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

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≥1.On the 2-sphere and 4-sphere,this was established recently by Chang and Wang.Our proof uses an alternative and elementary argument.