19], this allows to conclude that the analyzing vectors ϕ and ψ for A respectively B define the same space Ḟα p,q (A) = Ḟα p,q (B).…”

“…For p, q < ∞, the space S 0 (R d ) is a dense subspace of Ḟα p,q (A). This fact follows easily from the various atomic and molecular decompositions of Ḟα p,q (A), see, e.g., [4,6,18,19].…”

“…Lastly, let us mention an application of Theorem 1.2. In [18,19], we proved continuous maximal characterizations of anisotropic Triebel-Lizorkin spaces Ḟα p,q (A) and obtained new results on their molecular decomposition. These results were obtained under the additional assumption that the expansive matrix A is exponential, in the sense that A = exp(C) for some matrix C ∈ R d×d .…”

“…In [6] and the introduction of this paper, the spaces Ḟα ∞,q (A), q ∈ (0, ∞), are alternatively defined using the cube [0, 1] d instead of an expansive ellipsoid Ω A . However, it is easily seen that both conditions define the same space, see, e.g., [19,Lemma 2.2]. See also Theorem 3.1 below for the equivalent norm on Ḟα ∞,∞ (A) used in the introduction.…”

“…For this particular case, the Triebel-Lizorkin spaces provide a unifying scale of function spaces that encompasses, among others, the Lebesgue, Sobolev, Hardy and BMO spaces. The anisotropic Triebel-Lizorkin spaces Ḟα p,q (A) associated to a general expansive matrix A were first introduced in [6] and further studied in, e.g., [1,4,5,8,[18][19][20]. These anisotropic spaces are useful for the analysis of mixed homogeneity properties of functions and operators as the dilation structure allows different directions to be scaled by different dilation factors.…”