# Riesz operators and some spherical representations for hyperbolic groups

Adrien Boyer and Jean-Claude Picaud  Université de Paris - Paris - France
<EMAIL_ADDRESS>Université de Tours and Zürich University  jean-
<EMAIL_ADDRESS>To L. and N.

(Date: August 27, 2024)

###### Abstract.

We introduce the Riesz operator in the context of Gromov hyperbolic groups in
order to investigate a one parameter family of non unitary boundary Hilbertian
representations of hyperbolic groups. We prove asymptotic Schur’s relations,
the latter being the main result of this paper. Up to normalization, the Riesz
operator plays the role in the context of hyperbolic groups of the Knapp-Stein
intertwiner for complementary series for Lie groups. Assuming the positivity
of the Riesz operator, we define an analogue of complementary series for
hyperbolic groups and prove their irreducibility.

###### Key words and phrases:

dual system of boundary representations, irreducibility, Bader-Muchnik ergodic
theorems, complementary series.

###### 1991 Mathematics Subject Classification:

Primary 43; Secondary 22

##  1\. Introduction

###  1.1. Representations theory basics

Let $G$ be a group, $\mathcal{V}$ be a complex vector space and
$GL(\mathcal{V})$ be the group of complex invertible linear maps acting on
$\mathcal{V}$. _A linear representation_ $(\pi,\mathcal{V})$ of $G$ is a
homomorphism $\pi:G\rightarrow GL(\mathcal{V})$. A _Hilbertian representation_
$(\pi,\mathcal{V})$ of $G$ is a homomorphism $\pi:G\rightarrow
GL_{c}(\mathcal{V})$, where $\mathcal{V}$ is a complex Hilbert space and
$GL_{c}(\mathcal{V})$ denotes the group of bounded invertible operators. A
_unitary representation_ $(\pi,\mathcal{V})$ of $G$ is an Hilbertian
representation $\pi:G\rightarrow\mathcal{U}(\mathcal{V})$ where
$\mathcal{U}(\mathcal{V})$ the group of unitary operators acting on
$\mathcal{V}$.
Given $\langle\cdot,\cdot\rangle$ a $\mathbb{C}$-pairing for
$(\mathcal{V},\mathcal{W})$ two $\mathbb{C}$-vector spaces and a
representation $(\pi,\mathcal{V})$ of $G$, the contragredient representation
of $(\pi,\mathcal{V})$ with respect to $\langle\cdot,\cdot\rangle$ is defined
as $(\tilde{\pi},\mathcal{W})$ where for all
$(v,w)\in\mathcal{V}\times\mathcal{W}$ and for all $g\in G$:

(1.1) $\displaystyle\langle\pi(g)v,w\rangle=\langle
v,\tilde{\pi}(g^{-1})w\rangle.$

Given two _linear representations_ (_Hilbertian, resp. unitary_) of $G$ :
$(\pi_{1},\mathcal{V}_{1})$ and $(\pi_{2},\mathcal{V}_{2})$, an intertwining
operator is (_bounded, resp. isometric_) operator
$\mathcal{I}:\mathcal{V}_{1}\rightarrow\mathcal{V}_{2}$, satisfying for all
$g\in G$ :

(1.2) $\mathcal{I}\pi_{1}(g)=\pi_{2}(g)\mathcal{I}.$

We say that two _representations_ are equivalent if there exists an
intertwining operator that is an isomorphism in the category concerned. Given
a representation $(\pi,\mathcal{V})$, a subspace
$\mathcal{W}\subset\mathcal{V}$ is invariant by $\pi$ if
$\pi(\gamma)\mathcal{W}\subset\mathcal{W}$ for all $\gamma\in G$. A Hilbertian
representation $(\pi,\mathcal{V})$ is _irreducible_ if $\pi$ has no closed
proper invariant subspace.

###  1.2. General discussion about the dual of a group

A basic problem in harmonic analysis on a locally compact group $G$ is to
describe the set $\widehat{G}$ of all irreducible unitary representations, up
to equivalence, called the unitary dual of $G$. For example, in the setting of
reductive Lie groups this problem can be solved and we refer to the
introduction of [52] for more details in this context. In the setting of
countable discrete groups the situation is drastically different. Except for
virtually abelian groups, the dual space of a discrete group $\Gamma$ is not
even a Hausdorff space for the standard topology structure (see for instance
Glimm [33] and Thoma [53]). Quoting the authors of [17] (see also [28]) : “In
all other cases there is no natural Borel coding of $\widehat{\Gamma}$, i.e.
$\widehat{\Gamma}$ is not countably separated; for lack of a systematic
procedure of constructing all irreducible representations, a natural problem
is to construct large classes of irreducible representations.”
According to [27], this is of interest to understand the _non unitary-dual_ of
a group. In this paper we will construct a large class of irreducible
Hilbertian representations for non-elementary _Gromov hyperbolic groups_ [34]
and exhibit a subclass of those groups for which those irreducible non-unitary
Hilbertian representations can be unitarized.

###  1.3. Construction of representations for Gromov hyperbolic groups: the
spherical boundary representations

Let $\Gamma$ be a non-elementary discrete Gromov hyperbolic group. We will
consider $\delta$-hyperbolic metric spaces $(X,d)$, _roughly geodesic and
$\epsilon$-good_ (see Section 2.3 for the definitions and Section 1.4 for the
motivation) on which $\Gamma$ acts properly and cocompactly by isometries. Fix
such $(X,d)$, pick $o\in X$ and set $B(o,R)=\\{x\in X|d(o,x)<R\\}$. Recall
that the Gromov boundary $\partial X$ of $(X,d)$ is endowed with a family of
conformal 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

(1.3) $D:={Q\over\epsilon}$

where

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

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. In the context of
compact quotients of Hadamard manifolds $\nu_{o}$ is known as the _Patterson-
Sullivan_ measure. The _class_ of measures $[\nu_{o}]$ is invariant under the
action of $\Gamma$ and independent of the choice of $\epsilon$. Hence, it
induces a one parameter family of representations which we call _spherical
boundary representations,_ already studied in [10] and defined by :

(1.5)
$\pi_{t}(\gamma)v(\xi)=\bigg{(}\frac{d\gamma_{*}\nu}{d\nu}(\xi)\bigg{)}^{\frac{1}{2}+t}v(\gamma^{-1}\xi),\;\;\;\;\;(t\in\mathbb{R},\;v\in
L^{2}(\partial X,\nu_{o}))$

whose class depends _a priori_ on $(X,d)$.

The adjoint operator $\pi^{*}_{t}(\gamma)$ of $\pi_{t}(\gamma)$ is given for
any $\gamma\in\Gamma$ by:

(1.6) $\pi^{*}_{t}(\gamma)=\pi_{-t}(\gamma^{-1}).$

In the following, we will denote $\langle\cdot,\cdot\rangle$ the inner product
on $L^{2}(\partial X,\nu_{o})$, and if $\textbf{1}_{\partial X}$ is the
characteristic function of $\partial X$ we will consider _the spherical
function :_

(1.7)
$\phi_{t}:\gamma\in\Gamma\mapsto\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},\textbf{1}_{\partial X}\rangle.$

The representation $\pi_{0}$ is unitary and might be thought as an analog of
the endpoint of principal series for $SL(2,\mathbb{R})$. It appears in
different contexts as a boundary representation and it is also called quasi-
regular representation (or Koopman representation) - see [18], [3], [32],
[10], [12], [13], [15], [14], [29], [41] and [42] for boundary representations
and see [24] and [25] for other quasi-regular representations. More recently
in [19], boundary representations of hyperbolic groups have been used to
obtain results in the general setting of locally compact groups.

###  1.4. Intertwiner and Riesz operator

Define the operator for $t>0$:

(1.8) $\mathcal{I}_{t}(v)(\xi):=\int_{\partial
X}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta).$

We call this operator the Knapp-Stein operator associated with the metric $d$
in reference to [38]. It is a self-adjoint compact operator on $L^{2}(\partial
X,\nu_{o})$. The fundamental observation is that, when the metric space
$(X,d)$ is an _$\epsilon$ -good hyperbolic space_, (see Nica and Špakula [48]
and Section 2.3 below), the operator $\mathcal{I}_{t}$ intertwines $\pi_{t}$
and $\pi_{-t}$ on $L^{2}(\partial X,\nu_{o})$, namely for all
$\gamma\in\Gamma$ and for all $v\in L^{2}(\partial X,\nu_{o}):$

(1.9) $\mathcal{I}_{t}\pi_{t}(\gamma)v=\pi_{-t}(\gamma)\mathcal{I}_{t}v.$

In the following, it will be convenient to introduce the function

(1.10) $\displaystyle\sigma_{t}:\xi\in\partial
X\mapsto\mathcal{I}_{t}(\textbf{1}_{\partial
X})(\xi)\in\mathbb{R}^{+}\;\;\;\;(t>0).$

According to the previous notations, we define as well the spherical Riesz
operator of order $t>0$, the Riesz operator for short, as:

(1.11) $\mathcal{R}_{t}(v)(\xi):=\frac{1}{\sigma_{t}(\xi)}\int_{\partial
X}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta).$

###  1.5. Several representations

Let $\mathbb{C}[\Gamma]$ the group algebra and for each $t\in\mathbb{R}_{+}$
set :

(1.12) $\mathcal{F}_{t,\textbf{1}_{\partial
X}}:=\pi_{t}(\mathbb{C}[\Gamma])\subset L^{2}(\partial X,\nu_{o}).$

The mapping $(v,w)\mapsto\langle v,\mathcal{I}^{2}_{t}(w)\rangle$ is a
positive symmetric sesquilinear form on $\mathcal{F}_{t,\textbf{1}_{\partial
X}}$. Set $N_{t}:=\\{v\in\mathcal{F}_{t,\textbf{1}_{\partial
X}}|\mathcal{I}_{t}(v)=0\\}$. Endow the quotient space
$\mathcal{F}_{t,\textbf{1}_{\partial X}}/N_{t}$ with the inner product
$\langle\cdot,\mathcal{I}^{2}_{t}(\cdot)\rangle$. Let
$(\mathcal{K}_{t},\langle\cdot,\cdot\rangle_{\mathcal{K}_{t}})$ be the Hilbert
space completion of $\mathcal{F}_{t,\textbf{1}_{\partial X}}/N_{t}$ with
respect to $\langle\cdot,\mathcal{I}^{2}_{t}(\cdot)\rangle$. In an analogous
way, with the hypothesis that the Riesz operator is positive (and we will show
that this is equivalent for ${\mathcal{I}}_{t}$ to be positive), the bilinear
form $\langle\cdot,\mathcal{I}_{t}(\cdot)\rangle$ induces another Hilbert
space which will be denoted by
$(\mathcal{H}_{t},\langle\cdot,\cdot\rangle_{\mathcal{H}_{t}})$. We have:

######  Proposition 1.1.

If for some $t>0$ the operator $\mathcal{R}_{t}$ is positive, then
$(\pi_{t},\mathcal{H}_{t})$ is a unitary representation.

We shall study in the following the families of _Hilbertian representations_
$(\pi_{t},\mathcal{K}_{t})_{t>0}$ and _unitary representations_
$(\pi_{t},\mathcal{H}_{t})_{0<t\leq\frac{1}{2}}$ when the Riesz operator is
assumed to be positive (and we will see that a necessary condition for
positivity is $t\leq 1/2$). The latter are the analogous of complementary
series.

###  1.6. Results

Let $\Gamma$ be a non elementary Gromov hyperbolic group acting cocompactly
and properly on _a roughly geodesic, $\epsilon$-good, $\delta$-hyperbolic
space_ $(X,d)$. Let $\partial X$ be the Gromov boundary of $X$ and set
$\overline{X}:=X\cup\partial X$. Define for any $R>0$ and for any non-negative
integers $n\geq 1$, the spheres
$S^{\Gamma}_{n,R}:=\\{\gamma\in\Gamma|(n-1)R\leq d(o,\gamma o)<nR\\}$. The
cardinal of $S^{\Gamma}_{n,R}$ is denoted by $|S^{\Gamma}_{n,R}|$.
To write the next statements in a synthetic form, we adopt the following
notations : for $t>0$, we set $\mathcal{E}_{t,0}=L^{2}(\partial X,\nu_{o})$
and $\mathcal{E}_{t,2}=\mathcal{K}_{t}$. For the defintion of
$\mathcal{E}_{t,1}$, we need to assume that $0<t\leq{1\over 2}$ and that
$\mathcal{R}_{t}$ is positive. Under these assumptions we set
$\mathcal{E}_{t,1}=\mathcal{H}_{t}$. Then we can state our main result, the
following asymptotic Schur’s orthogonality relations :

######  Theorem 1.2.

For any $R>0$ large enough, there exists a sequence of measures
$\beta_{n,R}:\Gamma\rightarrow\mathbb{R}^{+}$, supported on
$S^{\Gamma}_{n,R}$, satisfying $\beta_{n,R}(\gamma)\leq C/|S^{\Gamma}_{n,R}|$
for some $C>0$ independent of $n$ such that for all $t,t^{\prime}>0$, for all
$i,j=0,1,2$, for all $f,g\in C(\overline{X})$, for all
$v,w\in\mathcal{E}_{t,i}$,
$v^{\prime},w^{\prime}\in\mathcal{E}_{t^{\prime},j}$:

$\displaystyle\lim_{n\to\infty}\sum_{\gamma\in
S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)f(\gamma
o)g(\gamma^{-1}o)\frac{\langle\pi_{t}(\gamma)v,w\rangle_{\mathcal{E}_{t,i}}}{\phi_{t}(\gamma)}$
$\displaystyle\frac{\overline{\langle\pi_{t^{\prime}}(\gamma)v^{\prime},w^{\prime}\rangle}_{\mathcal{E}_{t^{\prime},j}}}{\phi_{t^{\prime}}(\gamma)}$
$\displaystyle=\langle g_{|_{\partial
X}}\mathcal{R}_{t}(v),\mathcal{R}_{t^{\prime}}(v^{\prime})\rangle\overline{\langle\mathcal{I}^{i}_{t}(w),f_{|\partial
X}\mathcal{I}^{j}_{t^{\prime}}(w^{\prime})\rangle}.$

######  Remark 1.3.

In the following cases, $\beta_{n,R}$ is the uniform probability measure on
$S^{\Gamma}_{n,R}$ (namely
$\beta_{n,R}=\dfrac{1}{|S^{\Gamma}_{n,R}|}\sum_{\gamma\in
S^{\Gamma}_{n,R}}D_{\gamma o}$ where $D_{\gamma o}$ stands for the unit Dirac
mass at $\gamma o$):

  1. (1)

$\Gamma=\mathbb{F}_{r}$ is the (non abelian) free group on $r\geq 2$
generators acting on its standard Cayley graph with the standard left
invariant word metric.

  2. (2)

$\Gamma$ is the fundamental group of a compact negatively curved manifold
acting on $(X,d)$ its universal cover.

  3. (3)

$\Gamma$ is a non elementary discrete group of isometries acting cocompactly
on a CAT(-1) space with non arithmetic length spectrum.

######  Remark 1.4.

The asymptotic Schur relations have been studied as such for tempered
representations of Lie groups very recently in [26]. It is worth noting that
the case $(\mathcal{E}_{1,t},\mathcal{E}_{1,t^{\prime}})$ with
$t,t\leq\frac{1}{2}$ of the above theorem provides an example of Schur
relations for unitary representations that are _not tempered_ , namely the
complementary series.

We deduce three theorems à la Bader-Muchnik from the above theorem. The main
point is that the convergence (and the irreducibility) of $\pi_{t}$ does not
really depend on the Hilbert spaces on which $\Gamma$ acts via $\pi_{t}$.

######  Corollary 1.5.

For any $R>0$ large enough, there exists a sequence of measures
$\beta_{n,R}:\Gamma\rightarrow\mathbb{R}^{+}$, supported on
$S^{\Gamma}_{n,R}$, satisfying $\beta_{n,R}(\gamma)\leq C/|S^{\Gamma}_{n,R}|$
for some $C>0$ independent of $n$ such that for all $t>0$, for all $i=0,1,2$,
for all $f,g\in C(\overline{X})$, for all $v,w\in\mathcal{E}_{t,i}$:

$\sum_{\gamma\in S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)f(\gamma
o)g(\gamma^{-1}o)\frac{\langle\pi_{t}(\gamma)v,w\rangle_{\mathcal{E}_{t,i}}}{\phi_{t}(\gamma)}\to\langle
g_{|_{\partial X}}\mathcal{R}_{t}(v),\textbf{1}_{\partial
X}\rangle\overline{\langle w,f_{|_{\partial X}}\rangle}_{\mathcal{E}_{t,i}},$

as $n\to+\infty$.

From which we obtain the following results of irreducibility of Hilbertian
representations:

######  Corollary 1.6.

Let $t>0$.

  1. (1)

The Riesz operator is injective if and only if the Hilbertian representation
$(\pi_{t},L^{2}(\partial X,\nu_{o}))$ is irreducible.

  2. (2)

If the Riesz operator is not injective, then the Hilbert space $L^{2}(\partial
X,\nu_{o})$ splits as:

$L^{2}(\partial X,\nu_{o})=\mathcal{V}_{t}\oplus\mathcal{W}_{-t},$

where $(\pi_{t},\mathcal{V}_{t})$ and $(\pi_{-t},\mathcal{W}_{-t})$ are
Hilbertian representations. Moreover:

     * •

If $0<t<\frac{1}{2}$, the representations are $(\pi_{t},\mathcal{V}_{t})$ and
$(\pi_{-t},\mathcal{W}_{-t})$ are infinite dimensional.

     * •

Let $t>0$. The Hilbertian representation $(\pi_{-t},\mathcal{W}_{-t})$ is
irreducible and thus the quotient representation $(\pi_{t},L^{2}(\partial
X,\nu_{o})/\mathcal{V}_{t})$ is irreducible as the contragredient
representation of $(\pi_{-t},\mathcal{W}_{-t})$ with respect to the
$L^{2}$-pairing. Moreover $(\pi_{-t},\mathcal{W}_{-t})$ and
$(\pi_{-t^{\prime}},\mathcal{W}_{-t^{\prime}})$ are inequivalent for positive
real numbers $t\neq t^{\prime}$.

     * •

If $t=\frac{1}{2}$, then $(\pi_{-\frac{1}{2}},\mathcal{W}_{-\frac{1}{2}})$ and
$(\pi_{\frac{1}{2}},L^{2}(\partial X,\nu_{o})/\mathcal{V}_{\frac{1}{2}})$ are
the trivial representation.

######  Corollary 1.7.

Let $t>0$. The Hilbertian representation $(\pi_{t},\mathcal{K}_{t})$ is
irreducible as well as its contragredient representation with respect to the
$L^{2}$-pairing. Moreover $(\pi_{t},\mathcal{K}_{t})$ and
$(\pi_{t^{\prime}},\mathcal{K}_{t^{\prime}})$ are inequivalent for positive
real numbers $t\neq t^{\prime}$.

######  Corollary 1.8.

We have:

  1. (1)

It $t=\frac{1}{2}$, then $(\pi_{t},\mathcal{H}_{t})$ is the trivial
representation.

  2. (2)

If $t>\frac{1}{2}$, then $\mathcal{R}_{t}$ cannot be positive.

  3. (3)

If for some $t\in]0,\frac{1}{2}[$, the operator $\mathcal{R}_{t}$ is positive,
then $(\pi_{t},\mathcal{H}_{t})$ is an infinite dimensional irreducible
unitary representation and its contragredient representation with respect to
the $L^{2}$-pairing is irreducible as well. Moreover
$(\pi_{t},\mathcal{H}_{t})$ and $(\pi_{t^{\prime}},\mathcal{H}_{t^{\prime}})$
are inequivalent for positive real numbers $t\neq t^{\prime}$ in
$[0,\frac{1}{2}]$.

######  Remark 1.9.

In this paper, the families $(\pi_{t})_{t\in\mathbb{R}}$ defined in (1.5) (for
different Hilbert spaces) can be thought as one parameter non-unitary
deformations of $\pi_{o}$. The Hilbertian representations
$(\pi_{t},\mathcal{K}_{t})_{t\in\mathbb{R}^{*}_{+}}$ might be thought as an
Hilbertian analog of complementary series for general hyperbolic groups.

