Matrix-Weighted Besov Spaces Associated with Non-isotropic Dilations

Let $\alpha \in \mathbb{R}$, $p\in [1,\mathrm{\infty })$, $q\in (0,\mathrm{\infty }]$, $\mathbf{W}$ be a matrix weight, and $A$ be an expansive dilation on ${\mathbb{R}}^d$. In this paper, the authors firstly investigate and develop some aspects of homogeneous anisotropic Besov spaces ${\dot{B}}_{p,A}^{\alpha ,q}({\mathbb{R}}^d,\mathbf{W})$ and inhomogeneous anisotropic Besov spaces $B_{p,A}^{\alpha ,q}({\mathbb{R}}^d,\mathbf{W})$ theory in the matrix weight setting. Moreover, we show that these spaces are characterized by the magnitude of the $\phi$-transforms in appropriate sequence spaces. Notably, all these results remain novel even in the diagonal non-isotropic case (when $A=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_d)$ with $\{{\lambda }_j{\}}_{j=1}^d\subset \mathbb{C}$).