论文标题
较高维度的Krein-Feller操作员的光谱尺寸
Spectral dimensions of Krein-Feller operators in higher dimensions
论文作者
论文摘要
我们通过一种表单方法研究了Krein-Feller操作员的频谱尺寸。我们利用$ν$的光谱分区函数，如[Kesseböhmer和Niemann中所引入的，以适应自适应近似的精确渐近顺序。 2023年，Arxiv：2312.16644]，并且假设$ \ infty $ - $ν$的较低$ \ igimension超过$ D-2 $，我们将上诺伊曼的光谱尺寸确定为光谱分区功能的独特零，从而揭示了这些频谱和分数和分数数字的内在连接。我们表明，如果$ \ infty $ dimension $ν$的限制严格小于$ d-2 $，则表格方法会分解。给出了示例的关键案例，即$ν$的$ \ infty $ dimension等于$ d-2 $。我们在光谱分区函数上提供了其他规律性假设，确保诺伊曼光谱维度存在并与dirichlet光谱维度重合。提供了文献中先前治疗过的几个突出的例子，即绝对连续的措施和更常见的ahlfors-david常规措施，以及以前在文献中未接受过的示例，即具有或没有重叠的自我统一措施，为此，我们表明存在dirichlet和noumann频谱的存在，并且可以从$ l l^$}中获得。我们演示了如何使用我们的方法来获得ahlfors-david常规措施的上下渐近光谱界限。此外，我们为上诺伊曼光谱尺寸提供了尖锐的界限，以$ν$的支撑及其下部$ \ \ \ iftty $ dimension的上层尺寸。最后，我们举例说明了光谱维度不存在的示例。
We study the spectral dimensions of Krein-Feller operators for arbitrary for arbitrary finite Borel measures $ν$ on the $d$-dimensional unit cube ($d\geq2$) via a form approach. We make use of the spectral partition function of $ν$ as introduced in [Kesseböhmer and Niemann, Exact asymptotic order for adaptive approximations. 2023, arXiv:2312.16644] and, assuming that the lower $\infty$-dimension of $ν$ 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 $ν$ 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 $ν$ 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 $ν$ and its lower $\infty$-dimension. Finally, we give an example for which the spectral dimension does not exist.
