Given a matrix-weight $W$ in the Muckenhoupt class $\mathbf{ A}_p(\mathbb{ R}^n)$, $1\leq p<\infty$, we introduce corresponding vector-valued continuous and discrete $\alpha$-modulation spaces $M^{ s,\alpha}_{ p,q}(W)$ and $m^{ s,\alpha}_{ p,q}(W)$ and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of pseudo-differential operators on vector valued spaces is given.
Categories: math.FA