###  1.7. Comments on conditionally negative hyperbolic groups

An hyperbolic group $\Gamma$ may satisfy property $(T)$ or in an opposite
direction Haagerup property.
In the first case, the trivial representation (which is realized by
$(\pi_{\frac{1}{2}},\mathcal{H}_{\frac{1}{2}})$) is isolated and this is an
obstruction for the Riesz operator ${\mathcal{R}}_{t}$ to be positive in a
neighborhood of $1/2$. We suspect the existence of $\varepsilon>0$ depending
on $(X,d)$ such that $\mathcal{R}_{t}$ is positive on
$]0,\frac{1}{2}-\varepsilon]$ generalizing the positivity of
${\mathcal{R}}_{t}$ obtained for uniform lattices of $Sp(n,1)$ in [21] (see
Theorem 3.4.3 in [21] and also [39]).
In the second case, amongst several definitions, Haagerup property is
characterized by the existence of a proper and conditionally negative function
on $\Gamma$, that is a map $\psi\,:\,\Gamma\to\mathbb{R}_{+}$ satisfying for
all $n\geq 1$, for all $\gamma_{1},\cdots,\gamma_{n}\in\Gamma$ and for all
real numbers $\lambda_{1},\cdots,\lambda_{n}$ such that $\sum\lambda_{i}=0$,

$\sum_{i,j}\lambda_{i}\lambda_{j}\psi(\gamma_{j}^{-1}\gamma_{i})\leq 0.$

An example of such function is given by $\psi(\gamma)=d(x,\gamma x)$ when
$\Gamma$ acts by isometries on an hyperbolic space $(X,d)$, with the
additional hypothesis that $d$ is of conditionally negative type, that is
$\sum_{i,j}\lambda_{i}\lambda_{j}d(x_{i},x_{j})\leq 0$ for all
$x_{1},\cdots,x_{n}\in X$ and $\sum\lambda_{i}=0$. Then, the argument
appearing in [7] for CAT(-1) spaces can be extended to the hyperbolic
framework to deduce the positivity of ${\mathcal{R}}_{t}$ for all
$t\in]0,{1\over 2}]$. Namely :

######  Proposition 1.10.

If $\Gamma$ acts on a proper $\delta$-hyperbolic, roughly geodesic and
$\epsilon$-good space $(X,d)$ with compact quotient and $d$ as above, then
$\mathcal{R}_{t}$ is positive for all $0<t\leq\frac{1}{2}$.

We will see (Proposition 5.4) that if $\Gamma$ acts with compact quotient on
$(X,d)$ roughly geodesic, $\delta$-hyperbolic and $\epsilon$-good, positivity
of $({\mathcal{R}}_{t})_{t\in]0,{1\over 2}]}$ implies Haagerup property for
$\Gamma$. Conversely, it would be interesting to know if Haagerup property for
$\Gamma$ implies the existence of a $\Gamma$-hyperbolic space $(X,d)$ such
that the corresponding Riesz operators are positive.

###  1.8. Structure of the paper

In Section 2, we introduce all the geometrical concepts concerning the
hyperbolic spaces and groups. The notions of strongly hyperbolic space and
$\epsilon$-good spaces are discussed. We also recall equidistribution results
in the spirit of Margulis-Roblin. Then all the basics of Patterson-Sullivan
measure theory are developed. We start Section 3 by studying the properties of
the function $\sigma_{t}$ introduced in (1.10), we prove elementary properties
of the Riesz operator as well as for the intertwiner $\mathcal{I}_{t}$, such
as its compacity. We discuss the existence of a left invertible operator of
the intertwiner. In Section 4, we define and study the Hilbert spaces on which
the group acts. In particular we study their pairing structure. In Section 5,
we study and define the notion of spherical functions and make connections
with positive definite functions. We also study the positivity of
$\mathcal{I}_{t}$ assuming the metric on the space $X$ defines a negative
definite kernel. In Section 6 we prove spectral inequalities for $\pi_{t}$ and
we give several consequences of those inequalities as mixing properties of the
boundary representations. We also introduce the generalized Poisson transform
and the Riesz transform. Those operators are fundamental to establish all
convergence theorems. Section 7 is devoted to the proofs. Last, we add an
appendix providing a way of unitarization based on $\ell^{2}$-cohomology of
discrete groups.

### Acknowledgement

We are grateful to Nigel Higson to have strengthened our interest in
intertwining operator and would also like to thank Jean-Martin Paoli for
having suggested to study the square of the intertwining operator. We wish to
thank Vincent Millot for useful discussion and for the reference [50].
Eventually, we would like to thank Antoine Pinochet-Lobos for helpful comments
on a first version of this manuscript. Finally, we are grateful to Marc
Bourdon for discussions on Besov spaces and $\ell^{2}$-Betti numbers.
The first author thanks warmly Christophe Pittet for his support. The second
author is grateful to Institut für Mathematik Universität Zürich - specially
to Viktor Schröder and also for the stimulating ambiance in Dynamical Systems
seminar - for their kind hospitality during the past two years when this work
was partially completed.
The authors thank the MFO for RIP program providing excellent research
environment where this work has been initiating.

##  2\. Preliminaries

In the sequel we adopt a universal notation $C$ for constants independent of
other variables than the global geometry.

###  2.1. Hyperbolic metric spaces and boundaries

(A good reference for this section is [16]).
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

(2.1) $(x,y)_{o}\geq\min\\{(x,z)_{o},(z,y)_{o}\\}-\delta,$

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

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

We will call indifferently $(X,d)$ a $\delta$-hyperbolic space when the
constant $\delta$ needs to be precised and we will only consider in the sequel
_proper_ hyperbolic metric spaces (metric spaces for which closed balls are
compact).
A map $\phi:(X,d_{X})\to(Y,d_{Y})$ between metric spaces is a _quasi-isometry_
if there exist positive constants $L,\lambda>0$ so that

(2.3) $\frac{1}{L}d_{X}(x,y)-\lambda\leq d_{Y}(\phi(x),\phi(y))\leq
Ld_{X}(x,y)+\lambda.$

Within the class of _geodesic metric spaces_ , hyperbolicity is invariant
under quasi-isometries, which is not the case for arbitrary metric spaces.

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 (independant on the base point) is denoted by $\partial X$. 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

(2.4) $(\xi,\eta)_{o}:=\sup\liminf_{i,j}(a_{i},b_{j})_{o}$

(where the supremum is taken over au 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
(it is obviously continuous on $X\times X$ as the distance function to a point
is).

######  Proposition 2.1.

[16, 3.17 Remarks, p. 433].
Let $(X,d)$ be an hyperbolic metric space and fix a base point $o$ in $X$.

  1. (1)

For all $\xi,\eta$ in $\overline{X}$ there exist sequences $(a_{n})$ and
$(b_{n})$ such that $\xi=\lim a_{n}$, $\eta=\lim b_{n}$ and
$(\xi,\eta)_{o}=\lim_{n\rightarrow+\infty}(a_{n},b_{n})_{o}$.

  2. (2)

For all $\xi,\eta$ and $u$ in $\overline{X}$ by taking limits we still have

$(\xi,\eta)_{o}\geq\min{\\{(\xi,u)_{o},(u,\eta)_{o}\\}}-2\delta.$

  3. (3)

For all $\xi,\eta$ in $\partial X$ and all sequences $(a_{i})$ and $(b_{j})$
in X with $\xi=\lim a_{i}$ and $\eta=\lim b_{j}$, we have:

$(\xi,\eta)_{o}-2\delta\leq\liminf_{i,j}(a_{i},b_{j})_{o}\leq(\xi,\eta)_{o}.$

We refer to [32, 8.- Remarque, Chapitre 7, p. 122] for a proof of the
statement (3). The boundary $\partial X$ carries a family of _visual metrics_
, depending on $d$ and a real parameter $\epsilon>0$ denoted from now by
$d_{o,\epsilon}$. For $1<\epsilon\leq\frac{\log(2)}{4\delta}$ then
$d_{o,\epsilon}$ is a metric, relative to a base point o on $\partial X$
satisfying

(2.5) $e^{-\epsilon(\xi,\eta)_{o}-\epsilon c_{v}}\leq
d_{o,\epsilon}(\xi,\eta)\leq e^{-\epsilon(\xi,\eta)_{o}},$

for some $c_{v}>0$ such that $e^{-\epsilon c_{v}}=(3-2e^{\delta\epsilon}).$

We refer to [16, Chapter III-H pp. 435-437] for those visual metrics.

###  2.2. Hyperbolic groups

####  2.2.1. Definition

A group $\Gamma$ acts properly discontinuously on a proper metric space if for
every compact sets $K,L\subset X$, the set $\\{\gamma\in\Gamma\;;\;\gamma
K\cap L\neq\emptyset\\}$ is finite. The action is said to be geometric if
furthermore $\Gamma$ acts by isometries and $X/\Gamma$ is compact. A group
$\Gamma$ is said to be hyperbolic if it acts geometrically on a proper
hyperbolic space. 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 so called metric 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 geometrically is quasi-isometric to a (any of the) Cayley
graph(s) of a hyperbolic group. All constructions below will depend a priori
on hyperbolic metrics spaces $(X,d)$ on which $\Gamma$ acts geometrically and
we will consider as well non geodesic hyperbolic spaces.

####  2.2.2. Notation

As far as the geometry of $(X,d)$ or that of $\Gamma$ for its action as above
is concerned, we will fix the following notations :
Endow $\Gamma$ with the length function
$|\cdot|:\Gamma\rightarrow\mathbb{R}^{+}$ defined as $|\gamma|=d(o,\gamma o)$
for a basepoint $o$ and for any $\gamma\in\Gamma.$
The open (resp. closed) balls in $X$ are denoted by $B_{X}(x,R)$ (resp.
$\bar{B}_{X}(x,R))$ and define the spheres of $X$ associated with
$n\in\mathbb{N}$ and fixed $R>0$ as $S_{n,R}(o)=B_{X}(x,(n+1)R)\setminus
B_{X}(x,nR)$, and the corresponding spheres of $\Gamma$ as
$S^{\Gamma}_{n,R}(o):=\Gamma.o\cap S_{n,R}(o)$. In the following, we will use
the notation $S_{n,R}$ and $S^{\Gamma}_{n,R}$ rather than $S_{n,R}(o)$ and
$S^{\Gamma}_{n,R}(o)$, after having picked a base point $o$.

###  2.3. Roughly geodesic spaces, good and strong hyperbolic spaces and
Gromov boundary

The classical theory of $\delta$-hyperbolic spaces works under the assumption
that the spaces are geodesic. In general, the Gromov product associated with a
word metric on a Cayley graph of a Gromov hyperbolic group does not extend
continuously to the bordification. In the following we will need metrics $d$
on $\Gamma$ such that the Gromov product extends continuously to the
bordification : the price to pay is that $(\Gamma,d)$ won’t be geodesic but
only _roughly geodesic_ :

######  Definition 2.2.

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 $|t-s|-C_{X}\leq d(r(t),r(s))\leq|t-s|+C_{X}$ 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. If $\Gamma$
is a hyperbolic group endowed with a left invariant hyperbolic 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
[31, Section 3.1]. Hence a hyperbolic group acts geometrically on a proper,
roughly geodesic $\delta$-hyperbolic metric space.

######  Definition 2.3.

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

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 $B(\xi,r)$.

In [48] the authors introduce the notion of strong hyperbolicity that we
recall here:

######  Definition 2.4.

A metric space $(X,d)$ is _$\epsilon$ -strongly hyperbolic_ if for all
$x,y,z,o$ we have

$e^{-\epsilon(x,y)_{o}}\leq e^{-\epsilon(x,z)_{o}}+e^{-\epsilon(z,y)_{o}}.$

Then, the authors prove the following

######  Theorem 2.5.

(Nica-Špakula) An $\epsilon$-strongly hyperbolic space is a $\epsilon$-good,
$\frac{\log(2)}{\epsilon}$-hyperbolic space.

To sum up, we state the following theorem, that is a combination of results
due to Blachère, Haïssinsky and Matthieu [6] and of Nica and Špakula [48].

######  Theorem 2.6.

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

As indicated in the introduction, we will consider in the following geometric
actions of $\Gamma$ (hyperbolic) on roughly geodesic, $\epsilon$-good
hyperbolic spaces $(X,d)$. Two examples of such spaces are given 1) by the
group $\Gamma$ itself endowed with the Mineyev metric [44] and 2) again by
$\Gamma$ endowed with Green metrics $d_{\mu}$ associated with symmetric
finitely supported random walks $\mu$. Roughly geodesic property of the second
(family of) example is proved by Blachère, Haïssinsky and Matthieu in [6] and
Nica and Špakula in [48] prove that $(\Gamma,d_{\mu})$ is a
$\epsilon$-strongly hyperbolic space.

From now on, except if indicated, we consider $(X,d)$ a proper roughly
geodesic $\epsilon$-good $\delta$-hyperbolic space.

###  2.4. Busemann functions

In the context of $\epsilon$-good spaces, the Busemann function is defined as
:

(2.6) $\beta_{\cdot}(\cdot,\cdot):(\xi,x,y)\in\partial X\times X\times
X\mapsto\beta_{\xi}(x,y)=\lim_{z\to\xi}d(x,z)-d(z,y)\in\mathbb{R}.$

Note that:

(2.7) $\beta_{\xi}(x,y)=2(\xi,y)_{x}-d(x,y).$

Proposition 2.1 (3) implies that we have for all $\xi\in\partial X$ and for
all $x,y\in X$:

(2.8) $(\xi,y)_{x}\leq d(x,y),$

and thus

(2.9) $\beta_{\xi}(x,y)\leq d(x,y).$

The visual metrics $(d_{x,\epsilon})_{x\in X}$ of parameter $\epsilon>0$ on
the boundary satisfy the following conformal relation, that is for all $x,y\in
X$ and for all $\xi,\eta\in\partial X$ we have:

(2.10)
$d_{y,\epsilon}(\xi,\eta)=\exp\big{(}\frac{\epsilon}{2}(\beta_{\xi}(x,y)+\beta_{\eta}(x,y))\big{)}d_{x,\epsilon}(\xi,\eta).$

###  2.5. The Patterson-Sullivan measure

Recall that the volume growth of $\Gamma$-orbits on $(X,d)$ is controlled as

(2.11) $C^{-1}e^{QR}\leq|\Gamma.o\cap B(o,R)|\leq Ce^{QR},$

for some constant $C$ independent of $R$.
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 $\sharp\Lambda_{\Gamma}>2$ (and in
this case, $\sharp\Lambda_{\Gamma}=\infty$). In the following $\Gamma$ acts
geometrically so that $\Lambda_{\Gamma}=\partial X$.

The Patterson-Sullivan measure $\nu_{o}$ is quasi-invariant under the action
of $\Gamma$ on $\partial X$ and since we are working in the setting of
hyperbolic groups actings on $\epsilon$-good $\delta$-hyperbolic space we have
for $\nu_{o}$-almost every $\xi\in\partial X$:

(2.12) $\frac{d\gamma_{*}\nu_{o}}{d\nu_{o}}(\xi)=e^{\large
Q\beta_{\xi}(o,\gamma o)}.$

It turns out that the support of $\nu_{o}$ is in $\partial X$ and moreover
$\nu_{o}$ is Ahlfors regular of dimension $D$, which means that we have the
following estimate for the volumes of balls: there exists $C_{\nu}>0$ so that
for all $\xi\in\Lambda_{\Gamma}$ for all $r\leq Diam(\partial X)$:

(2.13) $C_{\nu}^{-1}r^{D}\leq\nu_{o}(B(r,\xi))\leq C_{\nu}r^{D}.$

Finally, the Patterson-Sullivan measure is ergodic for the action of $\Gamma$,
and thus unique up to a constant. The foundations of Patterson-Sullivan
measures theory are in the important papers [49] and [51] (see also [7],[18]
and [45] for more general results in the context of CAT(-1) spaces). These
measures are also called conformal densities. When considering hyperbolic
spaces (and discrete groups acting on it), analogous constructions give rise
to _quasi-conformal densities_. For a non-elementary hyperbolic group $\Gamma$
acting on a proper hyperbolic space, Coornaert [22, Théorème 8.3] proves the
existence of $\Gamma$-invariant quasi-conformal densities of dimension $Q$.
This construction has been extended to the case of roughly geodesic metric
spaces in [6].

###  2.6. Some properties on the boundary

####  2.6.1. Upper Gromov bounded by above

We say that a space $X$ is _upper Gromov bounded by above_ with respect to
$o$, if there exists a constant $M_{X}>0$ such that for all $x\in X$,

(2.14) $\sup_{\xi\in\partial X}(\xi,x)_{o}\geq d(o,x)-M_{X}.$

This definition appears in the work of Connell and Muchnik in [23] as well as
in the work of Garncarek on boundary unitary representations [31]. It allows
to choose a point in the boundary playing the role of the forward endpoint of
a geodesic starting at $o$ passing through $x$ like in the context of simply
connected Riemannian manifold of negative curvature.
In particular, every roughly geodesic metric space is upper Gromov bounded by
above.

Whenever $x\in X$ is given, we denote by $\hat{x}_{o}$ a point in the boundary
satisfying

(2.15) $(\hat{x}_{o},x)_{o}\geq d(o,x)-M_{X}.$

If $\gamma$ is an element of $\Gamma$, we denote by $\widehat{\gamma}_{o}$ the
point in the boundary defined as $\widehat{\gamma o}_{o}$ and
$\check{\gamma}_{o}$ the point $\widehat{\gamma^{-1}o}_{o}$.

####  2.6.2. Partitions of the boundary

Let $(X,d)$ be a roughly geodesic, $\epsilon$-good, $\delta$-hyperbolic space
and pick $o\in X$ a base point. We introduce a partition of the boundary as in
[10, Section 3.2]:

For $n\geq 2$, for $R>0$, for all $y\in S_{n,R}\subset X$ and for all
$k\in\\{0,\dots,n\\}$, define the Borel sets $A_{k,R}(o,y)$ of $\partial X$ as
:

(2.16) $A_{0,R}(o,y):=\\{\xi\in\partial
X|(\xi,y)_{o}<R\\},\;\;\;A_{n,R}(o,y):=\\{\xi\in\partial
X|nR\leq(\xi,y)_{o}\\}$

and for $k\in\\{1,\dots,n-1\\}:$

(2.17) $A_{k,R}(o,y):=\\{\xi\in\partial X|kR\leq(\xi,y)_{o}<(k+1)R\\}.$

In the following, we set $A_{k,R}(y)=A_{k,R}(o,y)$ by $A_{k,R}(y)$. For all
$y\in X$, the sets $(A_{k,R}(y))_{k=0,\dots,n}$ provide a partition of
$\partial X$.

If $y=\eta\in\partial X$, a similar partition
$\cup_{k\in\mathbb{N}}A_{k,R}(\eta))$ of $\partial X\setminus\\{\eta\\}$ is
defined where :

(2.18) $A_{0,R}(\eta):=\\{\xi\in\partial X|(\xi,\eta)_{o}<R\\}$

and for $k\in\mathbb{N}^{*}:$

(2.19) $A_{k,R}(\eta):=\\{\xi\in\partial X|kR\leq(\xi,\eta)_{o}<(k+1)R\\}.$

######  Lemma 2.7.

There exist $R>0$ and $C>0$ large enough such that for all $y$ in
$\overline{X}$ for all $k\in\mathbb{N}$:

$C^{-1}e^{-QkR}\leq\nu_{o}(A_{k,R}(y))\leq C^{-1}e^{-QkR}.$

We refer to [10, Lemma 3.5] for a proof of the above fact.

###  2.7. Equidistribution à la Roblin-Margulis

We refer to [15, Theorem 3.2] for a proof of the following theorem inspired by
Roblin’s Theorem [45, Théorème 4.1.1 ]. The unit Dirac mass centered at $x\in
X$ is denoted by $D_{x}$.

######  Theorem 2.8.

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

  1. (1)

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

$\beta_{n,R}(\gamma)\leq C/|S^{\Gamma}_{n,R}|.$

  2. (2)

We have the following convergence:

$\sum_{\gamma\in S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)D_{\gamma o}\otimes
D_{\gamma^{-1}o}\rightharpoonup\nu_{o}\otimes\nu_{o},$

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

######  Remark 2.9.

We deduce immediately from Theorem 2.8 the following equidistribution theorem:

$\sum_{\gamma\in S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)D_{\gamma
o}\rightharpoonup\nu_{o},$

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

We also need the “boundary version” of Theorem 2.8 (see [15, Lemma 3.1]):

######  Theorem 2.10.

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

  1. (1)

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

$\beta_{n,R}(\gamma)\leq C/|S^{\Gamma}_{n,R}|.$

  2. (2)

We have the following convergence :

$\sum_{\gamma\in
S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)D_{\hat{\gamma}_{o}}\otimes
D_{\check{\gamma}_{o}}\rightharpoonup\nu_{o}\otimes\nu_{o},$

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

##  3\.  The Riesz operators on the boundary of hyperbolic groups

###  3.1. The kernel $k_{t}$, the function $\sigma_{t}$.

To define the Riesz operators, we need to introduce the following kernel on
the boundary related to the visual metric. For any $t\in\mathbb{R}$, we
consider the symmetric positive kernel :

(3.1) $k_{t}:(x,y)\in\overline{X}\times\overline{X}\mapsto
e^{(1-2t)Q(x,y)_{o}}\in\mathbb{R}^{+}.$

The previous expression reads :

(3.2) $k_{t}(\xi,\eta)=\frac{1}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}.$

for $(\xi,\eta)\in\partial X\times\partial X\setminus Diag.$

For $A>0$, consider the open subset :

$U^{A}:={\\{(x,y)\in\overline{X}\times\overline{X}|(x,y)_{o}<A\\}},$

and the compact subset :

$U_{A}:={\\{(x,y)\in\overline{X}\times\overline{X}|(x,y)_{o}\geq A\\}}.$

Then define the truncatures of $k_{t}$ as :

(3.3)
$k^{A}_{t}:(x,y)\in\overline{X}\times\overline{X}\mapsto\textbf{1}_{U^{A}}(x,y)e^{(1-2t)Q(x,y)_{o}}\in\mathbb{R}^{+},$

and

(3.4)
$k_{t,A}:(x,y)\in\overline{X}\times\overline{X}\mapsto\textbf{1}_{U_{A}}(x,y)e^{(1-2t)Q(x,y)_{o}}\in\mathbb{R}^{+}.$

Define $\sigma_{t}$ on the whole space $\overline{X}$ as the following
positive measurable function :

(3.5) $\sigma_{t}:x\in\overline{X}\mapsto\int_{\partial
X}k_{t}(\xi,x)d\nu_{o}(\xi)\in\mathbb{R}^{+}\cup\\{+\infty\\}.$

In general $k_{t}$ is a singular kernel. Nevertheless, we will see that
$\sigma_{t_{|_{X}}}$ is continuous on $X$ for any $t\in\mathbb{R}$. If $t>0$,
we shall see in the following that $\sigma_{t}$ extends to a continuous
function on $\partial X$ and thus extend to a continuous function on the whole
space $\overline{X}$. To do so we shall also consider, for all $A>0$, both
following truncated functions :

(3.6) $\sigma^{A}_{t}:y\in\overline{X}\mapsto\int_{\partial
X}k^{A}_{t}(\xi,y)d\nu_{o}(\xi)=\int_{(\xi,y)<{A}}e^{(1-2t)Q(\xi,y)_{o}}d\nu_{o}(\xi)\in\mathbb{R}^{+},$

and

(3.7) $\sigma_{t,A}:y\in\overline{X}\mapsto\int_{\partial
X}k_{t,A}(\xi,y)d\nu_{o}(\xi)=\int_{(\xi,y)_{o}\geq
A}e^{(1-2t)Q(\xi,y)}d\nu_{o}(\xi)\in\mathbb{R}^{+}.$

First, we start by an elementary lemma.

