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 \(\alpha \in {\mathbb {R}}\), \(0<p<\infty \), \(0<q \le \infty \), with the matrix weight W belonging to the \(A_p\) class. For \(1<p<\infty \), we show that \(L^p(W) = \dot{F}^{0 2}_p(W)\), 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.

我们定义矢量值的矩阵加权函数空间\（\ dot {F} ^ {\ alpha q} _p（W）\）（齐次）和\（F ^ {\ alpha q} _p（W）\）\（{\ mathbb {R}} ^ n \）上的（不均匀），对于\（\ alpha {in \ mathbb {R}} \\），\（0 <p <\ infty \），\（0 < q \ le \ infty \），矩阵权重W属于\（A_p \）类。对于\（1 <p <\ infty \），我们证明\（L ^ p（W）= \ dot {F} ^ {0 2} _p（W）\），对于\（k \ in { \ mathbb {N}} \），即\（F ^ {k 2} _p（W）\）与矩阵加权Sobolev空间\（L ^ p_k（W）\）一致，从而获得\（L ^ p（W）\）和\（L ^ p_k（W）\）的Littlewood-Paley特征。我们证明，当且仅当向量值函数的小波或\（\ varphi \）变换系数属于相关序列时，它才属于\（\ dot {F} ^ {\ alpha q} _p（W）\）空格\（\ dot {f} ^ {\ alpha q} _p（W）\）。我们还根据减少与W关联的运算符来表征这些空间。