Spectral dimensions of Krein-Feller operators in higher dimensions

Abstract

We study the spectral dimensions of Krein-Feller operators for arbitrary for arbitrary finite Borel measures $\nu$ on the $d$-dimensional unit cube ($d\geq2$) via a form approach. We make use of the spectral partition function of $\nu$ as introduced in [Kesseb\"ohmer and Niemann, Exact asymptotic order for adaptive approximations. 2023, arXiv:2312.16644] and, assuming that the lower $\infty$-dimension of $\nu$ exceeds $d-2$, we identify the upper Neumann spectral dimension as the unique zero of the spectral partition function, thus revealing the intrinsic connection of these spectral and fractal-geometric quantities. We show that if the lower $\infty$-dimension of $\nu$ is strictly less than $d-2$, the form approach breaks down. Examples are given for the critical case, that is the lower $\infty$-dimension of $\nu$ equals $d-2$. We provide additional regularity assumptions on the spectral partition function, guaranteeing that the Neumann spectral dimension exists and coincides with the Dirichlet spectral dimension. Several prominent examples previously treated in the literature are provided, namely absolutely continuous measures and more generally Ahlfors-David regular measures, and examples not previously treated in the literature, namely self-conformal measures with or without overlaps, for which we show that both the Dirichlet and Neumann spectral dimensions exist and how they can be obtained from the $L^{q}$-spectrum of the measures. We demonstrate how our approach can be used to obtain upper and lower asymptotic spectral bounds for the case of Ahlfors-David regular measures. Moreover, we provide sharp bounds for the upper Neumann spectral dimension in terms of the upper Minkowski dimension of the support of $\nu$ and its lower $\infty$-dimension. Finally, we give an example for which the spectral dimension does not exist.