Sharp Hardy-Adams inequalities for bi-Laplacian on hyperbolic space of dimension four

We establish sharp Hardy-Adams inequalities on hyperbolic space ${\mathbb{B}}^4$ of dimension four. Namely, we will show that for any $\alpha >0$ there exists a constant $C_{\alpha }>0$ such that $${\int }_{{\mathbb{B}}^4}(e^{32{\pi }^2{u}^2}-1-32{\pi }^2{u}^2)dV=16{\int }_{{\mathbb{B}}^4}\frac{e^{32{\pi }^2{u}^2}-1-32{\pi }^2{u}^2}{(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^4}dx\le C_{\alpha }\mathrm{.}$$ for any $u\in C^{\infty}_{0}($\mathbb{B}$^{4})$ with $${\int }_{{\mathbb{B}}^4}(-{\mathrm{\Delta }}_{\mathbb{H}}-\frac{9}{4})(-{\mathrm{\Delta }}_{\mathbb{H}}+\alpha )u\cdot udV\le 1.$$ As applications, we obtain a sharpened Adams inequality on hyperbolic space ${\mathbb{B}}^4$ and an inequality which improves the classical Adams’ inequality and the Hardy inequality simultaneously. The later inequality is in the spirit of the Hardy-Trudinger-Moser inequality on a disk in dimension two given by Wang and Ye [37] and on any convex planar domain by the authors [26]. The tools of fractional Laplacian, Fourier transform and the Plancherel formula on hyperbolic spaces and symmetric spaces play an important role in our work.