Riesz capacity: Hausdorff measure and extremal ratios

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

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Fan, Qiuling

Contributors dc:contributor

• Laugesen, Richard
• Tyson, Jeremy
• Song, Renming
• Zharnitsky, Vadim

• Riesz capacity
• Hausdorff measure
• extremal ratios

Statement dc:rights

• Copyright 2025 Qiuling Fan

Language dc:language

en

Handle dc:identifier

https://hdl.handle.net/2142/132507

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/132507