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

Abstract

This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters p ∈ (0, ∞), q ∈ (0, ∞] and α ∈ R. The equivalent norm is defined in terms of the decay of wavelet coefficients, quantified by a Peetre-type space over a one-parameter dilation group. For the Banach space regime p, q ≥ 1, we use this characterization to prove the existence of frames and Riesz sequences of dual molecules for the Triebel-Lizorkin spaces; the atoms are obtained by translations and anisotropic dilations of a single function, where neither the translation nor dilation parameters are required to belong to a discrete subgroup. Explicit criteria for molecules are given in terms of smoothness, decay and moment conditions.

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

“…In a previous paper [20], we obtained characterizations of anisotropic Triebel-Lizorkin spaces Ḟα p,q , with α ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞], in terms of Peetre-type maximal functions and continuous wavelet transforms. In addition, as an application of these characterizations, it was shown that these spaces admit dual frames and Riesz sequences of molecules.…”

“…In contrast to the usual quasi-norms defining (anisotropic) Triebel-Lizorkin spaces Ḟα p,q for p < ∞ (see, e.g., [3][4][5]20]), the quantities (1.3) and (1.4) consider only averages over small scales. The quasi-norms (1.3) and (1.4) can therefore be considered as "localized versions" of the immediate analogue of the quasi-norms defining Ḟα p,q for p < ∞, which would lead to an unsatisfactory definition of Ḟα ∞,q , see [11,Section 5] and the references therein.…”

“…Maximal characterizations. As in [20], we assume additionally that the expansive matrix A ∈ GL(d, R) is exponential, in the sense that A = exp(B) for some matrix B ∈ R d×d . The power of A is then defined as A s := exp(sB) for s ∈ R.…”

“…The proof method of Theorem 1.1 is modeled on our proof of the maximal characterizations of Triebel-Lizorkin spaces Ḟα p,q with p ∈ (0, ∞) and q ∈ (0, ∞] given in [20]. In particular, it combines a sub-mean-value property of the Peetre-type maximal function (see Proposition 3.2) with maximal inequalities.…”

“…In particular, it combines a sub-mean-value property of the Peetre-type maximal function (see Proposition 3.2) with maximal inequalities. Despite the similarities in the approach, the calculations in the proof of Theorem 1.1 differ non-trivially from these in [20,Theorem 3.5] since only averages over small scales appear in the quasi-norms defining Ḟα ∞,q .…”