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

Abstract Let I I be a bounded interval of R {\mathbb{R}} and λ 1 ( I ) {\lambda }_{1}\left(I) denote the first eigenvalue of the nonlocal operator ( − Δ ) 1 4 {(-\Delta )}^{\tfrac{1}{4}} with the Dirichlet boundary. We prove that for any 0 ⩽ α < λ 1 ( I ) 0\leqslant \alpha \lt {\lambda }_{1}(I) , there holds sup u ∈ W 0 1 2 , 2 ( I ) , ‖ ( − Δ ) 1 4 u ‖ 2 2 − α ∥ u ∥ 2 2 ≤ 1 ∫ I e π u 2 d x < + ∞ , \mathop{\sup }\limits_{u\in {W}_{0}^{\frac{1}{2},2}(I),\Vert {\left(-\Delta )}^{\tfrac{1}{4}}u{\Vert }_{2}^{2}-\alpha {\parallel u\parallel }_{2}^{2}\le 1}\mathop{\int }\limits_{I}{e}^{\pi {u}^{2}}{\rm{d}}x\lt +\infty , 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.