Hardy-Littlewood-Sobolev inequality on hyperbolic space

Question

Let $I_\alpha = (-\Delta)^{-\alpha/2}$ be the Riesz potential on $\mathbb{R}^n$. The Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^n$ says $$||I_\alpha f||_{L^q} \leq C||f||_{L^p}$$ where $q = \frac{np}{n - \alpha p}$, $0 < \alpha < n$, $1 < p < q < \infty$.

I am curious whether such a statement (may be with modifications) holds on hyperbolic space $\mathbb{H}^n$. Furthermore, if there are corresponding statements on more general spaces, I would appreciate anything on that as well. Thanks a lot!

Answer 1

Yes, the Hardy-Littlewood-Sobolev inequality $\| I_\alpha (f) \|_q \le C \| f \|_p$ is true in the hyperbolic space $\mathbb{H}_n$. It is actually a consequence of a Varopoulos theorem. Varopoulos theorem says that if an operator $A$ is the generator of a Markov semigroup $e^{tA}$, then the ultracontractivity bound

$\| e^{tA} f \|_\infty \le \frac{C_1}{t^{n/2}} \| f \|_1 $

implies that $A$ satisfies the Hardy-Littlewood-Sobolev inequality $\| A^{-\alpha /2} f \|_q \le C_2 \| f \|_p$ where $q=\frac{np}{n-\alpha p}$. You can find the proof of the theorem when $A$ is a Laplace-Beltrami operator in a lecture on my blog

Proof of Varopoulos theorem

The proof for general operators $A$ can be found in Chapter 1 of the book Analysis and geometry on groups by Varopoulos,Saloff-Coste and Coulhon.

In the special case of the hyperbolic space, there is an explicit expression for the heat kernel $p(t,x,y)$ from which one can deduce that

$p(t,x,y)\le \frac{C}{t^{n/2}}$

and thus the ultracontractivity of the heat semigroup.

Answer 2

This paper: http://arxiv.org/abs/0708.0269 (Genquian Liu, 2007, revised 2013) seems to have relevant results and references.