Riesz operators and $L^{p}$-boundary representations for hyperbolic groups

TL;DR

This paper explores the relationship between Riesz operators and Lp-boundary representations of hyperbolic groups, establishing conditions for irreducibility based on the injectivity of these operators.

Contribution

It provides a new criterion linking Riesz operator injectivity to the irreducibility of boundary representations in hyperbolic groups.

Findings

Lp-boundary representations are irreducible iff Riesz operators are injective.

Established a characterization of irreducibility for hyperbolic group representations.

Connected operator theory with geometric group properties.

Abstract

We investigate Lp-boundary representations of hyperbolic groups. We prove that such representations are irreducible if and only if the corresponding Riesz operators are injective.

Equations

$$\displaystyle[\pi_{t}(\gamma)v](\xi)=\bigg{(}\frac{d\gamma_{*}\nu_{d}}{d\nu_{d}}(\xi)\bigg{)}^{\frac{1}{2}+t}v(\gamma^{-1}\xi).$$

$${\varphi }_t:\gamma \in \Gamma \mapsto ⟨{\pi }_t(\gamma ){\text{1}}_{\partial \Gamma },{\text{1}}_{\partial \Gamma }⟩\in {\mathbb{R}}^{+},$$

$${\sigma }_t:\xi \in \partial \Gamma \mapsto {\mathcal{I}}_t({\text{1}}_{\partial \Gamma })(\xi )\in {\mathbb{R}}^{+}(t>0).$$

$$\sum _{\gamma \in S_{n,R}^{\Gamma }}{\mu }_{n,R}(\gamma )f(\gamma )g({\gamma }^{-1})\frac{⟨{\pi }_t(\gamma )v,w⟩}{{\varphi }_t(\gamma )}\to ⟨g_{{|}_{\partial \Gamma }}{\mathcal{R}}_t(v),{\text{1}}_{\partial \Gamma }⟩⟨f_{{|}_{\partial \Gamma }},w⟩,$$

$$(x,y{)}_o\ge \min \{(x,z{)}_o,(z,y{)}_o\}-\delta ,$$

$$(x,y{)}_o=\frac{1}{2}(d(x,o)+d(y,o)-d(x,y)).$$

$$(\xi ,\eta {)}_o:=\sup \underset{i,j}{\mathrm{liminf}}(a_i,b_j{)}_o$$

$$d_{y,\epsilon}(\xi,\eta)=e^{\frac{\epsilon}{2}\big{(}\beta_{\xi}(x,y)+\beta_{\eta}(x,y)\big{)}}d_{x,\epsilon}(\xi,\eta),$$

$$(\xi ,x,y)\in \partial X\times X\times X\mapsto {\beta }_{\xi }(x,y):=\underset{n\to +\infty }{\lim }d(x,a_n)-d(y,a_n),$$

$${\beta }_{\xi }(x,y)=-d(x,y)+2(\xi ,y{)}_x.$$

$$|t-s|-C_X\le d(r(t),r(s))\le |t-s|+C_X$$

$$D:=\frac{Q}{ϵ}$$

$$Q=Q_{\Gamma ,d}:=\underset{R\to +\infty }{\mathrm{limsup}}\frac{1}{R}\log |\Gamma .o\cap B(o,R)|,$$

$$C^{-1}{r}^D\le {\nu }_o(B(r,\xi ))\le C{r}^D.$$

$$\underset{\xi \in \partial X}{\sup }(\xi ,x{)}_o\ge d(o,x)-M.$$

$$({\hat{x}}_o,x{)}_o\ge d(o,x)-M.$$

$$O_r(o,x):=\{\xi \in \partial X|(\xi ,x{)}_o\ge d(x,o)-r\}.$$

$$B({\hat{x}}_o,e^{-ϵ(d(o,x)-r+\delta )})\subset O_r(o,x)\subset B({\hat{x}}_o,e^{-ϵ(d(x,o)-r-\delta )}).$$

$$(\xi ,x{)}_o$$

$$\ge \min \{d(o,x)-r+\delta ,d(o,x)-M\}-\delta$$

$$=d(o,x)-r$$

$$\ge d(o,x)-r.$$

$$(\xi ,{\hat{x}}_o{)}_o$$

$$\ge \min \{d(x,o)-r,d(o,x)-M\}-\delta$$

$$\ge d(o,x)-r-\delta .$$

$$C^{-1}{e}^{-Qd(o,x)}\le {\nu }_o(O_r(o,x))\le C{e}^{-Qd(o,x)}.$$

$${\cup }_{\gamma \in S_{n,R}^{\Gamma }}{O}_r(o,\gamma o)=\partial X.$$

$$R>3(C_X+\kappa )$$

$$(\xi ,\gamma o{)}_o$$

$$\ge \min \{(n+1)R-3C_X,nR-\kappa \}-\delta$$

$$\ge nR-\kappa -\delta$$

$$\ge |\gamma |-R-\kappa -\delta ,$$

$$(\gamma o,z_{\xi }{)}_o$$

$$\ge \min \{|\gamma |-r,(n+1)R-3C_X\}-\delta$$

$$\ge \min \{nR-r,(n+1)R-3C_X\}-\delta$$

Full text

1. Introduction

Consider a non-elementary hyperbolic group $\Gamma$ endowed with an invariant metric $d$, satisfying some regularity assumptions, acting by measure class preserving transformations on its Gromov boundary $(\partial\Gamma,\nu_{d})$ equipped with $\nu_{d}$ the so-called Patterson-Sullivan measure associated with $d$. This yields to a one parameter family of isometric representations on $L^{p}$-spaces denoted by $(\pi_{t},L^{p}(\partial\Gamma,\nu_{d}))$, defined for almost every $\xi\in\partial\Gamma$ as

[TABLE]

The above representation satisfies $\|\pi_{t}(\gamma)v\|_{p}=\|v\|_{p}$ for all $v\in L^{p}(\partial\Gamma,\nu_{d})$ and for all $\gamma\in\Gamma$, where $p$ is such that $1/p=1/2+t$ with $-\frac{1}{2}<t<\frac{1}{2}$. We call these representations $L^{p}$-boundary representations of hyperbolic groups.

The boundary representation of hyperbolic groups is nothing but $\pi_{0}$ and has been intensively studied this last decade and might be seen, from a dynamical point of view, as a generalization of ergodicity see [2], [3], [22], [7], [8], [11], [13], [12], [21], [26] [27], and in [16]. In the papers [5], [6] and [10] the representations (1.1) have already been studied but rather as representations on Hilbert spaces. In this paper we focus on $L^{p}$-spaces. We characterize the irreducibility of $L^{p}$-boundary representations $(\pi_{t},L^{p}(\partial\Gamma,\nu_{d}))$ with $1/p=1/2+t$ (where $-1/2<t<1/2$) thanks to an intertwining operator associated with the metric $d$, denoted by $\mathcal{I}_{t}$ satisfying $\mathcal{I}_{t}\pi_{t}=\pi_{-t}\mathcal{I}_{t}$. We prove that this a bounded operator $\mathcal{I}_{t}:L^{p}(\partial\Gamma,\nu)\rightarrow L^{q}(\partial\Gamma,\nu)$ with $1/q=1/2-t$, defined only for $0<t<1/2$ (see Subsection 2.18). It already appears in the context of hyperbolic groups and Hilbert spaces in [10], [25] and in CAT(-1) spaces [5] .