######  Lemma 3.1.

Let $t\in\mathbb{R}$. For all $A>0$, the function $\sigma^{A}_{t}$ is
continuous on $\overline{X}$.

###### Proof.

Let $A>0$. For all $y\in\overline{X}$, the map $\xi\in\partial X\mapsto
k^{A}_{t}(\xi,y)$ is measurable and integrable: for all $(\xi,y)\in\partial
X\times\overline{X}$ we have $|k^{A}_{t}(\xi,y)|\leq e^{(1-2t)QA}.$ Since we
are working in the class of $\epsilon$-good spaces, the map
$y\in\overline{X}\mapsto k^{A}_{t}(\xi,y)$ is continuous for all
$\xi\in\partial X$. Hence for any $A>0$, by the Lebesgue continuity Theorem,
the function $y\in\overline{X}\mapsto\sigma^{A}_{t}(y)\in\mathbb{R}^{+}$ is
continuous on $\overline{X}$. ∎

######  Lemma 3.2.

Let $t>0$. The function $\sigma_{t_{|_{\partial X}}}$ is continuous on
$\partial X$.

###### Proof.

Let $A_{N}=NR$ with $N\in\mathbb{N}^{*}$ and $R$ large enough so that Lemma
2.7 is valid, namely there exists $C>0$ such that for all $\eta\in\partial X$
and for all $k\in\mathbb{N}$: $C^{-1}e^{-kQR}\leq\nu_{o}(A_{k,R}(\eta))\leq
Ce^{-kQR}$. Consider the sequence $(\sigma^{A_{N}}_{t})_{N\in\mathbb{N}}$ in
$C(\partial X)$. Use the partition of $\partial X$ defined above to write for
all $\eta\in\partial X$ and for $M>N:$

$\displaystyle\sigma^{A_{M}}_{t}(\eta)-\sigma^{A_{N}}_{t}(\eta)$
$\displaystyle=\int_{NR<(\xi,\eta)_{o}\leq
MR}e^{(1-2t)Q(\xi,\eta)_{o}}d\nu_{o}(\xi)$
$\displaystyle=\sum_{k=N}^{M-1}\int_{\xi\in
A_{k,R}(\eta)}e^{(1-2t)Q(\xi,\eta)_{o}}d\nu_{o}(\xi)$ $\displaystyle(\mbox{
For all $t>0$ })$ $\displaystyle\leq
C\sum^{M-1}_{k=N}\nu_{o}(A_{k,R}(\eta))e^{(1-2t)QkR}$ $\displaystyle\leq
C\sum^{M-1}_{k=N}e^{-2tQkR}$
$\displaystyle=Ce^{-2tQNR}\bigg{(}\dfrac{1-e^{-2tQ(M-N)R}}{1-e^{-2tQR}}\bigg{)}.$

Thus, $\|\sigma^{A_{M}}_{t}-\sigma^{A_{N}}_{t}\|_{\infty}\leq
Ce^{-2tQNR}\bigg{(}\dfrac{1-e^{-2tQ(M-N)R}}{1-e^{-2tQR}}\bigg{)}.$ Therefore,
the sequence $(\sigma^{A_{N}}_{t})_{N\in\mathbb{N}}$ is a Cauchy sequence in
$(C(\partial X),\|\cdot\|_{\infty})$.

Now, observe that for all $\eta\in\partial X$, the map $\xi\mapsto
k_{t}(\xi,\eta)$ is measurable and is in $L^{1}(\partial X,\nu_{o})$ for $t>0$
by [47, Lemma 5.3]. We give a proof of the above fact for the convenience of
the reader. We have for all $\eta\in\partial X$:

$\displaystyle\sigma_{t}(\eta)$ $\displaystyle=\int_{\partial
X}\frac{1}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)$
$\displaystyle=\sum^{+\infty}_{k=0}\int_{A_{k,R}(\eta)}e^{(1-2t)Q(\xi,\eta)_{o}}d\nu_{o}(\xi)$
$\displaystyle(\mbox{ For all $t>0$ })$ $\displaystyle\leq
C\sum^{+\infty}_{k=0}\nu_{o}(A_{k,R}(\eta))e^{(1-2t)QkR}$ $\displaystyle\leq
C\sum^{+\infty}_{k=0}e^{-2tQkR}$ $\displaystyle=C\dfrac{1}{1-e^{-2tQR}}.$

Hence, the map $\sigma_{t_{|_{\partial X}}}$ is in $L^{\infty}(\partial
X,[\nu_{o}])$.

This function is actually continuous : following the above computation, we
have

(3.8) $\displaystyle\|\sigma_{A_{N}}-\sigma_{t}\|_{\infty}\leq
Ce^{-2tQNR}\bigg{(}\dfrac{1}{1-e^{-2tQR}}\bigg{)}\to 0,$

as $N\to+\infty$. Since $(C(\partial X),\|\cdot\|_{\infty})$ is complete, we
deduce that $\sigma_{t_{|_{\partial X}}}$ is continuous on $\partial X$. ∎

######  Lemma 3.3.

Let $R>0$ large enough. There exists $C>0$, such that for all $t>0$, for all
$\eta\in\partial X$ and for all $A>0$, we have:

$\limsup_{y\to\eta}\sigma_{t,A}(y)\leq C\frac{e^{-2tQA}}{1-e^{-2tQR}}.$

###### Proof.

The proof is similar to that of the previous lemma : Set $A=NR$ with $R$ large
enough so that Lemma 2.7 holds. Consider the measurable function
$y\in\overline{X}\mapsto\sigma_{t,A}(y)\in\mathbb{R}^{+}$. We have for all
$y\in S_{M,R}$ and for $M>N:$

$\displaystyle\sigma_{t,A}(y)$
$\displaystyle=\int_{NR<(\xi,y)_{o}}e^{(1-2t)Q(\xi,y)_{o}}d\nu_{o}(\xi)$
$\displaystyle=\sum_{k=N}^{M}\int_{\xi\in
A_{k,R}(y)}e^{(1-2t)Q(\xi,y)_{o}}d\nu_{o}(\xi)$ $\displaystyle(\mbox{ For all
$t>0$ })$ $\displaystyle\leq C\sum^{M}_{k=N}\nu_{o}(A_{k,R}(y))e^{(1-2t)QkR}$
$\displaystyle\leq C\sum^{M}_{k=N}e^{-2tQkR}$
$\displaystyle=Ce^{-2tQNR}\bigg{(}\dfrac{1-e^{-2tQ(M-N)R}}{1-e^{-2tQR}}\bigg{)}.$

Therefore :

$\limsup_{y\to\eta}\sigma_{t,A}(y)\leq
C\bigg{(}\dfrac{e^{-2tQNR}}{1-e^{-2tQR}}\bigg{)},$

and the conclusion follows. ∎

######  Proposition 3.4.

The function $\sigma_{t}$ is continuous on $\overline{X}$, and moreover there
exists $R>0$ and $C>0$ such that for all $t>0$ and for all $\eta\in\partial X$
:

$C^{-1}\dfrac{1}{1-e^{-2tQR}}\leq\sigma_{t}(\eta)\leq
C\dfrac{1}{1-e^{-2tQR}}\cdot$

###### Proof.

Lemma 3.2 ensures that $\sigma_{t_{|_{\partial X}}}$ is continuous for $t>0$.
We also know that $\sigma_{t_{|_{X}}}$ is continuous. To prove that
$\sigma_{t}$ extends continuously to the boundary, we only have to prove that
whenever $y\to\eta\in\partial X$ for all $\eta\in\partial X$, we have
$\sigma_{t}(y)\to\sigma_{t}(\eta)$. Write for all $N\in\mathbb{N}:$

$\displaystyle|\sigma_{t}(y)-\sigma_{t}(\eta)|$
$\displaystyle\leq|\sigma_{t}(y)-\sigma^{A_{N}}_{t}(y)|+|\sigma^{A_{N}}_{t}(y)-\sigma^{A_{N}}_{t}(\eta)|+|\sigma^{A_{N}}_{t}(\eta)-\sigma_{t}(\eta)|$
$\displaystyle\leq|\sigma_{t,A_{N}}(y)|+|\sigma^{A_{N}}_{t}(y)-\sigma^{A_{N}}_{t}(\eta)|+\|\sigma^{A_{N}}_{t}-\sigma_{t}\|_{\infty}.$

Lemma 3.3 implies for all $N\in\mathbb{N}:$

$\limsup_{y\to\eta}|\sigma_{t}(y)-\sigma_{t}(\eta)|\leq\limsup_{y\to\eta}|\sigma^{A_{N}}_{t}(y)-\sigma^{A_{N}}_{t}(\eta)|+\|\sigma^{A_{N}}_{t}-\sigma_{t}\|_{\infty}.$

Lemma 3.1 implies for all $N\in\mathbb{N}:$

$\limsup_{y\to\eta}|\sigma_{t}(y)-\sigma_{t}(\eta)|\leq\|\sigma^{A_{N}}_{t}-\sigma_{t}\|_{\infty}.$

Eventually, use (3.8) in Lemma 3.2 to let $N\to\infty$ and to obtain:

$\lim_{y\to\eta}\sigma_{t}(y)=\sigma_{t}(\eta),$

and the proof is done. ∎

It is worth noting that, following the same method of computation used in the
above lemmas, we have :

######  Lemma 3.5.

For any $R$ large enough, there exists $C>0$ such that we have for all
$t\in\mathbb{R}^{*}$, for all $A>0$ and for all $\eta\in\partial X$ :

  1. (1)

$C^{-1}\frac{1-e^{-2tQA}}{1-e^{-2tQ}}\leq\sigma^{A}_{t}(\eta)\leq
C\frac{1-e^{-2tQA}}{1-e^{-2tQ}},$

  2. (2)

$C^{-1}\frac{e^{-2tQA}}{1-e^{-2tQ}}\leq\sigma_{t,A}(\eta)\leq
C\frac{e^{-2tQA}}{1-e^{-2tQ}}.$

######  Remark 3.6.

If $(X,d)$ is a rank one Riemmanian symmetric space of negative constant
curvature the function, $\sigma_{t}$ is $K$-invariant, where $K$ denotes here
the maximal compact subgroup viewed as the stabilizer of $o$ and since the
action of $K$ on the boundary is transitive thus $\sigma_{t}$ is constant.

We state a counting lemma related to $k_{t}$ that already appears in the
context of CAT(-1) space in [7], that is the discrete version of Lemma 3.3.

######  Lemma 3.7.

Let $R>0$ large enough and let $(\beta_{n,R})_{n\in\mathbb{N}}$ be the
sequence of weights provided by Theorem 2.8. There exist a constant $C>0$ and
$R>0$ large enough such that for all $A>0$ and for any $h\in\Gamma$ we have
for all $n$ large enough:

$\limsup_{n\to\infty}\sum_{\\{g\in S_{n,R}|(go,ho)\geq
A\\}}\beta_{n,R}(g)k_{t}(ho,go)\leq C\frac{e^{-2tQA}}{1-e^{-2tQR}}.$

###### Proof.

Set $R>0$ large enough. Fix $h\in\Gamma$. For all $n\in\mathbb{N}$ and for all
$k\in\mathbb{N}^{*}$, define $U_{k,R}(h)\subset S^{\Gamma}_{n,R}$ as :

$U_{k,R}(h):=\\{g\in S^{\Gamma}_{n,R}|(k-1)R<(go,ho)\leq kR\\}.$

Set $A=NR$ with $R$ large enough. By Lemma 4.3 in [31], there exists $C>0$
satisfying for all $k\leq n$ :

(3.9) $|U_{k,R}(h)|\leq Ce^{(n-k)RQ},$

and for $k\geq n$ :

$|U_{k,R}(h)|\leq C.$

Hence, we have for $n$ large enough :

$\displaystyle\sum_{\\{g\in S_{n}|(go,ho)\geq A\\}}\beta_{n,R}(g)k_{t}(ho,go)$
$\displaystyle=\sum^{n}_{k=N}\sum_{g\in
U_{k,R}(h)}\beta_{n,R}(g)e^{(1-2t)Q(go,ho)}$ $\displaystyle\leq
C\sum^{n}_{k=N}\big{(}\beta_{n,R}(g)\cdot|U_{k,R}(h)|\cdot
e^{Q(1-2t)k}\big{)}$ $\displaystyle\leq C\sum^{n}_{k=N}e^{-2tQRk},$

where the last inequality follows from (3.9) together with Item 1 of Theorem
2.8. Thus, we have for all $A=NR>0$:

$\limsup_{n\to+\infty}\sum_{\\{g\in S_{n}|(go,ho)\geq
A\\}}\beta_{n,R}(g)k_{t}(ho,go)\leq C\frac{e^{-2tQA}}{1-e^{-2tQR}},$

and the proof is done. ∎

###  3.2. Definition and elementary properties of the Riesz operator
$\mathcal{R}_{t}$ for $t>0$.

According to [36, Section 3.10], the standard Riesz potentials of a locally
integrable function $v$ on $\mathbb{R}^{n}$ of order $t$ is defined as :

(3.10)
$\mathcal{R}_{t}(v)(x)=\frac{1}{c_{t}}\int_{\mathbb{R}^{n}}\frac{v(y)}{|x-y|^{Q-t}}dy,$

where $Q=n$ and $c_{t}$ is a constant depending on $n$. According to [50,
Chapter 6, Section 2.1, Definition 6.13], the spherical Riesz potential
operator is defined for $t>0$ as :

(3.11)
$\mathcal{R}_{t}(v)(\xi)=\frac{1}{\sigma_{t}}\int_{\mathbb{S}^{n-1}}v(\eta)k_{t}(\xi,\eta)d\nu_{o}(\eta)=\frac{1}{\sigma_{t}}\int_{\partial
X}\dfrac{v(\eta)}{|\xi-\eta|^{Q-t}}d\nu_{o}(\eta),$

where $|\xi-\eta|$ denotes the euclidiean distance between two points
$\xi,\eta\in\mathbb{S}^{n-1}$, where $Q=n-1$ and

$\sigma_{t}=2^{t}\pi^{\frac{n-1}{2}}\frac{\Gamma(t)}{\Gamma(\frac{Q-t}{2})},$

where $\Gamma$ here stands for the $\Gamma$-function.
As we have seen in the introduction, in the context of a general hyperbolic
group acting on the compact metric measure space $(\partial
X,d_{o,\epsilon},\nu_{o})$ of conformal dimension $D$, the spherical Riesz
potential of order $t>0$ is defined as:

(3.12) $\mathcal{R}_{t}(v)(\xi):=\frac{1}{\sigma_{t}(\xi)}\int_{\partial
X}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta).$

In the mean time, we also have considered the operator :

(3.13) $\mathcal{I}_{t}(v)(\xi):=\int_{\partial
X}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta).$

We call this operator the Knapp-Stein operator (associated with a metric
$d_{o,\epsilon}$) in reference to [38] together with Proposition 3.16.

Associated to the function $\sigma_{t}$, we shall consider the multiplication
operator on $L^{2}(\partial X,\nu_{o})$ defined, for $t>0$ as :

(3.14) $M_{\sigma_{t}}:v\in L^{2}(\partial X,\nu_{o})\mapsto\sigma_{t}\cdot
v\in L^{2}(\partial X,\nu_{o}).$

Here are some properties that $M_{\sigma_{t}}$ satisfies:

  * •

$M_{\sigma_{t}}$ is a bounded operator and satisfies
$\|M_{\sigma_{t}}\|_{L^{2}\to L^{2}}=\|\sigma_{t}\|_{\infty},$ where
$\|\sigma_{t}\|_{\infty}$ denotes the essential supremum norm.

  * •

$M_{\sigma_{t}}$ is self-adjoint.

  * •

$M_{\sigma_{t}}$ is an invertible operator (by Proposition 3.4) such that
$M^{-1}_{\sigma_{t}}=M_{\sigma^{-1}_{t}}$.

Note that the relation between $\mathcal{R}_{t}$ and $\mathcal{I}_{t}$ is
given by

(3.15) $\mathcal{R}_{t}=M_{\sigma^{-1}_{t}}\mathcal{I}_{t}.$

The first properties of the Riesz potentials and the intertwiner
$\mathcal{I}_{t}$ for $t>0$ are in the next propositions.

######  Proposition 3.8.

Let $t>0$. The operators $\mathcal{R}_{t}$ and $\mathcal{I}_{t}$ are bounded
operators. More precisely we have :

$\|\mathcal{R}_{t}\|_{L^{2}\to
L^{2}}\leq\|\sigma_{t}\|_{\infty}\|\sigma^{-1}_{t}\|_{\infty}\mbox{ and
}\|\mathcal{I}_{t}\|_{L^{2}\to L^{2}}\leq\|\sigma_{t}\|_{\infty}.$

Moreover $\mathcal{I}_{t}$ is a self adjoint operator.

###### Proof.

This is based on the so-called Schur’s test. By Cauchy-Schwarz we obtain for
almost every $\xi\in\partial X:$

$\displaystyle|\mathcal{I}_{t}(v)(\xi)|$ $\displaystyle\leq\int_{\partial
X}\dfrac{|v(\eta)|}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta)$
$\displaystyle\leq\bigg{(}\int_{\partial
X}\dfrac{1}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\partial
X}\dfrac{|v^{2}(\eta)|}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)}^{\frac{1}{2}}.$

Then, we have:

$\displaystyle\|\mathcal{I}_{t}v\|^{2}_{2}$ $\displaystyle\leq\int_{\partial
X}\sigma_{t}(\xi)\bigg{(}\int_{\partial
X}\dfrac{|v^{2}(\eta)|}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)}d\nu_{o}(\xi)$
Proposition 3.5 $\displaystyle\leq\|\sigma_{t}\|_{\infty}\int_{\partial
X}\bigg{(}\int_{\partial
X}\dfrac{|v^{2}(\eta)|}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta)d\nu_{o}(\xi)\bigg{)}$
by Fubini’s Theorem
$\displaystyle\leq\|\sigma_{t}\|^{2}_{\infty}\bigg{(}\int_{\partial
X}|v^{2}(\eta)|d\nu_{o}(\eta)\bigg{)}$
$\displaystyle=\|\sigma_{t}\|^{2}_{\infty}\|v\|^{2}_{2}.$

Hence, we have $\|\mathcal{I}_{t}v\|_{2}\leq\|\sigma_{t}\|_{\infty}\|v\|_{2}.$
By the relation (3.15), we obtain:

$\displaystyle\|\mathcal{R}_{t}\|_{L^{2}\to L^{2}}$
$\displaystyle\leq\|\mathcal{I}_{t}\|_{L^{2}\to
L^{2}}\|M_{\sigma^{-1}_{t}}\|_{L^{2}\to L^{2}}$
$\displaystyle\leq\|\sigma_{t}\|_{\infty}\|\sigma^{-1}_{t}\|_{\infty}.$

Moreover, since $k_{t}$ is symmetric, Fubini’s Theorem implies that
$\mathcal{I}_{t}$ is self adjoint. ∎

The following statment will also be useful :

######  Lemma 3.9.

Let $t>0$. The Riesz operator $\mathcal{R}_{t}$ is positive if and only if the
Knapp-Stein operator $\mathcal{I}_{t}$ is positive.

