A note on the Moser-Trudinger inequality in Sobolev-Slobodeckij spaces in dimension one

Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$

“…On the Sobolev-Slobodeckij spaces W s,p (Ω) with sp = N, similar fractional Trudinger-Moser inequality is also proved by Parini-Ruf [25] when N ≥ 2 and Iula [11] when N = 1. In this case, the result is weaker and the inequality holds true only for 0 ≤ α < α * N,p for some (explicit) value α * N,p .…”

“…For the estimate (2.4), we refer to Iula [11] Proposition 2.2. For the estimate (2.5), we refer to [11] equation (35).…”

Remarks on the fractional Moser-Trudinger inequality

“…The above theorem has been extended by Iula [14] to the one dimensional case along with the very similar proof of that. Note that, in dimension n = 2 as already mentioned in [26], lim s→1 − (1 − s)α * s,2 = 2π 2 which is same as the optimal exponent α 2 = 4π in the local case, up to an appropriate constant which is truly appear in the study of asymptotic behavior of the semi-norm in the limiting case s → 1 − .…”

“…In 2015, Martinazzi [22] studied the fractional Adams inequality in a Besel potential space. Moreover, when Ω = R n the validity of the fractional Moser-Trudinger inequality verified by Iula [14] (n = 1) and Zhang [32] (n ≥ 2) and very recently, Thin [30] proved a singular version of fractional Moser-Trudinger inequality. We stated the result here of Zhang, only for the case required.…”

“…where f : R + → R + is such that f (t) → ∞ as t → ∞, holds true for the same exponents of the standard Moser-Trudinger inequality. In the one-dimensional case, first Iula [14] answered this question negatively. In the present paper, we will extend this result to a higher dimension followed by the same approach.…”