Our main result is the following:

Theorem 1**.**

For all $-1/2<t<1/2$ and $1/p=1/2+t$, the $L^{p}$-boundary representations $(\pi_{t},L^{p}(\partial\Gamma,\nu_{d}))$ corresponding to $d$ are irreducible if and only if the intertwiner $\mathcal{I}_{|t|}$ is injective.

Remark 1.1**.**

In particular, we prove that the representation $(\pi_{t},L^{p}/\ker\mathcal{I}_{t})$ is irreducible for $1/p=1/2+t$ with $0<t<1/2.$

We also deduce the following result in the context of rank one semisimple Lie groups. We do not know if the following theorem is present in the literature.

Theorem 2**.**

Let $G$ be a rank one semisimple Lie group of non compact type. Let $\Gamma$ be a lattice in $G$. The $L^{p}$-boundary representations of $\Gamma$ corresponding to $\nu$ the unique $K$-invariant probability measure on its Poisson-Furstenberg boundary $G/P$ are irreducible for all $1<p<+\infty$, where $K$ is the maximal compact subgroup and $P$ the minimal parabolic subgroup of $G$.

Indeed, we derive the above theorem from a generalizations of an ergodic theorem à la Bader-Muchnik for $L^{p}$-boundary representations, see Theorem 1.2 in the following.

Notation

Endow $\Gamma$ with the length function corresponding to $d$, $|\cdot|:\Gamma\rightarrow\mathbb{R}^{+}$ defined as $|\gamma|=d(1,\gamma)$ where $1$ is the identity element and $\gamma\in\Gamma.$

Let $S^{\Gamma}_{n,R}:=\{\gamma\in\Gamma|nR\leq|\gamma|<(n+1)R\}$ for $R>0$ and let $|S^{\Gamma}_{n,R}|$ be the cardinal of $S^{\Gamma}_{n,R}$.

As in [6], we recall the definition of a spherical function associated with $\pi_{t}$. This is the matrix coefficient:

[TABLE]

where $\textbf{1}_{\partial\Gamma}$ stands for the characteristic function of $\partial\Gamma$.

It will be convenient to also introduce the continuous function (see [10, Lemma 3.2])

[TABLE]

This is also a strictly positive function. Let $M_{\sigma^{-1}_{t}}$ be the corresponding multiplication operator, that is a bounded operator acting on any $L^{p}(\partial\Gamma,\nu_{d})$ for any $p>1$.

We denoted by $\mathcal{R}_{t}=M_{\sigma^{-1}_{t}}\mathcal{I}_{t}$ the Riesz operator as in [10].

Convergence results

We deduce the above theorems from a theorem à la Bader-Muchnik for $L^{p}$-boundary representations of hyperbolic groups. Surprisingly, this kind of theorem known in the Hilbertian context holds for $L^{p}$-spaces.

Theorem 1.2**.**

For $R>0$ large enough, there exists a sequence of measures $\mu_{n,R}:\Gamma\rightarrow\mathbb{R}^{+}$, supported on $S^{\Gamma}_{n,R}$, satisfying $\mu_{n,R}(\gamma)\leq C/|S^{\Gamma}_{n,R}|$ for some $C>0$ independent of $n$ such that for all $0<t<1/2$, for all $f,g\in C(\Gamma\cup\partial\Gamma)$, for all $v\in L^{p}(\partial\Gamma,\nu_{o})$ and $w\in L^{q}(\partial\Gamma,\nu_{o})$:

[TABLE]

as $n\to+\infty$.

Remark 1.3**.**

If $\mu$ denotes a finitely supported random walk on a non-elementary hyperbolic group $\Gamma$ and $d:=d_{\mu}$ denotes the corresponding Green metric, then $(\Gamma,d_{\mu})$ satisfies the assumptions of Theorem 1 and 1.2.

Acknowledgement

The first author would like to thank Ian Tice for useful discussions about Proposition 3.1 and Nigel Higson for comments on Theorem 2.

1.1. Structure of the paper

Section 2 contains preliminaries on $\delta$-hyperbolic spaces and hyperbolic groups, Patterson-Sullivan measures and equidistribution results, $L^{p}$-boundary representations as well as the definition of the intertwiner $\mathcal{I}_{t}$ for $t>0.$

In Section 3, we recall some basic facts in interpolation theory and some known results about spherical functions for hyperbolic groups. We prove that $\mathcal{I}_{t}$ is a bounded operator from $L^{p}(\partial\Gamma,\nu)$ to $L^{q}(\partial\Gamma,\nu)$ with $1/p=1/2+t$ and $1/q=1/2-t$ where $0<t<1/2$.

Section 4 is devoted to proofs of Theorem 1.2 and Theorem 1. In particular, a new tool we use is a $L^{p}$-version of Radial Property RD for $L^{p}$-boundary representations.

Section 5 is a discussion about the case of rank one globally symmetric spaces of non compact type and we provide a proof of Theorem 2.

2. Preliminaries on geometrical setting

2.1. The geometrical setting and the regularity assumptions of the metric

A nice reference is [14].

A metric space $(X,d)$ is said to be hyperbolic if there exists $\delta\geq 0$ and a111if the condition holds for some $o$ and $\delta$, then it holds for any $o$ and $2\delta$ basepoint $o\in X$ such that for any $x,y,z\in X$ one has

[TABLE]

where $(x,y)_{o}$ stands for the Gromov product of $x$ and $y$ from $o$, that is

[TABLE]

We consider proper hyperbolic metric space (the closed balls are compact).

A sequence $(a_{n})_{n\in\mathbb{N}}$ in $X$ converges at infinity if $(a_{i},a_{j})_{o}\rightarrow+\infty$ as $i,j$ goes to $+\infty$. Set $(a_{n})\sim(b_{n})\Leftrightarrow(a_{i},b_{j})_{o}\to\infty$ as $i,j\to\infty$ : it defines an equivalence relation and the set of equivalence classes (that does not depent on the base point) is denoted by $\partial X$ and is called the Gromov boundary of $X$. The topology on $X$ naturally extends to $\overline{X}:=X\cup\partial X$ so that $\overline{X}$ and $\partial X$ are compact sets. The formula

[TABLE]

(where the supremum is taken over all sequences $(a_{n}),(b_{n})$ which represent $\xi$ and $\eta$ respectively) allows to extend the Gromov product on $\overline{X}\times\overline{X}$ but in a non continuous way in general. Moreover the boundary $\partial X$ carries a family of visual metrics, depending on $d$ and a real parameter $\epsilon>0$ denoted from now by $d_{o,\epsilon}$. The metric space $(\partial X,d_{o,\epsilon})$ is a compact subspace of the bordification $\overline{X}:=\partial X\cup X$ (also compact) and the open ball centered at $\xi$ of radius $r$ with respect to $d_{o,\epsilon}$ will be denoted by $B(\xi,r)$.

It turns out that in general, the Gromov product does not extend continuously to the bordification, see for example [14, Example 3.16]. Following the authors of [31], we say that a hyperbolic space $X$ is $\epsilon$-good, where $\epsilon>0$, if the following two properties hold for each base point $o\in X$:

• •

The Gromov product $(\cdot,\cdot)_{o}$ on $X$ extends continuously to the bordification $X\cup\partial X$.

• •

The map $d_{o,\epsilon}:(\xi,\eta)\in\partial X\times\partial X\mapsto e^{-\epsilon(\xi,\eta)_{o}}$ is a metric on $\partial X$.

