Matrix-Weighted Besov Spaces Associated with Non-isotropic Dilations

Let $\alpha\in\mathbb{R}$, $p\in[1,\infty)$, $q\in(0,\infty]$, $\mathbf{W}$ 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 $\dot{B}^{\alpha,q}_{p,A}(\mathbb{R}^d,\mathbf{W})$ and inhomogeneous anisotropic Besov spaces $B^{\alpha,q}_{p,A}(\mathbb{R}^d,\mathbf{W})$ theory in the matrix weight setting. Moreover, we show that these spaces are characterized by the magnitude of the $\varphi$-transforms in appropriate sequence spaces. Notably, all these results remain novel even in the diagonal non-isotropic case (when $A = \mathrm{diag}(\lambda_1, \lambda_2, \ldots, \lambda_d)$ with $\{\lambda_j\}_{j=1}^d \subset \mathbb{C}$).