One-dimensional Dirichlet forms with prescribed Hölder regularity

Abstract

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 $(\mathcal{E},\mathcal{F})$.