Riesz capacity: Hausdorff measure and extremal ratios
Fan, Qiuling
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https://hdl.handle.net/2142/132507
Description
Title
Riesz capacity: Hausdorff measure and extremal ratios
Author(s)
Fan, Qiuling
Issue Date
2025-12-01
Director of Research (if dissertation) or Advisor (if thesis)
Laugesen, Richard
Doctoral Committee Chair(s)
Tyson, Jeremy
Committee Member(s)
Song, Renming
Zharnitsky, Vadim
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Riesz capacity
Hausdorff measure
extremal ratios
Language
eng
Abstract
Riesz capacity measures the size of a set in $\mathbb{R}^n$ in terms of a pairwise interaction kernel $|x-y|^{-p}$ with exponent $p
In the first part of the dissertation, the decay rate of Riesz capacity as the exponent $p$ increases to $n$ is shown to yield the Hausdorff measure of the set. The result applies to strongly rectifiable sets, and so in particular to submanifolds of Euclidean space. For strictly self-similar fractals, a one-sided decay estimate is found.
How does the capacity change when the exponent $p$ increases? A longstanding conjecture by P\'olya and Szeg\H{o} claims that the ball maximizes the ratio of $q$-capacity over $p$-capacity when $q>p>0$. In the second part of the dissertation, we investigate the capacity ratio when $p
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