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

TL;DR

This paper introduces new volumetric estimates for relative p-capacities in rotationally symmetric manifolds, enabling sharper embeddings and precise bounds on principal p-frequencies, advancing understanding in geometric analysis.

Contribution

It provides the first basic volumetric estimates for relative p-capacities and applies them to improve embeddings and frequency bounds in rotationally symmetric manifolds.

Findings

Sharp weak (p,q)-imbeddings established

Precise lower bounds for principal p-frequencies derived

Novel volumetric estimates for relative p-capacities introduced

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.