###### Proof.

Assume that $\mathcal{R}_{t}$ is positive. Then, the operator
$M_{\sigma^{1/2}_{t}}\mathcal{R}_{t}M_{\sigma^{1/2}_{t}}^{*}=M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{1/2}_{t}}$
is positive and thus
$sp(M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{1/2}_{t}})\subset\mathbb{R}^{+}$,
where $sp(\cdot)$ denotes the spectrum of an operator. Moreover
$sp(M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{1/2}_{t}})=sp(\mathcal{I}_{t}M_{\sigma^{1/2}_{t}}M_{\sigma^{-1/2}_{t}})=sp(\mathcal{I}_{t})$.
Thus $sp(\mathcal{I}_{t})\subset\mathbb{R}^{+}$ and the operator
$\mathcal{I}_{t}$ is positive.
Assume that $\mathcal{I}_{t}$ is positive. Then, the operator
$M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{-1/2}_{t}}^{*}=M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{-1/2}_{t}}$
is positive and thus
$sp(M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{-1/2}_{t}})\subset\mathbb{R}^{+}$.
Then
$sp(M_{\sigma^{-1/2}_{t}}\mathcal{I}_{t}M_{\sigma^{-1/2}_{t}})=sp(M_{\sigma^{-1}_{t}}\mathcal{I}_{t})$.
Since $M_{\sigma^{-1}_{t}}\mathcal{I}_{t}=\mathcal{R}_{t}$, thus the operator
$\mathcal{R}_{t}$ is positive. ∎

###  3.3. Truncature of operators

####  3.3.1. Truncature of $\mathcal{I}_{t}$

Let $t>0$. Define for any $A>0$, the bounded operator denoted by
$\mathcal{I}^{A}_{t}$ acting on a vector $v\in L^{2}(\partial X,\nu_{o})$ as :

$\mathcal{I}^{A}_{t}v(\xi)=\int_{\\{\eta|(\xi,\eta)_{o}<A\\}}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta),$

and define the bounded operator $\mathcal{I}_{A,t}$ acting on a vector $v\in
L^{2}(\partial X,\nu_{o})$ as :

$\mathcal{I}_{A,t}v(\xi)=\int_{\\{\eta|(\xi,\eta)_{o}\geq
A\\}}\frac{v(\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta).$

Observe

$\mathcal{I}_{t}=\mathcal{I}^{A}_{t}+\mathcal{I}_{A,t}.$

######  Lemma 3.10.

For any $R>0$ large enough there exists $C>0$ such that for all $t>0:$

$\|\mathcal{I}^{A}_{t}\|_{L^{2}\to L^{2}}\leq
C\cdot\frac{1-e^{-2tQA}}{1-e^{-2tQR}}$

and

$\|\mathcal{I}_{A,t}\|_{L^{2}\to L^{2}}\leq
C\cdot\frac{e^{-2tQA}}{1-e^{-2tQR}}.$

###### Proof.

Following the proof of Proposition 3.8, we obtain for all $A>0:$

$\|\mathcal{I}^{A}_{t}\|_{L^{2}\to L^{2}}\leq\|\sigma^{A}_{t}\|_{\infty}.$

Use now Lemma 3.5 to estimate $\|\sigma^{A}_{t}\|_{\infty}$: for $R>0$ large
enough there exists $C>0$ such that

$\|\mathcal{I}^{A}_{t}\|_{L^{2}\to L^{2}}\leq
C\cdot\frac{1-e^{-2tQA}}{1-e^{-2tQR}}.$

The proof dealing with $\mathcal{I}_{A,t}$ is exactly the same. ∎

######  Proposition 3.11.

For $t>0$, the operator $\mathcal{I}_{t}$ is compact from $C(\partial X)$ to
$C(\partial X)$ and from $L^{2}(\partial X)$ to $L^{2}(\partial X)$.

###### Proof.

We start by proving that $\mathcal{I}_{t}$ is a compact operator from
$C(\partial X)$ to $C(\partial X)$. Note that $\mathcal{I}_{t}(f)$ defines a
continuous function on the boundary with $f\in C(\partial X)$. Indeed, by the
arguments of Lemma 3.1, the function $\mathcal{I}^{A}_{t}(f)$ is continuous
for all $A>0$. Now, use

$\|\mathcal{I}^{A_{M}}_{t}(f)-\mathcal{I}^{A_{N}}_{t}(f)\|_{\infty}\leq
C\|f\|_{\infty}\|\sigma^{A_{M}}_{t}-\sigma^{A_{N}}_{t}\|_{\infty},$

to ensure that $\mathcal{I}^{A_{N}}_{t}(f)$ is a Cauchy sequence in
$C(\partial X)$. To compute the limit observe that

$\|\mathcal{I}_{t}(f)-\mathcal{I}^{A_{N}}_{t}(f)\|_{\infty}\leq\|f\|_{\infty}\|\sigma^{A}_{t}\|_{\infty}.$

To conclude the proof, use $(1)$ of Lemma 3.5 to obtain
$\mathcal{I}^{A_{N}}_{t}(f)\to\mathcal{I}_{t}(f)$ as $n\to\infty$, whenever
$t>0$.

Secondly, let us prove compacity of this operator : this is a standard
application of Ascoli-Arzela Theorem. Consider the subset

$S:=\\{\mathcal{I}_{t}(f)|f\in C(\partial X)\mbox{ such that
}\|f\|_{\infty}\leq 1\\}\subset C(\partial X).$

Apply Heine Theorem to the function $\sigma_{t}$. Pick $\varepsilon>0$. There
exists $\delta>0$ such that $|\sigma_{t}(\xi)-\sigma_{t}(\eta)|<\varepsilon$,
whenever $d_{o,\epsilon}(\xi,\eta)<\delta$. We have for all $f\in C(\partial
X)$ with $\|f\|_{\infty}\leq 1$ and for all $\xi,\eta\in\partial X$ with
$d_{o,\epsilon}(\xi,\eta)<\delta$ :

$|\mathcal{I}_{t}(f)(\xi)-\mathcal{I}_{t}(f)(\eta)|\leq|\sigma_{t}(\xi)-\sigma_{t}(\eta)|<\varepsilon.$

Hence, $S$ is equicontinuous and the Arzela-Ascoli Theorem implies that
$\mathcal{I}_{t}$ is compact for $t>0$.
Let us prove that $\mathcal{I}_{t}$ is a compact operator from $L^{2}\to
L^{2}$. Let $N>0$ and note that

$\mathcal{I}^{N}_{t}(v)(\xi)=\int_{\partial
X}v(\eta)h_{N}(\xi,\eta)d\nu_{o}(\eta),$

is a Hilbert-Schmidt operator since $h_{N}$ in $L^{2}(\partial X\times\partial
X,\nu_{o}\otimes\nu_{o})$ where :

$h_{N}(\xi,\eta)=\left\\{\begin{array}[]{ll}\frac{1}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}&\mbox{if
$(\xi,\eta)_{o}<N,$ }\\\ 0&\mbox{if not.}\end{array}\right.$

Hence, $\mathcal{I}^{N}_{t}$ is compact for all $N$ and moreover, Lemma 3.10
implies that

$\|\mathcal{I}_{t}-\mathcal{I}^{N}_{t}\|_{L^{2}\to
L^{2}}=\|\mathcal{I}_{N,t}\|_{L^{2}\to L^{2}}\to 0,$

as $N\to+\infty$. So, $\mathcal{I}_{t}$ is compact.

∎

###  3.4. A one parameter family of derivations

We consider a family of derivations on the commutative algebra of continuous
functions on the boundary $C(\partial X)$. The spaces $L^{2}(\partial
X,\nu_{o})$ and $L^{2}(\partial X\times\partial X,\nu_{o}\otimes\nu_{o})$
carry a natural structure of $C(\partial X)$ bi-module. For any
$t\in\mathbb{R}$, define the following derivation operators, a priori
unbounded from $L^{2}(\partial X)$ to $L^{2}(\partial X\times\partial
X,\nu_{o}\otimes\nu_{o})$ by :

(3.16) $\displaystyle\delta_{t}:v\in\mathcal{D}(\delta_{t})\cap C(\partial
X)\subset L^{2}(\partial X)$ $\displaystyle\mapsto\delta_{t}v\in
L^{2}(\partial X\times\partial X,\nu_{o}\otimes\nu_{o}),$

where

$[\delta_{t}v](\xi,\eta):=\frac{1}{d_{o,\epsilon}^{(\frac{1}{2}-t)D}(\xi,\eta)}\big{(}v(\xi)-v(\eta)\big{)}.$

and where the domain of $\delta_{t}$ is:

(3.17) $\mathcal{D}(\delta_{t})=\\{v\in L^{2}(\partial X)|\int_{\partial
X\times\partial
X}\frac{|v(\xi)-v(\eta)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)<+\infty\\}.$

We have for all $v,w\in C(\partial X)$:

(3.18) $\delta_{t}(v\cdot w)=v\cdot\delta_{t}(w)+\delta_{t}(v)\cdot w.$

Not that the inner derivation $\delta_{\frac{1}{2}}$ is nothing but :

(3.19) $\delta_{\frac{1}{2}}:v\in L^{2}(\partial X,\nu_{o})\mapsto
v\otimes\textbf{1}_{\partial X}-\textbf{1}_{\partial X}\otimes v\in
L^{2}(\partial X\times\partial X,\nu_{o}\otimes\nu_{o}),$

also called the “classical Dirichlet Jump”. In this paper, we focus only on
the case $t>0$. Then define for $t>0$

(3.20) $\Delta_{t}:=\frac{\delta_{t}^{*}\delta_{t}}{2}.$

The properties of those operators, related to the Riesz operators are in the
following proposition.

######  Proposition 3.12.

Let $t>0$.

  1. (1)

The operator $\delta_{t}$ is a bounded derivation from $L^{2}(\partial
X,\nu_{o})$ to $L^{2}(\partial X\times\partial X,\nu_{o}\otimes\nu_{o})$. And
$\delta^{*}_{t}:L^{2}(\partial X\times\partial
X,\nu_{o}\otimes\nu_{o})\rightarrow L^{2}(\partial X,\nu_{o})$ satisfies

$\delta^{*}_{t}F(\cdot):=\bigg{(}\int_{\partial
X}\frac{F(\cdot,\eta)-F(\eta,\cdot)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\cdot,\eta)}d\nu_{o}(\eta)\bigg{)}.$

  2. (2)

The operator $\Delta_{t}$ is a bounded and positive operator acting on
$L^{2}(\partial X,\nu_{o})$. Its expression reads as follows. For all $v\in
L^{2}(\partial X,\nu_{o})$, and for all $\xi\in\partial X$ :

$\Delta_{t}(v)(\xi)=\int_{\partial
X}\frac{v(\xi)-v(\eta)}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta).$

  3. (3)

We have:

$\mathcal{I}_{t}=M_{\sigma_{t}}-\Delta_{t}.$

###### Proof.

We prove (1). We have

$\displaystyle\|\delta_{t}v\|^{2}_{L^{2}\otimes L^{2}}$
$\displaystyle=\int_{\partial X\times\partial
X}\frac{|v(\xi)-v(\eta)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)$
$\displaystyle\leq\int_{\partial X\times\partial
X}\frac{|v(\xi)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)+\int_{\partial
X\times\partial
X}\frac{|v(\eta)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)$
$\displaystyle+2\bigg{(}\int_{\partial X\times\partial
X}\frac{|v(\xi)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)\bigg{)}^{\frac{1}{2}}\bigg{(}\int_{\partial
X\times\partial
X}\frac{|v(\eta)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)\bigg{)}^{\frac{1}{2}}$
(by the Fubini Theorem) $\displaystyle=4\int_{\partial
X}|v(\xi)|^{2}\sigma_{t}(\xi)d\nu_{o}(\xi)$ $\displaystyle\leq
4\|\sigma_{t}\|_{\infty}\|v\|^{2}_{2},$

hence $\delta_{t}$ is bounded from $L^{2}(\partial X,\nu_{o})$ to
$L^{2}(\partial X\times\partial X,\nu_{o}\otimes\nu_{o})$. Now observe for all
$v\in L^{2}(\partial X,\nu_{o})$ and $F\in L^{2}(\partial X\times\partial
X,\nu_{o}\otimes\nu_{o})$:

$\displaystyle\langle\delta_{t}v,F\rangle$ $\displaystyle=\int_{\partial
X\times\partial
X}\frac{(v(\xi)-v(\eta))\overline{F}(\xi,\eta)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)$
$\displaystyle=\int_{\partial X\times\partial
X}\frac{v(\xi)\overline{F}(\xi,\eta)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)-\int_{\partial
X\times\partial
X}\frac{v(\eta)\overline{F}(\xi,\eta)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)$
$\displaystyle=\int_{\partial X}v(\xi)\bigg{(}\int_{\partial
X}\frac{\overline{F}(\xi,\eta)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)}d\nu_{o}(\xi)-\int_{\partial
X}v(\eta)\bigg{(}\int_{\partial
X}\frac{\overline{F}(\xi,\eta)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)\bigg{)}d\nu_{o}(\eta)$
$\displaystyle=\int_{\partial X}v(\xi)\bigg{(}\int_{\partial
X}\frac{\overline{F}(\xi,\eta)-\overline{F}(\eta,\xi)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)}d\nu_{o}(\xi)$
$\displaystyle=\langle v,\delta^{*}_{t}F\rangle$

where for almost all $\xi\in\partial X$

$\delta^{*}_{t}F(\xi):=\bigg{(}\int_{\partial
X}\frac{F(\xi,\eta)-F(\eta,\xi)}{d^{(\frac{1}{2}-t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)\bigg{)},$

and (1) is proved.
For (2), write for $v\in L^{2}(\partial X,\nu_{o})$, and for all
$\xi\neq\eta\in\partial X:$

$\delta_{t}(v)(\xi,\eta)=\frac{1}{d_{o,\epsilon}^{(\frac{1}{2}-t)D}(\xi,\eta)}\big{(}v(\xi)-v(\eta)\big{)}.$

Using the expression of $\delta^{*}_{t}$ proved in $(\ref{1'''})$ we obtain
for almost all $\xi\in\partial X$:

$\delta^{*}_{t}(\delta_{t}v)(\xi)=2\int_{\partial
X}\frac{v(\xi)-v(\eta)}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta).$

Hence,

$\Delta_{t}v(\xi)=\int_{\partial
X}\frac{v(\xi)-v(\eta)}{d_{o,\epsilon}^{(1-2t)D}(\xi,\eta)}d\nu_{o}(\eta).$

Concerning (3), write for $v\in L^{2}(\partial X,\nu_{o})$:

$\displaystyle\mathcal{I}_{t}(v)(\xi)$ $\displaystyle=\int_{\partial
X}\frac{v(\eta)}{d^{(1-2t)Q}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)$
$\displaystyle=\int_{\partial
X}\frac{v(\eta)-v(\xi)}{d^{(1-2t)Q}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)+\int_{\partial
X}\frac{v(\xi)}{d^{(1-2t)Q}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\eta)$
$\displaystyle=-\Delta_{t}(v)(\xi)+\sigma_{t}(\xi)v(\xi),$

and the proof is complete. ∎

We close this section by a characterization of the positivity of
$\mathcal{R}_{t}$ is terms of spectral radius of $\Delta_{t}$. Consider

(3.21) $\widetilde{\Delta_{t}}:=M_{\sigma^{-1}_{t}}\Delta_{t}.$

It seems that studying precisely the eigenvalues of ${\mathcal{I}}_{t}$ is out
of reach in the framework of general hyperbolic groups. Nevertheless, we
obtain the following :

######  Proposition 3.13.

We have:

  1. (1)

The eigenspace $\\{v\in L^{2}(\partial X,\nu_{o})|\mathcal{R}_{t}v=v\\}$ is
nothing but the space of constant functions.

  2. (2)

The operator $\mathcal{R}_{t}$ is positive if and only if
$\|\widetilde{\Delta_{t}}\|_{L^{2}\to L^{2}}\leq 1$.

###### Proof.

We prove (1). Indeed pick $v\in L^{2}(\partial X,\nu_{o})$ such that
$\mathcal{R}_{t}v=v$. It follows by (3) of Proposition 3.12 that
$M_{\sigma^{-1}_{t}}\Delta_{t}v=0$. Thus $\Delta_{t}v=0$. Therefore
$\langle\Delta_{t}v,v\rangle=0$, which writes :

$\int_{\partial X\times\partial
X}\frac{|v(\xi)-v(\eta)|^{2}}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)=0.$

Hence, for almost every $(\xi,\eta)\in\partial X\times\partial X$ we have:

$|v(\xi)-v(\eta)|=0.$

Therefore, $v$ has to be constant.

The proof of (2) follows from (3) of Proposition 3.12.
If $\mathcal{R}_{t}$ is positive, then for all $v\in L^{2}(\partial
X,\nu_{o})$,
$\langle\mathcal{R}_{t}v,v\rangle=\|v\|^{2}-\langle\widetilde{\Delta_{t}}v,v\rangle\geq
0.$ Since $\widetilde{\Delta_{t}}$ is a positive operator, we have
$\|\widetilde{\Delta_{t}}\|_{L^{2}\to L^{2}}\leq 1.$
Now, if $\|\widetilde{\Delta_{t}}\|_{L^{2}\to L^{2}}\leq 1$ then for all $v\in
L^{2}(\partial X,\nu_{o})$,

$\langle\mathcal{R}_{t}v,v\rangle=\|v\|^{2}-\langle\widetilde{\Delta_{t}}v,v\rangle\geq\|v\|^{2}(1-\|\widetilde{\Delta_{t}}\|_{L^{2}\to
L^{2}})\geq 0,$

and the proof is done.

∎

######  Lemma 3.14.

The operator $\mathcal{R}_{t}$ goes to $0$ for the weak operator topology as
$t\to 0$.

###### Proof.

Observe that $\sup_{t>0}\|\mathcal{R}_{t}\|_{L^{2}\to L^{2}}<+\infty$. Hence,
the sequence $(\mathcal{R}_{t})_{t>0}$ converges when $t\to 0$ to some bounded
operator for the weak operator topology, up to subsequences. We shall compute
the weak limit $\lim_{t\to 0}\mathcal{R}_{t}$. It is not hard to see for all
$v,w$ Lipschitz functions that $\langle\widetilde{\Delta_{t}}v,w\rangle\to 0$
as $t\to 0$. Indeed, observe that for $R>0$ large enough, Proposition 3.4
provides some $C>0$ so that:

$\displaystyle\langle\widetilde{\Delta_{t}}v,w\rangle$ $\displaystyle\leq
C\|v\|_{Lip}\|w\|_{Lip}\big{(}1-e^{-2tQR}\big{)}\bigg{(}\int_{\partial
X\times\partial
X}\frac{d^{2}_{o,\epsilon}(\xi,\eta)}{d^{(1-2t)D}_{o,\epsilon}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta)\bigg{)}$
$\displaystyle\leq
C\|v\|_{Lip}\|w\|_{Lip}\big{(}1-e^{-2tQR}\big{)}\bigg{(}\frac{1}{1-e^{(-2tQ-2\epsilon)R}}\bigg{)},$

where the second inequality is based on the method of computations of Lemma
3.2. Eventually, let $t\to 0$ to conclude the proof. ∎

###  3.5. Inverse of $\mathcal{I}_{t}$ (and $\mathcal{R}_{t}$)

Finding a formula for the inverse of Riesz potential operators is a broad
subject. We refer to [50, Section 3, Ch. 6].
In this section, we prove a simple abstract formula for the inverse of the
Riesz potential operators on their image. To our knowledge, in the general
setting of general hyperbolic groups, almost nothing is known about the kernel
and the image of those operators. This general lemma is probably well known,
but we were not able to find a proof in the literature.

######  Lemma 3.15.

Let $T$ be a self adjoint compact operator acting on a Hilbert space $H$.
There exists a unbounded closed self adjoint operator $S:\overline{Im(T)}\to$
whose domain is $\mbox{Dom}(S)=Im(T)$ such that $S$ satisfies:

  1. (1)

$ST(u)=u$ for any $u\in H$ and $TS(v)=v$ for any $v\in Im(T)$.

  2. (2)

If $T$ is a positive operator, we can write $S=c\cdot I+D$ where
$D:\overline{Im(T)}\to H$ is an unbounded closed positive self adjoint
operator and $c$ a positive constant.

  3. (3)

$S^{2}T^{2}(u)=u$ for any $u\in H$ and $T^{2}S^{2}(v)=v$ for any $v\in Im(T)$.