The classical theory of $\delta$-hyperbolic spaces works under the assumption that the spaces are geodesic but to guarantee that the Gromov product extends continuously to the boundary, that is if two sequences $a_{n},b_{m}\in X\to\xi,\eta\in\partial X$, then the Gromov product satisfies $(a_{n},b_{m})_{o}\to(\xi,\eta)_{o}$, we shall work under the assumption of roughly geodesic spaces. In particular the conformal relation on the boundary holds: for all $x,y\in X$ and for all $\xi,\eta\in\partial X:$

[TABLE]

where the Busemann function $\beta_{\cdot}(\cdot,\cdot)$ is defined as

[TABLE]

where $(a_{n})_{n\in\mathbb{N}}$ represents $\xi.$ Recall that for all $\xi\in\partial X$ and $x,y\in X$

[TABLE]

A metric space $(X,d)$ is roughly geodesic if there exists $C=C_{X}>0$ so that for all $x,y\in X$ there exists a rough geodesic joining $x$ and $y$, that is map $r:[a,b]\subset\mathbb{R}\rightarrow X$ with $r(a)=x$ and $r(b)=y$ such that

[TABLE]

for all $t,s\in[a,b]$.

We say that two rough geodesic rays $r,r^{\prime}:[0,+\infty)\rightarrow X$ are equivalent if

$\sup_{t}d(r(t),r^{\prime}(t))<+\infty$. We write $\partial_{r}X$ for the set of equivalence classes of rough geodesic rays. When $(X,d)$ is a proper roughly geodesic space, $\partial X$ and $\partial_{r}X$ coincide.

2.2. Hyperbolic groups

For an introduction to theory of hyperbolic groups we refer to [24] and [23].

Recall that a group $\Gamma$ acts properly discontinuously on a proper metric space if for every compact sets $K,L\subset X$, the set $|\{\gamma\in\Gamma\;;\;\gamma K\cap L\neq\emptyset\}|<\infty$. A group $\Gamma$ is said to be hyperbolic if it acts by isometries on some proper hyperbolic metric space $(X,d)$ such that $X/\Gamma$ is compact. A hyperbolic group is necessarily finitely generated (by Švarc-Milnor’s lemma). For such $\Gamma$, any finite set of generators $\Sigma$ gives rise to a Cayley graph $({\mathcal{G}}(\Gamma,\Sigma),d_{\Sigma})$ whose set of vertices are the elements of $\Gamma$, linked by length-one edges if and only if they differ by an element of $\Sigma$. Every geodesic hyperbolic metric space $(X,d)$ on which $\Gamma$ acts by isometries properly discontinuously with compact quotient is quasi-isometric to a Cayley graph of a hyperbolic group. If $\Gamma$ is a hyperbolic group endowed with a left invariant metric quasi-isometric to a word metric, it turns out that the metric space $(\Gamma,d)$ is a proper roughly geodesic $\delta$-hyperbolic metric space, see for example [22, Section 3.1].

The limit set of $\Gamma$ denoted by $\Lambda_{\Gamma}$ is the set of accumulation points in $\partial X$ of an (actually any) orbit. Namely $\Lambda_{\Gamma}:=\overline{\Gamma.o}\cap\partial X$, with the closure in $\overline{X}$. We say that $\Gamma$ is non-elementary if $|\Lambda_{\Gamma}|>2$ (and in this case, $|\Lambda_{\Gamma}|=\infty$). If $\Gamma$ is non-elementary and if the action is cocompact then $\Lambda_{\Gamma}=\partial X$.

Eventually, note that a combination of results due to Blachère, Haïssinsky and Matthieu [4] and of Nica and Špakula [31] provides

Theorem 2.1**.**

A hyperbolic group acts by isometries, properly discontinuously and cocompactly on a proper roughly geodesic $\epsilon$-good $\delta$-hyperbolic space.

2.3. To sum up

We assume that the metric space we are considering satisfies the following conditions:

• •

The metric space $(X,d)$ is $\delta$-hyperbolic.

• •

The metric space $(X,d)$ proper.

• •

The metric space $(X,d)$ is roughly geodesic.

• •

The metric space $(X,d)$ $\epsilon$-good with some $\epsilon>0,$

and we let a non-elementary group $\Gamma$ act on $(X,d)$ under the following conditions:

• •

The action of $\Gamma$ is by isometries.

• •

The action is properly discontinuous.

• •

The action is cocompact.

In other words, the group $\Gamma$ is a non-elementary hyperbolic group and thus $\Gamma$ is infinite, discrete, countable and non-amenable.

2.4. The Patterson -Sullivan measure

Fix such $(X,d)$, pick an origin $o\in X$ and set $B(o,R)=\{x\in X|d(o,x)<R\}$. Consider a family of visual metrics $(d_{x,\epsilon})_{x\in X}$ associated with a parameter $\epsilon$. The compact metric space $(\partial X,d_{o,\epsilon})$ admits a Hausdorff measure of dimension

[TABLE]

where

[TABLE]

is the critical exponent of $\Gamma$ (w.r.t. its action on $(X,d)$). This $D$-Hausdorff measure is nonzero, finite, unique up to a constant, and denoted by $\nu_{o}$ when we normalize it to be a probability. The fundamental property we use is the Ahlfors regularity: the support of $\nu_{o}$ is in $\partial X$ and we say that $\nu_{o}$ is Ahlfors regular of dimension $D$, if we have the following estimate for the volumes of balls: there exists $C>0$ so that for all $\xi\in\Lambda_{\Gamma}$ for all $r\leq Diam(\partial X)$

[TABLE]

The class of measures $\nu_{o}$ is invariant under the action of $\Gamma$ and independent of the choice of $\epsilon$. We refer to [32], [33], [15] and [17] for Patterson-Sullivan measures theory.

2.5. Shadows and control of Busemann functions

Upper Gromov bounded by above

This assumption appears in the work of Connell and Muchnik in [19] as well as in the work of Garncarek on boundary unitary representations [22]. We say that a space $X$ is upper gromov bounded by above with respect to $o$, if there exists a constant $M>0$ such that for all $x\in X$ we have

[TABLE]

Morally, this definition allows us to choose a point in the boundary playing the role of the forward endpoint of a geodesic starting at $o$ passing through $x$ in the context of simply connected Riemannian manifold of negative curvature.

We denote by $\hat{x}_{o}$ a point in the boundary satisfying

[TABLE]

In particular, every roughly geodesic metric spaces are upper Gromov bounded by above (see for example [22, Lemma 4.1]).

2.5.1. Definition of shadows

Let $(X,d)$ be a roughly geodesic, $\epsilon$-good, $\delta$-hyperbolic space. Let $r>0$ and a base point $o\in X$. Define a shadow for any $x\in X$ denoted by $O_{r}(o,x)$ as

[TABLE]

Lemma 2.2**.**

Let $r>M+\delta$. Then

[TABLE]

Proof.

Assume $r>M+\delta$. For the left inclusion we have

[TABLE]

For the other inclusion

[TABLE]

∎

The above lemma combined with Ahlfors regularity of $\nu_{o}$ provides

Lemma 2.3**.**

There exists $C>0$ such that for any $x\in X$, and for $r>M+\delta$

[TABLE]

Here is a lemma dealing with a covering and the multiplicity of a covering by shadows of the boundary.

