Matrix weighted \(\alpha \)-modulation spaces

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