This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove where the doubling index is a notion of growth defined by This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are the notion of stable growth, and a multiscale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant imuprovement over the previous best‐known bound , which implied Nadirashvili's conjecture.

中文翻译：

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调和函数的节点体积几乎尖锐的下界


本文重点研究调和函数的增长与其零集的豪斯多夫测度之间的关系。让 是 和 中的实值调和函数。我们证明倍增指数是由下式定义的增长概念。这为 的零集的豪斯多夫测度提供了几乎尖锐的下界，推测它在 中是线性的。本文的新内容是稳定增长的概念，以及调和函数倍增指数分布下界的多尺度归纳技术。它比之前最著名的界限有了显着的改进，这暗示了纳迪拉什维利的猜想。