Lemma 2.4**.**

1. We have the two following properties:
2. (1)

For $R>0$ large enough, there exists $r>0$ such that

[TABLE] 3. (2)

For all $R,r>0$ large enough, there exists an integer $m$ such that for all $\xi\in\partial X$ we have for all $n\in\mathbb{N}$, $\sum_{\gamma\in S^{\Gamma}_{n,R}}\textbf{1}_{O_{r}(o,\gamma o)}(\xi)\leq m.$

Proof.

Let $\kappa$ be the diameter of a relatively compact fundamental domain of the action of $\Gamma$ on $X$ containing $o$. Set

[TABLE]

where $C_{X}$ is the constant coming from the assumption 2.6.

We prove (1). Let $\xi\in\partial X$ and consider $r_{o}$ a roughly geodesic representing $\xi$. Define $z_{\xi}:=r_{o}(nR+R/2)$. Hence, $nR\leq d(o,z_{\xi})<(n+1)R.$ Since the action is cocompact, there exists $\gamma\in\Gamma$ such that $d(\gamma o,z_{\xi})\leq\kappa.$ The choice of (2.13) ensures $nR\leq d(\gamma o,o)<(n+1)R.$ Therefore,

[TABLE]

and thus $\xi\in O_{r}(o,\gamma o)$ with $r=R+\kappa+\delta$.

We now prove (2). Take $r>0$ and $R$ satisfying (2.13). For any $\gamma\in S^{\Gamma}_{n,R}$ and for all $\xi\in O_{r}(o,\gamma o)$

[TABLE]

By definition of the Gromov product we deduce that $d(\gamma o,z_{\xi})\leq\rho$ where $\rho=2(R+r+\delta)$. Thus the set $\{\gamma\in S^{\Gamma}_{n,R}|O_{r}(o,\gamma o)\ni\xi\}$ is contained in $B(z_{\xi},\rho)\cap\Gamma.o$. Since the action is cocompact, by taking a positive constant $R^{\prime}$ bigger than $\kappa>0$, we obtain $|B(z_{\xi},\rho)\cap\Gamma.o|\leq|B(o,R^{\prime})\cap\Gamma.o|$ with $R^{\prime}=\rho-\kappa$. Set $m:=|B(o,R^{\prime})\cap\Gamma.o|$ to conclude the proof. ∎

Recall that there exists $M>0$, such that if $\gamma$ is an element of $\Gamma$, one can choose $\hat{\gamma}_{o}$ a point in $\partial X$ satisfying (2.11)

[TABLE]

Lemma 2.5**.**

There exists $C>0$ such that for all $\xi\in\partial X$ there exists $g_{\xi}\in S^{\Gamma}_{n,R}$ such that for all $\gamma\in S^{\Gamma}_{n,R}$, $\beta_{\xi}(o,\gamma o)\leq\beta_{\hat{\gamma}_{o}}(o,g_{\xi}o)+C.$

Proof.

Pick a point $\xi\in\partial X$. Consider a roughly geodesic $r_{o}$ starting at $o$ representing $\xi$ and choose a point $z_{\xi}$ on it such that $nR\leq d(z_{\xi},o)<(n+1)R$ (since $R$ is large enough). We have for $\gamma\in S^{\Gamma}_{n,R}$

[TABLE]

and therefore either $(\hat{\gamma o},z_{\xi})_{o}\leq(\gamma o,z_{\xi})_{o}+\delta$ or $(\gamma o,z_{\xi})_{o}\geq|\gamma|-M-\delta\geq d(o,z_{\xi})-M-R-\delta\geq(\hat{\gamma o},z_{\xi})_{o}-M-R-\delta.$ In other words,

[TABLE]

with $C_{\delta,M,R}=M+R+\delta.$ It follows that

[TABLE]

with $C^{\prime}=C_{X}+R+\delta+C_{\delta,M,R}>0$.

Equality (2.5) implies that $\beta_{\xi}(o,\gamma o)\leq\beta_{\hat{\gamma o}}(o,z_{\xi})+2C^{\prime}.$ Now, choose $R$ large enough so that there exists $g_{\xi}\in\Gamma$ with $d(g_{\xi}o,z_{\xi})\leq\kappa$ and $go\in S^{\Gamma}_{n,R}$, where $\kappa$ is the diameter of a fundamental domain of the action of $\Gamma$ on $X$ containing $o$. Then

[TABLE]

To conclude the proof write

[TABLE]

∎

2.6. Equidistribution à la Roblin-Margulis

The following theorem appears under this form for the first time in [13, Theorem 3.2] and has been inspired by results in [29] and [28]. The unit Dirac mass centered at $x\in X$ is denoted by $D_{x}$.

Theorem 2.6**.**

For any $R>0$ large enough, there exists a sequence of measures $\mu_{n,R}:\Gamma\rightarrow\mathbb{R}^{+}$ such that

1. (1)

There exists $C>0$ satisfying for all $n\in\mathbb{N}$ and all $\gamma\in S^{\Gamma}_{n,R}$ that

[TABLE] 2. (2)

We have the following convergence:

[TABLE]

as $n\to+\infty$, for the weak convergence in $C(\overline{X}\times\overline{X})$.*

2.7. $L^{p}$-representations

The expression (1.1) of $\pi_{t}$ defines an isometric $L^{p}$-representation of $\Gamma$ for the exponent

[TABLE]

with $0<t<1/2$. Denote its conjugate exponent

[TABLE]

Observe that the contragredient representation of $(\pi_{t},L^{p})$ is $(\pi_{-t},\overline{L^{q}})$ with respect to the (non-degenerate) pairing

[TABLE]

In particular, the adjoint operator $\pi^{*}_{t}(\gamma)$ of $\pi_{t}(\gamma)$ is given for any $\gamma\in\Gamma$ by

[TABLE]

2.8. An Intertwiner

Following [10], recall the definition of the operator $\mathcal{I}_{t}$ for $t>0$: for almost every $\xi\in\partial X$

[TABLE]

It has already been observed in [10] that $\mathcal{I}_{t}$ is a self-adjoint compact operator on $L^{2}(\partial X,\nu_{o})$. We will show that for $0<t<1/2$, the formula (2.18) defines $\mathcal{I}_{t}$ as a bounded operator from $L^{p}(\partial X,\nu_{o})$ to $L^{q}(\partial X,\nu_{o})$ with with $1/p=1/2+t$ and $1/q=1/2-t$, see Proposition 3.3. Moreover, working under the assumptions of $\epsilon$-good spaces guarantees that the operator $\mathcal{I}_{t}$ intertwines $\pi_{t}$ and $\pi_{-t}$ from $L^{p}(\partial X,\nu_{o})$ to $L^{q}(\partial X,\nu_{o})$ thanks to the relation (2.4), see [10, Proposition 3.17]. Namely, for all $\gamma\in\Gamma$ and for all $v\in L^{p}(\partial X,\nu_{o}):$

[TABLE]

It will be also useful to consider

[TABLE]

Observe that $\tilde{\sigma_{t}}$ restricted to $\partial X$ is nothing but $\sigma_{t}$ defined in (1.3). We recall that the function $\tilde{\sigma_{t}}$ si continuous on $\overline{X},$ see [10, Proposition 3.4 ].

3. Interpolation theory: Strong inequality of type $(p,q)$ for the intertwining operator and Application of Riesz-Thorin Theorem

The aims of this section is to provide some material of interpolation theory to prove the main result concerning the operator $\mathcal{I}_{t}$, with $0<t<1/2$ based on the weak-type Schur’s test. The connection of interpolation theory and Lorentz psaces with boundary representations are already in the paper [18] and [20, Chapter 6]. Note also the very recent work [25].

