# Sobolev spaces and uniform boundary representations

Kevin Boucher  and  Ján Špakula  School of Mathematical Sciences, University
of Southampton, Highfield, Southampton, SO17 1BJ, UK
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We prove uniform boundedness of certain boundary representations on
appropriate fractional Sobolev spaces $W^{s,p}$ with $p>1$ for arbitrary
Gromov hyperbolic groups. These are closed subspaces of $L^{p}$ and in
particular Hilbert spaces in the case $p=2$.

This construction allows us, for an appropriate choice of $p$, to approximate
the trivial representation through uniformly bounded representations. This
phenomenon does not have analogue in the setting of isometric representations
whenever the hyperbolic group considered has the Property (T) [4].

The key is the introduction of a notion of metrically conformal operator on a
metric space endowed with a conformal structure à la Mineyev [56] and a metric
analogue of the isomorphisms of Sobolev spaces induced by the Cayley transform
[3].

###### Key words and phrases:

hyperbolic groups, uniformly bounded representations, metric measure spaces,
fractional Sobolev spaces, Sobolev embedding, metric Cayley transform

###### 1991 Mathematics Subject Classification:

43A65, 20F67, 46E36

##  1\. Introduction

###  1.1. Landscape

The theory of spherical representations deals with representations realised as
spaces of ordinary functions over the flag manifold of semi-simple Lie groups.
In the broader context of geometric groups such as Gromov hyperbolic groups,
the flag manifold is substituted by a certain type of boundary. We refer to
these representations as boundary representations. This terminology was
popularised by Bader and Muchnik in [5], who investigated the irreducibility
of the Koopman representation of a negatively curved group on its visual
boundary.

Suppose that a group $\Gamma$ acts isometrically on a negatively curved space
$X$, and thus on its boundary $\partial X$ by quasi-Möbius transformations.
The boundary representations are then ones given by the formula

(1.1)
$\pi_{z}(g)\phi=\left(\tfrac{\mathrm{d}g_{*}\nu}{\mathrm{d}\nu}\right)^{z}\cdot
g^{*}\phi,$

where $z\in S=\\{w\in{\mathbb{C}}:0\leq\mathrm{Re}(w)\leq 1\\}$, $\nu$ is an
appropriate measure on $\partial X$, and $\phi\in\mathcal{D}_{z}$ for some
vector space $\mathcal{D}_{z}$ of functions on $\partial X$. We denote by
$g^{*}$ the induced action on functions, i.e.
$(g^{*}\phi)(\xi)=\phi(g^{-1}\xi)$. In this context, two natural categories
stand out: first, the _principal series_ , when $z=\frac{1}{2}+it$,
$t\in\mathbb{R}$, on $\mathcal{D}_{z}=L^{2}(\partial X,\nu)$; these are
unitary. Second, the _complementary series_ , for $z=s\in\mathbb{R}\cap S$,
which are our subject of interest. These do not have obvious unitary
structures, that is, a domain $\mathcal{D}_{s}$ with a Hilbert space norm
making $\pi_{s}$ unitary.

These two categories of representations were extensively studied and play an
important in harmonic analysis of semi-simple Lie groups [48, 47, 37]. One of
the motivations is the strong analogy between rank 1 lattices and hyperbolic
groups, and thus a potential extension of the theory to this broader class.
This was achieved for free groups by Figà-Talamanca and Piccardello, who were
able to develop an intrinsic theory of spherical representations in [29].

Little is known about the role of principal series and their properties for
general hyperbolic groups, and it is only recently that complementary series
have been investigated, cf. [10, 11, 15]. These representations, when
available, describe a path between a tempered representation (when
$z=\frac{1}{2}$) and the trivial representation (when $z$ approaches $0$ or
$1$). As such, Kazhdan’s Property (T) is a natural obstruction for their
existence for all $z\in[\frac{1}{2},1)$, and the largest admissible parameter,
$z$ in this interval, as a sort of distance from the trivial representation.

When we widen the context from unitary representations to _uniformly bounded_
representations, there is no such obvious restriction. The purpose of this
paper is to construct uniform structures111If $\pi$ is a linear representation
of a group $G$ on a complex vector space $V$, a _uniform structure_ is a norm
on $V$, such that the completion is a Hilbert space $H$, and such that $\pi$
extends to a uniformly bounded representation on $H$, i.e. $\sup_{g\in
G}\|\pi(g)\|<\infty$. for these representations regardless of Property (T).
This is motivated by the following conjectures:

######  Conjecture 1 (Spectral conjecture).

Let $\Gamma$ be a hyperbolic group. There exists $f:\Gamma\to\mathbb{R}_{+}$,
such that for every compact set $K\subset\Gamma$ and $\varepsilon>0$ there
exists a uniformly bounded representation $\pi_{K,\varepsilon}$ of $\Gamma$ on
a Hilbert space $H_{K,\varepsilon}$ without non-zero invariant vectors, and a
unit vector $v_{K,\varepsilon}\in H_{K,\varepsilon}$ such that

$\text{diam}(\pi_{K,\varepsilon}(K)v_{K,\epsilon})\leq\varepsilon,$

and for all $g\in\Gamma$ $\|\pi_{K,\varepsilon}(g)\|\leq f(g)$ 222 The last
condition on pointwise uniform boundedness over $K\subset\Gamma$ and
$\varepsilon>0$ implies that one can apply a limiting procedure (e.g. a direct
sum, or an ultraproduct) to a family of
$(\pi_{K,\varepsilon})_{K,\varepsilon}$ to obtain a single linear
representation $\pi$ of $\Gamma$ on a Hilbert space which does not have non-
zero invariant vectors, but almost has invariant vectors (in the language of
[4])..

For infinite groups with Property (T), to the best of our knowledge the
Conjecture is only known to be true for $\mathrm{Sp}(n,1)$, $n\geq 1$, due to
the work of Cowling [22, 23] using boundary representations. See also [46, 10,
11] for explicit boundary constructions. Note that higher rank lattices do not
satisfy this conjecture [25] even when the class of Banach spaces is extended
to $L^{p}$ (compare with Theorem 3).

######  Remark 1.1.

For a uniformly bounded representation $\pi$ of a group $\Gamma$, denote
$\|\pi\|=\sup_{g\in\Gamma}\|\pi(g)\|$. One may also consider a stricter
variant of Conjecture 1, where one would require a uniform bound (depending
only on $\Gamma$) on $\|\pi_{K,\varepsilon}\|$. Dooley’s work [27] implies
also this stronger statement for $Sp(n,1)$.

Conjecture 1 is closely related to another one:

######  Conjecture 2 (Attributed to Shalom, [62]).

For every hyperbolic group there exists a uniformly bounded representation on
a Hilbert space which admits a proper cocycle. Equivalently, every hyperbolic
group admits a uniformly Lipschitz proper action on a Hilbert space by affine
transformations.

Indeed, in the context of unitary representations, Schoenberg’s Theorem [7,
Prop. 2.11.1] explains that there is a correspondence between semi-groups of
unitary representations approaching the trivial one, and proper cocycles with
unitary coefficients, i.e. proper isometric actions on Hilbert spaces. As this
correspondence no longer exists in the uniformly bounded context, the two
conjectures might not be equivalent.

Clearly, a-T-menable hyperbolic groups (for example surface groups, and
generally hyperbolic groups acting geometrically on CAT(0) cube complexes)
satisfy Conjecture 2. Remarkably, this Conjecture has been recently proved
also for $\mathrm{Sp}(n,1)$, $n\geq 1$, by S. Nishikawa [60].

Another motivation to construct and study analogues of complementary series
for discrete groups comes from the research surrounding the Baum–Connes
conjecture [6, 32]. Roughly speaking, one of the ingredients in the Dirac-dual
Dirac method [45, 38] for proving the Conjecture is a path of representations
that connect the regular with the trivial representation. The purely
C*-algebraic methods ask for such a path through unitary representations and
Kazhdan’s Property (T) prevents its existence. There are two existing
approaches to bypass this problem. The first, pioneered and remarkably
successfully executed by Lafforgue [49, 50], is to step outside of the
C*-world, develop and use Banach KK-theory — this allows to use a path through
‘slowly growing representations’. The second, for $\mathrm{Sp}(n,1)$ by Julg
[42, 43, 44], uses uniform structures for the complementary series
representations [22, 23].

Uniform structures on the representations $\pi_{s}$ with $s\in[0,1]$
introduced above appear to be related to positive conformal operators and
their corresponding Sobolev spaces. It is known that those operators spectrum
are involved into important functional analytic inequality such as Lieb,
Sobolev and others [16, 51]. Our approach stress further this connection in a
purely metric setting and might be a source of interest beyond our prime
concern.

###  1.2. Results

We work in the setting of metric measure spaces. (The reader may want to keep
in mind an example of a Gromov boundary of a non-elementary hyperbolic group
or $\mathbb{R}^{n}$.) A _metric measure space_ is a triple $(Z,d,\nu)$, such
that $(Z,d)$ is a complete metric space without isolated points, and $\nu$ is
a Borel measure on $Z$. We always assume that it is _Ahlfors $D$-regular_ for
some $D\geq 0$, i.e.

$\nu(B(z,r))\asymp r^{D}\quad\text{ for $z\in Z$ and $0\leq
r<\text{diam}(Z)$.}$

In this case, $D$ is necessarily the Hausdorff dimension of $(Z,d)$ and $\nu$
is equivalent to the Hausdorff $D$-measure [53, Section 1.4.3]. We also need
to consider Ahlfors $D$-regular quasi-metric measure spaces, i.e. triples
$(Z,d,\nu)$ as above where $d$ is only a _quasi-metric_ 333Alternatively, a
_quasi-metric_ is a function satisfying the axioms of a metric with the
exception of the triangle inequality being relaxed to $d(x,y)\leq
K(d(x,z)+d(z,y))$ for some $K\geq 1$. However for the application to
hyperbolic groups it is sufficient to stick to genuine metrics throughout.,
i.e. a symmetric function $d:Z^{2}\to[0,\infty)$ for which there exists a
genuine metric $\rho\asymp d$.

For an Ahlfors $D$-regular quasi-metric measure space $(Z,d,\nu)$, $p,s\geq
0$, and a measurable function $\phi:Z\to\mathbb{C}$, we denote

(1.2) $[\phi]_{s,p}^{p}=_{\text{def}}\iint_{Z\times
Z}\frac{|\phi(\xi)-\phi(\eta)|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta).$

We define the _fractional $(s,p)$-Sobolev space_ $W^{p,s}(Z)$ to be

$W^{s,p}(Z)=\Big{\\{}\phi\in L^{p}(Z):[\phi]_{s,p}<\infty\Big{\\}},$

with the norm $\|\phi|W^{s,p}\|^{p}=\|\phi|L^{p}\|^{p}+[\phi]_{s,p}^{p}$ for
$\phi\in W^{s,p}(Z)$. The _homogeneous fractional Sobolev space_
$\accentset{\mbox{\large.}}{W}^{s,p}(Z)$ is defined as the completion of

$\\{\phi:Z\to\mathbb{C}:\phi\text{ is compactly supported, measurable and
$[\phi]_{s,p}^{p}$ finite }\\}$

with the semi-norm $[\cdot]_{s,p}$; see also Section 4. Note that when $p=2$,
these spaces are Hilbert spaces and when $Z=\mathbb{R}^{n}$ we obtain the
classical fractional Sobolev spaces [26].

Given a complete metric space $(Z,d)$ and $a\in Z$, we denote
$Z_{a}=Z\setminus\\{a\\}$ and

(1.3) $d_{a}(\xi,\eta)=_{\text{def}}\frac{d(\xi,\eta)}{d(\xi,a)d(\eta,a)}$

on $Z_{a}$. A _Cayley transform of $(Z,d)$ at $a$_ is any metric space
$(Z_{a},\rho_{a})$, such that $\rho_{a}$ induces the subspace topology on
$Z_{a}\subset Z$ and $\rho_{a}\asymp d_{a}$. This generalises the concept of a
stereographic projection within the context of metric spaces, cf. [56, Section
9]. By [36, Lemma 4.5], one can always find a metric $\rho_{a}$ bi-Lipschitz
to $d_{a}$, with Lipschitz constants independent of $a\in Z$. Moreover,
$d_{a}$ is already a metric whenever $(Z,d)$ is a Gromov boundary of a
strongly hyperbolic metric space444Any finitely generated Gromov hyperbolic
group admits a geometric action on strongly hyperbolic metric space, see [57,
59]. [56, Theorem 16]. (Mineyev refers to the collection
$\\{(Z_{a},d_{a})\\}_{a\in Z}$ as a “metric conformal structure”.) As detailed
later, Cayley transform preserves the Hausdorff dimension, and any choice of a
metric bi-Lipschitz to $d_{a}$ leads to equivalent fractional Sobolev spaces.

Our main result is the following:

######  Theorem 1.

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric measure space, $a\in
Z$, $t\in\mathbb{R}$, $s\in(0,1)$ and $p\in[1,\infty)$ with $sp<D$. Then the
map

(1.4)
$\Omega_{s+it,p}(a):W^{s,p}(Z)\rightarrow\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a});\quad\psi\mapsto
d^{\frac{2D}{p}-2s-2it}(a,\cdot)\psi|_{Z_{a}}$

is an isomorphism, with the inverse given by

(1.5)
$\Omega^{-1}_{s+it,p}(a):\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\rightarrow
W^{s,p}(Z);\quad\psi\mapsto[d^{-\frac{2D}{p}+2s+2it}(a,\cdot)\psi].$

where $[f]$ stands for the class of the $f$ in $L^{p}(Z)$. Moreover, the norms
of $\Omega_{s+it,p}(a)^{\pm 1}$ are bounded independently of $a\in Z$ and
$t\in\mathbb{R}$.

This result may be compared with the work of Astengo, Cowling, and DiBlasio
[3] for $\mathrm{Sp}(n,1)$, $n\geq 1$, as we obtain an analogous result in the
general setting of metric spaces. However the range of the parameter $s$ is
restricted to $(0,1)$. The analogy goes further in that the isomorphisms
$\Omega_{\cdot}$ come from the Radon–Nikodym derivative of the canonical map
$(Z_{a},d_{a},\nu_{a})\hookrightarrow(Z,d,\nu)$ which is — in our setting —
the inclusion.

