The aim of this work is to provide a representation of the multifractal Hausdorff and packing dimensions of compact sets in the Euclidean space in terms of the lower and upper local $$(q,\mu )$$ -dimensions of a probability measure on $$\mathbb {R}^n$$ , respectively. In addition, a relationship is established allowing to determine the Olsen’s multifractal functions by means of the corresponding measures’ versions. The multifractal functions are defined as the supremum over the lower and upper multifractal dimensions of all Borel probability measures.

中文翻译：

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奥尔森多重分形函数的一些新表征

这项工作的目的是提供欧几里得空间中紧集的多重分形 Hausdorff 和包装维数的表示，根据 $$(q,\mu)$$ 上局部概率测度的下限和上限维数$\mathbb {R}^n$$ ，分别。此外，建立了一个关系，允许通过相应的度量版本来确定奥尔森的多重分形函数。多重分形函数被定义为所有 Borel 概率测度的下和上多重分形维数的最高值。