###### Proof.

We prove Item (1). Since $T$ is a compact self adjoint operator, consider a
system of orthonormal vectors $(v_{i})$ associated with eigenvalues
$\big{(}\lambda_{i}(T)\big{)}_{i}$ such that
$|\lambda_{1}(T)|\geq|\lambda_{2}(T)|\geq\dots$ and write for all $u$ the
$\|\cdot\|_{H}$-convergence : $Tu=\sum^{+\infty}_{i=1}\lambda_{i}(T)\langle
u,v_{i}\rangle v_{i}$ . For all $n$, define the sequence of bounded operators
$S_{n}$ as :

$S_{n}(\cdot):=\sum_{k=1}^{n}\frac{1}{\lambda_{k}(T)}\langle\cdot,v_{i}\rangle
v_{i}.$

We shall see that for all $w\in Im(T)$, $\lim_{n\to\infty}S_{n}(w)$ exists for
the $\|\cdot\|_{H}$-convergence. Let $w\in Im(T)$. There exists an unique
$u\in H\ominus\ker(T)$ such that $w=T(u)$. Therefore write for all $n:$

$\displaystyle S_{n}(w)$ $\displaystyle=S_{n}T(u)$
$\displaystyle=S_{n}(\sum^{+\infty}_{i=1}\lambda_{i}(T)\langle u,v_{i}\rangle
v_{i})$ $\displaystyle=\sum_{k=1}^{n}\langle u,v_{k}\rangle v_{k}.$

Since $u\in H$, we have the following $\|\cdot\|_{H}$-convergence :
$S_{n}(w)\to u$. Thus, for all $w\in Im(T)$ define

$S(w):=\lim_{n\to+\infty}S_{n}(w),$

and moreover for all $u\in H\ominus\ker(T)$, we have $ST(u)=u$. By definition
of $S$, we have $Im(T)=\mbox{Dom}(S)$ and obviously $\mbox{Dom}(S)$ is dense
in $\overline{Im(T)}.$
Since $\lambda_{i}(T)$ goes to $0$ the operator $S$ is unbounded on $H$.
Furthermore $S$ is closed: let $w_{p}\in Im(T)$ converges strongly to $w$ and
$S(w_{p})$ converges strongly to $z$. Then, there exists a an unique $u_{p}\in
H\ominus\ker(T)$ so that $w_{p}=T(u_{p})$. Thus, $Sw_{p}=u_{p}$. So, $u_{p}$
converges strongly to $z$. Since $T$ is continuous we have $T(u_{p})\to
T(z)=w$, so $w\in Im(T)=\mbox{Dom}(S)$ and $Sw=z$.
We prove now Item (2) : Assume now that $T$ is positive and assume also that
$\|T\|_{H\to H}\leq 1$. Consider $p_{n}$ the orthogonal projection onto the
space $\bigoplus^{n}_{k=1}\ker(T-\lambda_{k}I)$. The definition of $S_{n}$
reads as follows :

$S_{n}(\cdot):=p_{n}(\cdot)+\sum_{k=1}^{n}\bigg{(}\frac{1}{\lambda_{k}(T)}-1\bigg{)}\langle\cdot,v_{k}\rangle
v_{k},$

and since $0\leq\lambda_{k}(T)\leq 1$ it follows that the sequence of
operators

$D_{n}(\cdot)=\sum_{k=1}^{n}\big{(}\frac{1}{\lambda_{k}(T)}-1\big{)}\langle\cdot,v_{k}\rangle
v_{k}$

is a sequence of positive operators. Besides, $p_{n}$ converges strongly to
the identity operator. Since $S$ is defined as the limit of $S_{n}$ on
$Im(T)$, we define for all $w\in Im(T)$,

$D(w):=\lim_{n\to+\infty}D_{n}(w)=\lim_{n\to+\infty}S_{n}(w)-p_{n}(w)=D(w).$

Note that $D$ is a unbounded positive closed operator. By construction, we
have $S=I+D$ on $Im(T)$.
If $\|T\|_{H\to H}$ is not bounded by $1$, perform the above construction with
$\dfrac{T}{\|T\|_{H\to H}}$ and thus $S=c\cdot I+D$ where $D$ is a unbounded
positive closed operator on $Im(T)$ and $c=\|T\|_{H\to H}>0$. And the proof is
done.

The proof of Item (3) is the same as the proof of Item (1). ∎

###  3.6. Riesz potentials as Intertwiners

####  3.6.1. The case of $SL(2,\mathbb{R})$

Following [46, Ch. V, Section IV], in the case of $SL(2,\mathbb{R})$ acting on
the boundary of the Poincaré disk identified to the circle endowed with the
Lebesgue measure $\frac{d\theta}{2\pi}$, the Riesz potentials intertwines
$\pi_{t}$ and $\pi_{-t}$ in the following manner: for $v\in
L^{2}(\partial\mathbb{D},\frac{d\theta}{2\pi}),$ and for almost every
$\omega\in\partial\mathbb{D}$

(3.22)
$\mathcal{I}_{t}(v)(\omega)=\frac{1}{2\pi}\int_{[0,2\pi]}\frac{v(\theta)}{\big{(}1-\cos(\theta-\omega)\big{)}^{\frac{1}{2}-t}}d\theta,$

and then for all $g\in SL(2,\mathbb{R})$

(3.23) $\mathcal{I}_{t}\pi_{t}(g)=\pi_{-t}(g)\mathcal{I}_{t}.$

####  3.6.2. The case of Gromov hyperbolic groups

We recall that the intertwining property of ${\mathcal{I}}_{t}$ relies on
strong hyperbolicity of $(X,d)$.

######  Proposition 3.16.

For any $t>0$, we have for all $\gamma\in\Gamma$,
$\mathcal{I}_{t}\pi_{t}(\gamma)=\pi_{-t}(\gamma)\mathcal{I}_{t}$ on
$L^{2}(\partial X,\nu_{o})$.

###### Proof.

Let $v\in L^{2}(\partial X,\nu_{o})$ and write
$\pi_{t}(\gamma)v=e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o)}v(\gamma^{-1}\xi)$. We have for all real numbers $s>0$ and $t\in\mathbb{R}$
and for almost all $\eta\in\partial X$

$\displaystyle(\mathcal{I}_{s}\pi_{t}(\gamma)v)(\eta)$
$\displaystyle=\int_{\partial X}\frac{e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o)}v(\gamma^{-1}\xi)}{d_{o,\epsilon}^{(1-2s)D}(\xi,\eta)}d\nu_{o}(\xi)$
$\displaystyle=\int_{\partial
X}\frac{v(\gamma^{-1}\xi)}{d_{o,\epsilon}^{(1-2s)D}(\xi,\eta)}e^{(\frac{1}{2}+t)Q\beta_{\gamma^{-1}\xi}(\gamma^{-1}o,o)}d\nu_{o}(\xi)$
$\displaystyle=\int_{\partial
X}\frac{v(\gamma^{-1}\xi)}{d_{o,\epsilon}^{(1-2s)D}(\xi,\eta)}e^{-(\frac{1}{2}+t)Q\beta_{\gamma^{-1}\xi}(o,\gamma^{-1}o)}d\nu_{o}(\xi)$
$\displaystyle=\int_{\partial
X}\frac{v(\xi)}{d_{o,\epsilon}^{(1-2s)D}(\gamma\xi,\eta)}e^{(\frac{1}{2}-t)Q\beta_{\xi}(o,\gamma^{-1}o)}d\nu_{o}(\xi)$
Thanks to (2.10) and (1.3) $\displaystyle=\int_{\partial
X}\frac{v(\xi)e^{(\frac{1}{2}-t)Q\beta_{\xi}(o,\gamma^{-1}o)}}{e^{(\frac{1}{2}-s)Q(\beta_{\xi}(o,\gamma^{-1}o)+\beta_{\gamma^{-1}\eta}(o,\gamma^{-1}o))}d_{o,\epsilon}^{(1-2s)D}(\xi,\gamma^{-1}\eta)}d\nu_{o}(\xi)$
$\displaystyle=\int_{\partial
X}\frac{v(\xi)e^{(s-t)Q\beta_{\xi}(o,\gamma^{-1}o)}}{d_{o,\epsilon}^{(1-2s)D}(\xi,\gamma^{-1}\eta)}d\nu_{o}(\xi)$
$\displaystyle=e^{(\frac{1}{2}-s)Q\beta_{\eta}(o,\gamma o)}\int_{\partial
X}\frac{v(\xi)e^{(s-t)Q\beta_{\xi}(o,\gamma^{-1}o)}}{d_{o,\epsilon}^{(1-2s)D}(\xi,\gamma^{-1}\eta)}d\nu_{o}(\xi)$
$\displaystyle=\pi_{-s}(\gamma)\mathcal{I}_{s}\big{(}M_{e^{(s-t)Q\beta_{\cdot}(o,\gamma^{-1}o)}}v\big{)}(\eta).$

Then the proof is done by taking $s=t$. ∎

##  4\. Boundary Dual system representations and Spherical functions

###  4.1. Dual system representation

We refer the reader to [37, Section 8] for a general discussion about the
notion of pairing. A $\mathbb{C}$-pairing is a pair of vector spaces
$(H_{1},H_{2})$ together with a non-degenerate bilinear form
$\langle\cdot,\cdot\rangle:H_{1}\times H_{2}\rightarrow\mathbb{C}$, that is
$\\{v\in H_{1}|\langle v,w\rangle=0,\forall w\in H_{2}\\}=0$ and $\\{w\in
H_{2}|\langle v,w\rangle=0,\forall v\in H_{1}\\}=0.$ We shall consider the
weak topology on $H_{1}$ generated by the linear forms $t_{w}:v\in
H_{1}\mapsto\langle v,w\rangle\in\mathbb{C}$ for all $w\in H_{2}$, denoted by
$\sigma(H_{1},H_{2})$ and the weak topology on $H_{2}$ denoted by
$\sigma(H_{2},H_{1})$.

We refer to [27, Section 2] for dual systems and we adapt this notion here for
a group rather than for an algebra.
A _dual system representation of $\Gamma$_ (called rather a “linear system
representation” in [27]) is the data of:

  1. (1)

A $\mathbb{C}$-pairing of Banach spaces :
$(H_{1},H_{2},\langle\cdot,\cdot\rangle).$

  2. (2)

Two representations $(\pi_{1},H_{1})$ and $(\pi_{2},H_{2})$ of $\Gamma$ such
that $\langle\pi_{1}(\gamma)v,\pi_{2}(\gamma)w\rangle=\langle v,w\rangle$ for
all $(v,w)\in H_{1}\times H_{2},$ for all $\gamma\in\Gamma$.

######  Remark 4.1.

If $H_{2}=H^{*}_{1}$, the representation $\pi_{2}$ is nothing but the
contragredient representation of $\pi_{1}$ for the natural dual pairing.

######  Example 4.2.

Consider the $\mathbb{C}$-pairing $(L^{2}(\partial
X,\nu_{o}),\overline{L^{2}(\partial X,\nu_{o})})$ with respect to the
$L^{2}$-inner product, where $\overline{L^{2}(\partial X,\nu_{o})}$ is the
conjugate Hilbert space of $L^{2}(\partial X,\nu_{o})$ equipped with the
conjugate action of $\mathbb{C}$ : for $\lambda\in\mathbb{C}$, and for $v\in
L^{2}(\partial X,\nu_{o})$ we have $\lambda\cdot v=\overline{\lambda}v.$ If
$\pi_{t}$ denotes the spherical boundary representation and the conjugate
representation $\overline{\pi_{-t}}$ satisfies:
$\overline{\pi_{-t}}(\gamma)(\lambda\cdot
v)=\lambda\cdot\overline{\pi_{-t}}(\gamma)(v).$ Hence, the representations
$(\pi_{t},L^{2}(\partial X,\nu_{o}))$ and
$(\overline{\pi_{-t}},\overline{L^{2}(\partial X,\nu_{o}))}$ are a dual linear
system of $\Gamma$.

More generally, in the following if $V$ denotes a vector space, then
$\overline{V}$ denotes its conjugate vector space.

We say that a dual system representations $(\pi_{1},H_{1})$ and
$(\pi_{2},H_{2})$ of $\Gamma$ is irreducible if $(\pi_{1},H_{1})$ has no
proper $\sigma(H_{1},H_{2})$-closed subspace stable by $\pi_{1}$,
_equivalently_ $(\pi_{2},H_{2})$ has no proper $\sigma(H_{2},H_{1})$-closed
subspace stable by $\pi_{2}$.

The aim of this section is to construct three dual system representations of
$\Gamma$ for which we will prove irreducibility.

###  4.2. Linear representations

We fix $t>0$. The algebraic representation defined in (1.12) provides another
algebraic representation of $\Gamma$ via $\pi_{-t}$, defined as follows :

(4.1)
$\mathcal{F}_{-t,\sigma_{t}}:=\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial
X}}).$

Note that for $t>0$, we have
$\mathcal{F}_{-t,\sigma_{t}}=\\{\pi_{-t}(f)\sigma_{t},f\in\mathbb{C}[\Gamma]\\}\subset
L^{2}(\partial X,\nu_{o})$. Since $\mathcal{I}_{t}$ intertwines $\pi_{t}$ and
$\pi_{-t}$ on $L^{2}(\partial X,\nu_{o})$, we obtain a representation given by
$(\pi_{-t},\mathcal{F}_{-t,\sigma_{t}})$.

###  4.3. A first Hilbertian dual system representations of $\Gamma$
associated with $\mathcal{I}_{t}$

Associated to the intertwiner $\mathcal{I}_{t}$, the Hilbert space
$L^{2}(\partial X,\nu_{o})$ decomposes as:

(4.2) $L^{2}(\partial
X,\nu_{o})=\ker(\mathcal{I}_{t})\oplus\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{2}}.$

We have no information in our setting about
$\mathcal{V}_{t}:=\ker(\mathcal{I}_{t})$. The closed subspace
$\mathcal{V}_{t}$ of $L^{2}(\partial X,\nu_{o})$ is a $\pi_{t}$
subrepresentation of $\Gamma$ although
$\mathcal{W}_{t}:=\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{2}}$ is a
$\pi_{-t}$ subrepresentation. It turns out that if $\mathcal{I}_{t}$ is not
injective, then neither $(\pi_{t},L^{2}(\partial X,\nu_{o}))$ nor
$(\overline{\pi_{-t}},\overline{L^{2}(\partial X,\nu_{o})})$ is irreducible.
Thus, we shall consider the representations $(\pi_{t},L^{2}(\partial
X,\nu_{o})/\mathcal{V}_{t})$ and
$(\overline{\pi_{-t}},\overline{\mathcal{W}_{t}})$. Notice that is a dual
system representations of $\Gamma$ with respect to the standard $L^{2}$-inner
product.

######  Example 4.3.

The dual space $L^{2}(\partial X,\nu_{o})/\mathcal{V}_{t}$ is nothing but
$\overline{\mathcal{W}_{t}}$. Hence the contragredient representation of
$(\pi_{t},L^{2}(\partial X,\nu_{o})/\mathcal{V}_{t})$ with respect to the
standard $L^{2}$-inner product is
$(\overline{\pi}_{-t},\overline{\mathcal{W}_{t}})$. Note that if
$\mathcal{I}_{t}$ is injective, we recover Example 4.2.

######  Remark 4.4.

The space $\mathcal{F}_{t,\textbf{1}_{\partial X}}$ endowed with
$\langle\cdot,\cdot\rangle$ is a pre-Hilbert space. Consider

(4.3) $\overline{\mathcal{F}_{t,\textbf{1}_{\partial
X}}}^{\|\cdot\|_{2}}=\mathcal{L}_{t},$

and this representation is well defined since there exists $C>0$ such that for
all $\gamma\in\Gamma$, and for all $v\in\mathcal{L}_{t}$, we have:

$\|\pi_{t}(\gamma)v\|_{2}\leq Ce^{Qt|\gamma|}\|v\|_{2}.$

######  Remark 4.5.

In the following, we will make use that $\mathcal{L}_{t}=L^{2}(\partial
X,\nu_{o})$ for $t>0$ and postpone the proof of this equality to Section 6
(Proposition 6.13).

###  4.4. A non unitary Hilbertian analog of complementary series

Since it seems that proving the positivity of $\mathcal{I}_{t}$ is a hard
question (even for some $0<t<\frac{1}{2}$), we rather consider the
sesquilinear form

$\langle\cdot,\cdot\rangle_{\mathcal{I}^{2}_{t}}:(v,w)\in\mathcal{F}_{t,\textbf{1}_{\partial
X}}\times\mathcal{F}_{t,\textbf{1}_{\partial X}}\mapsto\langle
v,\mathcal{I}^{2}_{t}(w)\rangle.$

Turn first $\mathcal{F}_{t,\textbf{1}_{\partial X}}$ into a pre-Hilbert space.
Consider $N_{t}:=\\{v\in\mathcal{F}_{t,\textbf{1}_{\partial
X}}|\mathcal{I}_{t}(v)=0\\}=\mathcal{V}_{t}\cap\mathcal{F}_{t,\textbf{1}_{\partial
X}}$. Observe that one can equip $\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}$ with $\langle\cdot,\cdot\rangle_{\mathcal{I}^{2}_{t}}$. Hence,
$(\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t},\langle\cdot,\cdot\rangle_{\mathcal{I}^{2}_{t}})$ becomes a pre-
Hilbert space. Eventually consider the Hilbert completion of
$\mathcal{F}_{t,\textbf{1}_{\partial X}}/N_{t}$ and define the Hilbert space

(4.4) $\overline{\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}}^{\|\cdot\|_{\mathcal{I}^{2}_{t}}}:=\mathcal{K}_{t},$

######  Lemma 4.6.

The Hilbertian representation $(\pi_{t},\mathcal{K}_{t})$ is well defined.

###### Proof.

First of all, note that $(\pi_{t},\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t})$ is still a linear representation of $\Gamma$ because
$(\pi_{t},N_{t})$ is a linear representation of $\Gamma$. Moreover we have:

$\displaystyle\|\pi_{t}(\gamma)v\|_{\mathcal{I}^{2}_{t}}$
$\displaystyle=\|\mathcal{I}_{t}(\pi_{t}(\gamma)v)\|$
$\displaystyle=\|\pi_{-t}(\gamma)\mathcal{I}_{t}(v)\|$ $\displaystyle\leq
Ce^{tQ|\gamma|}\|\mathcal{I}_{t}(v)\|$
$\displaystyle=Ce^{tQ|\gamma|}\|v\|_{\mathcal{I}^{2}_{t}},$

where the second equality follows from Proposition 3.16 and the above
inequality is a consequence of 6.1.

∎

We go on in this subsection by the following observation.

######  Lemma 4.7.

The left inverse $\mathcal{J}_{t}$ of $\mathcal{I}_{t}$ provided by Lemma 3.15
applied to $\mathcal{I}_{t}$ intertwines $\pi_{-t}$ and $\pi_{t}$ i.e.

$\pi_{t}\mathcal{J}_{t}=\mathcal{J}_{t}\pi_{-t}$

on $\mathcal{H}^{\prime}_{t}.$

###### Proof.

Let $v=\mathcal{I}_{t}(\pi_{t}(h)\textbf{1}_{\partial
X})\in\mathcal{F}_{-t,\sigma_{t}}=\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial
X}})$ with $h\in\mathbb{C}[\Gamma]$. We have

$\displaystyle\mathcal{J}_{t}\pi_{-t}(\gamma)v$
$\displaystyle=\mathcal{J}_{t}\mathcal{I}_{t}(\pi_{t}(\gamma)\pi_{t}(h)\textbf{1}_{\partial
X})$ $\displaystyle=(\pi_{t}(\gamma)\pi_{t}(h)\textbf{1}_{\partial X})$
$\displaystyle=(\pi_{t}(\gamma)\mathcal{J}_{t}\mathcal{I}_{t}\pi_{t}(h)\textbf{1}_{\partial
X})$ $\displaystyle=\pi_{t}(\gamma)\mathcal{J}_{t}v.$

We conclude the proof by using the fact $\mathcal{J}_{t}$ is closed. ∎

Endow now $\mathcal{F}_{t,\sigma_{t}}$ with the sesquilinear form
$\langle\cdot,\cdot\rangle_{\mathcal{J}^{2}_{t}}:=\langle\cdot,\mathcal{J}^{2}_{t}(\cdot)\rangle$
where $\mathcal{J}_{t}$ is the left inverse of $\mathcal{I}_{t}$. Note that
$(\mathcal{F}_{t,\sigma_{t}},\langle\cdot,\cdot\rangle_{\mathcal{J}^{2}_{t}})$
is a pre-Hilbert space.
Then turn $\mathcal{F}_{t,\sigma_{t}}$ into a Hilbert space by considering

$\mathcal{K}^{\prime}_{t}:=\overline{\mathcal{F}_{t,\sigma_{t}}}^{\|\cdot\|_{\mathcal{J}^{2}_{t}}}.$

######  Lemma 4.8.

The representation $(\pi_{-t},\mathcal{K}^{\prime}_{t})$ is well defined.

###### Proof.

We have for all $\gamma\in\Gamma$, for all $v\in\mathcal{K}_{t}$ :

$\displaystyle\|\pi_{-t}(\gamma)v\|_{\mathcal{J}^{2}_{t}}$
$\displaystyle=\|\mathcal{J}_{t}(\pi_{-t}(\gamma)v)\|$
$\displaystyle=\|\pi_{t}(\gamma)\mathcal{J}_{t}(v)\|$ $\displaystyle\leq
Ce^{tQ|\gamma|}\|\mathcal{J}_{t}(v)\|$
$\displaystyle=Ce^{tQ|\gamma|}\|v\|_{\mathcal{J}^{2}_{t}},$

where the second equality follows from Lemma 4.7 and the above inequality is a
consequence of 6.1.