In the same spirit as in [3], we use Theorem 1 to construct uniformly bounded
representations:
A self-homeomorphism of a metric space $(Z,d)$ is called _Möbius_ , if it
preserves cross-ratio, i.e. the quantity
$[x,y;z,w]=_{\text{def}}\frac{d(x,z)d(y,w)}{d(x,w)d(y,z)}$ for pairwise
distinct $x,y,z,w\in Z$. We denote
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ the locally compact,
second countable group of Möbius transformations of $(Z,d)$. Any
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ admits a _metric
derivative_ $|g^{\prime}|\in\operatorname{\mathrm{Lip}}_{+}(Z)$ (cf.
Preliminaries below). If $\nu$ is the $D$-Hausdorff measure on $(Z,d)$, then
$[\mathrm{d}{g_{*}\nu}/\mathrm{d}\nu](\xi)=|({g^{-1}})^{\prime}|^{D}(\xi)$ for
$\xi\in Z$, see e.g. [58, Section 4].

######  Theorem 2.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space where $\nu$ is
the Hausdorff $D$-measure on $(Z,d)$. Let $t\in\mathbb{R}$, $s\in(0,1)$ and
$p\in[1,\infty)$ with $sp<D$. For
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $\psi\in
W^{s,p}(Z)$, denote555The meaning for the parameter $s+it$ in this Theorem is
at odds with the notation ‘$z$’ used in the original formula (1.1).
Representation theory references often use (1.1), but for considerations in
this paper, the notation here is more natural. The two are related by
$s+it=\frac{2D}{p}(\frac{1}{2}-z)$.

(1.6) $\pi^{(p)}_{s+it}(g)\psi=|(g^{-1})^{\prime}|^{\frac{D}{p}-s-it}\cdot
g^{*}\psi.$

Then the representations $\pi^{(p)}_{s+it}$ of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on $W^{s,p}(Z)$ are well
defined and uniformly bounded, independently of $t\in\mathbb{R}$.

The above Theorem yields uniformly bounded boundary representations of Gromov
hyperbolic groups:

######  Example 1.2.

Let $\Gamma$ be a non-elementary Gromov hyperbolic group. Then $\Gamma$ acts
geometrically on a proper, roughly geodesic, strongly hyperbolic space
$(X,\rho)$. [57, 59]. Fix a basepoint $o\in X$. The Gromov boundary,
$Z=\partial X$, can be endowed with a visual metric $d$, such that $\Gamma$
acts on $(Z,d)$ by Möbius transformations [56, Section 5]. Furthermore,
$(Z,d)$ has finite Hausdorff dimension $D$. With the $D$-dimensional Hausdorff
measure $\nu$, $(Z,d,\nu)$ is an Ahlfors $D$-regular metric measure space
[21]. Hence the above Theorem yields uniform structures for the boundary
representations $\pi_{s+it}^{(p)}$ of $\Gamma$ for $s\in(0,1)$, $sp<D$,
$t\in\mathbb{R}$. Note that this corresponds to
$\max\\!\left(0,\frac{1}{2}-\frac{1}{D}\right)<\text{Re}(z)<\frac{1}{2}$ when
$p=2$ in the notation of (1.1).

In particular for $p=2$, we obtain a Hilbertian uniform structure for
$(\pi_{s})_{s}$ with $s\in(0,1)$ and $2s\leq D$, namely $W^{s,2}(Z)$. To the
best of our knowledge it is the first result that constructs uniformly bounded
representations which applies to arbitrary hyperbolic groups, including those
with the Property (T). Note that if $D\leq 2$, then such $\Gamma$ is
necessarily a-T-menable, as in this case $\Gamma$ acts properly by isometries
on an $L^{2}$-space by [58]. The infimum of all the constants $D\geq 0$ that
can appear in Example 1.2 is essentially Mineyev’s “hyperbolic dimension” [56,
Section 10].

Our method requires a choice of metric conformal operators (cf. Example 2.3),
and the ones we make in this paper has a limitation in their orders (see
Section 4). However, we believe that our approach will prove to be useful
beyond this work (cf. also Appendix A) to approach Conjectures 1 and 2.

Let us outline another consequence of our work. Property (T) does not prevent
existence of a proper action on a $L^{p}$-spaces in general: all hyperbolic
groups admit a proper isometric action on an $L^{p}$-space for $p\geq 2$ large
enough [64, 58, 12]. On the other hand, Bader, Furman, Gelander, and Monod [4]
proved that any locally compact second countable group $G$ with Property (T)
also satisfies the following rigidity statement: any linear isometric
representation $\rho$ of $G$ on a closed subspace of some $L^{p}$-space for
$p\not=4,6,8,\dots$ without non-zero invariant vectors does not almost have
invariant vectors, i.e. for some compact subset $K\subset G$ one has
$\inf_{\|v\|=1}\text{diam}(\rho(K)v)>0$. In other words, for Property (T)
groups, the trivial representation is isolated even among isometric
representations on subspaces of $L^{p}$-spaces. We show that for hyperbolic
groups, regardless of (T), one can approach the trivial representation through
uniformly bounded representations on closed subspaces of $L^{p}$:

######  Theorem 3.

Let $\Gamma$ be a non-elementary hyperbolic group, let $K\subset\Gamma$ be
compact, and let $(Z,d,\nu)$ be as in Example 1.2. Then for all
$p\in(D,\infty)$ and $s\in(0,\frac{D}{p})$, the uniformly bounded
representations $\pi^{(p)}_{s}$ on $W^{s,p}(Z)$ have no non-zero invariant
vectors, satisfy $\sup_{s\in(0,\frac{D}{p})}\|\pi_{s}^{(p)}(g)\|<\infty$ and

$\text{diam}(\pi_{s}^{(p)}(K){\bf 1})\xrightarrow{s\rightarrow\frac{D}{p}}0.$

It remains to observe that $W^{s,p}(Z)$ is a closed subspace of an
$L^{p}$-space (Proposition 4.4), and the constant function $\mathbf{1}$ is a
unit vector in $W^{s,p}(Z)$ for all $p\in[1,\infty)$ and $s\in(0,1)$.

This result can be interpreted as a positive answer to an $L^{p}$-version of
Conjecture 1.

We conclude this Section with a description of the dual representations in the
Hilbert space case, i.e. for the rest of this Section we assume that $p=2$,
and denote $H^{s}=W^{s,2}$. We denote the dual to the space $H^{s}$ by
$H^{-s}$. Similarly as for the free groups and in the smooth case, the norm on
$H^{-s}(Z)$ can be expressed in terms of potentials, using potential theory on
metric measure spaces [34, 39, 20].

Before stating the result, recall that the Knapp–Stein operator

(1.7)
$I_{s}(\phi)(\eta)=\int_{Z}\frac{\phi(\xi)}{d^{D-2s}(\xi,\eta)}\mathrm{d}\nu(\xi)$

was studied in [10] to describe a unitary structure for $\pi_{-s}^{(2)}$ when
it is positive definite. This is the case for example when $(Z,d)$ arises as a
boundary of a roughly geodesic strongly hyperbolic metric space $(X,\rho)$
where $\rho$ is conditionally negative definite, see [10]. See also [35] and
[15] for similar considerations.666Let us emphasize that in the existing work
relating to the Knapp–Stein operators as above (alternatively, one may see
this as working on the space of Bessel or Riesz potentials), positivity in
some form has to be _assumed_ , and Property (T) prevents that. The results in
this paper do not need such an assumption; we work in the ‘dual’ picture which
involves Sobolev spaces.

######  Theorem 4.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space where $\nu$ is
the Hausdorff $D$-measure on $(Z,d)$. Let $s\in(0,1)$ with $2s<D$. Then
$L^{2}(Z)$ is dense in $H^{-s}(Z)$, and there exists positive definite
integral operator with kernel $k$ on $L^{2}(Z)$ satisfying $k\asymp d^{-D+2s}$
such that the norm on $H^{-s}(Z)$ is given by:

$\left\|\phi|H^{-s}(Z)\right\|^{2}=\iint_{Z\times
Z}\\!\phi(\xi)\overline{\phi(\eta)}k(\xi,\eta)\,\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta),\quad\text{for
all }\phi\in L^{2}(Z).$

The representation $\pi_{-s}^{(2)}$ of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ defined by (1.6) extends
to a uniformly bounded representation on $H^{-s}(Z)$.

###  1.3. Organisation

In Section 2 we give an overview of our approach. Section 3 contains the
prerequisites. In Section 4, we define the fractional Sobolev spaces and show
their basic properties. Section 5 then concerns the fractional Sobolev
inequality. In Section 6 we prove Lorentz norm bounds of certain multipliers,
needed in Section 7 where we prove Theorem 1, namely that the operators
associated to the Cayley transform are isomorphisms of Sobolev spaces. In
Section 8 we apply this to show that the boundary representations are
uniformly bounded, i.e. Theorem 2. In Section 9 we prove Theorem 3: for
sufficiently large $p$, $\pi_{s}^{(p)}$ approach the trivial representation.
In Section 10 we use potential theory, when $p=2$, to describe the dual spaces
and dual representations in terms of kernel operators analogous to the
Knapp–Stein operator. In Appendix B we give calculations postponed from
Section 2. In Appendix A we clarify the groupoid picture, and finally in
Appendix C we indicate more formally how can one think of the metric conformal
bundle as a tangent space, and explain rescaling of operators.

###  1.4. Acknowledgements

The authors were supported (JS partially) by EPSRC Standard Grant
EP/V002899/1. JS would like to thank Bogdan Nica for introducting JS to this
subject, and for helpful conversations.

For the purpose of open access, the author has applied a CC BY public
copyright licence to any Author Accepted Manuscript version arising from this
submission.

##  2\. Outline of the construction

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric measure space, where
$\nu$ is the Hausdorff $D$-measure. We sketch our approach to (boundary)
representations in the case when $p=2$, $t=0$, and assuming the conformal
inversion $d_{a}$ from (1.3) is a genuine metric on $Z_{a}=Z\setminus\\{a\\}$.
We start by explaining the idea and framework around Theorem 2, which
eventually exposes the role of Theorem 1 in the story.

For readers familiar with the language of groupoids, the principal idea is the
following. Any uniformly bounded representation of an _amenable_ group can be
unitarised, i.e. written as a unitary representation conjugated by an
invertible. More generally, suppose that a group $G$ acts amenably on
$(Z,\nu)$. Starting with a uniformly bounded representation $\tau$ of $G$ on a
Hilbert space $H$, we extend it to a representation of the transformation
groupoid $G\ltimes Z$ (on the constant Hilbert bundle over $Z$ with fibre
$H$). Applying amenability of $G\ltimes Z$, it splits into a unitary
representation of the groupoid $G\ltimes Z$, and an equivariant cocycle over
$Z$ which encodes the analytic properties of $\tau$.

We proceed “backwards”: by describing a convenient (metric) bundle over $Z$,
and showing how a choice of a “conformal operator on the bundle” yields an
isometric (unitary) representation of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$ on an
appropriate Hilbert bundle. Finally, we show how to use the Cayley transform
to obtain a uniformly bounded cocycle, yielding uniformly bounded
representations of $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$.

The (metric) bundle we will consider is the collection $(Z_{a},d_{a})_{a\in
Z}$ (defined by (1.3); see also Preliminaries below). We can consider each
$(Z_{a},d_{a})$ as a cotangent space to $Z$ at $a$, see Appendix C.
Furthermore, $Z_{a}$ is an Ahlfors $D$-regular metric measure space when
endowed with the measure $\nu_{a}$ given by
$\mathrm{d}\nu_{a}(\xi)=d^{-2D}(a,\xi)\mathrm{d}\nu(\xi)$. The bundle carries
a natural (diagonal) action of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ by

$g(a,\xi)=(ga,g\xi)\in Z_{ga},\quad\text{ for all
}g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\text{ and }\xi\in
Z_{a},$

which induces an action of the groupoid
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$. For the
purposes of this paper, it is sufficient to consider
$(Z_{a},d_{a},\nu_{a})_{a\in Z}$ as simply a collection of metric measure
spaces. However, let us point out that we can understand it as a genuine
bundle by considering the total space

$TZ=_{\text{def}}Z\times Z\setminus\text{diagonal}\simeq(Z_{a},d_{a})_{a\in
Z}.$

This space carries a
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$-invariant measure $m$
given as $m=\int\nu_{a}\mathrm{d}\nu(a)$ (or
$\mathrm{d}m(\xi,\eta)=d^{-2D}(\xi,\eta)\,\mathrm{d}\nu(\xi)\,\mathrm{d}\nu(\eta)$),
which is uniquely determined when $Z$ is obtained from a geometric action of a
hyperbolic group, and called Bowen–Margulis–Sullivan measure [55], see also
[58, Subsection 7.1].

######  Remark 2.1.

The Koopman (or quasi-regular) representation from this diagonal action of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on $(TZ,m)$ can be
interpreted as a unitary representation of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$ on the Hilbert
bundle $(L^{2}(Z_{a},\nu_{a}))_{a\in Z}$ given by

$\phi\mapsto|g^{\prime}|^{-D}(a)g^{*}\phi\in
L^{2}(Z_{ga},\nu_{ga}),\quad\text{ for
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ and $\phi\in
L^{2}(Z_{a},\nu_{a})$.}$

For $s\geq 0$, we define a
$(\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z),s)$-conformal operator on
the bundle $TZ$ to be a collection $(\mathcal{D}_{a},T_{a})_{a\in Z}$ of
densely defined closed operators $T_{a}$ on $L^{2}(Z_{a},\nu_{a})$, such that
for all $g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ and
$\phi\in\mathcal{D}_{a}$, $g^{*}\phi\in\mathcal{D}_{ga}$ 777Recall that the
notation $g^{*}$ denotes the induced action of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on functions: for
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $a\in Z$, and
$\phi:Z_{a}\to\mathbb{C}$, we have $g^{*}\phi:Z_{ga}\to\mathbb{C}$ is given by
$(g^{*}\phi)(\xi)=\phi(g^{-1}\xi)$. and

$T_{ga}(g^{*}\phi)=|g^{\prime}|^{2s}(a)\,g^{*}(T_{a}\phi).$

We can think of $(\mathcal{D}_{a},T_{a})_{a\in Z}$ as a decomposition of a
single operator on $L^{2}(TZ,m)$.

######  Remark 2.2.

This definition is motivated by the following notion of metric conformality: a
densely defined operator $T$ on $L^{2}(Z)$, with a
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$-invariant domain
$\mathcal{D}(T)$, is called
_$(\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z),s,s^{\prime})$
-conformal_ (for $s,s^{\prime}\in\mathbb{R}$); or _$s$ -conformal_ if
$s^{\prime}=1-s$; if it satisfies (with notation of (1.1))

$T\pi_{s}(g)=\pi_{s^{\prime}}(g)T\quad\text{for
}g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\text{ and
}\phi\in\mathcal{D}(T).$

A fundamental example of an $s$-conformal operator is the Knapp–Stein operator
(1.7). The motivation for considering operators on bundles is a rescaling
phenomenon observed for conformal operators on $Z$, see Appendix C.

