Matrix-weighted Anisotropic Smoothness Spaces

Given a quasi-norm on ${\mathbb{R}}^d$ induced by a one-parameter dilation group, we consider matrix weights $W$ in an adapted Muckenhoupt class ${\mathbf{A}}_p$, $0<p<\mathrm{\infty }$, and use these weights to introduce and study anisotropic matrix-weighted smoothness spaces in both continuous and discrete settings. The spaces are constructed by means of a decomposition method in the frequency domain. We prove the equivalence of the continuous and discrete spaces using suitably adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and applications to the study of Fourier multipliers and pseudo-differential operators on vector-valued smoothness spaces are given.