Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix $\mathcal A_\infty$ Weights
Let $\alpha\in\mathbb{R}$, $p\in(0,\infty)$, and $q\in(0,\infty]$. In this article, we develop a theory of matrix-weighted anisotropic Besov spaces associated with an expansive matrix $A$ and an $\mathcal A_{p,\infty}$-matrix weight $W$. We first introduce the homogeneous spaces $\dot B_{p,q}^{\alpha}(A,W)$ and establish their $\varphi$-transform characterization. Then we construct counterexamples to show that the assumption $W\in\mathcal A_{p,\infty}$ in this characterization cannot be relaxed to $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$. The same counterexamples also show that this weaker condition $W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$ is insufficient to ensure the well-definedness of $\dot B_{p,q}^{\alpha}(A,W)$. Next we characterize $\mathcal A_{p,\infty}$-matrix weights via the rescaled maximal operator, which leads naturally to a new concept of the critical rescaling index that quantitatively captures the self-improving behavior of matrix weights. In terms of this index, we obtain optimal boundedness for almost diagonal operators on the associated sequence spaces $\dot b_{p,q}^{\alpha}(A,W)$. Based on these, we further establish the molecular characterization of $\dot B_{p,q}^{\alpha}(A,W)$ and some sharp boundedness results for pseudo-differential operators on these spaces.