Riesz capacity: Hausdorff measure and extremal ratios

Fan, Qiuling

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

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

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

Graduation Semester

2025-12

Type of Resource

Thesis

Handle URL

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

Copyright and License Information

Copyright 2025 Qiuling Fan

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