3.1. Lorentz spaces, interpolation and applications

We follow [18]. Let $(\Omega,\mu)$ be a measure space. If $f:\Omega\rightarrow\mathbb{C}$ is a measurable function then define the nonincreasing rearrangement of $f$

[TABLE]

The real function $f^{*}$ is a positive nonincreasing function, equimeasurable with $|f|$ and right continuous. Define then the norm, if $1<p,q<\infty$ as

[TABLE]

and $1<p<\infty$ with $q=\infty$:

[TABLE]

Define the Lorentz spaces for $1<p<+\infty$ and $1<q\leq+\infty$.

[TABLE]

Here are some useful facts:

1. (1)

$L^{p,p}=L^{p}.$ 2. (2)

$(L^{p,q})^{*}=L^{p^{\prime},q^{\prime}}.$ 3. (3)

$L^{p,q_{1}}\subset L^{p,q_{2}}$ with $q_{1}<q_{2}$, that is

[TABLE]

for some $C_{p,q_{1},q_{2}}>0.$

Here is the ** the fundamental tool, called the weak-type Schur’s test coming form interpolation theory.

Proposition 3.1**.**

Let $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$ be $\sigma$-finite measure spaces and let $1<p,q,r<\infty$ be such that

[TABLE]

Let $k:X\times Y\mapsto\mathbb{R}$ be a measurable function and suppose that there exists $A>0$ such that

[TABLE]

Therefore the formula $Tv(x)=\int_{Y}k(x,y)v(y)d\nu(y)$ defined a.e a function in $L^{q}(X,\mu)$ whenever $v$ is in $L^{p}(Y,\nu)$. And moreover for all $1\leq s\leq\infty$ there exists a constant $C=C_{p,q,r,s}>0$ depending on $p,q,r,s$ such that

[TABLE]

We refer to [34, Proposition 6.1] for a proof.

3.1.1. Analogs of homogenous functions on the boundary

Lemma 3.2**.**

Let $1<1/p<\infty.$ We have for all $\xi\in\partial X$

[TABLE]

Proof.

Let $s>0.$ We have for all $\xi\in\partial X$

[TABLE]

We obtain for all $\xi\in\partial X$ and for $t>0:$

[TABLE]

for a constant $C>0$ coming from the Ahlfors regularity property 2.9. It follows that $1/d^{pD}_{o,\epsilon}(\xi,\cdot)\in L^{1/p,\infty}.$ ∎

For $t>0$ we have for all $\xi\in\partial X$

[TABLE]

with $r=1/(1-2t).$ By symmetry, we have for all $\eta\in\partial X$ that

[TABLE]

as well. We obtain the following result:

Proposition 3.3**.**

Let $0<t<1/2$ and let $p,q$ such that $1/p=1/2+t$ and $1/q=1/2-t$. The operator $\mathcal{I}_{t}$ is bounded from $L^{p}(\partial X,\nu_{o})$ to $L^{q}(\partial X,\nu_{o})$.

Proof.

Note that $1/p+(1-2t)=1/2+t+(1-2t)=1+1/2-t=1+1/q.$ Proposition 3.1 implies that

[TABLE]

for all $1\leq s\leq\infty$. Pick $p\leq s=2/1-2t=q$. Thus $\|\mathcal{I}_{t}(v)\|_{L^{q,s}}=\|\mathcal{I}_{t}(v)\|_{L^{q}}$ and $\|v\|_{L^{p,s}}=\|v\|_{L^{p,q}}\leq C_{p,q}\|v\|_{L^{p,p}}=\|v\|_{L^{p}}$ by (3) to complete the proof. ∎

3.2. Consequences of spectral gap estimates and Riesz-Thorin theorem

3.2.1. Spherical functions on hyperbolic groups

As in [6], we recall the definition of a spherical function associated with $\pi_{t}$. This is the matrix coefficient:

[TABLE]

and introduce the function $\omega_{t}(\cdot)$ for $t\in\mathbb{R}^{*}$ defined as

[TABLE]

Note that $\omega_{t}(\cdot)$ is a positive function for all $t\in\mathbb{R}^{*}$ converging uniformly on compact sets of $\mathbb{R}$ to $\omega_{0}(x)=x$ as $t\to 0$, and $\omega_{-t}(x)=e^{2tQ}\omega_{t}(x)$.

In [6], the following estimates, called * Harish-Chandra Anker estimates*, naming related to [1] have been proved. There exists $C>0$, such that for any $t\in\mathbb{R}$, we have for all $\gamma\in\Gamma$

[TABLE]

Set for all $x\in\mathbb{R}$

[TABLE]

3.2.2. A $L^{p}$-spectral inequality

We briefly recall some facts. In [6] the following spectral inequality, generalizing the so called “Haagerup Property” or Property RD has been proved. Pick $R>0$ large enough. There exists $C>0$ such that for any $t\in\mathbb{R}$ and for all $f_{n}\in\mathbb{C}[\Gamma]$ supported in $S^{\Gamma}_{n,R}$, we have

[TABLE]

For $R>0$ large enough and for any $n\in\mathbb{N}^{*}$, consider $f_{n}=\sum_{\gamma\in S^{\Gamma}_{n,R}}\mu_{n,R}(\gamma)D_{\gamma}\in\mathbb{C}[\Gamma]$ supported in $S^{\Gamma}_{n,R}$. Note that (1) of Theorem 5.3 implies the existence of some positive constant $C>0$ such that for any $n\in\mathbb{N}^{*}$

[TABLE]

Lemma 2.4 (1) implies the existence of $C>0$ such that $|S^{\Gamma}_{n,R}|\leq Ce^{-QnR}$. From the lower bound of (3.3) together with above growth estimate we deduce the following “spectral gap”: there exists a constant $C>0$ such that for any $t\in\mathbb{R}$, we have for all $\gamma\in\Gamma,$ for all non negative integers $n$

[TABLE]

where $\widetilde{\phi_{t}}(nR)$ satisfies $C^{-1}\widetilde{\phi_{t}}(nR)\leq\phi_{t}(\gamma)\leq C\widetilde{\phi_{t}}(nR)$ for all $\gamma\in S^{\Gamma}_{n,R}$ where $C$ is a constant independent on $n$.

The aim of this subsection is to prove a $L^{p}$-version of the above inequality (3.6).

Although $\pi_{t}$ is an isometric action on $L^{p}$ with $1/p=1/2+t$, it defines also a representation $\pi_{t}:\Gamma\rightarrow\mathbb{GL}(L^{r})$ where $\mathbb{GL}(L^{r})$ stands for the group of bounded invertible linear operators acting on $L^{r}$. More precisely

Proposition 3.4**.**

For any $-1/2\leq t\leq 1/2$, for all $\gamma\in\Gamma$ the operator $\pi_{t}(\gamma)$ is bounded invertible operator on $L^{r}$ for all $1\leq r\leq\infty$ and moreover $\gamma\in\pi_{t}(\gamma)\in\mathbb{GL}(L^{r})$ is a group morphism.

Proof.

Pick $-1/2\leq t\leq 1/2$. Assume $1\leq r<\infty$. We have for all $\gamma\in\Gamma$ and for all $v\in L^{r}$

[TABLE]

where the inequality follows from the fact $|\beta_{\xi}(x,y)|\leq d(x,y)$ for all $\xi\in\partial X$ and for all $x,y\in X$.

