No 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.

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 .

Continuing previous work, this paper provides maximal characterizations of anisotropic Triebel-Lizorkin spaces $$\dot{\textbf{F}}^{\alpha }_{p,q}$$ F ˙ p , q α for the endpoint case of $$p = \infty $$ p = ∞ and the full scale of parameters $$\alpha \in \mathbb {R}$$ α ∈ R and $$q \in (0,\infty ]$$ q ∈ ( 0 , ∞ ] . In particular, a Peetre-type characterization of the anisotropic Besov space $$\dot{\textbf{B}}^{\alpha }_{\infty ,\infty } = \dot{\textbf{F}}^{\alpha }_{\infty ,\infty }$$ B ˙ ∞ , ∞ α = F ˙ ∞ , ∞ α is obtained. As a consequence, it is shown that there exist dual molecular frames and Riesz sequences in $$\dot{\textbf{F}}^{\alpha }_{\infty ,q}$$ F ˙ ∞ , q α .

This paper provides maximal function characterizations of anisotropic Triebel–Lizorkin spaces associated to general expansive matrices for the full range of parameters $$p \in (0,\infty )$$ p ∈ ( 0 , ∞ ) , $$q \in (0,\infty ]$$ q ∈ ( 0 , ∞ ] and $$\alpha \in {\mathbb {R}}$$ 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. As an application, the existence of dual molecular frames and Riesz sequences is obtained; the wavelet systems are generated 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 mild decay, moment, and smoothness conditions.