A derivation of the sharp Moser-Trudinger-Onofri inequalities from the fractional Sobolev inequalities

TL;DR

This paper derives sharp Moser-Trudinger-Onofri inequalities on spheres as limits of fractional Sobolev inequalities, providing a simpler proof that extends previous results to all dimensions.

Contribution

It introduces an elementary approach to derive these inequalities from fractional Sobolev inequalities for all dimensions, generalizing recent specific cases.

Findings

Established sharp inequalities on standard and CR spheres.

Extended previous results to all dimensions $n \,\geq\, 1$.

Provided an elementary proof method.

Abstract

We derive the sharp Moser-Trudinger-Onofri inequalities on the standard $n$-sphere and CR $(2n+1)$- sphere as the limit of the sharp fractional Sobolev inequalities for all $n\ge 1$. On the $2$-sphere and $4$-sphere, this was established recently by S.-Y. Chang and F. Wang. Our proof uses an alternative and elementary argument.