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 \[ \int_{\mathbb{B}^{4}}(e^{32\pi^{2} u^{2}}-1-32\pi^{2} u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32\pi^{2} u^{2}}-1-32\pi^{2} u^{2}}{(1-|x|^{2})^{4}}dx\leq C_{\alpha}. \] for any $u\in C^{\infty}_{0}(\mathbb{B}^{4})$ with \[ \int_{\mathbb{B}^{4}}\left(-\Delta_{\mathbb{H}}-\frac{9}{4}\right)(-\Delta_{\mathbb{H}}+\alpha)u\cdot udV\leq1. \] 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.