∎

Using the general theory of dual pairings, several properties are obvious and
well known. We sum up the properties of the dual pairs we shall consider, and
we give a proof of these facts.

######  Proposition 4.9.

Let $t>0$.

  1. (1)

The pair
$(\mathcal{K}_{t},\overline{\mathcal{K}^{\prime}_{t}},\langle\cdot,\cdot\rangle)$
is a $\mathbb{C}$-pairing.

  2. (2)

The topological dual of $\mathcal{K}_{t}$ is isomorphic and isometric to
$\overline{\mathcal{K}^{\prime}_{t}}$, i.e. the map

$\lambda:w\in\overline{\mathcal{K}^{\prime}_{t}}\mapsto\lambda_{w}(\cdot):=\langle\cdot,w\rangle\in(\mathcal{K}_{t})^{*}$

is an isomorphism and
$\|w\|_{\mathcal{K}^{{}^{\prime}}_{t}}=\sup_{\\{v\in\mathcal{K}_{t}|\|v\|_{\mathcal{K}_{t}=1}\\}}|\langle
v,w\rangle|$.

  3. (3)

We have the following embeddings for all $t>0$:

$\mathcal{K}^{\prime}_{t}\subset L^{2}(\partial
X,\nu_{o})\subset\mathcal{K}_{t}.$

  4. (4)

For all $t>0$, the intertwiner $\mathcal{I}_{t}$ defines a bounded operator
acting on $\mathcal{K}_{t}$.

  5. (5)

For all $t>0$, the operator $\mathcal{I}^{2}_{t}$ defines an isometry from
$\mathcal{K}_{t}\rightarrow\mathcal{K}^{\prime}_{t}$ and $\mathcal{J}^{2}_{t}$
defines an isometry from $\mathcal{K}^{\prime}_{t}\rightarrow\mathcal{K}_{t}$
such that
$\mathcal{I}^{2}_{t}\circ\mathcal{J}^{2}_{t}=I_{\mathcal{K}^{\prime}_{t}}$ and
$\mathcal{J}^{2}_{t}\circ\mathcal{I}^{2}_{t}=I_{\mathcal{K}_{t}}$.

###### Proof.

We give a proof of Item (1). Observe first that $\langle\cdot,\cdot\rangle$ is
a well defined bilinear form on $\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}\times\overline{\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial
X}})}$. Then, it remains to prove that the standard inner product is non
degenerate on $\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}\times\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial X}})$. On
one hand, if $v\in\mathcal{F}_{t,\textbf{1}_{\partial X}}$ satisfies $\langle
v,w\rangle=0$ for all $w\in\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial
X}})$ it follows, since $\mathcal{I}_{t}$ is continuous and since
$\mathcal{F}_{t,\textbf{1}_{\partial X}}$ is $\|\cdot\|_{2}$-dense in
$L^{2}(\partial X,\nu_{o})$, that
$v\perp\overline{Im(\mathcal{I}_{t})}^{\|\cdot\|_{2}}$. Hence,
$v\in\mathcal{V}_{t}\cap\mathcal{F}_{t,\textbf{1}_{\partial X}}=N_{t}$ and it
follows that $v=0$.
On the other hand, using again the fact that
$\mathcal{F}_{t,\textbf{1}_{\partial X}}$ is $\|\cdot\|_{2}$-dense in
$L^{2}(\partial X,\nu_{o})$, we obtain that if $w$ satisfies $\langle
v,w\rangle=0$ for all $v\in\mathcal{F}_{t,\textbf{1}_{\partial X}}/N_{t}$,
then $w=0$.
Observe now the continuity property given for
$(v,w)\in\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}\times\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial X}})$ by :

$\displaystyle|\langle v,w\rangle|$
$\displaystyle=|\langle\mathcal{J}_{t}\mathcal{I}_{t}v,w\rangle|$
$\displaystyle=|\langle\mathcal{I}_{t}v,\mathcal{J}_{t}w\rangle|$
$\displaystyle\leq\|v\|_{\mathcal{K}_{t}}\|w\|_{\mathcal{K}^{\prime}_{t}}.$

Hence, $\langle\cdot,\cdot\rangle$ is a well defined non-degenerate bilinear
form on $\mathcal{K}_{t}\times\overline{\mathcal{K}^{\prime}_{t}}$.

We prove now Item (2). For any $w\in\overline{\mathcal{K}^{\prime}_{t}}$, the
map $\lambda_{w}:v\mapsto\langle v,w\rangle$ defines an element of
$(\mathcal{K}_{t})^{*}$. We shall show that given
$\ell\in(\mathcal{K}_{t})^{*}$, there exists $w\in\mathcal{K}^{\prime}_{t}$
such that $\ell=\lambda_{w}$. Using the Riesz representation theorem there
exists $a\in\overline{\mathcal{K}_{t}}$ such that for all
$v\in\mathcal{K}_{t}$, $\ell(v)=\langle v,a\rangle_{\mathcal{K}_{t}}=\langle
v,\mathcal{I}^{2}_{t}{a}\rangle.$ It turns out that $\mathcal{I}^{2}_{t}{a}$
is an element of $\overline{\mathcal{K}^{\prime}_{t}}$ :

$\|\mathcal{I}^{2}_{t}{a}\|^{2}_{\mathcal{K}^{\prime}_{t}}=\langle\mathcal{I}^{2}_{t}{a},\mathcal{J}^{2}_{t}\mathcal{I}^{2}_{t}{a}\rangle=\langle\mathcal{I}^{2}_{t}{a},a\rangle=\langle\mathcal{I}_{t}{a},\mathcal{I}_{t}{a}\rangle=\|a\|_{\mathcal{K}_{t}}.$

Hence any element of $(\mathcal{K}_{t})^{*}$ is represented by an element of
$\overline{\mathcal{K}^{\prime}_{t}}$ and since the pairing is non degenerate,
$(\mathcal{K}_{t})^{*}$ is isomorphic and isometric to
$\overline{\mathcal{K}^{\prime}_{t}}$.
We give a proof of Item (3). The fact that $L^{2}(\partial X,\nu_{o})$ embeds
continuously in $\mathcal{K}_{t}$ follows from the continuity of
$\mathcal{I}_{t}$. We have for all $v\in L^{2}(\partial X,\nu_{o})$

$\|v\|_{\mathcal{K}_{t}}\leq C\|v\|,$

where the constant $C$ depends on $t.$ The other inclusion follows from Lemma
3.15. Since we are working with positive operators, the left inverse operator
$\mathcal{J}_{t}$ of $\mathcal{I}_{t}$ can be written $c\cdot I+D$ on its
domain, where $D$ is a positive operator and $c>0$ is a positive constant.
Hence for all $v\in\mathcal{K}^{\prime}_{t}$ we have
$\|v\|^{2}_{\mathcal{K}^{\prime}_{t}}=c^{2}\|v\|^{2}+R(v),$ where $R(v)$ is a
positive quadratic form. Thus,

$\|v\|_{\mathcal{K}^{\prime}_{t}}\geq c\|v\|,$

and the proof is done.

We prove now Item (4). We have for all $v\in\mathcal{K}_{t},$

$\displaystyle\|\mathcal{I}_{t}(v)\|_{\mathcal{K}_{t}}$
$\displaystyle=\langle\mathcal{I}^{2}_{t}(v),\mathcal{I}^{2}_{t}(v)\rangle$
$\displaystyle=\|\mathcal{I}_{t}(\mathcal{I}_{t}(v))\|$
$\displaystyle\leq\|\mathcal{I}_{t}\|_{L^{2}\to L^{2}}\|\mathcal{I}_{t}(v)\|$
$\displaystyle=\|\mathcal{I}_{t}\|_{L^{2}\to L^{2}}\|v\|_{\mathcal{K}_{t}}.$

We prove now Item (5). We have for all $v\in\mathcal{K}_{t},$

$\displaystyle\|\mathcal{I}^{2}_{t}(v)\|_{\mathcal{K}^{\prime}_{t}}$
$\displaystyle=\langle\mathcal{I}^{2}_{t}(v),\mathcal{J}^{2}_{t}\mathcal{I}^{2}_{t}(v)\rangle$
$\displaystyle=\langle\mathcal{I}^{2}_{t}(v),v\rangle$ $\displaystyle=\langle
v,\mathcal{I}^{2}_{t}(v)\rangle$ $\displaystyle=\|v\|_{\mathcal{K}_{t}}.$

Moreover for all $v\in\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial
X}})$, a dense subset of $\mathcal{K}^{\prime}_{t}$ we have

$\displaystyle\|\mathcal{J}^{2}_{t}(v)\|_{\mathcal{K}_{t}}$
$\displaystyle=\langle\mathcal{J}^{2}_{t}(v),\mathcal{I}^{2}_{t}\mathcal{J}^{2}_{t}(v)\rangle$
$\displaystyle=\langle\mathcal{J}^{2}_{t}(v),v\rangle$ $\displaystyle=\langle
v,\mathcal{J}^{2}_{t}(v)\rangle$
$\displaystyle=\|v\|_{\mathcal{K^{\prime}}_{t}}.$

Besides, Lemma 3.15 implies $\mathcal{J}^{2}_{t}\mathcal{I}^{2}_{t}(v)=v$ for
all $v\in\mathcal{F}_{t,\textbf{1}_{\partial X}}$ which is a dense subspace of
$\mathcal{K}_{t}$ and $\mathcal{I}^{2}_{t}\mathcal{J}^{2}_{t}(v)=v$ for all
$v\in\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial X}})$ which is a
dense subspace of $\mathcal{K^{\prime}}_{t}$. Use the fact that
$\mathcal{I}^{2}_{t}$ defines an isometry from
$\mathcal{K}_{t}\rightarrow\mathcal{K}^{\prime}_{t}$ and $\mathcal{J}^{2}_{t}$
defines an isometry from $\mathcal{K}^{\prime}_{t}\rightarrow\mathcal{K}_{t}$
to conclude the proof. ∎

###  4.5.  Unitary representations: an analog of complementary series

Assume that $0<t<\frac{1}{2}$ and $\mathcal{R}_{t}$ is positive. Therefore,
Lemma 3.9 implies that $\mathcal{I}_{t}$ is positive. Consider the
sesquilinear form

(4.5)
$\langle\cdot,\cdot\rangle_{\mathcal{I}_{t}}:(v,w)\in\mathcal{F}_{t,\textbf{1}_{\partial
X}}\times\mathcal{F}_{t,\textbf{1}_{\partial X}}\mapsto\langle
v,\mathcal{I}_{t}(w)\rangle\in\mathbb{C}.$

Let $L_{t}:=\\{v\in\mathcal{F}_{t,\textbf{1}_{\partial X}}|\langle
v,v\rangle_{\mathcal{I}_{t}}=0\\}.$ Consider the pre-Hilbert space
$\mathcal{F}_{t,\textbf{1}_{\partial X}}/L_{t}$ endowed with
$\langle\cdot,\cdot\rangle_{\mathcal{I}_{t}}$ and turn it into a Hilbert space
defined as:

(4.6) $\overline{\mathcal{F}_{t,\textbf{1}_{\partial
X}}/L_{t}}^{\|\cdot\|_{I_{t}}}:=\mathcal{H}_{t}.$

Now, we are ready to prove Proposition 1.1.

###### Proof.

We only have to prove that $(\pi_{t},\mathcal{H}_{t})$ is a unitary
representation. Let $\gamma\in\Gamma$ and $v,w\in\mathcal{H}_{t}$. We have:

$\displaystyle\langle\pi_{t}(\gamma)v,w\rangle_{\mathcal{H}_{t}}$
$\displaystyle=\langle\pi_{t}(\gamma)v,w\rangle_{\mathcal{I}_{t}}$
$\displaystyle=\langle\pi_{t}(\gamma)v,\mathcal{I}_{t}(w)\rangle$
$\displaystyle=\langle v,\pi_{-t}(\gamma^{-1})\mathcal{I}_{t}(w)\rangle$
$\displaystyle=\langle v,\mathcal{I}_{t}(\pi_{t}(\gamma^{-1})w)\rangle$
$\displaystyle=\langle v,\pi_{t}(\gamma^{-1})w\rangle_{\mathcal{H}_{t}},$

where the fourth equality follows from Proposition 3.16. And the proof is
done. ∎

Consider also the sesqilinear form

(4.7)
$\langle\cdot,\cdot\rangle_{\mathcal{J}_{t}}:(v,w)\in\mathcal{F}_{-t,\sigma_{t}}\times\mathcal{F}_{-t,\sigma_{t}}\mapsto\langle
v,\mathcal{J}_{t}(w)\rangle\in\mathbb{C},$

and define the Hilbert space:

(4.8)
$\overline{\mathcal{F}_{-t,\sigma_{t}}}^{\|\cdot\|_{\mathcal{J}_{t}}}:=\mathcal{H}^{\prime}_{t}.$

######  Lemma 4.10.

The representation $(\pi_{-t},\mathcal{H}^{\prime}_{t})$ is a well defined
unitary representation.

The proof of the above lemma follows from Lemma 4.7. The following is similar
to Proposition 4.2 but deals with the space $\mathcal{H}_{t}$ and
$\mathcal{H}^{\prime}_{t}$.

######  Proposition 4.11.

Let $0<t<\frac{1}{2}$. Assume that $\mathcal{I}_{t}$ is positive operator.

  1. (1)

The pair
$(\mathcal{H}_{t},\overline{\mathcal{H}^{\prime}_{t}},\langle\cdot,\cdot\rangle)$
is a $\mathbb{C}$-pairing.

  2. (2)

The topological dual of $\mathcal{H}_{t}$ is isomorphic and isometric to
$\overline{\mathcal{H}^{\prime}_{t}}$.

  3. (3)

We have the following embeddings

$\mathcal{H}^{\prime}_{t}\subset L^{2}(\partial
X,\nu_{o})\subset\mathcal{H}_{t}.$

  4. (4)

The intertwiner $\mathcal{I}_{t}$ defines a bounded operator acting on
$\mathcal{H}_{t}$.

  5. (5)

The intertwiner $\mathcal{I}_{t}$ defines an isometry from
$\mathcal{H}_{t}\rightarrow\mathcal{H}^{\prime}_{t}$ and $\mathcal{J}_{t}$
defines an isometry from $\mathcal{H}^{\prime}_{t}\rightarrow\mathcal{H}_{t}$
such that $\mathcal{I}_{t}\circ\mathcal{J}_{t}=I_{\mathcal{H}^{\prime}_{t}}$
and $\mathcal{J}_{t}\circ\mathcal{I}_{t}=I_{\mathcal{H}_{t}}$.

###### Proof.

To deal with the case of $\mathcal{H}_{t}$, use $\mathcal{I}^{1/2}_{t}$ and
$\mathcal{J}^{1/2}_{t}$ instead of $\mathcal{I}_{t}$ and $\mathcal{J}_{t}$ in
the proof of Proposition 4.2.

∎

###  4.6. Basic properties of those Hilbertian representations

We enumerate and sum up in the following, several properties of the Hilbertian
representations we are studying.

######  Proposition 4.12.

Let $t>0$.

  1. (1)

The intertwiner $\mathcal{I}_{t}$ extends to an isometry from
$\mathcal{K}_{t}$ to $\mathcal{W}_{-t}$.

  2. (2)

If $0<t<\frac{1}{2}$ the representations $(\pi_{t},\mathcal{K}_{t})$ and
$(\pi_{-t},\mathcal{W}_{t})$ are infinite dimensional.

  3. (3)

The representations $(\pi_{\frac{1}{2}},\mathcal{K}_{\frac{1}{2}})$ and
$(\pi_{-\frac{1}{2}},\mathcal{W}_{-\frac{1}{2}})$ are nothing but the trivial
representation.

  4. (4)

The representations $(\pi_{t},\mathcal{K}_{t})$ and
$(\overline{\pi_{-t}},\overline{\mathcal{K}^{\prime}_{t}})$ are a dual system
representation with respect to the $L^{2}$-inner product.

And if $\mathcal{I}_{t}$ is positive then :

  5. (5)

It implies $t\leq{1\over 2}$.

  6. (6)

If $0<t<\frac{1}{2}$, the representations $(\pi_{t},\mathcal{H}_{t})$ and
$(\pi_{-t},\mathcal{H}_{t})$ are infinite dimensional unitary representations.

  7. (7)

The representations $(\pi_{\frac{1}{2}},\mathcal{H}_{\frac{1}{2}})$ is nothing
but the trivial representation.

  8. (8)

The representations of $(\pi_{t},\mathcal{H}_{t})$ and
$(\overline{\pi_{-t}},{\mathcal{H}^{\prime}_{t}})$ are a dual system
representation of $\Gamma$ with respect to the $L^{2}$-inner product.

###### Proof.

We first prove Item (1). The operator
$\mathcal{I}_{t}:\mathcal{F}_{t,\textbf{1}_{\partial
X}}\rightarrow\mathcal{F}_{-t}\cdot\sigma_{t}$ extends as an isometry from
$\mathcal{K}_{t}\rightarrow\mathcal{W}_{-t}$. Indeed for all
$v\in\mathcal{F}_{t,\textbf{1}_{\partial X}}$ we have

$\|\mathcal{I}_{t}(v)\|_{2}=\|v\|_{\mathcal{K}_{t}}.$

Hence, $\mathcal{I}_{t}$ extends to an isometry from
$\overline{\mathcal{F}_{t,\textbf{1}_{\partial
X}}}^{\|\cdot\|_{\mathcal{K}_{t}}}$ to
$\overline{\mathcal{F}_{-t}\cdot\sigma_{t}}^{\|\cdot\|_{2}}$.
We prove Item (2): the vector space $\mathcal{F}_{t,\textbf{1}_{\partial
X}}/N_{t}$ is isomorphic to
$\mathcal{I}_{t}(\mathcal{F}_{t,\textbf{1}_{\partial X}})$ which is infinite
dimensional as a $\pi_{-t}$ representation of $\Gamma$ by Proposition 6.5.
Hence $\mathcal{K}_{t}$ and $\mathcal{W}_{t}$ are infinite dimensional Hilbert
spaces.
For Item (3), observe that $\mathcal{I}_{\frac{1}{2}}v=\big{(}\int_{\partial
X}vd\nu_{o}\big{)}\textbf{1}_{\partial X},$ that is the orthogonal projection
onto the one dimensional space of constant functions. Hence,
$\mathcal{W}_{-\frac{1}{2}}$ is a one dimensional vector space on which
$\pi_{-\frac{1}{2}}$ acts trivially. Observe also that
$\ker\mathcal{I}_{\frac{1}{2}}=N_{\frac{1}{2}}=L^{2}_{0}(\partial X)$, where
$L^{2}_{0}(\partial X)=L^{2}(\partial
X,\nu_{o})\ominus\mathbb{C}\textbf{1}_{\partial X}$. Therefore,
$(\pi_{\frac{1}{2}},\mathcal{K}_{\frac{1}{2}})$,
$(\pi_{\frac{1}{2}},L^{2}(\partial X,\nu_{o}){/}\mathcal{V}_{\frac{1}{2}})$
are the trivial representations.
For Item (4), notice that
$(\mathcal{K}_{t},\overline{\mathcal{K}^{\prime}_{t}},\langle\cdot,\cdot\rangle)$
is a $\mathbb{C}$-pairing with respect to $\langle\cdot,\cdot\rangle$.
Moreover we have for all $\gamma\in\Gamma$ and for any
$(v,w)\in\mathcal{K}_{t}\times\mathcal{K}^{\prime}_{t}$ that
$\langle\pi_{t}(\gamma)v,\overline{\pi_{-t}}(\gamma)w\rangle=\langle
v,w\rangle$.
For Item (5), see Proposition 5.2. For proving Item (6), (7) and (8) use
analogous arguments of the proofs of Item (2), (3) and (4).

∎

##  5\. Obstruction and an assumption for positivity

We start this section by an obstruction of positivity of $\mathcal{I}_{t}$ for
$t>\frac{1}{2}$.

###  5.1. Spherical functions on hyperbolic groups

Asymptotic behavior of matrix coefficients’ functions :

$\phi_{t}\,:\,\gamma\in\Gamma\mapsto\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},\textbf{1}_{\partial X}\rangle$

is a central question in representation theory and harmonic analysis on Lie
groups (see for example [30]). The above functions are also called spherical,
as they can be estimated by functions of length of $\gamma$ (see below). For
instance, in the Lie group framework, $\phi_{0}$ is Harish-Chandra’s function
$\Xi$ and has been estimated first by the eponymous author and then by Anker
who gave an optimal lower bound. In the context of hyperbolic groups one can
prove the following estimates ([28], [18], [31], [11]), called _Harish-Chandra
Anker estimates_ referring to [1] : there exists $C>0$ such that for all
$\gamma\in\Gamma:$

(5.1) $\displaystyle
C^{-1}(1+|\gamma|)e^{-\frac{1}{2}Q|\gamma|}\leq\Xi(\gamma)\leq
C(1+|\gamma|)e^{-\frac{1}{2}Q|\gamma|}).$

The family of functions $(\omega_{t})_{t\in\mathbb{R}^{*}}$ defined by

(5.2) $\omega_{t}(x)=\frac{2\sinh\big{(}tQx\big{)}}{e^{2tQ}-1},$

are positive, converges uniformly on compact sets to $x\mapsto x$ and satisfy
$\omega_{-t}(x)=e^{2tQ}\omega_{t}(x)$. It allows to extend Harish-Chandra
Anker estimates for all representations $\pi_{t}$ (see [10]).