For the case $r=\infty$, we have for all $\gamma\in\Gamma$ and for all $v\in L^{\infty}$ that

[TABLE]

Hence, for all $\gamma\in\Gamma$ the operator $\pi_{t}(\gamma)$ is bounded invertible operator on $L^{r}$ for all $1\leq r\leq\infty$. The cocycle property of the Radon-Nikodym derivative implies that $\pi_{t}$ is a morphism and thus $\pi^{-1}_{t}(\gamma)=\pi_{t}(\gamma^{-1})$. ∎

In order to prove a $L^{p}$-version of Inequality (3.6) we need the following crucial lemma.

Lemma 3.5**.**

Let $R>0$ large enough. For any $n\in\mathbb{N}$, set $f_{n}=\sum_{\gamma\in S^{\Gamma}_{n,R}}\mu_{n,R}(\gamma)D_{\gamma}\in\mathbb{C}[\Gamma]$. Consider $\pi_{t}(f_{n})$ as an operator from $L^{\infty}\to L^{\infty}$ with $t\in[-1/2,1/2]$. The exists $C_{\infty}>0$ such that for all $t\in[-1/2,1/2]$ and for all $n\in\mathbb{N}$

[TABLE]

Proof.

Let $t\in\mathbb{R}$ such that $-1/2\leq t\leq 1/2$ and $R>0$. Consider the sequence of functions $(G_{n})_{n}$ defined for each $n$ as

[TABLE]

Consider also

[TABLE]

Let $F\in L^{\infty}(\partial X,\nu_{o})$. We have for every $\xi\in\partial X$ and for all $n\in\mathbb{N}:$

[TABLE]

and thus:

[TABLE]

In other words, we shall prove

[TABLE]

Pick $\xi\in\partial X$. Lemma 2.5 implies that one can choose $R>0$ large enough such that there exist $C>0$ and $g_{\xi}\in S_{n,R}$ satisfying for all $\gamma\in S^{\Gamma}_{N,R}$

[TABLE]

Furthermore, the right inclusion of Lemma 2.2 implies that there exists $C^{\prime}>0$ such that for all $\eta\in O_{r}(o,\gamma o)$

[TABLE]

It follows a “quasi mean-value property” that reads as follows

[TABLE]

where $C_{Q,t}=e^{(\frac{1}{2}+t)Q+C^{\prime}}.$ Therefore, using an absorbing constant $C$ independent on $t\in[-1/2,1/2]$ we obtain

[TABLE]

where the first inequality follows Lemma 2.5, the second inequality follows from Lemma 2.3 combined with the growth of $|S^{\Gamma}_{n,R}|$ and the last inequality follows from the finite multiplicity of the covering $\cup_{\gamma\in S^{\Gamma}_{n,R}}O_{r}(o,\gamma o)$ proved in Lemma 2.4.

The estimates of the spherical functions (3.3) applied to $g_{\xi}\in S^{\Gamma}_{n,R}$ together with the above inequality imply for almost every $\xi\in\partial X$ and for all $n\in\mathbb{N}:$

[TABLE]

Thus

[TABLE]

as required.

The above method applied to $\check{G}_{n}$ implies

[TABLE]

∎

Eventually we obtain the $L^{p}$-version of Radial property RD for $L^{p}$-boundary representations.

Theorem 3.6**.**

Let $R>0$ be large enough and let $r\in[1,\infty]$ such that $0\leq 1/r\leq 1$. There exists $C>0$ such that for any $-\frac{1}{2}\leq t\leq\frac{1}{2}$, such that for all $n\in\mathbb{N}$ with $f_{n}=\sum_{\gamma\in S^{\Gamma}_{n,R}}\mu_{n,R}(\gamma)D_{\gamma}\in\mathbb{C}[\Gamma]$ supported on $S^{\Gamma}_{n,R}$ we have:

[TABLE]

And thus we have

[TABLE]

Remark 3.7**.**

It is worth noting that the above theorem can be viewed as a $L^{p}$-version of Radial Property RD for $L^{p}$-boundary representations of hyperbolic groups.

Proof.

The proof is based on Riesz-Thorin Theorem. We shall prove that for any $t\in[-1/2,1/2]$ and for each $n\in\mathbb{N}$ the operator $\pi_{t}(f_{n})$ viewed as an operator from $L^{1}\to L^{1}$ and as an operator from $L^{\infty}\to L^{\infty}$ is uniformly bounded with respect to $n$.

The second point follows from Lemma 3.5. We prove now the first point. First, observe that $\pi_{t}(f_{n})$ preserves the cone of positive functions since $f_{n}$ is positive and $\pi_{t}$ itself preserves the cone of positive functions. By decomposing a function $v$ into real an imaginary parts and positive and negative functions it is enough to find a bound for positive functions. Assume $v\geq 0$.

[TABLE]

where the first inequality follows from the proof of Lemma 3.5. Therefore Riesz-Thorin theorem implies that $\pi_{t}(f_{n})$ defines a bounded operator from $L^{r}$ to $L^{r}$ for any $r$ such that $1/r\in[0,1]$. ∎

4. Proofs

The proof of Theorem 1.2 relies on three steps.

Proof.

Let $0<t<\frac{1}{2}$.

Step 1: Uniform boundedness. Consider for $R>0$ and for all non-negative integer $n$ the function $f_{n}$ supported on $S^{\Gamma}_{n,R}$ defined as:

[TABLE]

Note that this a weighted version of the function $\sum_{\gamma\in S^{\Gamma}_{n,R}}\mu_{n,R}(\gamma)D_{\gamma}$ by the spherical function $\phi_{t}(\gamma).$ The $L^{p}$-spectral inequality of Theorem 3.6 together with the fact that there exists $C>0$ such that for all $\gamma\in S^{\Gamma}_{n,R}$, $C^{-1}\widetilde{\phi_{t}}(nR)\leq\phi_{t}(\gamma)$ imply

[TABLE]

Set $K:=\sup_{n}\|\pi_{t}(f_{n})\|_{L^{p}\to L^{p}}$. Given $f,g\in C(\overline{X})$ we have for all $n\in\mathbb{N}$

[TABLE]

By Banach-Alaoglu-Bourbaki Theorem and since on reflexive spaces the weak topology and the weak*-topology coincide, the limit

[TABLE]

exists for all $v\in L^{p}$, for all $w\in L^{q}$ and for all $f,g\in C(\overline{X})$, up to extraction.

Step 2: Computation of the limit.

We already know by [10] that we have the desired result for $v,w$ in a dense subspace of $L^{p},L^{q}$ spaces (e.g. that for all $v,w\in Lip(\partial X)$ and $f,g\in C(\overline{X})$)

[TABLE]

Step 3: Conclusion.

Assume that $0<t<1/2$. Therefore $\mathcal{I}_{t}$ and thus $\mathcal{R}_{t}$ are continuous on $L^{p}$ with $1/p=1/2+t$. The limit above together with the uniform bound of Step 1 imply eventually that for $f,g\in C(\overline{X})$ and for $(v,w)\in L^{p}\times L^{q}$:

[TABLE]

∎

4.1. Proof of irreducibility

To prove irreducibility of representations our main tool is Theorem 1.2.

