A1 Refereed original research article in a scientific journal
Uniformly Perfect Sets, Hausdorff Dimension, and Conformal Capacity
Authors: Rainio Oona, Sugawa Toshiyuki, Vuorinen, Matti
Publisher: Springer Nature
Publication year: 2024
Journal: Journal of Geometric Analysis
Journal name in source: JOURNAL OF GEOMETRIC ANALYSIS
Article number: ARTN 154
Volume: 34
Issue: 6
ISSN: 1050-6926
eISSN: 1559-002X
DOI: https://doi.org/10.1007/s12220-024-01599-5
Web address : https://link.springer.com/article/10.1007/s12220-024-01599-5
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/393346031
Using the definition of uniformly perfect sets in terms of convergent sequences, we apply lower bounds for the Hausdorff content of a uniformly perfect subset E of Rn to prove new explicit lower bounds for the Hausdorff dimension of E. These results also yield lower bounds for capacity test functions, which we introduce, and enable us to characterize domains of Rn with uniformly perfect boundaries. Moreover, we show that an alternative method to define capacity test functions can be based on the Whitney decomposition of the domain considered.
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