The Fractal Laplacian and Multifractal Quantities

Triebel, Hans

doi:10.1007/978-3-0348-7891-3_11

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…For Dirichlet boundary conditions, in [HLN06] a sufficient condition in terms of the maximal asymptotic direction of the L q -spectrum of ν has been established, as provided in (♠), which ensures a compact embedding of the relevant Sobolev space into L 2 ν . We would like to note that Triebel already stated this condition implicitly in 1997 in [Tri97] and in 2003 (see [Tri03;Tri04]) he indicated that there is a subtle connection between the multifractal concept of the L q -spectrum and analytic properties of the associated 'fractal' operators.…”

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confidence: 99%

Spectral dimensions of Krein-Feller operators in higher dimensions

Kesseböhmer¹,

Niemann²

2022

Preprint

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We study the spectral dimensions of Kreȋn-Feller operators for finite Borel measures ν on the d-dimensional unit cube via a form approach. We introduce the notion of the spectral partition function of ν, and assuming that the lower ∞-dimension of ν exceeds d − 2, we show that the upper spectral Neumann dimension coincides with the unique zero of the spectral partition function. We show that if the lower ∞-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 ∞-dimension of ν equals d − 2, such that for one case the form approach breaks down, another case, where the operator is well defined but we have no discrete set of eigenvalues, and for the third case, where the spectral dimension exists. We provide additional regularity assumptions on the spectral partition function, guaranteeing that the Neumann spectral dimension exists and may coincide with the Dirichlet spectral dimension. We provide examples-namely absolutely continuous measures, Ahlfors-David regular measure, and self-conformal measures with or without overlaps-for which the spectral partition function is essentially given by its L q -spectrum and both the Dirichlet and Neumann spectral dimensions exist. Moreover, we provide general bounds for the upper Neumann spectral dimension in terms of the upper Minkowski dimension of the support of ν and its lower ∞-dimension. Finally, we give an example for which the spectral dimension does not exist. Contents FB 3 -

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confidence: 99%