######  Example 2.3.

Suppose that $X$ is a $k$-regular tree for some integer $k\geq 3$, and fix a
vertex $o\in X$. The boundary $Z=\partial X$ is the set of infinite geodesic
paths starting from $o$. For two such paths, $\xi,\eta\in Z$, denote
$\langle\xi,\eta\rangle_{o}$ the length of the longest common subpath of $\xi$
and $\eta$ (also known as the Gromov product of $\xi$ and $\eta$). Then
$d(\xi,\eta)=\exp(-\langle\xi,\eta\rangle)$ is a metric on $Z$ of Hausdorff
dimension $D=\ln(k-1)$. Let $\nu$ be the $D$-Hausdorff measure on $(Z,d)$. For
$s\geq 0$ and $a\in Z$, consider

(2.1)
$\Delta_{a}^{s}\phi(\xi)=-\int_{Z_{a}}\frac{\phi(\eta)-\phi(\xi)}{d_{a}^{D+2s}(\xi,\eta)}\,\mathrm{d}\nu_{a}(\eta),$

where $\phi:Z_{a}\to\mathbb{C}$ is locally constant, zero near $a$, and
$\xi\in Z_{a}$. Then $(\Delta_{a}^{s})_{a\in Z}$ (with appropriate domains) is
a $(\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z),s)$-conformal operator;
see Calculation B.1.

Given an $s$-conformal operator $(\mathcal{D}_{a},T_{a})_{a\in Z}$ as above,
suppose further that each $T_{a}$ is a positive operator. Then the expression
$\left(T_{a}\phi,\phi\right)_{L^{2}(Z_{a})}$ defines a Hilbert space norm on
$\mathcal{D}_{a}$; denote by $\mathcal{H}_{T_{a}}$ its completion. For
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ and $a\in Z$, denote

(2.2)
$\Pi_{s}(g;a):\mathcal{H}_{T_{a}}\to\mathcal{H}_{T_{ga}},\quad\Pi_{s}(g;a)\phi=|g^{\prime}|^{\frac{D}{2}-s}(a)\,g^{*}\phi.$

Then each $\Pi_{s}(g;a)$ is well defined and unitary (Calculation B.2), and
consequently $\Pi_{s}$ is a unitary representation of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$, see (8.2) and
Appendix A. In other words, there is a correspondence between unitary
structures for $\Pi_{s}$ and positive definite $s$-conformal operators over
$TZ$.

If we further assume that each $T_{a}$ is an infinitesimal generator of a
Dirichlet form, we can describe the Hilbert spaces $\mathcal{H}_{T_{a}}$ in
terms of these forms. In particular, the fractional Laplacians
$(\Delta^{s}_{a})$ from Example 2.3 are associated to the “energy functionals”

$\mathcal{E}_{a}(\psi)=[\psi]_{s,2}^{2}=\iint\frac{|\psi(\xi)-\psi(\eta)|^{2}}{d_{a}^{D+2s}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\mathrm{d}\nu_{a}(\eta)\quad\text{for
$\psi\in\mathcal{D}_{a}$},$

leading to the homogeneous fractional Sobolev spaces
$\accentset{\mbox{\large.}}{W}^{s,2}(Z_{a})$ from the Introduction. In the
remainder of the paper, we consider the forms $[\cdot]_{s,p}$ for $s\in(0,1)$
on general Ahlfors $D$-regular metric measure spaces, and keep the notation
$\Delta_{a}^{s}$ for the infinitesimal generator of $[\cdot]_{s,2}$.

To obtain a representation of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ from $\Pi_{s}$, we
additionally need an equivariant cocycle, that is, a linear map
$c_{s}(b,a):\accentset{\mbox{\large.}}{W}^{s,2}(Z_{a})\to\accentset{\mbox{\large.}}{W}^{s,2}(Z_{b})$
for all $a,b\in Z$, satisfying the relations
$c_{s}(b,a)c_{s}(a,d)=c_{s}(b,d)$, $c_{s}(a,a)=\mathrm{Id}$, and

(2.3) $c_{s}(gb,ga)\Pi_{s}(g;a)=\Pi_{s}(g;b)c_{s}(b,a)$

for all $a,b,d\in Z$,
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$. Given such a cocycle
$c_{s}$ and $a\in Z$, the formula

(2.4) $\pi_{a,s}(g)=c_{s}(a,ga)\,\Pi_{s}(g;a)$

defines a representation of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on
$\accentset{\mbox{\large.}}{W}^{s,2}(Z_{a})$, which is uniformly bounded if
$c_{s}$ is uniformly bounded, i.e. if $\|c_{s}(\cdot,\cdot)\|$ is a bounded
function on $Z\times Z$.

######  Remark 2.4.

It follows from equivariance that $c(b,a)^{-1}\pi_{b,s}c(b,a)=\pi_{a,s}$ for
all $a\in Z$, and thus any choice of $a\in Z$ leads to a similar
representation.

In our case, we obtain the cocycle from the metric derivative of the
identification of $Z_{a}$ and $Z_{b}$ through the Cayley transforms; namely we
let

(2.5) $c_{s}(b,a)=\Omega_{s,2}(b)\circ\Omega_{s,2}^{-1}(a)\quad\text{for
}a,b\in Z.$

By Theorem 1, $c_{s}$ is a uniformly bounded cocycle.

${\accentset{\mbox{\large.}}{W}^{s,2}(Z_{a})}$${\accentset{\mbox{\large.}}{W}^{s,2}(Z_{ga})}$${W^{s,2}(Z)}$$\scriptstyle{c_{s}(a,ga)}$$\scriptstyle{\Pi_{s}(g;a)}$$\scriptstyle{\Omega_{s,2}^{-1}(ga)}$$\scriptstyle{\Omega_{s,2}(a)}$

The covariance relation (2.3) follows from a direct calculation (see 8.3)
showing that for $g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ we
actually have

(2.6) $\pi_{s}(g)=\Omega_{s,2}^{-1}(ga)\Pi_{s}(g;a)\,\Omega_{s,2}(a)$

(as operators on $W^{s,2}(Z)$), where $\pi_{s}$ is the original boundary
representation, given by
$\pi_{s}(g)\phi=|(g^{-1})^{\prime}|^{\frac{D}{2}-s}\cdot g^{*}\phi$.
Consequently, it is uniformly bounded on $W^{s,2}(Z)$.

###  2.1. A note about the general case

The $L^{p}$-version ($p\geq 2$) of the above construction is obtained by
considering non-linear $(p-1)$-homogeneous operators, namely the
$p$-fractional Laplacian, associated to the energy functional $[\cdot]_{s,p}$
from (1.2). As our approach deals primarily with the energy functionals, the
arguments remain the same as in the Hilbert space case.

##  3\. Preliminaries

###  3.1. Notation

For non-negative real-values functions $f,g$, we write $f\prec g$ if $\exists
c>0$ with $f\leq cg$. We also write $f\asymp g$ when $f\prec g$ and $g\prec
f$. In practice the constant(s) $c$ may depend on some of the parameters
appearing in the formulas (e.g. $p$, $s$, $D$); this will be clear from the
context, or specified as in $\asymp_{s,p}$.

If $(Y,d)$ is a metric space, we denote the balls by $B(x,r)=\\{y\in
Y:d(x,y)<r\\}$. We denote by $C(Y)$ the space of continuous functions
$\phi:Y\to\mathbb{C}$, by $C_{0}(Y)\subseteq C(Y)$ the subspace of functions
vanishing at infinity, and $C_{c}(Y)$ the subspace of the compactly supported
ones. We consider the latter two endowed with the supremum norm. We denote by
$\operatorname{\mathrm{Lip}}(Y)\subseteq C(Y)$ the set of Lipschitz functions,
i.e. functions $\phi:Y\to\mathbb{C}$ for which there exists $L\geq 0$ with
$|\phi(x)-\phi(y)|\leq Ld(x,y)$ for all $x,y\in Y$.

###  3.2. Metric measure spaces and Möbius maps

Let $(Z,d,\nu)$ be a metric measure space, that is, a complete metric space
$(Z,d)$ without isolated points, and a Borel measure $\nu$ on $Z$. We will
exclusively work with ones which are also Ahlfors $D$-regular, i.e. for which
we have $\nu(B(x,r))\asymp r^{D}$ for $0\leq r<\text{diam}(Z)$ and $x\in Z$.
It follows from [53, Section 1.4.3] that $D$ is then the Hausdorff dimension
of $(Z,d)$ and $\nu$ is equivalent to the $D$-dimensional Hausdorff measure on
$(Z,d)$. In the context of representations, we need a tighter relation between
$d$ and $\nu$, namely that $\nu$ _is_ the $D$-dimensional Hausdorff measure,
as explained below.

Recall that a Möbius map is a function $g:(Z,d)\to(Y,d^{\prime})$ between
metric space which preserves cross-ratios, i.e. the quantity
$[x,y;z,w]=\frac{d(x,z)d(y,w)}{d(x,w)d(y,z)}$, whenever this expression makes
sense. We denote by $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ the
group of self-homeomorphisms of $(Z,d)$. For any
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ there exists a
_metric derivative_ of $g$, that is, a Lipschitz function
$|g^{\prime}|:Z\to(0,\infty)$ satisfying the _geometric mean value property_

(3.1)
$d^{2}(g\xi,g\eta)=|g^{\prime}|(\xi)\,|g^{\prime}|(\eta)\,d^{2}(\xi,\eta)\qquad\text{for
all }g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z),\,\xi,\eta\in Z.$

see e.g. [58, Lemmas 6, 7] (cf. also [63]). Now, if $\nu$ is the
$D$-dimensional Hausdorff measure which is also Ahlfors regular, $g$ preserves
the class of $\nu$ and (a power of) $|g^{\prime}|$ gives the Radon–Nikodym
derivative of the action (see e.g. [58, Lemma 8]):

(3.2)
$\mathrm{d}\nu(g\xi)=|g^{\prime}|^{D}(\xi)\,\mathrm{d}\nu(\xi)\qquad\text{for
all }g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\text{ and }\xi\in
Z.$

We further have a cocycle formula

(3.3)
$|(gh)^{\prime}|(\xi)=|g^{\prime}|(h\xi)\,|h^{\prime}|(\xi)\qquad\text{for all
}g,h\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\text{ and }\xi\in
Z.$

It follows that the formula (1.6) in Theorem 2 defines a group homomorphism,
and (up to adjusting the labelling of the parameters) agrees with (1.1).
Finally, the above formulas imply that we have the following formal adjoint
relation (with respect to the $L^{2}$-pairing) for
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$:

(3.4) $\displaystyle\pi_{z}^{(p)}(g)^{*}=\pi_{-\overline{z}}^{(q)}(g^{-1}).$

As defined in the Introduction, for $a\in Z$ we consider the _Cayley
transform_ at $a$: denote

(3.5) $Z_{a}=Z\setminus\\{a\\},\quad
d_{a}(\xi,\eta)=\frac{d(\xi,\eta)}{d(\xi,a)d(\eta,a)},\quad\mathrm{d}\nu_{a}(\xi)=d^{-2D}(a,\xi)\mathrm{d}\nu(\xi).$

Then $d_{a}$ is a (complete) quasi-metric on $Z_{a}$, i.e. there exists a
metric $\rho_{a}$ on $Z_{a}$ satisfying
$\frac{1}{4}d_{a}(\xi,\eta)\leq\rho_{a}(\xi,\eta)\leq d_{a}(\xi,\eta)$ for all
$\xi,\eta\in Z_{a}$ by [36, Lemma 4.5]. However, as we need exact formulas
(below), rather than passing to $\rho_{a}$, we continue to work with quasi-
metric (measure) spaces. In particular, the notation $B(x,r)$ for balls in
$Z_{a}$ will denote the “quasi-balls” with respect to $d_{a}$. (As pointed out
in the Introduction, when $Z$ is the Gromov boundary of a strongly hyperbolic,
quasi-geodesic, proper metric space [59], then $d_{a}$ is complete metric
space by [56, Theorem 16].)

Denote by $\mathcal{C}_{a}:(Z_{a},d_{a})\to(Z,d)$ the inclusion. It is a
cross-ratio preserving map [56, Proposition 18], and its metric derivative is

(3.6) $|\mathcal{C}_{a}^{\prime}|(\xi)=d^{-2}(\xi,a).$

Turning to measures, the formula (3.5) for $\nu_{a}$ defines a Borel measure
on $Z_{a}$, which is Ahlfors $D$-regular, i.e. $\nu_{a}(B(x,r))\asymp r^{D}$
for $x\in Z_{a}$, $0\leq r$, and the constants in ‘$\asymp$’ are uniform over
$a\in Z$.

To be able to refer to spaces like these, let us state that by an _Ahlfors
$D$-regular quasi-metric measure space_ we shall mean a triple $(Y,d,\nu)$,
where $d$ is a complete quasi-metric on $Y$, $\nu$ is a Borel measure on
$(Y,d)$, and $\nu(B(y,r))\asymp r^{D}$ for $y\in Y$, $0\leq r<\text{diam}(Y)$.
Note that the topology on $Y$ depends only on the equivalence class of $d$,
and likewise Ahlfors regularity remains the same (possibly adjusting the
$\asymp$-constants) under replacing either/both $d$ and $\nu$ by equivalent
quasi-metrics or measures.