######  Proposition 5.1.

There exists $C>0$, such that for any $t\in\mathbb{R}$, we have for all
$\gamma\in\Gamma:$

$C^{-1}e^{-\frac{1}{2}Q|\gamma|}\big{(}1+\omega_{|t|}(|\gamma|)\big{)}\leq\phi_{t}(\gamma)\leq
Ce^{-\frac{1}{2}Q|\gamma|}\big{(}1+\omega_{|t|}(|\gamma|)\big{)}.$

The following definition of spherical functions seems to be more appropriate
in the context of general hyperbolic groups and coincides of course with the
standard one in the framework of semisimple Lie groups.
Pick $t\in]0,\frac{1}{2}]$ and set

(5.3)
$\varphi_{t}:\Gamma\mapsto\frac{1}{\|\sigma_{t}\|_{1}}\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},\mathcal{I}_{t}(\textbf{1}_{\partial X})\rangle\in\mathbb{R}.$

which is well defined since for $t>0$ the operator $\mathcal{I}_{t}$ is well
defined. We obtain the following characterization :

######  Proposition 5.2.

If $t$ is not in $]0,\frac{1}{2}]$, the intertwiner $\mathcal{I}_{t}$ cannot
be positive.

###### Proof.

Let $t>0$. Let $f=\sum_{g}c_{g}D_{g}$ be a finitely supported function in
$\mathbb{C}[\Gamma]$ and let
$v=\pi_{t}(f)=\sum_{g}c_{g}\pi_{t}(g)\textbf{1}_{\partial
X}\in\mathcal{F}_{t,\textbf{1}_{\partial X}}$. Consider
$\gamma\in\Gamma\mapsto\varphi_{t}(\gamma)$ and observe that :

$\displaystyle\sum_{g,h}\overline{c_{h}}c_{g}\varphi_{t}(h^{-1}g)$
$\displaystyle=\frac{1}{\|\sigma_{t}\|_{1}}\sum_{g,h}\overline{c_{h}}c_{g}\langle\pi_{t}(h^{-1}g)\textbf{1}_{\partial
X},\mathcal{I}_{t}(\textbf{1}_{\partial X})\rangle$
$\displaystyle=\frac{1}{\|\sigma_{t}\|_{1}}\sum_{g,h}\overline{c_{h}}c_{g}\langle\pi_{t}(h^{-1})\pi_{t}(g)\textbf{1}_{\partial
X},\mathcal{I}_{t}(\textbf{1}_{\partial X})\rangle$
$\displaystyle=\frac{1}{\|\sigma_{t}\|_{1}}\sum_{g,h}\overline{c_{h}}c_{g}\langle\pi_{t}(g)\textbf{1}_{\partial
X},\pi_{-t}(h)\mathcal{I}_{t}(\textbf{1}_{\partial X})\rangle$
$\displaystyle=\frac{1}{\|\sigma_{t}\|_{1}}\sum_{g,h}\overline{c_{h}}c_{g}\langle\pi_{t}(g)\textbf{1}_{\partial
X},\mathcal{I}_{t}(\pi_{t}(h)\textbf{1}_{\partial X})\rangle$
$\displaystyle=\frac{1}{\|\sigma_{t}\|_{1}}\langle
v,\mathcal{I}_{t}(v)\rangle,$

Hence, $\varphi_{t}$ is of positive type if and only if $\mathcal{I}_{t}$ is
positive on $L^{2}(\partial X,\nu_{o})$, since
$\mathcal{F}_{t,\textbf{1}_{\partial X}}$ is dense in $L^{2}(\partial
X,\nu_{o})$ by Proposition 6.13. Write

$\varphi_{t}(\cdot)=\frac{1}{\|\sigma_{t}\|_{1}}\langle\pi_{t}(\cdot)\textbf{1}_{\partial
X},\sigma_{t}\rangle,$

and use Proposition 3.4 to obtain the existence of $C>0$ such that for any
$t>0$ and for all $\gamma\in\Gamma:$

$C^{-1}\phi_{t}(\gamma)\leq\varphi_{t}(\gamma)\leq C\phi_{t}(\gamma).$

Hence $\varphi_{t}$ is unbounded if $t>\frac{1}{2}$ since $\phi_{t}$ is
unbounded, and thus $\varphi_{t}$ cannot be of positive type [5, (ii)
Proposition C.4.2]. Thus, the intertwiner $\mathcal{I}_{t}$ cannot be
positive. ∎

######  Remark 5.3.

Given a function $\phi$ of positive type on $\Gamma$, there exists a triple
$(\pi_{\phi},H_{\phi},\Omega_{\phi})$ such that $\pi_{\phi}$ is unitary cyclic
representation on $H_{\phi}$, that is
$H_{\phi}=\overline{\pi_{\phi}(\Gamma)\Omega_{\phi}}$, satisfying
$\phi(\cdot)=\langle\pi_{\phi}(\cdot)\Omega_{\phi},\Omega_{\phi}\rangle.$
This is the so called Gelfand-Naimark-Segal construction (GNS for short). We
refer to [5, Theorem C.4.10] for more details.
Perform the GNS construction with $\pi_{\varphi_{t}}$ for
$t\in[0,\frac{1}{2}]$. It turns out that the representations
$(\pi_{\varphi_{t}},H_{\varphi_{t}})$ and $(\pi_{t},\mathcal{H}_{t})$ are
unitarily equivalent.

Let $\Gamma$ be a discrete group acting geometrically on a proper roughly
geodesic $\delta$-hyperbolic $(X,d)$. Assume that $\mathcal{I}_{t}$ is
positive for $0<t\leq\frac{1}{2}$. Consider the unitary representations
$(\pi_{t},\mathcal{H}_{t})$. We have:

######  Proposition 5.4.

The group $\Gamma$ satisfies Haagerup property.

###### Proof.

Consider the sequence of spherical functions

$\varphi_{t_{n}}:\gamma\in\Gamma\mapsto\frac{1}{\|\sigma_{t_{n}}\|_{1}}\langle\pi_{t_{n}}(\gamma)\textbf{1}_{\partial
X},\mathcal{I}_{t_{n}}(\textbf{1}_{\partial X})\rangle\in\mathbb{R},$

with $t_{n}=\frac{1}{2}-\frac{1}{n+2}$ for all $n\in\mathbb{N}^{*}$. Observe
that

  1. (1)

$(\varphi_{t_{n}})_{n\in\mathbb{N}^{*}}$ is in $C_{0}(\Gamma)$ by Proposition
5.1.

  2. (2)

$\varphi_{t_{n}}(e)=1$ for all $n\in\mathbb{N}^{*}.$

  3. (3)

$\varphi_{t_{n}}\to 1$ as $n\to\infty$ on every finite subsets of $\Gamma$.

  4. (4)

$(\varphi_{t_{n}})_{n\in\mathbb{N}^{*}}$ are positive definite functions on
$\Gamma$ since $\mathcal{I}_{t}$ is assumed to be positive for
$0<t\leq\frac{1}{2}$.

Using [21, (2) of Theorem 2.1.1] we deduce that $\Gamma$ satisfies Haagerup
property. ∎

###  5.2. Equidistribution results

Let $t\in\mathbb{R}$ and consider the measure:

(5.4) $dm_{o,t}(\xi,\eta):=k_{t}(\xi,\eta)d\nu_{o}(\xi)d\nu_{o}(\eta)\in
C^{*}(\overline{X}\times\overline{X}).$

Let $A>0$ and consider the truncated _continuous_ kernel $k_{t,A}$
approximating $k_{t}$ on $\overline{X}\times\overline{X}$ defined as:

$k_{t,A}:(x,y)\in\partial X\times\partial
X\mapsto\left\\{\begin{array}[]{ll}e^{(1-2t)Q(x,y)_{o}}&\mbox{if
$(x,y)_{o}\in[0,A]$ }\\\ e^{(1-2t)QA}(A+1-(x,y)_{o})&\mbox{if
$A<(x,y)_{o}<A+1.$}\\\ 0&\mbox{if
$(x,y)_{o}\in]A+1,+\infty[.$}\end{array}\right.$

Thus define

(5.5) $dm^{A}_{o,t}(\xi,\eta):=k_{t,A}(\xi,\eta)d\nu_{o}(\xi)d\nu_{o}(\eta)\in
C^{*}(\overline{X}\times\overline{X}).$

######  Lemma 5.5.

For all $t>0$, we have the following uniform convergence: $m^{t,A}_{o}\to
m^{t}_{o}$ as $A\to\infty.$

###### Proof.

For all $F\in C^{*}(\overline{X}\times\overline{X})$ we have:

$\displaystyle|m^{t}_{o}(F)-m^{t,A}_{o}(F)|$
$\displaystyle=\bigg{|}\int_{\partial X\times\partial
X}F(\xi,\eta)k_{t}(\xi,\eta)-k_{t,A}(\xi,\eta)d\nu_{o}(\xi)d\nu_{o}(\eta)\bigg{|}$
$\displaystyle\leq C\|F\|_{\infty}\int_{(\xi,\eta)_{o}\geq
A+1}\frac{1}{d^{(1-2t)Q}_{o}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta).$

Hence,

$\|m_{o,t}-m^{A}_{o,t}\|_{C^{*}(\partial X\times\partial X)}\leq
C\int_{(\xi,\eta)_{o}\geq
A+1}\frac{1}{d^{(1-2t)Q}_{o}(\xi,\eta)}d\nu_{o}(\xi)d\nu_{o}(\eta).$

Using Inequality 3.8 of Lemma 3.3, let $A\to+\infty$ to conclude the proof. ∎

Define now for $R>0$ large enough :

(5.6) $m^{A}_{o,t,N,R}:=\sum_{g,h\in
S^{\Gamma}_{N,R}}\beta_{N,R}(g)\beta_{N,R}(h)k_{t,A}(go,ho)D_{go}\otimes
D_{ho}\in C^{*}(\overline{X}\times\overline{X}).$

Since $R$ is fixed, we write $m^{A}_{o,t,N}$ for $m^{A}_{o,t,N,R}$. Consider
the two sequences of measures defined by :

(5.7) $m_{o,N,R}:=\sum_{g,h\in S^{\Gamma}_{N,R}\times
S^{\Gamma}_{N,R}}\beta_{N,R}(g)\beta_{N,R}(h)D_{go}\otimes D_{ho}\in
C^{*}(\overline{X}\times\overline{X}).$

and, for $t\in\mathbb{R}:$

(5.8) $m^{t}_{o,N,R}:=\sum_{g,h\in
S^{\Gamma}_{N,R}}\beta_{N,R}(g)\beta_{N,R}(h)e^{(1-2t)Q(go,ho)_{o}}D_{go}\otimes
D_{ho}\in C^{*}(\overline{X}\times\overline{X}).$

######  Lemma 5.6.

For all $R>0$ large enough, the sequence $(m_{o,N,R})_{N\in\mathbb{N}}$
converges to $\nu_{o}\otimes\nu_{o}$ with respect to the weak* convergence on
$C(\overline{X}\times\overline{X})$.

###### Proof.

Since for all non negative integers $k$ and for all $g\in S_{k,R}$, we have
$\beta_{k}(g)\leq\frac{C}{|S_{k,R}|}$, it follows that

$\sup_{N}\|m_{o,N,R}\|_{C(\overline{X}\times\overline{X})^{*}}<+\infty.$

Thus, up to subsequences, it follows that $(m_{o,N,R})_{N\in\mathbb{N}}$ has a
limit with respect to the weak* topology on
$C(\overline{X}\times\overline{X})$. But on the space generated by pure
tensors written as $f\otimes g$ with $f,g\in C(\overline{X})$, we have
$m_{o,N,R}(f\otimes g)\to(\nu_{o}\otimes\nu_{o})(f\otimes g)$ as $N\to+\infty$
by applying Remark 2.9 for $R>0$ large enough. Hence, we deduce that for all
$F\in C(\overline{X}\times\overline{X})$ we have:
$m_{o,N,R}(F)\to\nu_{o}\otimes\nu_{o}(F)$ as $N\to+\infty$.

∎

####  5.2.1. Convergence of measures

######  Lemma 5.7.

Let $R>0$ large enough. For any $t<\frac{1}{2}$, we have the following weak*
convergence for all $A>0$ $m^{A}_{o,t,N}\to m^{A}_{o,t}$ as $N\to\infty$ in
$C^{*}(\overline{X}\times\overline{X}).$

###### Proof.

The proof is a direct application of Lemma 5.6 since the map
$(x,y)\in\overline{X}\times\overline{X}\mapsto k_{t,A}(x,y)$ is continuous. ∎

######  Lemma 5.8.

Assume that $t<\frac{1}{2}$. We have the following uniform convergence:

$\lim_{A\to+\infty}\sup_{N}\|m_{o,t,N}-m^{A}_{o,t,N}\|_{C^{*}(\overline{X}\times\overline{X})}\to
0.$

###### Proof.

Let $F\in C(\overline{X}\times\overline{X})$. We have:

$\displaystyle|m^{t}_{o,N}(F)-m^{t,A}_{o,N}(F)|$ $\displaystyle\leq
2\|F\|_{\infty}\sum_{\\{g,h\in
S_{N}|(go,ho)>A\\}}\beta_{N,R}(g)\beta_{N,R}(h)e^{(1-2t)Q(go,ho)}.$

For all $h\in S_{N,R}$, Lemma 3.7 provides a constant $C>0$ such that:

$\sum_{\\{g\in
S^{\Gamma}_{N,R}|(go,ho)>A\\}}\beta_{N,R}(g)e^{(1-2t)Q(go,ho)}\leq
C\frac{e^{-2tQA}}{1-e^{-2tQR}}.$

Therefore:

$\|m_{o,t,N}-m^{A}_{o,t,N}\|_{C^{*}(\overline{X}\times\overline{X})}\leq
C\frac{e^{-2tQA}}{1-e^{-2tQR}}.$

Let $A\to+\infty$ to conclude the proof. ∎

We obtain the nice equidistribution result in the context of hyperbolic groups
obtained in [7, Section 4] for CAT(-1) spaces:

######  Proposition 5.9.

For all $t<\frac{1}{2}$ we have:

$m_{o,t,N}\to m_{o,t},$

as $N\to+\infty$ for the weak* convergence of
$C^{*}(\overline{X}\times\overline{X})$

###### Proof.

For all $F\in C(\overline{X}\times\overline{X})$ such that $\|F\|_{\infty}\leq
1$ and for all non negative integers $N$:

$\displaystyle|m_{o,t,N}(F)-m_{o,t}(F)|$
$\displaystyle\leq|m_{o,t,N}(F)-m^{A}_{o,t,N}(F)|+|m^{A}_{o,t,N}(F)-m^{A}_{o,t}(F)|+|m^{A}_{o,t}(F)-m_{o,t}(F)|$
$\displaystyle\leq\sup_{N}\|m_{o,t,N}-m^{A}_{o,t,N}\|+|m^{A}_{o,t,N}(F)-m^{A}_{o,t}(F)|+\|m^{A}_{o,t}-m_{o,t}\|.$

Take the limit over $N$ applying Lemma 5.7 for the second term of the above
inequality, and take the limit over $A$ applying Lemma 5.8 to the first term
and Lemma 5.5 to the last term to finish the proof. ∎

###  5.3. Application: positivity of $\mathcal{I}_{t}$ for
$0<t\leq\frac{1}{2}$

We prove Proposition 1.10. Recall that the main assumption of this statement
is conditional negativity of the distance. We suspect the latter can be
relaxed, as it is a local and global geometric assumption and not only a
macroscopic one like Gromov hyperbolicity - hence unsatisfactory.

###### Proof.

Let $0<t<\frac{1}{2}$. Since the distance defines a conditionnaly negative
kernel, the Gromov product

$(g,h)\in\Gamma\times\Gamma\mapsto(go,ho)_{o}\in\mathbb{R}^{+},$

is positive definite, see [5, Lemma C.3.1]. It follows, [5, Proposition
C.1.6], that the kernel $k_{t}$ viewed on $\Gamma$

$(g,h)\in\Gamma\times\Gamma\mapsto e^{(1-2t)Q(go,ho)_{o}}\in\mathbb{R}^{+},$

is positive definite for $t<\frac{1}{2}$ on $\Gamma$. Let $v$ be continuous
functions in $C(\partial X)$ and extend $v$ to a continuous functions denoted
by $\hat{v}$ in $C(\overline{X})$. Pick $R>0$ large enough. We have for all
$N\in\mathbb{N}$:

$m_{o,t,N}(\hat{v}\otimes\overline{\hat{v}})=\sum_{g,h\in
S^{\Gamma}_{N,R}}\hat{v}(g)\overline{\hat{v}}(h)k_{t}(g,h)\geq 0.$

Let $N\to+\infty$ and Proposition 5.9 implies:

$m_{o,t}(\hat{v}\otimes\overline{\hat{v}})\geq 0.$

Finally, observe that for all $t\leq\frac{1}{2}$ and for all continuous
functions $v\in C(\partial X):$

$m_{o,t}(\hat{v}\otimes\overline{\hat{v}})=\langle
v,\mathcal{I}_{t}(v)\rangle\geq 0.$

We deduce that $\mathcal{I}_{t}$ is positive on $L^{2}(\partial X,\nu_{o})$.
By Lemma 3.9, we deduce that $\mathcal{R}_{t}$ is positive for
$0<t<\frac{1}{2}$.
The case $t=\frac{1}{2}$ gives
$\mathcal{I}_{\frac{1}{2}}(v)=\mathcal{R}_{\frac{1}{2}}(v)=\langle
v,\textbf{1}_{\partial X}\rangle\textbf{1}_{\partial X}$, that is the
orthogonal projection onto the space of constant functions which is a positive
operator.

∎

##  6\. Consequences of spectral gap estimates

Using duality arguments, we provide several spectral estimates based on [10,
Theorem 1.2] for the dual system representations studied in this paper. The
results can be viewed as a generalization of the result of [40] to the setting
of Gromov hyperbolic groups. Notice that the representations $\pi_{t}$ are
typical examples of _slow growth representations._

###  6.1. A spectral inequality

The main result of [10] is the following spectral inequality, generalizing the
so called “Haagerup Property” or Property RD. Let $R>0$ large enough. There
exists $C>0$ such that for any $t\in\mathbb{R}$ and for all
$f_{n}\in\mathbb{C}[\Gamma]$, whose support is in $S^{\Gamma}_{n,R}$, we have:

(6.1) $\|\pi_{t}(f_{n})\|_{L^{2}\to L^{2}}\leq
C\omega_{|t|}(nR)\|f_{n}\|_{\ell^{2}},$

(see (5.2) for the definition of $\omega_{t}$). We deduce the following
spectral gap estimate :

######  Proposition 6.1.

Let $R>0$ large enough. 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 $n$ and for $f_{n}=\sum_{\gamma\in
S^{\Gamma}_{n,R}}\beta_{n,R}(\gamma)D_{\gamma o}$:

$\|\pi_{t}(f_{n})\|_{L^{2}\to L^{2}}\leq C\phi_{t}(nR).$

###### Proof.

Pick $R>0$ large enough such that Inequality (6.1) is valid. Note that
$\|f_{n}\|^{2}_{\ell^{2}}=\sum_{\gamma\in
S^{\Gamma}_{n,R}}\beta^{2}_{n,R}(\gamma)$. But we know that there exsits $C>0$
such that

$C^{-1}e^{-QnR}\leq\beta_{n,R}(\gamma)\leq Ce^{-QnR},$

for all $\gamma\in S^{\Gamma}_{n,R}$. This leads to the estimates of the norm
as

$C^{-1}e^{-\frac{1}{2}QnR}\leq\|f_{n}\|_{\ell^{2}}\leq Ce^{-\frac{1}{2}QnR}.$

Hence, for all unit vectors $v,w\in L^{2}(\partial X,\nu_{o})$ we have:

$\displaystyle|\langle\pi_{t}(f_{n})v,w\rangle|$ $\displaystyle\leq
C\omega_{|t|}(nR)\|f_{n}\|_{\ell^{2}}$ $\displaystyle\leq
C\omega_{|t|}(nR)e^{-\frac{1}{2}QnR},$

where the first inequality follows from Inequality (6.1). Observe now that the
lower bound of Proposition 5.1 implies for all unit vectors $v,w\in
L^{2}(\partial X,\nu_{o})$:

$|\langle\pi_{t}(f_{n})v,w\rangle|\leq C\phi_{t}(nR).$

And the proof is done by the definition of the operator norm. ∎

####  6.1.1. For Hilbertian complementary series

We state a “Hilbertian version” of the previous theorem.

