Littlewood-Paley theory for matrix-weighted function spaces

We define the vector-valued, matrix-weighted function spaces (F) over dot(p)(alpha q)(W) (homogeneous) and F-p(alpha q)(W) (inhomogeneous) on R-n, for alpha is an element of R, 0 < p < infinity, 0 < q <= infinity, with the matrix weight W belonging to the A(p) class. For 1 < p < infinity, we show that L-p(W) = (F) over dot(p)(alpha q)(W), and, for k is an element of N, that F-p(k2)(W) coincides with the matrix-weighted Sobolev space Lkp(W), thereby obtaining Littlewood-Paley characterizations of L-p(W) and L-k(p)(W). We show that a vector-valued function belongs to (F) over dot(p)(alpha q)(W) if and only if its wavelet or phi-transform coefficients belong to an associated sequence space (F) over dot(p)(alpha q)(W). We also characterize these spaces in terms of reducing operators associated to W.