Sharp Adams and Hardy-Adams inequalities of any fractional order on hyperbolic spaces of all dimensions

Abstract:

We established in our recent work [Adv. Math. 319 (2017), pp. 567–598] the sharp Hardy-Adams inequalities for the bi-Laplacian-Beltrami operator $(-\Delta _{\mathbb {H}})^2$ on the hyperbolic space $\mathbb {B}^4$ in dimension four and for $\frac {n}{2}$th (integer) power of the Laplacian operator $(-\Delta _{\mathbb {H}})^{\frac {n}{2}}$ on $\mathbb {B}^n$ of any even dimension $n\ge 4$ in [Adv. Math., 2018]. The proofs of these inequalities rely on the special structures of the integer ($\frac {n}{2}$th) power and the even dimension $n$. These are the borderline cases of the sharp higher order Hardy-Sobolev-Maz’ya inequalities by the authors [Amer. J. Math., 2019]. Thus, it remains open if such sharp Hardy-Adams inequalities hold for hyperbolic spaces $\mathbb {B}^n$ of any odd dimension $n$ and any fractional power $\alpha$ of the Laplacian $(-\Delta _{\mathbb {H}})^{\alpha }$. The purpose of this paper is to settle the remaining cases completely.

One of the main purposes of this paper is to establish sharp Adams type inequalities on Sobolev spaces $W^{\alpha ,\frac {n}{\alpha }}(\mathbb {B}^{n})$ of any positive fractional order $\alpha <n$ on the hyperbolic spaces $\mathbb {B}^{n}$ of any dimension $n$. The second main purpose of our paper is to establish sharp Hardy-Adams inequalities of any fractional order $\alpha$ on hyperbolic spaces $\mathbb {B}^n$ of any dimension $n$. Such results were only recently obtained by the authors when $n\ge 4$ and $n$ is even. Fourier analysis on hyperbolic spaces play an important role in establishing the Adams and Hardy-Adams inequalities of any fractional order derivatives on hyperbolic spaces of any dimension. We also establish the fractional order Sobolev embedding theorems on hyperbolic spaces which are of their own independent interest.