Let \(B_g(1)\subset \mathbb {H}^2\) be the unit geodesic ball and \(W_0^{1,2}\) be the standard Sobolev space. Let \(\lambda _1(B_g(1))\) be the first Dirichlet eigenvalue in \(B_g(1)\) associated to the Laplace–Beltrami operator \(-\Delta _g\) on \(\mathbb {H}^2\) . In this paper, we establish a sharp Moser–Trudinger inequality involving the Adimurthi–Duret term by a delicate blow-up analysis. Specifically, we prove that \(\begin{aligned} \sup _{u\in W_0^{1,2}(B_g(1)),\,\Vert \nabla _gu\Vert _2=1}\int _{B_g(1)}e^{4\pi (1+\alpha \Vert u\Vert _2^2)u^2}\,d\textrm{Vol}_g<\infty \end{aligned}\) if and only if \(\alpha <\lambda _1(B_g(1))\) . We also prove the existence of extremal functions for the above inequality when \(\alpha \) is sufficiently small, which, to our knowledge, is the first result on the extremal function of Moser–Trudinger inequality involving Adimurthi–Druet term in Hyperbolic space. Besides, we extend the above results to the entire space \(\mathbb {H}^2\) partially, namely, we prove that \(\begin{aligned} \sup _{u\in W^{1,2}(\mathbb {H}^2),\,\Vert \nabla _g u\Vert _2=1}\int _{\mathbb {H}^2}\left( e^{4\pi (1+\alpha \Vert u\Vert _2^2)u^2}-1\right) \,d\textrm{Vol}_g\end{aligned}\) is finite if \(\alpha <1/4\) , and is infinite if \(\alpha >1/4\) . If \(\alpha =1/4\) , we derive a different Adimurthi–Druet type result using an effective scaling argument, which is established by Chen et al. in [9] and [10].
