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

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 ${\mathbb{R}}^n$ and $w\in {\mathcal{A}}_{\infty }(A)$ a Muckenhoupt ${\mathcal{A}}_{\infty }$-weight with respect to $A$. In this article, the authors first characterize the weighted anisotropic Triebel–Lizorkin space $F_{p,q}^{\alpha }(A;w)$ 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 $F_{p,q}^{\alpha }(A;w)$ with smoothness order $\alpha \in (0,2{\zeta }_{-})$ via a Lusin-area function involving the difference between $f(x)$ and its ball average

where $b:=|\det A|,\sigma (A)$ denotes the set of all eigenvalues of $A$,

$\rho$ denotes the step homogeneous quasi-norm associated with $A$ and, for any $k\in \{1,2,\dots \}$ and $x\in {\mathbb{R}}^n$, $B_{\rho }(x,b^{-k}):=\{y\in {\mathbb{R}}^n:\rho (x-y)<b^{-k}\}$.