Littlewood-Paley Theory for Matrix-Weighted Function Spaces

Abstract

We define the vector-valued, matrix-weighted function spaces ${\dot{F}}_p^{\alpha q}(W)$ (homogeneous) and $F_p^{\alpha q}(W)$ (inhomogeneous) on ${\mathbb{R}}^n$, for $\alpha \in \mathbb{R}$, $0<p<\infty$, $0<q\le \infty$, with the matrix weight $W$ belonging to the $A_p$ class. For $1<p<\infty$, we show that $L^p(W)={\dot{F}}_p^{02}(W)$, and, for $k\in \mathbb{N}$, that $F_p^{k2}(W)$ coincides with the matrix-weighted Sobolev space $L_k^p(W)$, thereby obtaining Littlewood-Paley characterizations of $L^p(W)$ and $L_k^p(W)$. We show that a vector-valued function belongs to ${\dot{F}}_p^{\alpha q}(W)$ if and only if its wavelet or $\phi$-transform coefficients belong to an associated sequence space ${\dot{f}}_p^{\alpha q}(W)$. We also characterize these spaces in terms of reducing operators associated to $W$.