We prove a local two-weight Poincaré inequality for cubes using the sparse domination method that has been influential in harmonic analysis. The proof involves a localized version of the Fefferman–Stein inequality for the sharp maximal function. By establishing a local-to-global result in a bounded domain satisfying a Boman chain condition, we show a two-weight p-Poincaré inequality in such domains. As an application we show that certain nonnegative supersolutions of the p-Laplace equation and distance weights are p-admissible in a bounded domain, in the sense that they support versions of the p-Poincaré inequality.

中文翻译：

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稀疏支配方法在有界域中的加权范数不等式

我们使用对谐波分析有影响的稀疏控制方法证明了立方体的局部二重Poincaré不等式。证明涉及针对急剧的极大函数的费弗曼-斯坦不等式的局部化形式。通过在满足Boman链条件的有界域中建立局部到全局结果，我们证明了在此类域中的两个权重p - Poincaré不等式。作为应用，我们证明了p -Laplace方程和距离权重的某些非负超解在有界域中是p-允许的，因为它们支持p- Poincaré不等式的形式。