Matrix weighted modulation spaces

Given a matrix-weight $W$ in the Muckenhoupt class ${\mathbf{A}}_p({\mathbb{R}}^n)$, $1\le p<\mathrm{\infty }$, we introduce corresponding vector-valued continuous and discrete $\alpha$-modulation spaces $M_{p,q}^{s,\alpha }(W)$ and $m_{p,q}^{s,\alpha }(W)$ 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.