Some remarks on the Sobolev inequality in Riemannian manifolds

TL;DR

This paper explores Sobolev and Hardy inequalities on noncompact Riemannian manifolds, linking geometric properties like isoperimetric profiles and p-hyperbolicity to optimal inequalities, including Euclidean best constants.

Contribution

It introduces a new approach combining isoperimetric profiles and Hardy inequalities to analyze Sobolev inequalities on manifolds with specific geometric conditions.

Findings

Established weighted Minerbe's type estimates in manifolds with p-hyperbolicity.

Revealed the connection between isoperimetric profiles and optimal Hardy inequalities.

Recovered the best Sobolev constant in Euclidean space.

Abstract

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe's type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the $p$-hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.