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Zeitschrift für Analysis und ihre Anwendungen


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Volume 39, Issue 1, 2020, pp. 1–26
DOI: 10.4171/ZAA/1648

Published online: 2020-01-24

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

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 second part of two works of the authors on the same topic. Let $A_{\vec{a}}$ be the matrix diag$\{2^{a_1},\ldots,2^{a_n}\}$, with $\vec{a}:=(a_1,\ldots, a_n)\in (0,\infty)^n$, and let $w\in\mathcal{A}_\infty(A_{\vec{a}})$ be a Muckenhoupt $\mathcal{A}_\infty$-weight with respect to $A_{\vec{a}}$. In this article, the authors characterize the weighted anisotropic Triebel–Lizorkin space $F^\alpha_{p,q}(A_{\vec{a}};\,w)$ with smoothness order $\alpha\in(0,2\zeta_-)$ in terms of the Lusin-area function and the Littlewood–Paley $g_\lambda^*$-function, defined via 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 $b:=|\mathrm {det} A_{\vec{a}}|$, $\sigma(A_{\vec{a}})$ denotes the set of all eigenvalues of $A_{\vec{a}}$, $$\lambda_-\in(1,\min\{|\lambda|: \lambda\in\sigma(A_{\vec{a}})\}],\quad \zeta_-:=\log_b\lambda_-.$$ Further, $\rho$ denotes the step homogeneous quasi-norm associated with $A_{\vec{a}}$ 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)

Keywords: Anisotropic weighted Triebel–Lizorkin space, ball average, Lusin-area function, $g_\lambda^*$-function

Liu Jun, Yang Dachun, Yuan Wen: Littlewood–Paley Characterizations of Weighted Anisotropic Triebel–Lizorkin Spaces via Averages on Balls II. Z. Anal. Anwend. 39 (2020), 1-26. doi: 10.4171/ZAA/1648