Intermediate dimensions of measures: Interpolating between Hausdorff and Minkowski dimensions

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2025-09-08 DOI:10.1016/j.jmaa.2025.130039

Nicolas E. Angelini , Ursula M. Molter , Jose M. Tejada

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引用次数: 0

Abstract

In this paper, we define a family of dimensions for Borel measures that lie between the Hausdorff and Minkowski dimensions for measures, analogous to the intermediate dimensions of sets. Previously, Hare et al. in [11] defined families of dimensions that interpolate between the Minkowski and Assouad dimensions for measures. Additionally, Fraser, in [8] introduced an additional family of dimensions that interpolate between the Fourier and Sobolev dimensions of measures. Our results address a “gap” in the study of dimension interpolation for measures, almost completing the spectrum of intermediate dimensions for measures: from Hausdorff to Assouad dimensions. Furthermore, Theorem 3.13 can be interpreted as a “reverse Frostman” lemma for intermediate dimensions. We also obtain a capacity-theoretic definition that enables us to estimate the intermediate dimensions of pushforward measures by projections.

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测度的中间维度：Hausdorff和Minkowski维度之间的插值

在本文中，我们定义了一组位于测度的Hausdorff维和Minkowski维之间的Borel测度的维，类似于集合的中间维。之前，Hare等人在b[11]中定义了在Minkowski维和Assouad维之间插入度量的维族。此外，Fraser在[8]中引入了一个额外的维族，它在傅里叶维和索博列夫维之间进行插值。我们的研究结果解决了测度测度维数插值研究中的一个“空白”，几乎完成了测度测度的中间维数谱：从Hausdorff维到Assouad维。更进一步，定理3.13可以解释为中间维的“逆Frostman”引理。我们还获得了一个能力理论定义，使我们能够通过预测来估计推进措施的中间维度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.