We close the subsection with calculation of the transformation rules for the
action of $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on the (metric
conformal bundle) $(Z_{a},d_{a},\nu_{a})_{a\in Z}$. For
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $a\in Z$, and all
$\xi,\eta\in Z_{a}$, we have

(3.7) $\displaystyle d_{ga}(g\xi,g\eta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Z_a-d_a-nu_a-
defn}}}{{=}}\frac{d(g\xi,g\eta)}{d(g\xi,ga)d(g\eta,ga)}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:GMV}}}{{=}}\frac{|g^{\prime}|^{\frac{1}{2}}(\xi)\,|g^{\prime}|^{\frac{1}{2}}(\eta)}{|g^{\prime}|^{\frac{1}{2}}(\xi)\,|g^{\prime}|^{\frac{1}{2}}(\eta)\,|g^{\prime}|(a)}\cdot\frac{d(\xi,\eta)}{d(\xi,a)d(\eta,a)}=\frac{d_{a}(\xi,\eta)}{|g^{\prime}|(a)},$

and similarly

(3.8)
$\displaystyle\mathrm{d}\nu_{ga}(g\xi)\stackrel{{\scriptstyle\eqref{eq:Z_a-
d_a-nu_a-
defn}}}{{=}}\frac{\mathrm{d}\nu(g\xi)}{d^{2D}(ga,g\xi)}\stackrel{{\scriptstyle\eqref{eq:change-
of-
var}\eqref{eq:GMV}}}{{=}}\frac{|g^{\prime}|^{D}(\xi)\mathrm{d}\nu(\xi)}{|g^{\prime}|^{D}(\xi)\,|g^{\prime}|^{D}(a)\,d^{2D}(a,\xi)}\stackrel{{\scriptstyle\eqref{eq:Z_a-
d_a-nu_a-defn}}}{{=}}\frac{\mathrm{d}\nu_{a}(\xi)}{|g^{\prime}|^{D}(a)}.$

###  3.3. Lorentz spaces

Recall that Lorentz spaces [52, 41] are function spaces that interpolate
between $L^{p}$-spaces. We recall the results we will need; for general theory
see e.g. textbook [33]. Let $(Z,\nu)$ be a non-atomic $\sigma$-finite measure
space. For a measurable $f:Z\to\mathbb{C}$, denote its non-increasing
rearrangement by $f^{*}:\mathbb{R}^{+}\to\mathbb{R}^{+}$,
$f^{*}(t)=\inf\\{s>0\mid\nu(\\{|f|>s\\})\leq t\\}$. For $1\leq p,q\leq\infty$,
define the _$(p,q)$ –Lorentz norm_ by

$\left\|f\,|L(Z;p,q)\right\|=\left\|t\mapsto
t^{1/p}f^{*}(t)\,|L^{q}(\tfrac{\mathrm{d}t}{t})\right\|.$

We write also $L(p,q)$ when the measure space is clear from the context. One
has that $L(p,p)=L^{p}$, and $L(p,\infty)$ is the _weak $L^{p}$-space_.

######  Proposition 3.1 ([33, Proposition 1.4.9]).

If $1\leq q\leq p<\infty$, then

$\left\|f\,|L(p,q)\right\|=p^{1/q}\left\|s\mapsto
s\cdot\nu(\\{|f|>s\\})^{1/p}\,|L^{q}(\tfrac{\mathrm{d}s}{s})\right\|.$

######  Proposition 3.2 (Inclusions [33, Proposition 1.4.10]).

For $1\leq p\leq\infty$ and $1\leq q<r\leq\infty$, one has $L(p,q)\subseteq
L(p,r)$.

######  Proposition 3.3 (Hölder-type inequality [33, Theorem 1.4.16]).

Let $p,p^{\prime},q,q^{\prime}\in[1,\infty]$ satisfy
$\frac{1}{p}+\frac{1}{p^{\prime}}=1=\frac{1}{q}+\frac{1}{q^{\prime}}$. Then
for $f\in L(p,q)$ and $h\in L(p^{\prime},q^{\prime})$, we have

$\int|f(\xi)h(\xi)|d\xi\leq\|f\mid L(p,q)\|\cdot\|h\mid
L(p^{\prime},q^{\prime})\|.$

######  Lemma 3.4 ([33, Remark 1.4.7]).

Let $p\in[1,\infty)$, $a\geq 1$. For any $u\in L(pa,a)$:

$\left\||u|^{a}\mid L(p,1)\right\|=\left\|u\mid L(ap,a)\right\|^{a}.$

##  4\. Fractional Sobolev spaces

For this section, let $(Y,d,\nu)$ be an Ahlfors $D$-regular quasi-metric
measure space. Fix also $1\leq p<\infty$ and $s\in(0,\infty)$.

Given a measurable function $\phi:Y\to\mathbb{C}$, recall from (1.2) that its
_Gagliardo seminorm_ is given by:

(4.1) $[\phi]_{s,p}=\left(\iint_{Y\times
Y}\frac{|\phi(\xi)-\phi(\eta)|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)\right)^{1/p}.$

Recall from the Introduction that the _fractional Sobolev space_ $W^{s,p}(Y)$
on $(Y,d,\nu)$ is the function space $W^{s,p}(Y)=\\{\phi\in
L^{p}(Y,\nu):[\phi]_{s,p}<\infty\\}$, endowed with norm
$\|\phi\,|\,W^{s,p}(Y)\|^{p}=\|\phi\,|\,L^{p}\|^{p}+[\phi]_{s,p}^{p}$, also
called the Gagliardo, or Aronszajn, or Slobodeckij norm [26, Section 2]. We
also need the homogeneous fractional Sobolev space
$\accentset{\mbox{\large.}}{W}^{s,p}(Y)$, which is defined as the completion
of the set of compactly supported, measurable functions $\phi:Z\to\mathbb{C}$
with $[\phi]_{s,p}<\infty$ in the semi-norm $[\cdot]_{s,p}$. Note that for
instance constant functions have zero Gagliardo seminorm. For a discussion
about homogeneous Sobolev spaces in the context of $\mathbb{R}^{n}$, see [17]
and references therein.

######  Remark 4.1.

When $Y=\mathbb{R}^{n}$ ($n\geq 1$) with the Euclidean metric and Lebesgue
measure, $W^{s,p}(\mathbb{R}^{n})=\\{0\\}$ for $s\geq 1$, see the discussion
after [26, Proposition 2.2]. In this case, for $s\in(0,1)$, we obtain the
fractional Sobolev spaces that interpolate between $L^{p}(\mathbb{R}^{n})$ and
the Sobolev spaces $W^{1,p}(\mathbb{R}^{n})$ of $L^{p}$-functions with
$p$-integrable gradients [26].

On the other hand if $Y$ is totally disconnected, for instance a finite
dimensional vector space over the $q$-adic numbers ($q$ prime) viewed as a
metric space and its corresponding Hausdorff measure, then $W^{s,p}(Y)$ is
dense in $L^{p}(Y)$ for all $s>0$ [8].

######  Remark 4.2.

Note that both $W^{s,p}(Y)$ and $\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ both
contain compactly supported Lipschitz functions. This is a classical
consequence of $D$-Ahlfors regularity: The value of $[\phi]_{s,p}$ for any
compactly supported Lipschitz function $\phi:Y\to\mathbb{C}$ is bounded from
above by an integral for the form $\iint_{A\times
A}C/d^{\alpha}(\xi,\eta)\,\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$ for some
$0\leq\alpha<D$, a bounded set $A\subset Y$ and a constant $0\leq C<\infty$.
Any such integral is finite, cf. for example Lemma 6.1.

Consequently, $W^{s,p}(Y)$ is dense in $L^{p}(Y)$, and both $C_{c}(Y)\cap
W^{s,p}(Y)$ and $C_{c}(Y)\cap\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ are dense
in $C_{0}(Y)$. In fact, they are also dense in $W^{s,p}(Y)$ and
$\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ respectively, i.e. our Sobolev spaces
are _regular_. This has been communicated to us by D. Gerontogiannis (private
communication; a trace theory argument).

######  Proposition 4.3.

Let $(Y,d,\nu)$ be an Ahlfors $D$-regular quasi-metric measure space,
$s\in(0,\infty)$, and $p\in[1,\infty)$. Then $W^{s,p}(Y)$, and
$\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ when $Y$ is unbounded, are complete.
Consequently, they are Banach spaces.

###### Proof.

Let $(u_{n})_{n}$ be a Cauchy sequence in $W^{s,p}(Y)$. Then $(u_{n})_{n}$ is
Cauchy in $L^{p}$, and thus converges to some $u\in L^{p}$ in $L^{p}$-norm.
Consequently, one can extract a subsequence, $(v_{n})_{n}$, which converges
pointwise to $u$. Thus by Fatou’s Lemma, we have for any $n\in\mathbb{N}$

$\displaystyle[v_{n}-u]^{p}_{s,p}$
$\displaystyle=\iint\frac{|(v_{n}(\xi)-u(\xi))-(v_{n}(\eta)-u(\eta))|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\xi\mathrm{d}\eta$
$\displaystyle=\iint\liminf_{m}\frac{|(v_{n}(\xi)-v_{m}(\xi))-(v_{n}(\eta)-v_{m}(\eta))|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\xi\mathrm{d}\eta$
$\displaystyle\leq\liminf_{m}\iint\frac{|(v_{n}(\xi)-v_{m}(\xi))-(v_{n}(\eta)-v_{m}(\eta))|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\xi\mathrm{d}\eta$
$\displaystyle=\liminf_{m}\,[v_{n}-v_{m}]_{s,p}^{p}.$

Consequently, using the triangle inequality for the Gagliardo seminorm, we
have

$\displaystyle[u_{n}-u]_{s,p}$
$\displaystyle\leq[u_{n}-v_{n}]_{s,p}+[v_{n}-u]_{s,p}$
$\displaystyle\leq[u_{n}-v_{n}]_{s,p}+\liminf_{m}[v_{n}-v_{m}]_{s,p}\xrightarrow{n\rightarrow\infty}0.$

Thus $u\in W^{s,p}(Y)$ and $u_{n}\to u$ in $W^{s,p}(Y)$.

For $\accentset{\mbox{\large.}}{W}^{s,p}(Y)$, the claim follows from an
analogous argument, first applying the Fractional Sobolev Inequality (Theorem
5) which provides a continuous embedding of
$\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ into $L^{p^{*}}(Y)$ for
$p^{*}=\frac{pD}{D-sp}$ when $Y$ is unbounded. ∎

######  Proposition 4.4.

Let $(Y,d,\nu)$ be an Ahlfors $D$-regular quasi-metric measure space,
$s\in(0,\infty)$, and $p\in[1,\infty)$. Then both $W^{s,p}(Y)$, and
$\accentset{\mbox{\large.}}{W}^{s,p}(Y)$ when $Y$ is unbounded, are closed
subspaces of an $L^{p}$-space. Consequently, they are reflexive.

###### Proof.

Define a measure on $Y\times Y$ by
$\mathrm{d}\mu(\xi,\eta)=d^{-D-sp}(\xi,\eta)\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$.
By standard arguments, this is a $\sigma$-finite Borel measure on $Y\times Y$.
For a measurable $\phi:Y\to\mathbb{C}$, denote
$D(\phi)(\xi,\eta)=\phi(\xi)-\phi(\eta)$. Then the linear maps

$\displaystyle T$ $\displaystyle:W^{s,p}(Y)\to L^{p}(Y)\oplus L^{p}(Y\times
Y;\mu),$ $\displaystyle\phi\mapsto(\phi,D(\phi))$ $\displaystyle D$
$\displaystyle:\accentset{\mbox{\large.}}{W}^{s,p}(Y)\to L^{p}(Y\times Y;\mu)$

are isometries (just from the definitions of the norms), and thus have closed
ranges. ∎

##  5\. A tight fractional Sobolev inequality

The purpose of this section is to prove the following Theorem. In this
section, the symbols $\prec,\asymp$ mean ‘up to a multiplicative constant
depending only on the space, $s$, and $p$’.

The results in this section are stated for Ahlfors regular metric measure
spaces, but the conclusion(s) follow for Ahlfors $D$-regular quasi-metric
measure spaces as well, replacing a metric $d$ by an equivalent quasi-metric
$\rho$ only requires possibly adjusting the constants hidden in ‘$\prec$’.

######  Theorem 5.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space (not
necessarily bounded), $s\in(0,\infty)$, and $p\geq 1$, such that $sp<D$.
Denote $p^{*}=\frac{pD}{D-sp}$. Then for any compactly supported measurable
function $f:Z\to\mathbb{C}$ we have

(5.1)
$\left\|f|L(p^{*},p)\right\|^{p}\prec\nu(Z)^{\frac{-sp}{D}}\left\|f|L^{p}\right\|^{p}+\iint_{Z^{2}}\frac{\left|f(\xi)-f(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\,\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta).$

There is an extensive literature about fractional Sobolev inequality in the
context of (open subsets of) $\mathbb{R}^{n}$; often having $\|f|L^{p^{*}}\|$
on the left-hand side, and without the $L^{p}$ term on the right-hand side
(which also disappears in (5.1) when $Z$ is unbounded). In view of the
continuous inclusion $L(p^{*},p)\subseteq L(p^{*},p^{*})=L^{p^{*}}$ [33,
Proposition 1.4.10], the $L^{p^{*}}$-version follows from the Lorentz
$L(p^{*},p)$-version. This tighter Lorentz version is also well known in the
context of $\mathbb{R}^{n}$, see e.g. [61].

For a proof, we build on a proof of a fractional Sobolev inequality for
$\mathbb{R}^{n}$ in [26, Theorem 6.5]. This proof uses very little analysis in
$\mathbb{R}^{n}$, and we can use large parts of it without modification. We
opted not to reproduce the complete proof here: the original is very well
written and detailed, and we do recommend that the reader consults the
original. We outline the main steps, and the parts of the proof that we need
to modify to our context.

Let us outline the main structure of the proof from [26, Chapter 6].

  1. (1)

One can assume that $f\in L^{\infty}(Z)$ using a standard argument, using [26,
Lemma 6.4]. This carries over verbatim.

  2. (2)

One “discretizes”: For $k\in\mathbb{Z}$, denote $A_{k}=\\{|f|>2^{k}\\}$ and
$a_{k}=\nu(A_{k})$. Then by Proposition 3.1 we have

$\displaystyle\left\|f|L(p^{*},p)\right\|^{p}$
$\displaystyle=p^{*}\int_{0}^{\infty}s^{p}\,\nu\left(|f|>s\right)^{{p}/{p^{*}}}\tfrac{\mathrm{d}s}{s}$
$\displaystyle\leq
p^{*}\sum_{k\in\mathbb{Z}}\int_{2^{k}}^{2^{k+1}}s^{p-1}a_{k}^{{p}/{p^{*}}}\mathrm{d}s=\tfrac{p^{*}\left(2^{p}-1\right)}{p}\cdot\sum_{k\in\mathbb{Z}}2^{kp}a_{k}^{p^{*}/p}.$

This is in fact slightly more straightforward than the original proof, as [26]
aims for $\|f|L^{p*}\|$ instead (cf. steps to obtain [26, (6.24)]).

