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Zeitschrift für Analysis und ihre Anwendungen


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Volume 38, Issue 4, 2019, pp. 397–418
DOI: 10.4171/ZAA/1643

Published online: 2019-10-23

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

Jun Liu[1], Dachun Yang[2] and Wen Yuan[3]

(1) Beijing Normal University, China
(2) Beijing Normal University, China
(3) Beijing Normal University, China

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 $F^\alpha_{p,q}(A;\,w)$ 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 $F^\alpha_{p,q}(A;\,w)$ with smoothness order $\alpha\in(0,2\zeta_-)$ via a Lusin-area function involving the difference between $f(x)$ and its ball average $$B_{b^{-k}}f(x):=\frac1{|B_\rho(x,b^{-k})|}\int_{B_\rho(x,b^{-k})}f(y)\,dy,\quad \forall\,x\in\mathbb{R}^n,\ \forall\,k\in\{1,2,\ldots\},$$ where $b:=|\mathrm {det} A|, \sigma(A)$ denotes the set of all eigenvalues of $A$, $$\lambda_-\in(1,\min\{|\lambda|: \lambda\in\sigma(A)\}],\quad \zeta_-:=\log_b\lambda_-,$$ $\rho$ denotes the step homogeneous quasi-norm associated with $A$ and, for any $k\in\{1,2,\ldots\}$ and $x\in\mathbb{R}^n$, $B_\rho(x,b^{-k}):=\{y\in\mathbb{R}^n: \rho(x-y) < b^{-k}\}$.

Keywords: Anisotropic weighted Triebel–Lizorkin space, expansive matrix, ball average, Peetre maximal function, Lusin-area function

Liu Jun, Yang Dachun, Yuan Wen: Littlewood–Paley Characterizations of Weighted Anisotropic Triebel–Lizorkin Spaces via Averages on Balls I. Z. Anal. Anwend. 38 (2019), 397-418. doi: 10.4171/ZAA/1643