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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!

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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.

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  • $\begingroup$ Dear Prof. Baudoin, I have a question about the heat kernel on Hyperbolic space. If n>3, then isn't the behavior of the heat kernel different for small and large t? To be more precise, according to Davies's result (see 'heat kernel bounds on hyperbolic space and kleinian groups' plms.oxfordjournals.org/content/s3-57/1/182.short) the heat kernel behaves like t^{-3/2} near zero and t^{-n/2} near infinity. $\endgroup$ – Tomas Oct 13 '16 at 14:39
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This paper: http://arxiv.org/abs/0708.0269 (Genquian Liu, 2007, revised 2013) seems to have relevant results and references.

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