Matrix-Weighted Besov Spaces Associated with Non-isotropic Dilations
Xiong Liu, Wenhua Wang
TL;DR
The paper develops a comprehensive theory of matrix-weighted Besov spaces in the setting of anisotropic dilations by $A$, defining both the homogeneous $\dot{B}_{p,A}^{\alpha,q}(\mathbb{R}^d,\mathbf{W})$ and inhomogeneous $B_{p,A}^{\alpha,q}(\mathbb{R}^d,\mathbf{W})$ spaces and characterizing them via $\varphi$-transforms in corresponding sequence spaces. It proves boundedness of the forward and inverse $\varphi$-transforms, introduces molecular and reducing-operator frameworks, and establishes equivalence of norms across test-function choices, providing a robust link between function and sequence spaces. The work further extends to inhomogeneous spaces and develops an almost diagonal operator calculus, showing that operators with suitably decaying matrix coefficients act boundedly on the Besov spaces under the $A_p$ matrix-weight condition. Collectively, these results generalize classical isotropic and scalar-weighted theories to a broad anisotropic, matrix-weighted context, with novelty even in the diagonal non-isotropic case and broad implications for harmonic analysis and PDEs in weighted, non-isotropic settings.
Abstract
Let $α\in\mathbb{R}$, $p\in[1,\infty)$, $q\in(0,\infty]$, $\mathbf{W}$ 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 $\dot{B}^{α,q}_{p,A}(\mathbb{R}^d,\mathbf{W})$ and inhomogeneous anisotropic Besov spaces $B^{α,q}_{p,A}(\mathbb{R}^d,\mathbf{W})$ 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 $A = \mathrm{diag}(λ_1, λ_2, \ldots, λ_d)$ with $\{λ_j\}_{j=1}^d \subset \mathbb{C}$).