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

TL;DR

This paper extends the theory of anisotropic Triebel-Lizorkin spaces by providing maximal characterizations at the endpoint case and establishing dual molecular frames and Riesz sequences.

Contribution

It offers new maximal characterizations for endpoint cases of anisotropic Triebel-Lizorkin spaces and introduces dual molecular frames and Riesz sequences.

Findings

Maximal characterizations for $p= obreak \infty$ in anisotropic Triebel-Lizorkin spaces.

Peetre-type characterization of anisotropic Besov space.

Existence of dual molecular frames and Riesz sequences.

Abstract

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