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

Abstract

This article is the second part of two works of the authors on the same topic. Let $A_{\vec{a}}$ be the matrix diag$\{2^{a_1},\dots ,2^{a_n}\}$, with $\vec{a}:=(a_1,\dots ,a_n)\in (0,\infty {)}^n$, 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 $F_{p,q}^{\alpha }(A_{\vec{a}};w)$ with smoothness order $\alpha \in (0,2{\zeta }_{-})$ in terms of the Lusin-area function and the Littlewood–Paley $g_{\lambda }^{\ast }$-function, defined via the difference between $f(x)$ and its ball average

where $b:=|\det A_{\vec{a}}|$, $\sigma (A_{\vec{a}})$ denotes the set of all eigenvalues of $A_{\vec{a}}$,

Further, $\rho$ denotes the step homogeneous quasi-norm associated with $A_{\vec{a}}$ and, for any $k\in \{1,2,\dots \}$ and $x\in {\mathbb{R}}^n$, $B_{\rho }(x,b^{-k}):=\{y\in {\mathbb{R}}^n:\rho (x-y)<b^{-k}\}$. As applications, the authors obtain a series of characterizations for weighted anisotropic Triebel–Lizorkin spaces $F_{p,\infty }^{\alpha }(A_{\vec{a}};w)$ via pointwise inequalities involving ball averages.