A Derivation of the Sharp Moser–Trudinger–Onofri Inequalities from the Fractional Sobolev Inequalities

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 Chang and Wang. Our proof uses an alternative and elementary argument.