Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix ${\mathcal{A}}_{\infty }$ Weights

Abstract

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 $\phi$-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.

96 pages