# Littlewood–Paley Characterizations of Weighted Anisotropic Triebel–Lizorkin Spaces via Averages on Balls I

### Jun Liu

Beijing Normal University, China### Dachun Yang

Beijing Normal University, China### Wen Yuan

Beijing Normal University, China

## Abstract

This article is the first part of two works of the authors on the same topic. Let $A$ be a general expansive matrix on $R_{n}$ and $w∈A_{∞}(A)$ a Muckenhoupt $A_{∞}$-weight with respect to $A$. In this article, the authors first characterize the weighted anisotropic Triebel–Lizorkin space $F_{p,q}(A;w)$ in terms of Peetre maximal functions or Lusin-area functions defined via Fourier analytical tools. As an application, the authors also establish a characterization of $F_{p,q}(A;w)$ with smoothness order $α∈(0,2ζ_{−})$ via a Lusin-area function involving the difference between $f(x)$ and its ball average

where $b:=∣detA∣,σ(A)$ denotes the set of all eigenvalues of $A$,

$ρ$ denotes the step homogeneous quasi-norm associated with $A$ and, for any $k∈{1,2,…}$ and $x∈R_{n}$, $B_{ρ}(x,b_{−k}):={y∈R_{n}:ρ(x−y)<b_{−k}}$.

## Cite this article

Jun Liu, Dachun Yang, Wen Yuan, Littlewood–Paley Characterizations of Weighted Anisotropic Triebel–Lizorkin Spaces via Averages on Balls I. Z. Anal. Anwend. 38 (2019), no. 4, pp. 397–418

DOI 10.4171/ZAA/1643