Spectral dimensions of Kreĭn–Feller operators and L-spectra

Kesseböhmer, Marc; Niemann, Aljoscha

doi:10.1016/j.aim.2022.108253

Cited by 8 publications

(5 citation statements)

References 32 publications

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“…The starting point of our investigations was the observation that for particular measures the quantization dimension D 1 (ν) of order r = 1 is connected to the upper spectral dimension s ν of the Kreȋn-Feller operator associated to ν for d = 1 as determined in [KN22c] via the identity D 1 (ν) = s ν / (1 − s ν ). Indeed, for r 1, we also expect similar connections to higher dimensional polyharmonic operators as considered in [KN22a].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…It is easy to construct purely atomic measures such that q r = 0 for all r > 0 and β ν (0) > 0, see [KN22c]. In this case it turns out that the upper quantization dimension is also 0.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…We recall some of the results of [KN22c] and in particular recall the newly introduced notion of partition functions with respect to a non-negative, monotone set function J defined on the dyadic cubes D. In the context of the present paper we are mainly concerned with the particular choice, for a ≥ 0,…”

Section: Partition Functions and L Q -Spectramentioning

confidence: 99%

“…Recall, for s > 0 we let Q s denote the cube centered and parallel with respect to Q such that Λ(Q) = s −d Λ Q s , s > 0. The following lemma was implicitly used in [KN22c].…”

Section: Application To the Quantization Dimensionmentioning

confidence: 99%

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Quantization dimensions of compactly supported probability measures via Rényi dimensions

Kesseböhmer¹,

Niemann²,

Zhu³

2022

Preprint

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We provide a full picture of the upper quantization dimension in terms of the Rényi dimension, in that we prove that the upper quantization dimension of order r > 0 for an arbitrary compactly supported Borel probability measure ν is given by its Rényi dimension at the point q r where the L q -spectrum of ν and the line through the origin with slope r intersect. In particular, this proves the continuity of r → D r (ν) as conjectured by Lindsay (2001). This viewpoint also sheds new light on the connection of the quantization problem with other concepts from fractal geometry in that we obtain a one-to-one correspondence of the upper quantization dimension and the L q -spectrum restricted to (0, 1). We give sufficient conditions in terms of the L q -spectrum for the existence of the quantization dimension. In this way we show as a byproduct that the quantization dimension exists for every Gibbs measure with respect to a C 1 -self-conformal iterated function system on R d without any assumption on the separation conditions as well as for inhomogeneous self-similar measures under the inhomogeneous open sets condition. Some known general bounds on the quantization dimension in terms of other fractal dimensions can readily be derived from our new approach, some can be improved.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…It is easy to construct purely atomic measures such that q r = 0 for all r > 0 and β ν (0) > 0, see [KN22c]. In this case it turns out that the upper quantization dimension is also 0.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Section: Partition Functions and L Q -Spectramentioning

confidence: 99%

“…Recall, for s > 0 we let Q s denote the cube centered and parallel with respect to Q such that Λ(Q) = s −d Λ Q s , s > 0. The following lemma was implicitly used in [KN22c].…”

Section: Application To the Quantization Dimensionmentioning

confidence: 99%

See 2 more Smart Citations

Quantization dimensions of compactly supported probability measures via Rényi dimensions

Kesseböhmer¹,

Niemann²,

Zhu³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…These Laplacians, as well as their generalizations, have been studied extensively in connection with fractal geometry, such as existence of an orthonormal basis of eigenfunctions, spectral dimension and spectral asymptotics, eigenvalues and eigenfunctions, eigenvalue estimates, differential equations, nodal inverse problems, wave equations and wave speed, heat equation and heat kernel estimates, etc. (see [4,7,12,16,[20][21][22][23][24][25]32,35,36,50,51,[53][54][55][56][57]62,63,69] and references therein). We remark that Freiberg and her coauthors defined a class of Laplacians that are more general in that Lebesgue measure is replaced by a more general measure (see, e.g., [21,23]).…”

Section: Introductionmentioning

confidence: 99%

Krein-Feller operators on Riemannian manifolds and Minkowski spaces

Ngai¹,

Ouyang²

2023

Preprint

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For a bounded open set Ω in a complete n-dimensional oriented Riemannian manifold and a positive finite Borel measure µ with support contained in Ω, we define an associated Dirichlet Laplacian ∆ µ by assuming the Poincaré inequality. We obtain sufficient conditions for the Laplacian to have compact resolvent and in this case, we prove the Hodge theorem for functions, which states that there exists an orthonormal basis of L 2 (Ω, µ) consisting of eigenfunctions of ∆ µ , the eigenspaces are finite-dimensional, and the eigenvalues of −∆ µ are real, countable, and increasing to infinity. One of these sufficient conditions is that the L ∞ -dimension dim ∞ (µ) of µ is greater than n − 2. The main tools we use are the Toponogov theorem and the Rauch comparison theorem. We study the condition dim ∞ (µ) > n − 2 for self-similar and self-conformal measures. Results in this paper extend analogous ones by Hu et al. in [J. Funct. Anal. 239 (2006), 542-565], which are established for measures on R n . We also study the d'Alembertian µ = d 2 /dt 2 − ∆ µ on the Minkowski space R 1,n .

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