Local Matrix Muckenhoupt Weights and Quantitative Weighted Inequalities Achieving Global Best Known Exponents

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 $-\mathrm{\Delta }+m^{2}I$ with $m\in (0,\mathrm{\infty })$ being large enough, whose quantitative bound when $p=2$ is precisely $[W{]}_{{\mathcal{A}}_2^{\mathrm{loc}}(r)}^{\frac{3}{2}}$.