  3. (3)

As in [26] one then applies a Lemma about sequences of numbers, [26, Lemma
6.2], to further get

$\|f|L(p^{*},p)\|\leq
C\sum_{k\in\mathbb{Z},a_{k}\not=0}2^{kp}a_{k+1}a_{k}^{-\frac{sp}{n}}.$

The final (delicate) part is showing that the right-hand side above bounds the
right-hand side of the desired inequality (5.1) from below; an analogue of
[26, Lemma 6.3]: this is Lemma 5.1 below. Modulo this Lemma, the proof of
Theorem 5 is done. ∎

######  Lemma 5.1.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space, $f\in
L^{\infty}(Z)$ with compact support, $s\in(0,1)$, $p\geq 1$, $sp<D$. For
$k\in\mathbb{Z}$, denote $A_{k}=\\{|f|>2^{k}\\}$ and $a_{k}=\nu(A_{k})$. Then

$\iint_{Z^{2}}\frac{\left|f(\xi)-f(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)+\nu(Z)^{-{sp}/{D}}\left\|f|L^{p}\right\|^{p}\succ\sum_{\begin{subarray}{c}k\in\mathbb{Z}\\\
a_{k}\not=0\end{subarray}}2^{kp}a_{k+1}a_{k}^{-sp/D}.$

###### Proof.

We slightly modify the proof of [26, Lemma 6.3]; we keep the notation
consistent with that proof, and focus on the parts which need adjustments. The
proof starts by discretizing as before, i.e. for $k\in\mathbb{Z}$ denoting
$A_{k}=\\{|f|>2^{k}\\}$ and $a_{k}=\nu(A_{k})$. Further subdivide into
$D_{k}=\\{2^{k+1}\geq|f|>2^{k}\\}$ and denote $d_{k}=\nu(D_{k})$. For brevity,
denote $K=\nu(Z)^{-\frac{sp}{D}}$. We finish the preparation by observing that

(5.2)
$\sum_{i\in\mathbb{Z}}d_{i}2^{pi}\leq\sum_{i\in\mathbb{Z}}\int_{D_{i}}|f(\xi)|^{p}\mathrm{d}\nu(\xi)=\left\|f|L^{p}\right\|^{p}.$

For $(x,y)\in D_{i}\times D_{j}$ with $j\leq i-2$ one has $|f(x)-f(y)|\geq
2^{i-1}$, and thus

$\displaystyle\sum_{j\in\mathbb{Z}\atop j\leq
i-2}\int_{D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(y)$
$\displaystyle\geq 2^{p(i-1)}\sum_{j\in\mathbb{Z}\atop j\leq
i-2}\int_{D_{j}}\frac{\mathrm{d}\nu(y)}{d(x,y)^{D+sp}}$
$\displaystyle=2^{p(i-1)}\int_{Z\setminus
A_{i-1}}\frac{\mathrm{d}\nu(y)}{d(x,y)^{D+sp}}$ (5.3) $\displaystyle\geq
c_{0}2^{pi}a_{i-1}^{-sp/n}-c_{1}K2^{pi}$

for suitable constants $c_{0},c_{1}>0$ by Lemma 5.2 below. Now we execute the
main scheme of the proof of [26, Lemma 6.3]: obtain the analogues of
(6.15)–(6.17) using our (5.3). For any $i\in\mathbb{Z}$ we thus have

(5.4) $\sum_{j\in\mathbb{Z}\atop j\leq i-2}\iint_{D_{i}\times
D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(x)\mathrm{d}\nu(y)\geq
c_{0}2^{pi}a_{i-1}^{-sp/n}d_{i}-c_{1}K2^{pi}d_{i}.$

Denoting $S=\sum_{i\in\mathbb{Z},a_{i-1}\not=0}2^{pi}a_{i-1}^{-sp/D}d_{i}$ as
in [26, (6.12)], summing over $i$, and using (5.2), we further get

(5.5) $\sum_{i\in\mathbb{Z}\atop a_{i-1}\not=0}\sum_{j\in\mathbb{Z}\atop j\leq
i-2}\iint_{D_{i}\times
D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(x)\mathrm{d}\nu(y)\geq
c_{0}S-c_{1}K\|f|L^{p}\|^{p}.$

Coming back to (5.4) and noting that
$d_{i}=a_{i}-\sum_{\ell\in\mathbb{Z},\,\ell\geq i+1}d_{\ell}$, we also get

(5.6) $\sum_{j\in\mathbb{Z}\atop j\leq i-2}\iint_{D_{i}\times
D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(x)\mathrm{d}\nu(y)\\\
\geq
c_{0}\left(2^{pi}a_{i-1}^{-sp/D}a_{i}-\textstyle\sum_{\ell\in\mathbb{Z},\,\ell\geq
i+1}2^{pi}a_{i-1}^{-sp/D}d_{\ell}\right)-c_{1}K2^{pi}d_{i}.$

Before combining the above, recall that in [26, (6.14)], the authors show that
$S\geq\sum_{i\in\mathbb{Z},\,a_{i-1}\not=0}\sum_{\ell\in\mathbb{Z},\,\ell\geq
i+1}2^{pi}a_{i-1}^{-sp/D}d_{\ell}$. Using this, (5.2), and (5.5) in what
follows, we start by summing over $i$ in (5.6):

$\displaystyle\sum_{i\in\mathbb{Z}\atop a_{i-1}\not=0}$
$\displaystyle\sum_{j\in\mathbb{Z}\atop j\leq i-2}\iint_{D_{i}\times
D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(x)\mathrm{d}\nu(y)$
$\displaystyle\geq c_{0}\left[\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}2^{pi}a_{i-1}^{-sp/D}a_{i}-\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}\sum_{\ell\in\mathbb{Z}\atop\ell\geq
i+1}2^{pi}a_{i-1}^{-sp/D}d_{\ell}\right]-c_{1}K\\!\\!\\!\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}\\!\\!\\!2^{pi}d_{i}$ $\displaystyle\geq
c_{0}\left[\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}2^{pi}a_{i-1}^{-sp/D}a_{i}-S\right]-c_{1}K\|f|L^{p}\|^{p}$
$\displaystyle\geq c_{0}\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}2^{pi}a_{i-1}^{-sp/D}a_{i}-\sum_{i\in\mathbb{Z}\atop
a_{i-1}\not=0}\sum_{j\in\mathbb{Z}\atop j\leq i-2}\iint_{D_{i}\times
D_{j}}\frac{|f(x)-f(y)|^{p}}{d(x,y)^{D+sp}}\mathrm{d}\nu(x)\mathrm{d}\nu(y)$
$\displaystyle\hskip 10.00002pt-2c_{1}K\|f|L^{p}\|^{p}.$

The rest of the proof proceeds as in [26, Lemma 6.1]: we bring the negative
terms from the right-hand side to the left-hand side, adjust the constants,
and use the discretization by $D_{i}\times D_{j}$ and symmetry (cf. [26,
(6.19)]) to obtain the required inequality. ∎

In the above proof, we have used the following analogue of [26, Lemma 6.1].

######  Lemma 5.2.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space. Then for any
$s\in(0,1)$, $p\geq 1$, and $E\subset Z$ with $\nu(E)<\infty$, and any $\xi\in
Z$, we have

$\int_{Z\setminus E}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}\geq
C\nu(E)^{-sp/D}-\nu(Z)^{-sp/D}$

for a suitable constant $C=C(Z,sp)>0$, which is uniformly bounded for $sp$ in
any fixed compact interval in $(0,\infty)$.

###### Proof.

The proof follows the one of [26, Lemma 6.1]. Denote by $c\geq 1$ the constant
from Ahlfors regularity, i.e. we have $c^{-1}r^{D}\leq\nu(B(\xi,r))\leq
cr^{D}$ for all $\xi\in Z$ and $0\leq r\leq\text{diam}(Z)$. Hence, with
$\alpha=c^{1/D}$, we have $\nu(B(\xi,\alpha r))\geq c^{-1}\alpha^{D}r^{D}\geq
r^{D}$. Denote $B=B(\xi,\alpha r)$ with $r=\nu(E)^{\frac{1}{D}}$. Then
$\nu(B)\geq\nu(E)$, and so

$\displaystyle\nu\left((Z\setminus E)\cap B\right)$
$\displaystyle=\nu(B)-\nu(E\cap B)\geq\nu(E)-\nu(E\cap
B)=\nu\left(E\cap(Z\setminus B)\right).$

It follows that

$\displaystyle\int_{Z\setminus
E}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}$
$\displaystyle=\int_{(Z\setminus E)\cap
B}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}+\int_{(Z\setminus
E)\cap(Z\setminus B)}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}$
$\displaystyle\geq(\alpha r)^{-D-sp}\nu\left((Z\setminus E)\cap
B\right)+\int_{(Z\setminus E)\cap(Z\setminus
B)}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}$ $\displaystyle\geq(\alpha
r)^{-D-sp}\nu\left(E\cap(Z\setminus B)\right)+\int_{(Z\setminus
E)\cap(Z\setminus B)}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}$
$\displaystyle\geq\int_{E\cap(Z\setminus
B)}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}+\int_{(Z\setminus
E)\cap(Z\setminus B)}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)}$
$\displaystyle=\int_{Z\setminus
B}\frac{\mathrm{d}\nu(\eta)}{d^{D+sp}(\xi,\eta)};$ with Lemma 5.3 below, we
continue: $\displaystyle\geq C\nu(B)^{-sp/D}-\nu(Z)^{-sp/D}$
$\displaystyle\geq Cc^{-sp/D}\alpha^{-sp}r^{-sp}-\nu(Z)^{-sp/D}$
$\displaystyle\geq C^{\prime}\nu(E)^{-sp/D}-\nu(Z)^{-sp/D}.\qed$

The following Lemma was used in the above proof; however it will be useful
elsewhere as well.

######  Lemma 5.3.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular quasi-metric measure space. Then for
any $0<\alpha$, any $\xi\in Z$ and $0\leq r\leq\text{diam}(Z)$, we have

$\int_{Z\setminus
B(\xi,r)}\frac{\mathrm{d}\nu(\eta)}{d^{D+\alpha}(\xi,\eta)}+\nu(Z)^{-\alpha/D}\asymp_{\alpha}\nu(B(\xi,r))^{{-\alpha}/{D}}.$

###### Proof.

Using the layer cake formula, we calculate

$\displaystyle\int_{Z\setminus
B(\xi,r)}\frac{\mathrm{d}\nu(\eta)}{d^{D+\alpha}(\xi,\eta)}$
$\displaystyle=\int_{\text{diam}(Z)^{-D-\alpha}}^{r^{-D-\alpha}}\nu\left(\\{d^{-D-\alpha}(\xi,\cdot)>t\\}\right)\mathrm{d}t$
$\displaystyle=\int_{\text{diam}(Z)^{-D-\alpha}}^{r^{-D-\alpha}}\nu\left(\\{d(\xi,\cdot)<t^{-1/(D+\alpha)}\\}\right)\mathrm{d}t$
$\displaystyle\asymp\int_{\text{diam}(Z)^{-D-\alpha}}^{r^{-D-\alpha}}t^{-D/(D+\alpha)}\mathrm{d}t=\tfrac{D+\alpha}{\alpha}\left[t^{\frac{\alpha}{D+\alpha}}\right]_{\text{diam}(Z)^{-D-\alpha}}^{r^{-D-\alpha}}$
$\displaystyle=\tfrac{D+\alpha}{\alpha}\left(r^{-\alpha}-\text{diam}(Z)^{-\alpha}\right).$

The conclusion now follows as
$r^{-\alpha}\asymp_{\alpha}\nu(B(\xi,r))^{-\alpha/D}$ and
$\text{diam}(Z)^{-\alpha}\asymp_{\alpha}\nu(Z)^{-\alpha/D}$. ∎

##  6\. Geometric control

In this section, the aim is to prove Propositions 6.4 and 6.6, i.e. Lorentz
norm bounds of certain multipliers that will be used in the proof of
boundedness of the operators induced by the Cayley transform in the next
Section.

######  Lemma 6.1.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular quasi-metric measure space. Then for
any $0<\alpha<D$, $\xi\in Z$, and $0\leq r\leq\text{diam}(Z)$, we have

$\int_{B(\xi,r)}\frac{\mathrm{d}\nu(\eta)}{d^{\alpha}(\xi,\eta)}\asymp\tfrac{\alpha}{D-\alpha}r^{D-\alpha}.$

######  Remark 6.2.

When $\alpha\leq 0$, we still have a (trivial) upper bound

$\int_{B(\xi,r)}d^{-\alpha}(\xi,\eta)\mathrm{d}\nu(\eta)\prec r^{D-\alpha}.$

###### Proof.

Using the layer cake formula, we have

$\displaystyle\int_{B(\xi,r)}\frac{\mathrm{d}\nu(\eta)}{d^{\alpha}(\xi,\eta)}$
$\displaystyle=\int_{r^{-\alpha}}^{\infty}\nu\left(\\{d^{-\alpha}(\xi,\cdot)>t\\}\right)\mathrm{d}t$
$\displaystyle=\int_{r^{-\alpha}}^{\infty}\nu\left(\\{d(\xi,\cdot)<t^{-1/\alpha}\\}\right)\mathrm{d}t$
$\displaystyle\asymp\int_{r^{-\alpha}}^{\infty}t^{-D/\alpha}\mathrm{d}t=\tfrac{\alpha}{\alpha-D}\left[t^{(\alpha-D)/\alpha}\right]_{r^{-\alpha}}^{\infty}=\tfrac{\alpha}{D-\alpha}r^{D-\alpha}.\qed$

######  Remark 6.3.

Let us record the classical observation that in an Ahlfors $D$-regular quasi-
metric measure space $(Z,d,\nu)$, for $\alpha\geq 0$ we have that the function
$\eta\mapsto d^{-\alpha}(\xi,\eta)$ belongs to $L(D/\alpha,\infty)$ for any
$\xi\in Z$, with the Lorentz norm bounded uniformly in $\xi\in Z$. This
follows from $\nu\left(\left\\{\zeta\in
Z:d^{-\alpha}(\xi,\zeta)>t\right\\}\right)\prec t^{-D/\alpha}$ for $t\geq 0$.

######  Proposition 6.4.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space. Let
$\sigma+it\in\mathbb{C}$, $s\in(0,1)$, and $1\leq p<\infty$, be such that
$\sigma\geq 0$, $\sigma p<D$, and $sp<D$. Then

$\int_{Z}\frac{\left|1-\big{[}\frac{d(a,\eta)}{d(a,\xi)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\prec_{\sigma,t,s,p}d^{-sp}(a,\eta),$

for $a,\eta\in Z$. Consequently, by Remark 6.3,

$\left\|\int_{Z}\frac{\left|1-\big{[}\frac{d(a,\cdot)}{d(a,\xi)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\cdot)}\mathrm{d}\nu(\xi)\left|L\left(Z;\tfrac{D}{sp},\infty\right)\right.\right\|$

is uniformly bounded for $a\in Z$.

