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.
Categories: math.AP math.CA