Essential $p$-capacity-volume estimates for rotationally symmetric manifolds

TL;DR

This work develops volume-capacity estimates for relative $p$-capacities on rotationally symmetric manifolds, revealing a dichotomy between vanishing and nonvanishing regimes governed by the comparison parameter $\alpha$ to the critical value $1-\frac{p}{n}$. It provides sharp, endpoint-aware weak $(p,q)$-embeddings and explicit lower bounds for the principal $p$-frequency $\lambda_{1,p}(\Omega)$, all anchored by isoperimetric and capacitary inequalities realized via capacitary potentials of geodesic balls. The results unify scaling-invariant capacity estimates with isoperimetric geometry to yield precise constants $c_{n,p}$ and asymptotics across $p\in[1,\infty]$, with sharpness demonstrated in the Euclidean case. Overall, the paper advances a capacitary-volume framework for geometric analysis on rotationally symmetric spaces and connects it to fundamental spectral inequalities.

Abstract

Given $p\in [1,\infty]$, this article presents the novel basic volumetric estimates for the relative $p$-capacities with their applications to finding not only the sharp weak $(p,q)$-imbeddings but also the precise lower bounds of the principal $p$-frequencies, which principally live in the rotationally symmetric manifolds.

Theorems & Definitions (10)

• Theorem 1.1
• proof
• Theorem 2.1
• proof
• Remark 2.2
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 4.1
• proof