2026-09-23T07:21:29Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/132507 2026-02-20 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2026-02-19 without embargo terms The student, Qiuling Fan, accepted the attached license on 2025-11-19 at 04:41. The student, Qiuling Fan, submitted this Dissertation for approval on 2025-11-19 at 04:47. This Dissertation was approved for publication on 2025-12-01 at 14:40. DSpace SAF Submission Ingestion Package generated from Vireo submission #22891 on 2026-02-19 at 18:24:59 Riesz capacity: Hausdorff measure and extremal ratios Fan, Qiuling 2025-12-01 Laugesen, Richard Tyson, Jeremy Song, Renming Zharnitsky, Vadim Riesz capacity Hausdorff measure extremal ratios eng 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 2025-12 Thesis https://hdl.handle.net/2142/132507 Copyright 2025 Qiuling Fan Mathematics Mathematics University of Illinois Urbana-Champaign Ph.D. Dissertation