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.