二元势理论与德拉姆函数
Dyadic potential theory and de Rham functions
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中文总结 AI 辅助
研究由两个递增分式线性变换驱动的德拉姆函数方程，证明解的存在唯一性定理，确定捕获区域，表明解是乘法二元容量的归一化累积容量函数，还用二元里斯容量锐化了奇异测度的豪斯多夫维数估计。
中文摘要 AI 辅助
我们研究由两个递增的分式线性变换驱动的德拉姆函数方程。主要目的是将相关解的奇点理论与二叉树上的二元势理论联系起来。首先证明了在全范围线性分式数据中递增、左连续解的存在唯一性定理，并确定了解连续的参数空间中的捕获区域。对于一大类参数，表明德拉姆解是乘法二元容量的归一化累积容量函数。这给出了莫比乌斯德拉姆系统的势理论模型。然后通过用二元里斯容量代替豪斯多夫维数来锐化冈村对与解相关的奇异测度的豪斯多夫维数估计。特别地，证明了承载全质量或零质量的博雷尔集大小的容量上下界，并得到了冈村奇点准则的容量锐化。
英文摘要
We study de Rham functional equations driven by two increasing fractional linear transformations. Our main purpose is to relate the singularity theory of the associated solutions to dyadic potential theory on the binary tree. We first prove an existence and uniqueness theorem for increasing, left-continuous solutions in the full range of linear fractional data, and identify the trapping region in parameter space where the solution is continuous. For a large class of parameters we show that the de Rham solution is the normalized cumulative capacitary function of a multiplicative dyadic capacity. This gives a potential-theoretic model for Möbius de Rham systems. We then sharpen Okamura's Hausdorff-dimensional estimates for the singular measure associated with the solution by replacing Hausdorff dimension with dyadic Riesz capacities at the upper endpoint of Okamura's theorem.
