AI 中文总结
本文研究局部 Muckenhoupt 矩阵权，建立局部极大、分数积分等算子的定量加权有界性，达到全局最佳指数，并应用于 Schrödinger 算子的 Riesz 变换。
AI 中文摘要
本文给出了局部 Muckenhoupt 矩阵权 $W$ 的各种实变量性质，并建立了若干算子在局部矩阵加权 Lebesgue 空间 $L^p(W)$ 上的定量有界性，包括局部（分数）极大算子、局部分数积分算子、局部 Haar 平方函数以及具有指数衰减的 Calderón--Zygmund 算子。对于局部极大算子，当 $p\in(1,2]$ 时我们获得了尖锐的定量界；而对于局部分数积分算子，我们获得了与全局最佳已知指数匹配的定量界，其标量情形已知是尖锐的。所采用的关键策略包括给出局部矩阵权的一个新的延拓性质（该性质可阐明其与全局权的关系）和一个最优尺度提升性质（该性质可平衡所考虑的权与算子的局部性）。作为应用，我们建立了与 Schrödinger 算子 $-\Delta+m^2I$（其中 $m\in(0,\infty)$ 足够大）相关联的 Riesz 变换在 $L^p(W)$ 上的定量有界性，当 $p=2$ 时其定量界恰为 $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$。
英文摘要
In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón--Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-Δ+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.
