Sharp Riesz capacity inequalities and the Pólya--Szegő conjecture

Mengwei Su; Qiang Sun; Ge Xiong

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arXiv · 2610.04811

Sharp Riesz capacity inequalities and the Pólya--Szegő conjecture

Mengwei Su·Qiang Sun·Ge Xiong

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Abstract

For $n\geq2$ and $p\in(n-2,n)$, we prove that closed balls maximize the ratio of Riesz $p$-capacity to Newtonian capacity among all compact subsets of $\mathbb{R}^n$. This resolves the classical Pólya--Szegő capacity ratio conjecture and its higher-dimensional extensions. For $n\geq4$, a sharp inequality for the ratio of Newtonian capacity to logarithmic capacity is established. All equality cases are characterized.

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BibTeX RIS

Mengwei Su, Qiang Sun, Ge Xiong. 2026-11-03. Sharp Riesz capacity inequalities and the Pólya--Szegő conjecture. https://arxiv.org/abs/2610.04811

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