发表机构
Universitat Autònoma de Barcelona（巴塞罗那自治大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了加倍测度下Bennewitz--Lewis定理的局部定量形式，通过二进工具和引导引理，同时推导出反向Hölder不等式与局部Gehring引理。
AI 中文摘要
设$\mu$是$\mathbb R^d$中支撑为$X$的加倍Radon测度。我们证明了在此设定下Bennewitz--Lewis定理的一个局部且定量的形式：若Radon测度$\nu$在$V\subset X$的有界相对开集上满足具有足够小常数的Bennewitz--Lewis条件，则$\nu|_V\ll\mu|_V$，且其密度在$V$的每个紧子集$K$上满足反向Hölder不等式。在此过程中，我们发展了这些论证所依赖的二进工具——通过邻居定义的三重立方体、基$\mathcal{D}\cup3\mathcal{D}$以及Whitney分解——并收集了该基上的Muckenhoupt型权类。同样的机制产生了关于扩大球的局部Gehring引理：两个定理均由一个单一的引导引理推导，该引理对$p=1$和$p>1$各陈述一次。
英文摘要
Let $μ$ be a doubling Radon measure in $\mathbb R^d$ with support $X$. We prove a local and quantitative form of the Bennewitz--Lewis theorem in this setting: if a Radon measure $ν$ satisfies the Bennewitz--Lewis condition on a bounded relatively open set $V\subset X$ with a small enough constant, then $ν|_V\llμ|_V$ and its density satisfies a reverse Hölder inequality on every compact subset $K\subset V$. Along the way we develop the dyadic toolkit these arguments run on --- triple cubes defined through neighbours, the basis $\mathcal{D}\cup3\mathcal{D}$, and a Whitney decomposition --- and we collect the Muckenhoupt-type classes of weights over that basis. The same machinery yields a local Gehring lemma for enlarged balls: both theorems are deduced from a single bootstrapping lemma, stated once for $p=1$ and $p>1$.
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