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Spectral dimensions of Krein-Feller operators in higher dimensions

Marc Kesseböhmer, Aljoscha Niemann

Abstract

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.

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 on the -dimensional unit cube () via a form approach. We make use of the spectral partition function of as introduced in [Kesseböhmer and Niemann, Exact asymptotic order for adaptive approximations. 2023, arXiv:2312.16644] and, assuming that the lower -dimension of exceeds , 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 -dimension of is strictly less than , the form approach breaks down. Examples are given for the critical case, that is the lower -dimension of equals . 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 -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 and its lower -dimension. Finally, we give an example for which the spectral dimension does not exist.
Paper Structure (34 sections, 45 theorems, 181 equations, 2 figures)

This paper contains 34 sections, 45 theorems, 181 equations, 2 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Introduction and background
  3. Preliminaries
  4. Main results
  5. Regularity results
  6. General bounds in terms of fractal dimensions
  7. Special examples and spectral asymptotic bounds
  8. Absolutely continuous measures
  9. Ahlfors–David regular measure
  10. Self-conformal measures
  11. Some technical prerequisites
  12. Stein extensions
  13. Min-Max principle
  14. Smoothing methods
  15. Partition functions and $L^{q}$-spectra
  16. ...and 19 more sections

Key Result

Theorem 1.1

Suppose there exists a non-negative, monotone and uniformly vanishing set function $\mathfrak{J}$ on $\mathcal{D}$, such that for all $Q\in\mathcal{D}$ and all $u\in\mathcal{C}_{b}^{\infty}\left(\overline{Q}\right)$ with $\int_{Q}u\;\mathrm d\Lambda=0$, we have Then $N^{N}\leq M_{\mathfrak{J}}$ and in particular, $\overline{s}^{N}\leq\overline{h}_{\mathfrak{J}}$ and $\underline{s}^{N}\leq\underli

Figures (2)

  • Figure 1.1: Partition function $\tau^{N}$ in dimension $d=3$ for the self-similar measure $\nu$ supported on the Sierpiński tetraeder with all four contraction ratios equal $1/2$ and with probability vector $\left(0.36,0.36,0.2,0.08\right)$. Natural bounds for $\overline{s}^{N}=q^{N}$ in this setting are the zeros of the dashed line $x\mapsto-x\left(\tau^{N}\left(0\right)-1\right)+\tau^{N}\left(0\right)$ and the dotted line $x\mapsto\left(1-x\right)\left(\dim_{\infty}\left(\nu\right)-1\right)+1$ as given in cor:upper_spectralDimGeneralUpper/lowerBound. In this case $\tau^{N}\left(0\right)=\overline{\dim}_{M}\left(\nu\right)=2$ and $\dim_{\infty}\left(\nu\right)=-\log\left(0.36\right)/\log\left(2\right)=1.47\ldots$
  • Figure 8.1: Support of the densities $f_{i}$ of $\nu_{i}$, $i=1,2,3$ with peak singularity concentrated in one corner for the examples discussed in sec:The-critical-case.

Theorems & Definitions (98)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 88 more