发表机构
University of Jyväskylä; University of St Andrews（于韦斯屈莱大学; 圣安德鲁斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造了一维强局部正则Dirichlet型，使其定义域中Hölder连续函数的上确界指数可任意指定为(0,1]中的值，并通过Radon测度的上Hausdorff维数刻画该指数。
AI 中文摘要
我们研究标准单位区间上的一类强局部、正则Dirichlet型。我们的工作旨在记录先前文献中未被注意到的某些Hölder正则性性质。特别地，作为我们的主要结果，我们证明对于每个$\delta \in (0,1]$，存在一个度量测度空间$(X,d,\mu)$，配备$L^2(\mu)$上的强局部、正则Dirichlet型$(\mathcal{E},\mathcal{F})$，使得$\delta$是使得Dirichlet型的定义域$\mathcal{F}$包含非常值的$\alpha$-Hölder连续函数的$\alpha \in (0,1]$的上确界。据我们所知，此前仅对$\delta = 1$已知此类例子。在我们的构造中，$\delta$的值由用于定义$(\mathcal{E},\mathcal{F})$的某个Radon测度的上Hausdorff维数来刻画。
英文摘要
We study a class of strongly local, regular Dirichlet forms on the standard unit interval. The aim of our work is to record some Hölder regularity properties that have not been noted in the prior literature. In particular, as our main result, we show that for every $δ\in (0,1]$ there exists a metric measure space $(X,d,μ)$ equipped with a strongly local, regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on $L^2(μ)$ with the property that $δ$ is the supremum of $α\in (0,1]$ for which the domain $\mathcal{F}$ of the Dirichlet form contains a non-constant $α$-Hölder continuous function. To the best of our knowledge, such examples were previously known only for $δ= 1$. In our construction, the value $δ$ is characterized by the upper Hausdorff dimension of a certain Radon measure that is used to define $(\mathcal{E},\mathcal{F})$.
Comments24 pages, comments are welcome!
