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

Given $p\in [1,\mathrm{\infty }]$, 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.