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arXiv · 2305.16723

Uniformly perfect sets, Hausdorff dimension, and conformal capacity

Oona Rainio·Toshiyuki Sugawa·Matti Vuorinen
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Abstract

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 $\mathbb{R}^n$ 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 $\mathbb{R}^n$ 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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Oona Rainio, Toshiyuki Sugawa, Matti Vuorinen. 2023-05-26. Uniformly perfect sets, Hausdorff dimension, and conformal capacity. https://arxiv.org/abs/2305.16723

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