The aim of this paper is three-fold. Firstly, we establish the sharp Adams inequality in the Lorentz–Sobolev space \(W^m L^{\frac{n}{m},q}({\mathbb {H}}^n)\) defined in the hyperbolic space \(\begin{aligned} \sup _{u\in W^mL^{\frac{n}{m},q}({\mathbb {H}}^n),\, \Vert \nabla _g^m u\Vert _{\frac{n}{m},q}\le 1} \int _{{\mathbb {H}}^n} \Phi _{\frac{n}{m},q}\big (\beta _{n,m}^{\frac{q}{q-1}} |u|^{\frac{q}{q-1}}\big ) dV_g <\infty \end{aligned}\) where \(q \in (1,\infty )\) if m is even, and \(1< q \le n/m\) if m is odd. Secondly, we improve the previous inequality by proving that for any \(q\ge 2n/(n-1)\) if m is even and \(2n/(n-1) \le q \le \frac{n}{m}\) if m is odd, it holds \(\begin{aligned} \sup _{u\in W^mL^{\frac{n}{m},q}({\mathbb {H}}^n),\, \Vert \nabla _g^m u\Vert _{\frac{n}{m},q}^q -\lambda \Vert u\Vert _{\frac{n}{m},q}^q \le 1} \int _{{\mathbb {H}}^n} \Phi _{\frac{n}{m},q}\big (\beta _{n,m}^{\frac{q}{q-1}} |u|^{\frac{q}{q-1}}\big ) dV_g <\infty \end{aligned}\) for any \(0<\lambda <C(n,m,n/m)^q\) where \(C(n,m,n/m)^q\) is the sharp Lorentz–Poincaré constant in the hyperbolic space. Thirdly, we establish the sharp Hardy–Adams inequality in the unit ball \({\mathbb {B}}^n\) for \(n\ge 2m+1\) , and \(q \ge 2n/(n-1)\) if m is even and \(2n/(n-1) \le q \le n/m\) if m is odd \(\begin{aligned} \sup _{u\in W^mL^{\frac{n}{m},q}({\mathbb {H}}^n),\, \Vert \nabla _g^m u\Vert _{\frac{n}{m},q}^q -C(n,m,\frac{n}{m})^q \Vert u\Vert _{\frac{n}{m},q}^q \le 1} \int _{B_n} e^{\beta _{n,m}^{\frac{q}{q-1}} |u|^{\frac{q}{q-1}}} dx <\infty . \end{aligned}\) Our Hardy–Adams inequality generalizes the Hardy–Moser–Trudinger inequality to the higher order derivatives, and the Hardy–Adams inequality of Li, Lu, Yang in \(W^{\frac{n}{2},2}({\mathbb {H}}^n)\) to the Lorentz–Sobolev spaces. Our approach relies on the non-increasing symmetric rearrangement technique and the sharp Lorentz–Sobolev type inequalities in the hyperbolic space previously studied by the author.
