Best Sobolev constants in the presence of sharp Hardy terms in Euclidean and hyperbolic space

TL;DR

This paper calculates optimal constants for Hardy-Sobolev inequalities with sharp Hardy terms across Euclidean and hyperbolic spaces, considering interior and boundary singularities, advancing understanding of these inequalities in geometric contexts.

Contribution

It provides explicit computations of best Sobolev constants for Hardy-Sobolev inequalities in multiple geometric settings with singularities.

Findings

Computed best Sobolev constants in Euclidean space with interior singularity.

Determined optimal constants in hyperbolic space with interior singularity.

Analyzed boundary singularities in Euclidean domains.

Abstract

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.