###### Proof.

Fix $a,\eta\in Z$. Denote

$U(\eta)=B(\eta,d(a,\eta)/2),\quad U(a)=B(a,d(a,\eta)/2).$

Consider $\xi\in U(a)$. Then
$d(a,\xi)<\frac{1}{2}d(a,\eta)\leq\frac{1}{2}(d(a,\xi)+d(\xi,\eta))$, and
hence $d(a,\xi)\leq 2d(\xi,\eta)$. Next, we have $d(a,\eta)\leq
d(a,\xi)+d(\xi,\eta)\leq 3d(\xi,\eta)$. Conversely, $d(\xi,\eta)\leq
d(a,\xi)+d(a,\eta)\leq\frac{3}{2}d(a,\eta)$. Summarising, we have

(6.1) $\xi\in U(a)\implies d(a,\xi)\prec d(a,\eta)\asymp d(\xi,\eta).$

Symmetrically, we have

(6.2) $\xi\in U(\eta)\implies d(\xi,\eta)\prec d(a,\eta)\asymp d(a,\xi).$

Finally, in $Z\setminus(U(a)\cup U(\eta))$, a similar calculation with the
triangle inequality gives

(6.3) $d(a,\eta)\leq 2d(a,\xi)\asymp d(\xi,\eta).$

The triangle inequality888It is at this point where a genuine metric is
required; (6.4) may not hold for a quasi-metric. also implies that

(6.4)
$\left|1-\tfrac{d(a,\xi)}{d(a,\eta)}\right|\leq\tfrac{d(\xi,\eta)}{d(a,\eta)}.$

We now estimate the integral in the statement of the Proposition on three
subsets of $Z$.

On $W=Z\setminus(U(a)\cup U(\eta))$, we have that
$\frac{d(a,\eta)}{d(a,\xi)}\leq 2$, and using Lemma 5.3 (as $sp<D$), we
estimate

$\int_{W}\frac{\left|1-\big{[}\frac{d(a,\eta)}{d(a,\xi)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\\\
\prec_{\sigma,p}\int_{Z\setminus
U(\eta)}\frac{1}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\prec_{s,p}\nu(U(\eta))^{\frac{-sp}{D}}\asymp\frac{1}{d^{sp}(a,\eta)}.$

On $U(a)$ using Lemma 6.1 (as $\sigma p<D$), and (6.1), we have

$\int_{U(a)}\frac{\left|1-\big{[}\frac{d(a,\eta)}{d(a,\xi)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\\\
\prec_{s,p}\frac{\nu(U(a))}{d^{D+sp}(a,\eta)}+\frac{d^{\sigma
p}(a,\eta)}{d^{D+sp}(a,\eta)}\int_{U(a)}\frac{1}{d^{\sigma
p}(a,\xi)}\mathrm{d}\nu(\xi)\prec_{\sigma,p}\frac{1}{d^{sp}(a,\eta)}.$

Finally, we deal with $U(\eta)$. The Mean Value Theorem implies that
$|1-x^{\sigma+it}|\prec_{\sigma,t}|1+x|$ for $x\in[\frac{2}{3},3]$. Using this
together with (6.2), (6.4), and Lemma 6.1, we have

$\int_{U(\eta)}\frac{\left|1-\big{[}\frac{d(a,\eta)}{d(a,\xi)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\prec_{\sigma,t}\int_{U(\eta)}\frac{\left|1-\frac{d(a,\eta)}{d(a,\xi)}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\\\
\leq\int_{U(\eta)}\frac{\mathrm{d}\nu(\xi)}{d^{p}(a,\xi)d^{D-(1-s)p}(\xi,\eta)}\prec_{\sigma,t,s,p}\frac{1}{d(a,\eta)^{sp}}.\qed$

######  Lemma 6.5.

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric measure space and
consider $(Z_{a},d_{a},\nu_{a})$ as defined in (3.5). Let $0<\alpha\leq D$ and
$k\geq 1$. Then

$d^{k\alpha}(\cdot,a)\in L\Big{(}Z_{a};\tfrac{D}{\alpha},\infty\Big{)}$

with norm bounded uniformly on $a\in Z$.

###### Proof.

We need to check that $t\mapsto
t^{D/\alpha}\cdot\nu_{a}\left(\left\\{d^{k\alpha}(\cdot,a)>t\right\\}\right)$
is bounded. We can assume $t<\text{diam}(Z)^{k\alpha}$. We calculate

$\displaystyle\nu_{a}\left(\left\\{d^{k\alpha}(\cdot,a)>t\right\\}\right)$
$\displaystyle=\int_{\\{d^{k\alpha}(\cdot,a)>t\\}}d^{-2D}(\xi,a)\,\mathrm{d}\nu(\xi)$
$\displaystyle\leq\int_{0}^{t^{-(2D)/(k\alpha)}}\nu\left(\left\\{d^{-2D}(\cdot,a)>s\right\\}\right)\mathrm{d}s$
$\displaystyle=\int_{0}^{t^{-(2D)/(k\alpha)}}\nu\left(\left\\{d(\cdot,a)<s^{-1/(2D)}\right\\}\right)\mathrm{d}s$
$\displaystyle\prec\int_{0}^{t^{-(2D)/(k\alpha)}}s^{-1/2}\mathrm{d}s=2t^{-\frac{D}{k\alpha}}.$

Multiplying the final expression by $t^{D/\alpha}$ yields power function with
exponent $\frac{(k-1)D}{k\alpha}\geq 0$, which is bounded as
$t<\text{diam}(Z)^{k\alpha}<\infty$. ∎

######  Proposition 6.6.

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric measure space and
consider $(Z_{a},d_{a},\nu_{a})$ as defined in (3.5). Let
$\sigma+it\in\mathbb{C}$, $s\in(0,1)$, $1\leq p<\infty$ be such that
$0\leq\sigma$, $\sigma p<D$, and $sp<D$. Then

$\int_{Z_{a}}\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\prec_{\sigma,t,s,p}d^{sp}(a,\eta),$

for $a,\eta\in Z$. Consequently, by Lemma 6.5,

$\left\|\int_{Z_{a}}\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\cdot)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\cdot)}\mathrm{d}\nu_{a}(\xi)\left|L\Big{(}Z_{a};\tfrac{D}{sp},\infty\Big{)}\right.\right\|$

is uniformly bounded for $a\in Z$.

###### Proof.

As in Proposition 6.4, let $U(\eta)=\\{\xi\in Z_{a}\mid
d(\xi,\eta)<d(a,\eta)/2\\}$ and $U(a)=\\{\xi\in Z_{a}\mid
d(\xi,a)<d(a,\eta)/2\\}$. We proceed again by estimating the integral on three
subsets of $Z$. The first step of all three calculations is

(6.5)
$\int\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\,\mathrm{d}\nu_{a}(\xi)\\\
=\int\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d^{D+sp}(\xi,\eta)}\,d^{-D+sp}(a,\xi)\,d^{D+sp}(a,\eta)\,\mathrm{d}\nu(\xi).$

First, on $W=Z\setminus(U(a)\cup U(\eta))$, we use (6.3), $sp<D$, $\sigma
p<D$, and Lemma 5.3:

$\int_{W}\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\,\mathrm{d}\nu_{a}(\xi)\\\
\prec_{\sigma,s,p}\int_{Z\setminus
U(a)}\frac{\mathrm{d}\nu(\xi)}{d^{D-sp}(a,\xi)}+d^{D+sp-\sigma
p}(a,\eta)\int_{Z\setminus U(a)}\frac{\mathrm{d}\nu(\xi)}{d^{2D-\sigma
p}(a,\xi)}\\\ \prec_{\sigma,s,p}d^{sp}(a,\eta).$

On $U(a)$, we use (6.1) and Lemma 6.1 (or Remark 6.2):

$\displaystyle\int_{U(a)}$
$\displaystyle\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\,\mathrm{d}\nu_{a}(\xi)\prec_{s,p}\int_{U(a)}\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d^{D-sp}(a,\xi)}\,\mathrm{d}\nu(\xi)$
$\displaystyle\prec_{p}\int_{U(a)}\frac{1}{d^{D-sp}(a,\xi)}\,\mathrm{d}\nu(\xi)+d^{-\sigma
p}(a,\eta)\int_{U(a)}\frac{1}{d^{D-(s+\sigma)p}(a,\xi)}\,\mathrm{d}\nu(\xi)$
$\displaystyle\prec_{s,p,\sigma}d^{sp}(a,\eta).$

We now deal with $U(\eta)$. Using the Mean Value Theorem as in the proof of
Proposition 6.4, (6.2), (6.4), and Lemma 6.1 (noting that $1-s>0$), we obtain

$\displaystyle\int_{U(\eta)}$
$\displaystyle\frac{\left|1-\big{[}\frac{d(a,\xi)}{d(a,\eta)}\big{]}^{\sigma+it}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\,\mathrm{d}\nu_{a}(\xi)$
$\displaystyle\prec_{\sigma,t}\int_{U(\eta)}\frac{\left|1-\frac{d(a,\xi)}{d(a,\eta)}\right|^{p}}{d^{D+sp}(\xi,\eta)}\,d^{-D+sp}(a,\xi)\,d^{D+sp}(a,\eta)\,\mathrm{d}\nu(\xi)$
$\displaystyle\leq
d^{2sp-p}(a,\eta)\int_{U(\eta)}\frac{\mathrm{d}\nu(\xi)}{d^{D-(1-s)p}(\xi,\eta)}\asymp_{\sigma,s,p}d^{sp}(a,\eta).\qed$

##  7\. The Cayley transform: Proof of Theorem 1

In this section, we complete the proof of Theorem 1 in the Introduction; the
two bounds are proved in Theorems 6 and 7 below.

######  Theorem 6.

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric measure space, and for
each $a\in Z$ we consider $(Z_{a},d_{a},\nu_{a})$ as defined in (3.5). Let
$1\leq p<\infty$, $t\in\mathbb{R}$, and $s\in(0,1)$ with $sp<D$. Denote
$z=s+it$. The operator

$\displaystyle\Omega_{z,p}(a)$ $\displaystyle:W^{s,p}(Z)\subset
L^{p}(Z)\rightarrow\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a}),$
$\displaystyle\Omega_{z,p}(a)\phi(\xi)$
$\displaystyle=d^{\frac{2D}{p}-2z}(a,\xi)\phi(\xi)\quad\text{for }\phi\in
W^{s,p}(Z),\xi\in Z_{a}.$

is well defined and bounded, uniformly in $a\in Z$ and $t\in\mathbb{R}$.

###### Proof.

Take $\phi\in W^{s,p}(Z)$. We calculate

$\displaystyle\left\|\Omega_{z,p}(a)\phi|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\right\|^{p}$
$\displaystyle=\iint_{Z^{2}_{a}}\frac{\left|\Omega_{z,p}(a)\phi(\xi)-\Omega_{z,p}(a)\phi(\eta)\right|^{p}}{d^{D+sp}_{a}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\mathrm{d}\nu_{a}(\eta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:C_a-deriv-
formula}}}{{=}}\iint_{Z^{2}_{a}}\frac{\left|d^{\frac{2D}{p}-2z}(\xi,a)\phi(\xi)-d^{\frac{2D}{p}-2z}(\eta,a)\phi(\eta)\right|^{p}}{d^{D+sp}_{a}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\mathrm{d}\nu_{a}(\eta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Z_a-d_a-nu_a-
defn}}}{{=}}\iint_{Z^{2}}\frac{\left|d^{2\left(\frac{D}{p}-z\right)}(\xi,a)\phi(\xi)-d^{2\left(\frac{D}{p}-z\right)}(\eta,a)\phi(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\cdot$
$\displaystyle\qquad\qquad\qquad\cdot\frac{1}{d^{\left(\frac{D}{p}-s\right)p}(\xi,a)}\frac{1}{d^{\left(\frac{D}{p}-s\right)p}(\eta,a)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
$\displaystyle=\iint_{Z^{2}}\frac{\left|\frac{d^{{D}/{p}-z}(\xi,a)}{d^{{D}/{p}-z}(\eta,a)}\phi(\xi)-\frac{d^{{D}/{p}-z}(\eta,a)}{d^{{D}/{p}-z}(\xi,a)}\phi(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta).$

Next, using the estimate: $|x+y+z|^{p}\leq 3^{p-1}(|x|^{p}+|y|^{p}+|z|^{p})$
for all $x,y,z\in{\mathbb{C}}$, we further have

$\iint_{Z^{2}}\frac{\left|\frac{d^{{D}/{p}-z}(\xi,a)}{d^{{D}/{p}-z}(\eta,a)}\phi(\xi)-\frac{d^{{D}/{p}-z}(\eta,a)}{d^{{D}/{p}-z}(\xi,a)}\phi(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)\\\
\leq 3^{p-1}\left\|\phi|W^{s,p}(Z)\right\|^{p}+2\cdot
3^{p-1}\int_{Z}\,\int_{Z}\frac{\left|1-\frac{d^{{D}/{p}-z}(\eta,a)}{d^{{D}/{p}-z}(\xi,a)}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\cdot|\phi(\eta)|^{p}\mathrm{d}\nu(\eta).$

Using Lorentz–Hölder inequality (Proposition 3.3), Lemma 3.4, the geometric
control Proposition 6.4 (as $p\cdot\left(D/p-s\right)<D$), and the fractional
Sobolev inequality (Theorem 5), we have

$\displaystyle\int_{Z}\,\int_{Z}$
$\displaystyle\frac{\left|1-\frac{d^{{D}/{p}-z}(\eta,a)}{d^{{D}/{p}-z}(\xi,a)}\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\cdot|\phi(\eta)|^{p}\mathrm{d}\nu(\eta)$
$\displaystyle\leq\left\|\int_{Z}\frac{\left|1-\frac{d^{{D}/{p}-z}(\cdot,a)}{d^{{D}/{p}-z}(\xi,a)}\right|^{p}}{d^{D+sp}(\xi,\cdot)}d\xi\left|L\left(\tfrac{D}{sp},\infty\right)\right.\right\|\cdot\left\||\phi|^{p}\left|L\left(\tfrac{D}{D-sp},1\right)\right.\right\|$
$\displaystyle\prec_{z,p}\left\|\phi\left|L\left(\tfrac{pD}{D-sp},p\right)\right.\right\|^{p}\prec_{s,p}\left\|\phi|W^{s,p}(Z)\right\|^{p}.$

