This article is the first part of two works of the authors on the same topic. Let be a general expansive matrix on and a Muckenhoupt -weight with respect to . In this article, the authors first characterize the weighted anisotropic Triebel–Lizorkin space 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 with smoothness order via a Lusin-area function involving the difference between and its ball average
where denotes the set of all eigenvalues of ,
denotes the step homogeneous quasi-norm associated with and, for any and , .
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. 397–418DOI 10.4171/ZAA/1643