Real-Variable Characterizations and Their Applications of Matrix-Weighted Triebel--Lizorkin Spaces

TL;DR

This paper provides new characterizations of matrix-weighted Triebel-Lizorkin spaces using classical harmonic analysis tools and proves boundedness of Fourier multipliers under generalized conditions, advancing understanding of weighted function spaces.

Contribution

It introduces novel characterizations of matrix-weighted Triebel-Lizorkin spaces and establishes Fourier multiplier boundedness using properties of matrix weights, differing from classical approaches.

Findings

Characterizations via Peetre maximal, Lusin area, and Littlewood-Paley functions.

Boundedness of Fourier multipliers under generalized Hörmander condition.

Utilization of matrix weight doubling property and reducing operators.

Abstract

Let $\alpha\in\mathbb R$, $q\in(0,\infty]$, $p\in(0,\infty)$, and $W$ be an $A_p(\mathbb{R}^n,\mathbb{C}^m)$-matrix weight. In this article, the authors characterize the matrix-weighted Triebel-Lizorkin space $\dot{F}_{p}^{\alpha,q}(W)$ via the Peetre maximal function, the Lusin area function, and the Littlewood-Paley $g_{\lambda}^{*}$-function. As applications, the authors establish the boundedness of Fourier multipliers on matrix-weighted Triebel-Lizorkin spaces under the generalized H\"ormander condition. The main novelty of these results exists in that their proofs need to fully use both the doubling property of matrix weights and the reducing operator associated to matrix weights, which are essentially different from those proofs of the corresponding cases of classical Triebel-Lizorkin spaces that strongly depend on the Fefferman-Stein vector-valued maximal inequality on Lebesgue spaces.