To finish the proof, we need to show that $\Omega_{z,p}(a)\phi$ is a limit (in
$[\cdot]_{s,p}$) of compactly supported functions on $Z_{a}$. First, we can
assume that $\phi$ (and hence also $\Omega_{z,p}(a)\phi$) is a bounded
function, by considering cut-offs of $\phi$, by the argument at the beginning
of the proof of [26, Theorem 6.5]. Next, we prove the following statement: if
$\psi:Z_{a}\to\mathcal{C}$ is a bounded measurable function with
$[\psi]_{s,p}<\infty$, then it is a $[\cdot]_{s,p}$-limit of boundedly
supported ones. For this, fix $o\in Z_{a}$, and consider functions
$f_{R}:Z_{a}\to[0,1]$ for $R>0$, such that $f_{R}|_{B(o,R)}\equiv 1$,
$f_{R}|_{Z_{a}\setminus B(o,R+1)}\equiv 0$, and $f_{R}$ are all $K$-Lipschitz
for some $C\geq 0$. For example, one can take

$f_{R}(\xi)=\frac{d_{a}(\xi,Z_{a}\setminus
B(o,R+1))}{d_{a}\\!\left(\xi,\overline{B(o,R)}\right)+d_{a}(\xi,Z_{a}\setminus
B(o,R+1))},\quad\xi\in Z_{a}.$

Observe that for $\xi,\eta\in Z_{a}$, we have

$\left|\psi(\xi)f_{R}(\xi)-\psi(\eta)f_{R}(\eta)\right|\leq|\psi(\xi)-\phi(\eta)|+K\|\psi|L^{\infty}\|\cdot
d_{a}(\xi,\eta).$

Consequently (cf. Remark 4.2),

(7.1) $[\psi\cdot f_{R}]_{s,p}\leq[\psi]_{s,p}+C\|\psi|L^{\infty}\|<\infty,$

independently of $R>0$. Consider now functions $F_{R}:Z_{a}\times
Z_{a}\to\mathbb{C}$, defined by

$F_{R}(\xi,\eta)=(\psi\cdot f_{R}-\psi)(\xi)-(\psi\cdot
f_{R}-\psi)(\eta),\quad\xi,\eta\in Z_{a}.$

Then $F_{R}\to 0$ pointwise for $R\to\infty$. Furthermore
$|F_{R}(\xi,\eta)|\leq 2|\psi(\xi)-\phi(\eta)|+K\|\psi|L^{\infty}\|\cdot
d_{a}(\xi,\eta)$, and the function on the right-hand side belongs to
$L^{p}\\!\left(Z_{a}\times
Z_{a},\frac{\mathrm{d}\nu_{a}\mathrm{d}\nu_{a}}{d^{D+sp}}\right)$ (similarly
to (7.1) above). Hence, by Lebesgue’s Dominated Convergence Theorem, $F_{R}\to
0$ in $L^{p}\\!\left(Z_{a}\times
Z_{a},\frac{\mathrm{d}\nu_{a}\mathrm{d}\nu_{a}}{d^{D+sp}}\right)$; in other
words, $\psi\cdot f_{R}\to\psi$ in $[\cdot]_{s,p}$. ∎

######  Theorem 7.

With the same assumptions and notation as in Theorem 6, the operator

$\displaystyle\Omega^{-1}_{z,p}(a)$
$\displaystyle:\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\rightarrow
W^{s,p}(Z)\subset L^{p}(Z)$ $\displaystyle\Omega^{-1}_{z,p}(a)\phi$
$\displaystyle=\left[d^{-\frac{2D}{p}+2z}(a,\cdot)\phi\right]\in L^{p}(Z)$

is well defined and bounded, uniformly in $a\in Z$ and $t\in\mathbb{R}$.

###### Proof.

Let $\phi\in\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})$ be measurable and
compactly supported.

Using Lorentz–Hölder inequality (Proposition 3.3), Lemma 3.4, Lemma 6.5, and
the fractional Sobolev inequality (Theorem 5):

$\displaystyle\left\|\Omega^{-1}_{z,p}(a)\phi\,|L^{p}(Z)\right\|^{p}$
$\displaystyle=\int_{Z}\left|d^{-\frac{2D}{p}+2z}(\xi,a)\phi(\xi)\right|^{p}\mathrm{d}\nu(\xi)\stackrel{{\scriptstyle\eqref{eq:Z_a-
d_a-nu_a-
defn}}}{{=}}\int_{Z_{a}}d^{2sp}(\xi,a)|\phi(\xi)|^{p}\mathrm{d}\nu_{a}(\xi)$
$\displaystyle\leq\left\|d^{2sp}(\cdot,a)\left|L\left(Z_{a};\tfrac{D}{sp},\infty\right)\right.\right\|\cdot\left\|\phi\left|L\left(Z_{a};\tfrac{pD}{D-sp},p\right)\right.\right\|^{p}$
$\displaystyle\prec_{s,p}\left\|\phi\left|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\right.\right\|^{p}.$

To bound the second term in $W^{s,p}(Z)$-norm, we calculate as in the proof of
Theorem 6:

$\displaystyle\iint_{Z^{2}}\frac{\left|\Omega_{z,p}^{-1}(a)\phi(\xi)-\Omega^{-1}_{z,p}(a)\phi(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:C_a-deriv-
formula}}}{{=}}\iint_{Z^{2}}\frac{\left|d^{-\frac{2D}{p}+2z}(\xi,a)\phi(\xi)-d^{-\frac{2D}{p}+2z}(\eta,a)\phi(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Z_a-d_a-nu_a-
defn}}}{{=}}\iint_{Z_{a}^{2}}\frac{\left|d^{-2\left(\frac{D}{p}+z\right)}(\xi,a)\phi(\xi)-d^{-2\left(\frac{D}{p}+z\right)}(\eta,a)\phi(\eta)\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\cdot$
$\displaystyle\qquad\qquad\qquad\cdot
d^{\left(\frac{D}{p}-s\right)p}(\xi,a)d^{\left(\frac{D}{p}-s\right)p}(\eta,a)\,\mathrm{d}\nu_{a}(\xi)\mathrm{d}\nu_{a}(\eta)$
$\displaystyle=\iint_{Z_{a}^{2}}\frac{\left|\frac{d^{{D}/{p}-z}(\eta,a)}{d^{{D}/{p}-z}(\xi,a)}\phi(\xi)-\frac{d^{{D}/{p}-z}(\xi,a)}{d^{{D}/{p}-z}(\eta,a)}\phi(\eta)\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\,\mathrm{d}\nu_{a}(\eta)$
$\displaystyle\leq
3^{p-1}\left\|\phi\left|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\right.\right\|^{p}+2\cdot
3^{p-1}\int_{Z}\,\int_{Z}\frac{\left|1-\frac{d^{{D}/{p}-z}(\xi,a)}{d^{{D}/{p}-z}(\eta,a)}\right|^{p}}{d_{a}^{D+sp}(\xi,\eta)}\mathrm{d}\nu_{a}(\xi)\cdot|\phi(\eta)|^{p}\mathrm{d}\nu_{a}(\eta)$
$\displaystyle\prec_{z,p}\left\|\phi\left|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\right.\right\|^{p},$

where the last inequality follows the same Lorentz norm estimate as in the
proof of Theorem 6, now using the other geometric control Proposition 6.6. ∎

##  8\. Representations: Proof of Theorem 2

In this Section, we deduce Theorem 2 from Theorem 1 much like we explained in
Section 2.

Let us recall the setup of Theorem 2. Let $(Z,d)$ be a complete metric space
without isolated points, let $\nu$ be the $D$-dimensional Hausdorff measure on
$Z$, and suppose that $(Z,d,\nu)$ is Ahlfors $D$-regular. Let
$t\in\mathbb{R}$, $s\in(0,1)$ and $p\in[1,\infty)$ with $sp<D$. Denote
$z=s+it$. Recall that we are considering the bundle
$(Z_{a},d_{a},\nu_{a})_{a\in Z}$ defined by (3.5).

For $a\in Z$, and $g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, we
check that the operators $\Pi_{w}^{(p)}(g;a):W^{s,p}(Z_{a})\to
W^{s,p}(Z_{ga})$,
$\Pi^{(p)}_{z}(g;a)\phi=|g^{\prime}|^{\frac{D}{p}-z}(a)\,g^{*}\phi$, are
isometries.

(8.1)
$\left\|\Pi_{z}^{(p)}(g;a)\phi\left|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{ga})\right.\right\|^{p}\\\
\begin{aligned} &\stackrel{{\scriptstyle\eqref{eq:defn-of-[]sp-
intro}}}{{=}}\iint_{Z_{ga}}\frac{\left|\Pi_{z}^{(p)}(g;a)\phi(\xi)-\Pi_{z}^{(p)}(g;a)\phi(\eta)\right|^{p}}{d_{ga}^{D+sp}(\xi\eta)}\,\mathrm{d}\nu_{ga}(\xi)\mathrm{d}\nu_{ga}(\eta)\\\
&=|g^{\prime}|^{D-sp}(a)\cdot\iint_{Z_{ga}}\frac{\left|\phi(\omega)-\phi(\tau)\right|^{p}}{d_{ga}^{D+sp}(g\omega,g\tau)}\,\mathrm{d}\nu_{ga}(g\omega)\mathrm{d}\nu_{ga}(g\tau)\\\
&\stackrel{{\scriptstyle\eqref{eq:GMV-for-d_a}\eqref{eq:change-of-var-
nu_a}}}{{=}}|g^{\prime}|^{D-sp}(a)\cdot\iint_{Z_{a}}\frac{\left|\phi(\omega)-\phi(\tau)\right|^{p}}{d_{a}^{D+sp}(\omega,\tau)}\cdot\frac{|g^{\prime}|^{D+sp}(a)}{|g^{\prime}|^{2D}(a)}\,\mathrm{d}\nu_{a}(\omega)\mathrm{d}\nu_{a}(\tau)\\\
&=\left\|\phi\left|\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})\right.\right\|^{p}.\end{aligned}$

Furthermore, they satisfy the relation of being an (isometric) representation
of $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$, namely that
$\Pi_{z}^{(p)}(gh;a)=\Pi_{z}^{(p)}(g;ha)\circ\Pi_{z}^{(p)}(h;a)$ for $a\in Z$,
$g,h\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$. Indeed, for
$\phi\in\accentset{\mbox{\large.}}{W}^{s,p}(Z_{a})$, $\xi\in Z_{gha}$ we have

(8.2) $\displaystyle\begin{aligned}
\Pi_{z}^{(p)}(gh;a)\phi(\xi)&=|(gh)^{\prime}|^{\frac{D}{p}-z}(a)\phi(h^{-1}g^{-1}\xi)\\\
&\stackrel{{\scriptstyle\eqref{eq:metric-deriv-
cocycle}}}{{=}}|g^{\prime}|^{\frac{D}{p}-z}(ha)\,|h^{\prime}|^{\frac{D}{p}-z}(a)\phi(h^{-1}g^{-1}\xi)\\\
&=|g^{\prime}|^{\frac{D}{p}-z}(ha)\,\Pi_{z}^{(p)}(h;a)\,\phi(g^{-1}\xi)\\\
&=\Pi_{z}^{(p)}(g;ha)\,\Pi_{z}^{(p)}(h;a)\,\phi(\xi).\end{aligned}$

We now calculate for $g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$,
$\phi\in W^{s,p}(Z)$, and $a,\xi\in Z$,

(8.3) $\Omega^{-1}_{z,p}(ga)\Pi_{z}^{(p)}(g;a)\,\Omega_{z,p}(a)\,\phi(\xi)\\\
\begin{aligned} &\stackrel{{\scriptstyle\eqref{eq:C_a-deriv-
formula}\eqref{eq:Pi_s_g_a-
defn}}}{{=}}d^{-\frac{2D}{p}+2z}(\xi,ga)\cdot|g^{\prime}|^{\frac{D}{p}-z}(a)\cdot
d^{\frac{2D}{p}-2z}(g^{-1}\xi,a)\cdot\phi(g^{-1}\xi)\\\
&\stackrel{{\scriptstyle\eqref{eq:GMV}}}{{=}}d^{-\frac{2D}{p}+2z+\frac{2D}{p}-2z}(g^{-1}\xi,a)\cdot|g^{\prime}|^{-\frac{D}{p}+z}(g^{-1}\xi)\cdot\phi(g^{-1}\xi)\\\
&\stackrel{{\scriptstyle\eqref{eq:metric-deriv-
cocycle}}}{{=}}|(g^{-1})^{\prime}|^{\frac{D}{p}-z}(\xi)\cdot\phi(g^{-1}\xi)=\pi_{z}^{(p)}(g)\phi(\xi).\end{aligned}$

We see that we recover our boundary representation as
$\pi_{z}^{(p)}(g)=\Omega_{z,p}^{-1}(ga)\circ\Pi_{z}^{(p)}(g;a)\circ\Omega_{z,p}(a)$,
showing that it is uniformly bounded on $W^{s,p}(Z)$ by Theorem 1. This
finishes the proof of Theorem 2.

######  Remark 8.1.

The calculation (8.3) also implies that the cocycle
$c_{z,p}(a,b)=\Omega_{z,p}(b)\circ\Omega^{-1}_{z,p}(a)$ (for $a,b\in Z$) is
covariant for the representation $\Pi_{z}^{(p)}$ of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)\ltimes Z$, i.e. the
relation (2.3), as the final expression in (8.3) then does not depend on $a$.
Consequently, we may view $\pi_{z}^{(p)}$ as conjugate (by the operators
$\Omega_{z,p}$) to uniformly bounded representations of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on $W^{s.p}(Z_{a})$, for
any $a\in Z$, given by $g\mapsto c_{z,p}(a,ga)\Pi_{z}^{(p)}(g;a)$. This is
analogous to the “noncompact picture” for representations of Lie groups.

######  Remark 8.2.

By Proposition 4.4, the (uniformly bounded) representations $\pi_{z}^{(p)}$ of
$\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ from Theorem 2 are
conjugate to uniformly bounded representations on closed subspaces of an
$L^{p}$-space.

######  Remark 8.3.

Continuing with the setup and notation of Theorem 2, the fractional Sobolev
Inequality (Theorem 5) shows that the spaces $W^{s,p}(Z)$ are continuously
embedded in the Lorentz function spaces $L(Z;p^{*},p)$, with
$p^{*}=\frac{pD}{D-sp}$. The dual of $L(Z;p^{*},p)$ (with respect to the
$L^{2}$-pairing) is another Lorentz space,
$L\\!\left(Z;\frac{qD}{D+sq},q\right)$, where $1/p+1/q=1$, see e.g. [33,
Theorem 1.4.16(vi)]. Consequently, the dual of $W^{s,p}(Z)$ contains this
Lorentz space as a dense subspace, and straightforward calculations show that
the “dual” representations $\pi_{-\overline{z}}^{(q)}$ are uniformly bounded
on the dual of $W^{s,p}(Z)$ (cf. [29, p.53, p.102]).

##  9\. Almost invariant vectors: Proof of Theorem 3

We start with an equicontinuity Lemma, which follows from the proof of [58,
Lemma 7], and [58, Proposition 14].

