# Littlewood-Paley Theory for Matrix-Weighted Function Spaces

## TL;DR

This paper develops Littlewood-Paley theory for matrix-weighted function spaces, providing characterizations and equivalences with Sobolev spaces and wavelet coefficients, extending harmonic analysis tools to matrix weights.

## Contribution

It introduces new matrix-weighted function spaces and establishes Littlewood-Paley characterizations, connecting them with Sobolev spaces and wavelet transforms.

## Key findings

- Equivalence of $L^p(W)$ and $ ext{F}^{0 2}_p(W)$ for $1<p<
$
- Identification of $F^{k 2}_p(W)$ with matrix-weighted Sobolev spaces
- Characterization of spaces via wavelet and $$-transform coefficients

## Abstract

We define the vector-valued, matrix-weighted function spaces $\dot{F}^{\alpha q}_p(W)$ (homogeneous) and $F^{\alpha q}_p(W)$ (inhomogeneous) on $\mathbb{R}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q \leq \infty$, with the matrix weight $W$ belonging to the $A_p$ class. For $1<p<\infty$, we show that $L^p(W) = \dot{F}^{0 2}_p(W)$, and, for $k \in \mathbb{N}$, that $F^{k 2}_p(W)$ coincides with the matrix-weighted Sobolev space $L^p_k(W)$, thereby obtaining Littlewood-Paley characterizations of $L^p(W)$ and $L^p_k (W)$. We show that a vector-valued function belongs to $\dot{F}^{\alpha q}_p(W)$ if and only if its wavelet or $\varphi$-transform coefficients belong to an associated sequence space $\dot{f}^{\alpha q}_p(W)$. We also characterize these spaces in terms of reducing operators associated to $W$.