######  Theorem 6.2.

Let $R>0$ be large enough. There exists $C>0$, such that for any $t>0$, for
all $f_{n}\in\mathbb{C}[\Gamma]$ supported on $S^{\Gamma}_{n,R}$, we have:

(6.2) $\displaystyle\|\pi_{t}(f_{n})\|_{\mathcal{K}_{t}\to\mathcal{K}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\omega_{t}(nR),$

and

(6.3)
$\displaystyle\|\pi_{-t}(f_{n})\|_{\mathcal{K}^{\prime}_{t}\to\mathcal{K}^{\prime}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\omega_{t}(nR).$

###### Proof.

We know by (6.1) that there exists $C>0$ such that for any
$t\in\mathbb{R}^{*}_{+}$, for all non-negative integers $n$ with
$f_{n}\in\mathbb{C}[\Gamma]$ supported in $S^{\Gamma}_{n,R}$, and for all
$v,w\in L^{2}(\partial X,\nu_{o})$:

$|\langle\pi_{-t}(f_{n})v,w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{2}\|w\|_{2}\omega_{-t}(nR).$

Using Lemma 4.11, we obtain for all $v,w\in\mathcal{K}_{t}$ on one hand:

$\displaystyle|\langle\pi_{t}(f_{n})v,\mathcal{I}^{2}_{t}(w)\rangle|$
$\displaystyle=|\langle\mathcal{I}_{t}\pi_{t}(f_{n})v,\mathcal{I}_{t}(w)\rangle|$
$\displaystyle=|\langle\pi_{-t}(f_{n})\mathcal{I}_{t}v,\mathcal{I}_{t}(w)\rangle|$
$\displaystyle\leq
C\|f_{n}\|_{\ell^{2}}\|\mathcal{I}_{t}v\|_{2}\|\mathcal{I}_{t}w\|_{2}\omega_{|-t|}(nR)$
$\displaystyle=C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{K}_{t}}\|w\|_{\mathcal{K}_{t}}\omega_{t}(nR).$

On the other hand obtain for all $v,w\in\mathcal{K}^{\prime}_{t}$:

$\displaystyle|\langle\pi_{-t}(f_{n})v,\mathcal{J}^{2}_{t}(w)\rangle|$
$\displaystyle=|\langle\mathcal{J}_{t}\pi_{-t}(f_{n})v,\mathcal{J}_{t}(w)\rangle|$
$\displaystyle=|\langle\mathcal{J}_{t}\pi_{-t}(f_{n})v,\mathcal{J}_{t}(w)\rangle|$
$\displaystyle=|\langle\pi_{t}(f_{n})\mathcal{J}_{t}(v),\mathcal{J}_{t}(w)\rangle|$
$\displaystyle\leq
C\|f_{n}\|_{\ell^{2}}\|\mathcal{J}_{t}v\|_{2}\|\mathcal{J}_{t}w\|_{2}\omega_{t}(nR)$
$\displaystyle=C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{K}^{\prime}_{t}}\|w\|_{\mathcal{K}^{\prime}_{t}}\omega_{t}(nR)$
$\displaystyle=C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{K}^{\prime}_{t}}\|w\|_{\mathcal{K}^{\prime}_{t}}\omega_{|-t|}(nR),$

where the third equality follows from Lemma 4.7 and the proof is done. ∎

####  6.1.2. For unitary complementary series

We assume in this section that there exists $0<t<\frac{1}{2}$ such that
$\mathcal{I}_{t}$ is a positive operator. In this case, the representations
$(\pi_{t},\mathcal{H}_{t})$ and $(\pi_{-t},\mathcal{H}^{\prime}_{t})$ are
unitary and we obtain the following spectral estimates :

######  Theorem 6.3.

Let $R>0$ large enough. There exists $C>0$ such that for any
$0<t<\frac{1}{2}$, for all $f_{n}\in\mathbb{C}[\Gamma]$ supported on
$S^{\Gamma}_{n,R}$, we have:

$\|\pi_{t}(f_{n})\|_{\mathcal{H}_{t}\to\mathcal{H}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\omega_{t}(nR),$

and

$\|\pi_{-t}(f_{n})\|_{\mathcal{H}^{\prime}_{t}\to\mathcal{H}^{\prime}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\omega_{t}(nR).$

###### Proof.

Pick $R>0$ large enough. For all $v,w\in L^{2}(\partial X,\nu_{o})$ and for
all $f_{n}$ supported on $S^{\Gamma}_{n,R}$, we have first:

$\langle\pi_{t}(f_{n})v,w\rangle=\langle v,\pi_{-t}(\check{f_{n}})w\rangle,$

where $\check{f}(\gamma)=\overline{f}(\gamma^{-1})$ for all $\gamma\in\Gamma$.
The spectral estimate (6.1) implies

$|\langle v,\pi_{-t}(\check{f_{n}})w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{2}\|w\|_{2}\omega_{t}(nR),$

because $\|\check{f_{n}}\|_{\ell^{2}}=\|f_{n}\|_{\ell^{2}}$. Using Item 3) of
Lemma 4.11 we obtain:

$|\langle v,\pi_{-t}(\check{f_{n}})w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}^{\prime}_{t}}\|w\|_{\mathcal{H}^{\prime}_{t}}\omega_{t}(nR).$

Note that for all $w\in\mathcal{H}^{\prime}_{t}$, the map
$v\in\mathcal{H}^{\prime}_{t}\mapsto\langle
v,\pi_{-t}(\check{f_{n}})w\rangle\in\mathbb{C}$ defines an element of
$(\mathcal{H}^{\prime}_{t})^{*}$. Therefore, Lemma 4.11 Item 1) provides for
all $w\in\mathcal{H}^{\prime}_{t}$:

$\|\pi_{-t}(\check{f_{n}})w\|_{\mathcal{H}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\|w\|_{\mathcal{H}^{\prime}_{t}}\omega_{t}(nR).$

Then, apply the above inequality for $w=\mathcal{I}_{t}(v)$ with
$v\in\mathcal{H}_{t}$, Lemma 4.11 Item 5) ensures that
$w\in\mathcal{H}^{\prime}_{t}$, to obtain:

$\|\pi_{-t}(\check{f_{n}})\mathcal{I}_{t}(v)\|_{\mathcal{H}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}_{t}}\omega_{t}(nR).$

Note now :

$\|\pi_{-t}(\check{f_{n}})\mathcal{I}_{t}(v)\|_{\mathcal{H}_{t}}=\|\mathcal{I}_{t}\pi_{t}(\check{f_{n}})(v)\|_{\mathcal{H}_{t}}=\|\pi_{t}(\check{f_{n}})(v)\|_{\mathcal{H}^{\prime}_{t}}.$

Hence, for all $v\in\mathcal{H}_{t}$ we have:

$\|\pi_{t}(\check{f_{n}})v\|_{\mathcal{H}^{\prime}_{t}}\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}_{t}}\omega_{t}(nR).$

The duality argument of Lemma 4.11 Item 1) implies for all unit vectors
$w\in\mathcal{H}_{t}$:

$\|\pi_{t}(\check{f_{n}})v\|_{\mathcal{H}^{\prime}_{t}}\geq|\langle\pi_{t}(\check{f_{n}})v,w\rangle|$

Eventually, we obtain the existence of $C>0$ such that for all
$v,w\in\mathcal{H}_{t}$

(6.4) $\displaystyle|\langle\pi_{t}(\check{f_{n}})v,w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}_{t}}\|w\|_{\mathcal{H}_{t}}\omega_{t}(nR).$

To conclude, use the fact that $\mathcal{I}_{t}$ defines a bounded operator on
the Hilbert space $\mathcal{H}_{t}$ and thus:

$|\langle\pi_{t}(\check{f_{n}})v,\mathcal{I}_{t}(w)\rangle|\leq
C\|\mathcal{I}_{t}\|^{\frac{1}{2}}_{L^{2}\to
L^{2}}\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}_{t}}\|w\|_{\mathcal{H}_{t}}\omega_{t}(nR).$

Since $\|\check{f_{n}}\|_{\ell^{2}}=\|f_{n}\|_{\ell^{2}}$, we finally have:

$\|\pi_{t}(f_{n})\|_{\mathcal{H}_{t}\to\mathcal{H}_{t}}\leq
C\|\mathcal{I}_{t}\|^{\frac{1}{2}}_{L^{2}\to
L^{2}}\|f_{n}\|_{\ell^{2}}\omega_{t}(nR).$

Furthermore, observe by Lemma 4.11 Item 5), that for all $v\in\mathcal{H}_{t}$
and $w\in\mathcal{H}^{{}^{\prime}}_{t}$ we have:

$\displaystyle|\langle\pi_{t}(f_{n})v,w\rangle|$
$\displaystyle=|\langle\pi_{t}(f_{n})v,\mathcal{I}_{t}\mathcal{J}_{t}w\rangle|$
$\displaystyle=|\langle\mathcal{I}_{t}\pi_{t}(f_{n})v,\mathcal{J}_{t}w\rangle|$
$\displaystyle=|\langle\pi_{-t}(f_{n})\mathcal{I}_{t}(v),\mathcal{J}_{t}w\rangle|.$

the above observation together with Inequality 6.4 imply for all
$v\in\mathcal{H}_{t}$ and $w\in\mathcal{H}^{{}^{\prime}}_{t}$:

$|\langle\pi_{-t}(f_{n})\mathcal{I}_{t}(v),\mathcal{J}_{t}w\rangle|=|\langle\pi_{t}(f_{n})v,w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v\|_{\mathcal{H}_{t}}\|w\|_{\mathcal{H}_{t}}\omega_{t}(n).$

Now, write $v=\mathcal{J}_{t}(v^{\prime})\in\mathcal{H}_{t}$ with
$v^{\prime}\in\mathcal{H}^{\prime}_{t}$ to obtain, again using Item 3 Lemma
4.11, the existence of $C>0$ such that for all
$v^{\prime},w\in\mathcal{H}^{\prime}_{t}$:

$|\langle\pi_{-t}(f_{n})v^{\prime},\mathcal{J}_{t}w\rangle|\leq
C\|f_{n}\|_{\ell^{2}}\|v^{\prime}\|_{\mathcal{H}^{\prime}_{t}}\|w\|_{\mathcal{H}^{\prime}_{t}}\omega_{t}(nR).$

And that is the second desired inequality.

∎

###  6.2. Some mixing properties of the representations $\pi_{t}$ for
$-\frac{1}{2}<t<\frac{1}{2}$

The notion of mixing and weak mixing properties has been intensively used in
ergodic theory, unitary representation theory, operator algebras. See for
example [5, p. 280 for definitions and Proposition A.1.12]. The main point
here is to define a useful notion of weak mixing property ensuring that the
_non unitary_ representations we are dealing with do not contain finite
dimensional subrepresentations.

######  Definition 6.4.

Let $(\langle\cdot,\cdot\rangle,H_{1},H_{2})$ a $\mathbb{C}$-pairing and let
$(\pi_{1},H_{1})$ and $(\pi_{2},H_{2})$ a dual system representation of
$\Gamma$ on Hilbert spaces. We say that the above dual system representation
of $\Gamma$ is weak mixing if for all finite subsets $F_{1}\subset H_{1}$,
$F_{2}\subset H_{2}$ and for all $\epsilon>0$ there exists $\gamma\in\Gamma$
such that $|\langle\pi_{1}(\gamma)v,w\rangle\langle
v^{\prime},\pi_{2}(\gamma)w^{\prime}\rangle|<\epsilon$ for all $v,w\in F_{1}$
with $v^{\prime},w^{\prime}\in F_{2}.$

######  Proposition 6.5.

If $t\in]-\frac{1}{2},\frac{1}{2}[$, then the Hilbertian dual system
representation of $\Gamma$ given by $(\pi_{t},L^{2}(\partial X,\nu_{o}))$ and
$(\pi_{-t},\overline{L^{2}(\partial X,\nu_{o})})$ is weak mixing.

###### Proof.

Let $F\subset\mathcal{H}:=L^{2}(\partial X,\nu_{o})$ and
$\overline{F}\subset\overline{\mathcal{H}}:=\overline{L^{2}(\partial
X,\nu_{o})}$ be a finite subsets of vectors of $\mathcal{H}$ and
$\overline{\mathcal{H}}$ and $\epsilon_{0}>0$ such that for all
$\gamma\in\Gamma$ we have
$|\langle\pi_{t}(\gamma)v,w\rangle\langle\pi_{-t}(\gamma)v^{\prime},w^{\prime}\rangle|\geq\epsilon_{0},$
for each pair of unit vectors $v,w\in F$ and
$v^{\prime},w^{\prime}\in\overline{F}$ . Inequality 6.1 implies for $R>0$
large enough that for all non negative integers $n$ and for all $v,w\in F$ and
$v^{\prime},w^{\prime}\in\overline{F}$:

$\sum_{\gamma\in
S^{\Gamma}_{n,R}}|\langle\pi_{t}(\gamma)v,w\rangle\langle\pi_{-t}(\gamma)v^{\prime},w^{\prime}\rangle|\leq
Ce^{2tQnR},$

where $C>0$ does depend on $t$. It follows, by considering the growth of
spheres $S^{\Gamma}_{n,R}$ that for all $n$:

$\epsilon_{0}e^{QnR}\leq Ce^{2|t|QnR}.$

and since $|t|<\frac{1}{2}$, the above inequality is absurd for $n$ large
enough. Hence the dual system representation provided by $\pi_{t}$ is weak
mixing for $-\frac{1}{2}<t<\frac{1}{2}$. ∎

We deduce the following property well known in the case of unitary
representations.

######  Proposition 6.6.

For $t\in]-\frac{1}{2},\frac{1}{2}[$, the representations
$(\pi_{t},L^{2}(\partial X,\nu_{o}))$ cannot have a finite dimensional
subrepresentations.

###### Proof.

Let $t\in]-\frac{1}{2},\frac{1}{2}]$ and $\mathcal{V}\subset L^{2}(\partial
X,\nu_{o})$ be a finite dimensional and $\pi_{t}$-invariant subspace. Let
$(v_{1},\dots,v_{m})$ an orthonormal basis of $\mathcal{V}$ of dimension $m$.
Let $v\in\mathcal{V}$ an unit vector. Write for all $\gamma\in\Gamma$:

$\displaystyle\pi_{t}(\gamma)v$
$\displaystyle=\sum^{m}_{i=1}\langle\pi_{t}(\gamma)v,v_{i}\rangle v_{i}.$

Observe that

$\langle\pi_{t}(\gamma)v,\pi_{-t}(\gamma)v\rangle=\|v\|^{2}=1.$

Now write for all $\gamma\in\Gamma$ and for all unit vectors
$v\in\mathcal{V}$:

(6.5) $\displaystyle 1=\langle\pi_{t}(\gamma)v,\pi_{-t}(\gamma)v\rangle$
$\displaystyle=\sum^{m}_{i=1}\langle\pi_{t}(\gamma)v,v_{i}\rangle\langle
v_{i},\pi_{-t}(\gamma)v\rangle.$

Now pick $0<\epsilon<\frac{1}{m}$. Therefore, Proposition 6.5 provides the
existence of $\gamma$ such that
$\bigg{|}\langle\pi_{t}(\gamma)v,v_{i}\rangle\langle
v_{i},\pi_{-t}(\gamma)v\rangle\bigg{|}<\epsilon$. Hence,

$\bigg{|}\sum^{m}_{i=1}\langle\pi_{t}(\gamma)v,v_{i}\rangle\langle
v_{i},\pi_{-t}(\gamma)v\rangle\bigg{|}<1,$

and thus equality (6.5) is absurd. ∎

As a corollary of this mixing property, we have:

######  Corollary 6.7.

If $0<t<\frac{1}{2}$, then the compact operator $\mathcal{I}_{t}$ has not
finite rank.

###### Proof.

Proposition 3.16 implies that the space $(\pi_{-t},\mbox{Im }\mathcal{I}_{t})$
is a subrepresentation of $(\pi_{-t},L^{2}(\partial X,\nu_{o}))$. Use
Proposition 6.5 to obtain that $\mathcal{I}_{t}$ is not a finite rank
operator. ∎

###  6.3. The Poisson transform and the Riesz transform

####  6.3.1. Decay of matrix coefficients

Let $A>0$. Define a subset of the boundary for $x,y\in X$ as

(6.6) $\mathcal{O}_{A}(x,y):=\\{\xi\in\partial X|(\xi,y)_{x}\geq A\\}.$

Here is the fundamental lemma concerning the matrix coefficients :

######  Lemma 6.8.

Let $t>0$ ; there exists $C$ such that for all $w\in Lip(\partial X)$ and for
all $A>0$ and for all $\gamma\in\Gamma$:

$\frac{\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},|w-w(\hat{\gamma})\textbf{1}_{\partial X}|\rangle}{\phi_{t}(\gamma)}\leq
C\|w\|_{Lip}\big{(}(\epsilon_{t}(|\gamma|)e^{2tQA}+e^{-A}\big{)},$

where $\epsilon_{t}(x)=O_{+\infty}(e^{-\alpha x})$ for some $\alpha>0$.

###### Proof.

Decompose the integral in two terms as follows:

$\displaystyle\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},|w-w(\hat{\gamma})\textbf{1}_{\partial X}|\rangle$
$\displaystyle=\int_{\mathcal{O}_{A}(o,\gamma
o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi)$ $\displaystyle+\int_{\partial
X\backslash\mathcal{O}_{A}(o,\gamma o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi).$

For the first term, we have:

$(\hat{\gamma},\xi)_{o}\geq\min\\{(\hat{\gamma},\gamma o)_{o},(\gamma
o,\xi)_{o})\\}-\delta.$

On one hand, if $(\hat{\gamma},\gamma o)_{o}\geq(\gamma o,\xi)_{o}$, then
$(\hat{\gamma},\xi)_{o}\geq(\gamma o,\xi)_{o}-\delta$ and so we have:

$\int_{\mathcal{O}_{A}(o,\gamma o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi)\leq
e^{-A+\delta}\|w\|_{Lip}\phi_{t}(\gamma),$

on the other hand if $(\hat{\gamma},\gamma o)_{o}\leq(\gamma o,\xi)_{o}$, then
$(\hat{\gamma},\xi)_{o}\geq(\hat{\gamma},\gamma o)_{o}-\delta$ and so using
Inequality (2.15) we obtain:

$\int_{\mathcal{O}_{A}(o,\gamma o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi)\leq
e^{-|\gamma|+\delta+M_{X}}\|w\|_{Lip}\phi_{t}(\gamma),$

for some $M_{X}>0$. Hence, we have:

$\int_{\mathcal{O}_{A}(o,\gamma o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi)\leq
C\|w\|_{Lip}(e^{-|\gamma|+\delta+M_{X}}+e^{-A+\delta})\phi_{t}(\gamma).$

Concerning the second term we follow the discretization process used, for
example, in Lemma 3.5. We have:

$\displaystyle\int_{\partial X\backslash\mathcal{O}_{A}(o,\gamma
o)}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma
o))}|w(\xi)-w(\hat{\gamma})|d\nu_{o}(\xi)$
$\displaystyle\leq\|w\|_{Lip}\int_{\partial
X\backslash\mathcal{O}_{A}(o,\gamma
o))}e^{(\frac{1}{2}+t)Q\beta_{\xi}(o,\gamma o)}d\nu_{o}(\xi)$
$\displaystyle=\|w\|_{Lip}e^{-(\frac{1}{2}+t)Q|\gamma|}\int_{\partial
X\backslash U_{A}}e^{(1+2t)Q(\xi,\gamma o)_{o}}d\nu_{o}(\xi)$
$\displaystyle\leq
C\cdot\|w\|_{Lip}e^{-(\frac{1}{2}+t)Q|\gamma|}\frac{1-e^{2tQA}}{1-e^{2tQ}}$
$\displaystyle\leq
C\cdot\|w\|_{Lip}\frac{e^{-tQ|\gamma|}}{\sinh(tQ)}e^{2tQA}\phi_{t}(\gamma),$

where the last inequality follows from the left-hand side of the lowed bound
(5.1). Therefore, gathering both terms we obtain:

$\displaystyle\langle\pi_{t}(\gamma)\textbf{1}_{\partial
X},|w-w(\hat{\gamma})\textbf{1}_{\partial X}|\rangle$ $\displaystyle\leq
C\|w\|_{Lip}\bigg{(}\frac{e^{-tQ|\gamma|}}{\sinh(tQ)}e^{2tQA}+e^{-|\gamma|+\delta+M_{X}}+e^{-A+\delta}\bigg{)}\phi_{t}(\gamma)$
$\displaystyle\leq
C\|w\|_{Lip}\bigg{(}\big{(}\frac{e^{-tQ|\gamma|}}{\sinh(tQ)}+e^{-|\gamma|}\big{)}e^{2tQA}+e^{-A}\bigg{)}\phi_{t}(\gamma)$
$\displaystyle=C\|w\|_{Lip}\bigg{(}\epsilon_{t}(|\gamma|)e^{2tQA}+e^{-A}\bigg{)}\phi_{t}(\gamma).$

where for $x\in\mathbb{R}_{+}$:

$\epsilon_{t}(x):=\frac{e^{-tQx}}{\sinh(tQ)}+e^{-x},$

and the proof is done. ∎

The following lemma extracts the decay of the matrix coefficient corresponding
to the second term in the above proof.

######  Lemma 6.9.

Let $t,A>0$. There exists $C$ such that for all $w\in C(\partial X)$ and for