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We derive the sharp Moser–Trudinger–Onofri inequalities on the standard n-sphere and CR $$(2n+1)$$(2n+1)-sphere as the limit of the sharp fractional Sobolev inequalities for all $$n\ge 1$$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.

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