带矩阵$\mathcal A_\infty$权的各向异性Besov空间的实变刻画及其应用
Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix $\mathcal A_\infty$ Weights
AI总结：
本文建立带$\mathcal A_{p,\infty}$矩阵权的各向异性Besov空间理论，给出其$\varphi$变换与分子刻画，构造反例说明权条件的最优性，引入临界重标指数并得到相关算子的精确有界性。
AI中文摘要：
设$\alpha\in\mathbb{R}$，$p\in(0,\infty)$，$q\in(0,\infty]$。本文建立了与膨胀矩阵$A$及$\mathcal A_{p,\infty}$矩阵权$W$相关的矩阵加权各向异性Besov空间理论。我们首先引入齐次空间$\dot B_{p,q}^\alpha(A,W)$，并建立其$\varphi$变换刻画。随后构造反例表明，该刻画中$W\in\mathcal A_{p,\infty}$的假设不能放宽为$W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$；这些反例同时证明，更弱的条件$W\in\bigcup_{r\in(0,\infty)}\mathcal A_r$不足以保证$\dot B_{p,q}^\alpha(A,W)$的良定性。接着我们通过重标极大算子刻画$\mathcal A_{p,\infty}$矩阵权，自然引出临界重标指数的新概念，用以定量刻画矩阵权的自提升性质。借助该指数，我们得到了相关序列空间$\dot b_{p,q}^\alpha(A,W)$上几乎对角算子的最优有界性。在此基础上，我们进一步建立了$\dot B_{p,q}^\alpha(A,W)$的分子刻画，以及伪微分算子在这些空间上的若干精确有界性结果。
英文摘要：
Let $α\in\mathbb{R}$, $p\in(0,\infty)$, 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 $\dot B_{p,q}^α(A,W)$ and establish their $φ$-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 $\dot B_{p,q}^α(A,W)$. 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 $\dot b_{p,q}^α(A,W)$. Based on these, we further establish the molecular characterization of $\dot B_{p,q}^α(A,W)$ and some sharp boundedness results for pseudo-differential operators on these spaces.
