We study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri's inequality for the unit disk in R 2 . Moreover we extend the analysis of [1] and [8] considering Adimurthi-Druet type functionals on compact surfaces with conical singularities and discussing the existence of extremals for such functionals.

We discuss some recent results by Parini and Ruf on a Moser-Trudinger type inequality in the setting of Sobolev-Slobodeckij spaces in dimension one. We push further their analysis considering the inequality on the whole R and we give an answer to one of their open questions.

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Let m ≥ 2 be an integer. For any open domain Ω ⊂ R 2m , non-positive function ϕ ∈ C ∞ (Ω) such that ∆ m ϕ ≡ 0, and bounded sequence (V k ) ⊂ L ∞ (Ω) we prove the existence of a sequence of functions (u k ) ⊂ C 2m−1 (Ω) solving the Liouville equation of order 2m (−∆) m u k = V k e 2mu k in Ω, lim sup k→∞ Ω e 2mu k dx < ∞, and blowing up exactly on the set S ϕ := {x ∈ Ω : ϕ(x) = 0}, i.e. lim k→∞ u k (x) = +∞ for x ∈ S ϕ and lim k→∞ u k (x) = −∞ for x ∈ Ω \ S ϕ , thus showing that a result of Adimurthi, Robert and Struwe is sharp. We extend this result to the boundary of Ω and to the case Ω = R 2m . Several related problems remain open.

We show a sharp fractional Moser-Trudinger type inequality in dimension 1, i.e. for an interval I b R, p 2 (1, 1) and some1 2 on I. This extends to the fractional case some previous results proven by Adimurthi for the Laplacian and the p-Laplacian operators.Finally with a technique of Ruf we show a fractional Moser-Trudinger inequality on R. MSC 2010. 26A33, 35R11, 35B33.