Lemma 4.1**.**

Let $0<t<\frac{1}{2}$. The vector $\textbf{1}_{\partial X}$ is cyclic for $(\pi_{t},L^{p})$ and the vector $\sigma_{t}$ is cyclic for $(\pi_{-t},\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}})$.

Proof.

Theorem 1.2 implies for all $w\in L^{q}$ the following convergence and for all continuous functions $f\in C(\overline{X})$:

[TABLE]

Now, given a function $f\in C(\partial X)$ consider $\tilde{f}\in C(\overline{X})$ such that $\tilde{f}_{|_{\partial X}}=f$. Set

[TABLE]

Hence, $f_{n}\to f$ with respect to the weak topology of $L^{p}$. Therefore, since $C(\partial X)$ is dense in $L^{p}$ and since the closure of the weak topology and the $\|\cdot\|_{p}$ coincides, the vector $\textbf{1}_{\partial X}$ is cyclic for $L^{p}$.

We prove now that $\sigma_{t}$ is cyclic for $\pi_{-t}$. Recall that for all $v\in L^{p}$ we have $\mathcal{I}_{t}(v)\in L^{q}$ by Proposition 3.3. Hence for all $v\in L^{p}$, with the same notation of (4.2) we have

[TABLE]

Since $\mathcal{I}_{t}$ is self adjoint, for all $v\in L^{p}$

[TABLE]

where $\mathcal{I}_{t}(f_{n})\in\pi_{-t}(\mathbb{C}[\Gamma])\sigma_{t}.$

Hence

[TABLE]

Eventually, using the density of $C(\partial X)$ in $L^{p}$ and the continuity of $\mathcal{I}_{t}$ we deduce

[TABLE]

∎

Proof of Theorem 1..

First of all, since $\mathcal{I}_{t}$ is continuous from $L^{p}$ to $L^{q}$, the subspace $\ker\mathcal{I}_{t}$ is a closed invariant subspace of $(\pi_{t},L^{p})$. Thus, if $\mathcal{I}_{t}$ is non injective, then $(\pi_{t},L^{p})$ is not irreducible.

We shall prove now that $\mathcal{I}_{t}$ is injective then $(\pi_{t},L^{p})$ is irreducible for $1/p=1/2+t.$ Since $\mathcal{I}_{t}$ is a continuous operator from $L^{p}\to L^{q}$, a standard result in Banach spaces theory asserts that the dual space of $L^{p}/\ker\mathcal{I}_{t}$ is the space $\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}}$, where the the weak closure for $L^{q}$ is the same as the $\|\cdot\|_{q}$-closure (see [30, Chapter 3]). Hence, the dual representation of $(\pi_{t},L^{p}/\ker\mathcal{I}_{t})$ is $(\pi_{-t},\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}})$. We shall prove that $(\pi_{-t},L^{q})$ is irreducible to obtain irreducibility of $(\pi_{t},L^{p}/\ker\mathcal{I}_{t})$.

Recall that $\pi_{-t}(f)\sigma_{t}=\mathcal{I}_{t}(\pi_{t}(f)\textbf{1}_{\partial X})$, for all $f\in\mathbb{C}[\Gamma]$ and the function $\sigma_{t}$ is cyclic in $\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}}$ for $\pi_{-t}$ by Lemma 4.1.

Now, let $0\neq K\subset\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}}\subset L^{q}(\partial X,\nu_{o})$ a $L^{q}$-closed subspace invariant by $\pi_{-t}$. Let $R>0$ large enough. For any $w\in K\subset L^{q}$ define for all $n\in\mathbb{N}$, the vector:

[TABLE]

where $\tilde{\sigma_{t}}$ has been defined in (2.20). Theorem 1.2 implies for all $v\in L^{p}(\partial X,\nu_{o})$ that as $n\to+\infty$:

[TABLE]

and the above convergence reads as follows with respect to the weak topology on $L^{q}:$

[TABLE]

Since $K$ is closed we have that $\langle\textbf{1}_{\partial X},w\rangle\sigma_{t}\in K.$ So, since $\sigma_{t}$ is cyclic, it is sufficient to show that there exists $0\neq w\in K$ such that

[TABLE]

Assume this is not the case: for all $w\in K$ we have $\langle\textbf{1}_{\partial X},w\rangle=0$. We would have that for all $\gamma\in\Gamma$ that $\langle\textbf{1}_{\partial X},\pi_{-t}(\gamma^{-1})w\rangle=0=\langle\pi_{t}(\gamma)\textbf{1}_{\partial X},w\rangle.$ And therefore, since $\textbf{1}_{\partial X}$ is cyclic for $(\pi_{t},L^{p}(\partial X,\nu_{o}))$ it implies that

[TABLE]

Since the pairing is non-degenerate then $K=\{0\}.$

Hence, if $K\neq\{0\}$ then $K$ contains $\sigma_{t}$ and thus it has to be $\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{q}}$ and the proof is done.

∎

5. Application to rank one semisimple Lie groups

Let $G$ be a connected semisimple Lie group with finite center and let $\mathfrak{g}$ be its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ and let $\mathfrak{k}$ be its Lie algebra. Let $\mathfrak{p}$ be the orthogonal complement of $\mathfrak{k}$ in $\mathfrak{g}$ relative to the Killing form $B$. Among the abelian sub-algebras of $\mathfrak{g}$ contained in the subspace $\mathfrak{p}$, let $\mathfrak{a}$ be a maximal one. We assume $\dim\mathfrak{a}=1$, i.e. the real rank of $G$ equals to $1$ (in particular $G$ is not compact). Let $\Sigma\subset\mathfrak{a}^{*}$ be the root system associated to $(\mathfrak{g},\mathfrak{a})$. Let

[TABLE]

be the root space of $\alpha\in\Sigma$. Recall that $\Sigma=\{-\alpha,\alpha\}$ or $\Sigma=\{-2\alpha,-\alpha,\alpha,2\alpha\}$ where $\alpha$ is a positive root ($\alpha\in\Sigma$ is positive if and only if $\alpha(H)>0$ for all $H\in\mathfrak{a}^{+}$). If $m_{1}=\dim\mathfrak{g}_{\alpha}$ and $m_{2}=\mathfrak{g}_{2\alpha}$ denoted by $\rho=\frac{1}{2}(m_{1}+2m_{2})\alpha$ be the half sum of positive roots. Let $H_{0}$ be the unique vecteur in $\mathfrak{a}$ such that $\alpha(H_{0})=1$. Hence, $\mathfrak{a}=\{tH_{0},t\in\mathbb{R}\}$ and $\mathfrak{a}_{+}=\{H\in\mathfrak{a}|\alpha(H)>0\}$ is identified with the open set of strictly positive real numbers. Let $\mathfrak{n}$ the nilpotent Lie algebra defined as the direct sum of root spaces of positive roots:

[TABLE]

Let $A=\exp(\mathfrak{a})$, $A^{+}=\exp(\mathfrak{a}^{+})$ and $N=\exp(\mathfrak{n})$. Let $G=KAN$ be the Iwasawa decomposition and $K\overline{A^{+}}K$ the Cartan decomposition defined by $\mathfrak{a}^{+}$ where $\overline{A^{+}}$ denoted the closure $A^{+}$. Let $Z(A)$ be the centralizer of $A$ in $G$ and $M=Z(A)\cap K$. The group $M$ normalizes $N$. Let $P=MAN$ be the minimal parabolic subgroup of $G$ associated to $\mathfrak{a}^{+}$. Let $\nu$ be the unique Borel regular $K$-invariant probability measure on the Furstenberg-Poisson boundary $G/P$ that is quasi-invariant under the action $G$ (we refer to [BDV, Appendix B] for a general discussion). Let

