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

TL;DR

This paper characterizes anisotropic Triebel-Lizorkin spaces using maximal functions and wavelet coefficients, providing new insights into their structure and applications to dual frames and Riesz sequences.

Contribution

It introduces maximal function characterizations for anisotropic Triebel-Lizorkin spaces with general expansive matrices and develops criteria for molecules without discrete subgroup restrictions.

Findings

Maximal function characterizations for all parameters

Existence of dual molecular frames and Riesz sequences

Wavelet systems generated by single functions with continuous parameters

Abstract

This paper provides maximal function characterizations of anisotropic Triebel-Lizorkin spaces associated to general expansive matrices for the full range of parameters $p \in (0,\infty)$, $q \in (0,\infty]$ and $\alpha \in \mathbb{R}$. 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.