Improved Sobolev inequality under constraints

TL;DR

This paper presents a new proof of an improved Sobolev inequality on spheres under moment constraints and extends the result to higher order moments, explicitly determining constants in specific cases.

Contribution

It provides a novel proof of Aubin's inequality under constraints and generalizes it to higher order moments, with explicit constants for second order moments.

Findings

New proof of Aubin's improved Sobolev inequality on spheres

Extension to higher order moments constraints

Explicit determination of constants in second order moments case

Abstract

We give a new proof of Aubin's improvement of the Sobolev inequality on $\mathbb{S}^{n}$ under the vanishing of first order moments of the area element and generalize it to higher order moments case. By careful study of an extremal problem on $\mathbb{S}^{n}$, we determine the constant explicitly in the second order moments case.