[TABLE]

be the associated $L^{p}$-boundary representation of $G$ and defined the corresponding spherical function

[TABLE]

The corresponding globally symmetric space of non compact type of $G$ is $G/K$ endowed with a $G$-invariant Riemannian metric denoted by $d$ induced by the Killing form on $\mathfrak{g}\ /\mathfrak{k}$ identified with the tangent space of $G/K$ at the point $o=eK$. A flat of dimension $k$ is defined as the image of a map $\mathbb{R}^{k}\rightarrow G/K$ locally isometric. The rank of $G$ is the largest dimension of a flat subspace of $G/K$. The rank one globally symmetric spaces of non compact type are classified as follows: there are the real hyperbolic spaces $H^{n}(\mathbb{R})$, the complex hyperbolic spaces $H^{n}(\mathbb{C})$, the quaternionic hyperbolic spaces $H^{n}(\mathbb{H})$ for $n\geq 2$ and the exceptional hyperbolic space that is the $2$-dimensional octonionic hyperbolic space $H^{2}(\mathbb{O})$.

If $(X,d)$ is one the above hyperbolic space, it is a CAT(-1) space and in particular it is a proper geodesic $\delta$-hyperbolic space and fits of course in the class of spaces 2.3 for $\epsilon=1$. One can therefore consider its Gromov boundary $\partial X$ or equivalently the geometric boundary of $X$. The group $G$ acts by isometries on $(X,d)$ and its discrete subgroup acts properly discontinuously on $X$. Assume that $\Gamma$ is lattice (uniform or non-uniform) and perform the Patterson-Sullivan construction associated to $(\Gamma,d)$ with the base point $o=eK\in X$ to obtain a measure supported on $\partial X$ denoted by $\nu_{o}$. The Hausdorff dimension of $\nu_{o}$ is the critical exponent $Q_{\Gamma}$ of $\Gamma$ that coincides with the volume growth of the corresponding hyperbolic spaces $Q_{G}=m_{1}+2m_{2}$.

A geodesic ray starting at the origin can be represented using Cartan decomposition as $c(t)=ke^{tH_{0}}\cdot o=kMe^{tH_{0}}\cdot o$ where $t\in\mathbb{R}_{+}$ and $k\in K$. Then the Furstenberg-Poisson boundary $G/P$ can be identified with the geometric boundary $\partial X$ in the case of rank one symmetric space. Indeed, one can identify $G/P=K/M$ thanks to the Iwasawa decomposition $KAN$. It turns out that the Patterson-Sullivan measure $\nu_{o}$ associated with a lattice $\Gamma$ supported on $\partial X$ of dimension $Q_{\Gamma}$ coincides with the unique $K$-invariant measure $\nu$ on $G/P$.

Thus, the $L^{p}$-boundary representation of $\Gamma$ is nothing but the restriction of $\rho_{t}$ to $\Gamma$

[TABLE]

with $p$ such that $1/p=1/2+t$ with $0<t<1/2$. Since $\Gamma$ might be a non-uniform lattice, the results obtained above dealing with hyperbolic groups do not apply to $\Gamma$. Nevertheless we have the exact analog of Lemma 3.5.

In the following, for $R>0$ and for any $n\in\mathbb{N}$, the spheres $S^{\Gamma}_{n,R}$ are defined with respect to the length function $|\gamma|:=d(o,\gamma o)$ corresponding to the Riemannian metric $d$ on the symmetric space $G/K$. Moreover, one can take for $\mu_{n,R}$ , for $R>0$ and for any $n\in\mathbb{N}$ the standard average

[TABLE]

Lemma 5.1**.**

Let $\Gamma$ be a lattice in $G$. Let $R>0$. For all $t\in[-1/2,1/2]$ and $n\in\mathbb{N}$ set $f_{n}=\frac{1}{|S^{\Gamma}_{n,R}|}\sum_{\gamma\in S^{\Gamma}_{n,R}}D_{\gamma}\in\mathbb{C}[\Gamma]$. Consider $\pi_{t}(f_{n})$ as an operator from $L^{\infty}\to L^{\infty}.$ The exists $C_{\infty}>0$ depending on $R$ such that for all $t\in[-1/2,1/2]$

[TABLE]

Proof.

Indeed the proof follows the same ideas of [9, Proposition 3.2] and [12, Section 2.5] ∎

Therefore, following exactly the same method of the proof of Theorem 1.2, we obtain

Theorem 5.2**.**

Let $\Gamma$ be a lattice in a rank one connected semisimple Lie group with finite center $G$. Let $R>0$ be large enough and let $r\in[1,\infty]$ such that $0\leq 1/r\leq 1$. There exists $C>0$ such that for any $-\frac{1}{2}\leq t\leq\frac{1}{2}$, such that for all $n\in\mathbb{N}$ with $f_{n}=\frac{1}{|S^{\Gamma}_{n,R}|}\sum_{\gamma\in S^{\Gamma}_{n,R}}D_{\gamma}\in\mathbb{C}[\Gamma]$ supported on $S^{\Gamma}_{n,R}$ we have:

[TABLE]

And thus we have

[TABLE]

The equidistribution theorem needed in (2.6) reads as follows in the context of lattices. We refer to [29] for the next results in a more general setting.

Theorem 5.3**.**

Let $\Gamma$ be a lattice in a rank one connected semisimple Lie group with finite center $G$. For any $R>0$, we have the following convergence:

[TABLE]

as $n\to+\infty$, for the weak convergence in $C(\overline{X}\times\overline{X})$.*

Eventually we obtain

Theorem 5.4**.**

Let $\Gamma$ be a lattice in a rank one connected semisimple Lie group with finite center $G$. For $R>0$ large enough, for all $0<t<1/2$, for all $f,g\in C(\overline{X})$, for all $v\in L^{p}(\partial X,\nu_{o})$ and $w\in L^{q}(\partial X,\nu_{o})$:

[TABLE]

as $n\to+\infty$.

5.1. Intertwining operators

The restriction of $\rho_{t}$ to $K$ on $L^{2}(K/M,\nu_{o})$ is given by

[TABLE]

since $\nu_{o}$ is $K$-invariant and note that the representation does not depend on $t$ anymore and provides a unitary representation of $K$. Therefore the intertwining relations 2.19 reads as follows: for all $k\in K$ and for all $t>0$

[TABLE]

Peter-Weyl Theorem implies that

[TABLE]

where $V_{n}$ are finite dimensional irreducible unitary representation of $K$. Therefore, Schur’s lemma implies that there exists a sequence of scalars $(\lambda_{n})_{n\geq 0}$ such that $\mathcal{I}_{t}$ restricted to $V_{n}$ is a scalar operator as follows $\mathcal{I}_{t|_{V_{n}}}=\lambda_{n}Id_{|_{V_{n}}}$ with $\lambda_{n}\neq 0$ for all $n\geq 0$. We deduce that $\mathcal{I}_{t}$ is injective viewed as an operator acting on $L^{2}$ and therefore it is injective as an operator from $L^{p}$ to $L^{q}$. Apply Theorem 5.4 to lattices in rank one semisimple Lie groups and use exactly the same arguments of Theorem 1 to complete the proof of Theorem 2.