Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals

Let I be a bounded interval of ℝ and λ1(I) denote the first eigenvalue of the nonlocal operator (-Δ)1/4 with the Dirichlet boundary. We prove that for any 0 ≤ α < λ1(I), there holds (Equation Presented) and the supremum can be attained. The method is based on concentration-compactness principle for fractional Trudinger-Moser inequality, blow-up analysis for fractional elliptic equation with the critical exponential growth and harmonic extensions.