######  Lemma 9.1.

Let $(Z,d)$ be a complete metric space without isolated points. Let
$K\subset\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ be a compact
subset of $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$. Then there
exists $\lambda>0$ such that all functions $|g^{\prime}|$ for $g\in K$ are
$\lambda$-Lipschitz.

Next, we show one part of Theorem 3, namely that $\pi_{s}^{(p)}$ have almost
invariant vectors (for an appropriate $s$).

######  Proposition 9.2.

Let $(Z,d,\nu)$ be a bounded Ahlfors $D$-regular metric space, where $\nu$ is
the Hausdorff $D$-measure on $(Z,d)$. Let
$K\subset\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ be a compact set,
$\varepsilon>0$, and $p>\max(D,1)$. Then there exists
$s_{0}\in(0,\frac{D}{p})$ such that for all $s\in(s_{0},\frac{D}{p})$ we have

$\sup_{g\in K}\left\|\pi_{s}^{(p)}(g)\mathbf{1}-\mathbf{1}\mid
W^{s,p}(Z)\right\|\leq\varepsilon.$

###### Proof.

Take a compact $K\subset\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$.
Then there exists $c>0$ such that $c^{-1}\mathbf{1}\leq|(g^{-1})^{\prime}|\leq
c\mathbf{1}$ for all $g\in K$. It follows that
$|(g^{-1})^{\prime}|^{\frac{D}{p}-s}\xrightarrow{s\rightarrow\frac{D}{p}}\mathbf{1}$
uniformly on $Z$ and $K$. As a consequence

$\displaystyle\left\|\pi_{s}^{(p)}(g)\mathbf{1}-\mathbf{1}\mid
L^{p}(Z)\right\|^{p}$
$\displaystyle=\int_{Z}\left||(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\xi)-1\right|^{p}\mathrm{d}\nu(\xi)\xrightarrow{s\rightarrow\frac{D}{p}}0.$

This deals with the $L^{p}$-part of the norm on $W^{s,p}(Z)$.

Denote $\Phi(t)=t^{\frac{D}{p}-s}$ for $t>0$. Using the Mean Value Theorem and
Lemma 9.1, we estimate for $\xi,\eta\in Z$:

$\left||(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\xi)-|(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\eta)\right|\\\
\begin{aligned}
&=\left|\Phi\\!\left(|(g^{-1})^{\prime}|(\xi)\right)-\Phi\\!\left(|(g^{-1})^{\prime}|(\eta)\right)\right|\\\
&\leq\left({\tfrac{D}{p}-s}\right)\cdot\max_{t\in[c^{-1},c]}\left(t^{\frac{D}{p}-s-1}\right)\cdot\left||(g^{-1})^{\prime}|(\xi)-|(g^{-1})^{\prime}|(\eta)\right|\\\
&\leq\left({\tfrac{D}{p}-s}\right)\cdot\max_{t\in[c^{-1},c]}\left(t^{\frac{D}{p}-s-1}\right)\cdot\lambda\cdot
d(\xi,\eta)\\\ &\prec_{K,p}\left({\tfrac{D}{p}-s}\right)\cdot
d(\xi,\eta).\end{aligned}$

Consequently,

$\displaystyle\left[\pi_{s}^{(p)}(g)\mathbf{1}-\mathbf{1}\right]_{s,p}$
$\displaystyle=\iint\frac{\left||(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\xi)-|(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
$\displaystyle\prec_{K,p}\left({\tfrac{D}{p}-s}\right)^{p}\iint\frac{1}{d^{D+sp-p}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
$\displaystyle\leq\left({\tfrac{D}{p}-s}\right)^{p}\iint\frac{1}{d^{D-(p-D)}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)\xrightarrow{s\rightarrow\frac{D}{p}}0,$

by Lemma 6.1, as $p-D>0$. ∎

The second part of Theorem 3 is that $\pi_{s}^{(p)}$ have no non-zero
invariant vectors. For this we use amenability of the boundary action.

######  Definition 9.3.

An action of a countable discrete group $\Gamma$ on a compact Hausdorff space
$Y$ is (topologically) _amenable_ , if there exists a sequence of
continuous999We think of $\operatorname{\mathrm{Prob}}(\Gamma)$ as a convex
subset of $\ell^{1}(\Gamma)$, endowed with the weak-* topology. maps
$f_{n}:Y\rightarrow\operatorname{\mathrm{Prob}}(\Gamma)$ such that for every
$g\in\Gamma$

$\lim_{n\to\infty}\sup_{y\in Y}\|f_{n}(g^{-1}y)-g^{*}(f_{n}(y))\|_{1}=0.$

The following is classical, cf. e.g. [18, Chapter 5, Exercise 4.4].

######  Proposition 9.4.

Let $\Gamma\curvearrowright Y$ be an amenable action of a discrete group
$\Gamma$ on a compact space $Y$. Then $\Gamma$ is amenable if (and only if)
$Y$ admits a $\Gamma$-invariant probability measure.

Boundary actions of hyperbolic groups are amenable: Adams proved these actions
are measurewise amenable (for any quasi-invariant measure on the boundary)
[1], and subsequently Anantharaman–Delaroche and Renault [2, Chapter 3] shown
that this implies topological amenability in the above sense (see also [2,
Appendix B]).

######  Theorem (Adams).

The action of a countable discrete hyperbolic group $\Gamma$ on its Gromov
boundary is amenable.

######  Proposition 9.5.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space. Let
$\Gamma<\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ be a non-amenable
group, such that the action of $\Gamma\curvearrowright Z$ is (topologically)
amenable. Then $\pi^{(p)}_{s}$ does not have non-zero invariant vectors in
$W^{s,p}(Z)$.

###### Proof.

We proceed by contradiction: suppose that there exists $0\neq\psi\in
W^{s,p}(Z)$ which is $\pi^{(p)}_{s}$-invariant. Then $A=\\{|\psi|\neq 0\\}$ is
a $\Gamma$-invariant non-empty subset of $Z$.

Denoting $p^{*}=\frac{pD}{D-sp}$, the classical Sobolev inequality (i.e.
Theorem 5 combined with the inclusion $L(Z;p^{*},p)\subset L^{p^{*}}\\!(Z)$)
implies that $\psi\in L^{p^{*}}\\!(Z)$. Furthermore
$\|\psi|L^{p^{*}}\|\not=0$.

Expanding $\pi^{(p)}_{s}(g)\psi=\psi$, we have

$\displaystyle|(g^{-1})^{\prime}|^{\frac{D}{p}-s}\psi(g^{-1}\xi)$
$\displaystyle=\psi(\xi)$
$\displaystyle\iff|(g^{-1})^{\prime}|^{D}|\psi|^{p^{*}}\\!(g^{-1}\xi)$
$\displaystyle=|\psi|^{p^{*}}\\!(\xi)$ $\displaystyle\iff
g^{*}(|\psi|^{p^{*}}\\!(\xi)\mathrm{d}\nu(\xi))$
$\displaystyle=|\psi|^{p^{*}}\\!(\xi)\mathrm{d}\nu(\xi)$

for all $g\in\Gamma$ and for $\nu$-almost every $\xi\in Z$. In other words,
the probability measure
$\frac{1}{\|\psi|L^{p^{*}}\|}|\psi|^{p^{*}}\mathrm{d}\nu$ on $Z$ is
$\Gamma$-invariant. This is a contradiction with Proposition 9.4. ∎

The final part of Theorem 3 is a bound on $\|\pi_{s}^{(p)}(g)\|$ independent
of $s\in(0,\frac{D}{p})$, but allowed to depend on $g\in\Gamma$. This is
provided by the following Proposition, where we use the notation
$\|\psi\|_{\infty}$ for the $L^{\infty}$-norm of $\psi$.

######  Proposition 9.6.

Let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure space. For
$p\in(1,\infty)$ with $p>D$, there exist $C,A>0$ such that:

$\|\pi_{s}^{(p)}(g)\|_{W^{s,p}(Z)\to W^{s,p}(Z)}\leq
C\left\||g^{\prime}|\right\|_{\infty}^{A}\left\||(g^{-1})^{\prime}|\right\|_{\infty}^{A}$

for every $s\in(0,\frac{D}{p})$ and
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$.

###### Proof.

Let $g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$. We start by
observing that $\min_{\xi\in
Z}|g^{\prime}|(\xi)=\||(g^{-1})^{\prime}|\|_{\infty}$ by (3.3), and hence by
the proof of [58, Lemma 7], the Lipschitz constant of $|g^{\prime}|^{1/2}$ is
bounded by a constant multiple of
$\||g^{\prime}|\|_{\infty}\cdot\||(g^{-1})^{\prime}|\|_{\infty}^{1/2}$ .
Applying the Mean Value Theorem and (3.3) again implies that the Lipschitz
constant $L$ of the function $|g^{\prime}|^{-\frac{D}{p}+s}(\cdot)$ is bounded
by a constant multiple of
$\||g^{\prime}|\|_{\infty}^{2}\cdot\||(g^{-1})^{\prime}|\|_{\infty}$ (remember
$s$ has bounded range).

Take $\phi\in W^{s,p}(Z)$, and calculate:

$\displaystyle\left[\pi_{s}^{(p)}(g)\phi\right]_{s,p}^{p}$
$\displaystyle=\iint_{Z}\frac{\left||(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\xi)\phi(g^{-1}\xi)-|(g^{-1})^{\prime}|^{\frac{D}{p}-s}(\eta)\phi(g^{-1}\eta)\right|^{p}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)$
(using (3.3), and denoting $\omega=g^{-1}\xi$, $\tau=g^{-1}\eta$)
$\displaystyle=\iint_{Z}\frac{\left||g^{\prime}|^{-\frac{D}{p}+s}(\omega)\phi(\omega)-|g^{\prime}|^{-\frac{D}{p}+s}(\tau)\phi(\tau)\right|^{p}}{d^{D+sp}(g\omega,g\tau)}\mathrm{d}\nu(g\omega)\mathrm{d}\nu(g\tau)$
(applying (3.1),(3.2), and bounding by the supremum of $|g^{\prime}|$)
$\displaystyle\leq\||g^{\prime}|\|_{\infty}^{D-sp}\iint_{Z}\frac{\left||g^{\prime}|^{-\frac{D}{p}+s}(\omega)\phi(\omega)-|g^{\prime}|^{-\frac{D}{p}+s}(\tau)\phi(\tau)\right|^{p}}{d^{D+sp}(\omega,\tau)}\mathrm{d}\nu(\omega)\mathrm{d}\nu(\tau)$
(inserting $\pm|g^{\prime}|^{-\frac{D}{p}+s}(\tau)\phi(\omega)$, using
$|a+b|^{p}\leq 2^{p-1}(|a|^{p}+|b|^{p})$, and $s>0$) $\displaystyle\leq
2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\int_{Z}|\phi(\omega)|^{p}\int_{Z}\frac{\left||g^{\prime}|^{-\frac{D}{p}+s}(\omega)-|g^{\prime}|^{-\frac{D}{p}+s}(\tau)\right|^{p}}{d^{D+sp}(\omega,\tau)}\mathrm{d}\nu(\tau)\mathrm{d}\nu(\omega)$
$\displaystyle\quad+2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\iint_{Z}\frac{|g^{\prime}|^{-D+sp}(\tau)\cdot|\phi(\omega)-\phi(\tau)|^{p}}{d^{D+sp}(\omega,\tau)}\mathrm{d}\nu(\omega)\mathrm{d}\nu(\tau)$
(using $0<s<\frac{D}{p}$; that $L$ is the Lipschitz constant of
$|g^{\prime}|^{-\frac{D}{p}+s}(\cdot)$; and (3.3) again) $\displaystyle\leq
2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\,L^{p}\int_{Z}|\phi(\omega)|^{p}\int_{Z}\frac{1}{d^{2D-p}(\omega,\tau)}\mathrm{d}\nu(\tau)\mathrm{d}\nu(\omega)$
$\displaystyle\quad+2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\||(g^{-1})^{\prime}|\|_{\infty}^{D-sp}\iint_{Z}\frac{\left|\phi(\omega)-\phi(\tau)\right|^{p}}{d^{D+sp}(\omega,\tau)}\mathrm{d}\nu(\omega)\mathrm{d}\nu(\tau)$
(using Lemma 6.1 since $2D-p<D$ ) $\displaystyle\leq
C_{1}(p)\,2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\,L^{p}\|\phi\|_{p}^{p}+2^{p-1}\||g^{\prime}|\|_{\infty}^{D}\||(g^{-1})^{\prime}|\|_{\infty}^{D}[\phi]_{s,p}^{p}$
$\displaystyle\leq
C_{2}(p)\||g^{\prime}|\|_{\infty}^{D+2p}\||(g^{-1})^{\prime}|\|_{\infty}^{p}\|\phi|W^{s,p}(Z)\|^{p}.$

To finish the proof, we estimate the $L^{p}$-part of the norm on $W^{s,p}(Z)$:

$\displaystyle\left\|\pi_{s}^{(p)}(g)\phi|L^{p}\right\|^{p}$
$\displaystyle=\int_{Z}|(g^{-1})^{\prime}|^{D-sp}(\xi)\left|\phi(g^{-1}\xi)\right|^{p}\mathrm{d}\nu(\xi)$
$\displaystyle\leq\left\||(g^{-1})^{\prime}|\right\|_{\infty}^{D}\int_{Z}|\phi(\omega)|^{p}|g^{\prime}|^{D}(\omega)\mathrm{d}\nu(\omega)$
$\displaystyle\leq\left\||(g^{-1})^{\prime}|\right\|_{\infty}^{D}\left\||g^{\prime}|\right\|_{\infty}^{D}\left\|\phi|L^{p}\right\|^{p}.\qed$

##  10\. Potential theory and dual representations when $p=2$

In this section, we prove Theorem 4.

For this section, let $(Z,d,\nu)$ be an Ahlfors $D$-regular metric measure
space. Let $s\in(0,1)$.

Denote $\mathcal{E}_{s}=[\cdot]_{s,2}$. We want to consider $\mathcal{E}_{s}$
as a regular Dirichlet form on $L^{2}(Z)$ [30]; this is explained e.g. in [40,
p. 251]. The only potential issue is the domain of $\mathcal{E}_{s}$. As
mentioned in Section 4, the set $O=W^{s,2}(Z)\cap C_{0}(Z)$ contains compactly
supported Lipschitz functions; the question is whether $O$ is dense in
$W^{s,2}(Z)$ with respect to
$\langle\cdot,\cdot\rangle_{L^{2}}+\mathcal{E}_{s}[\cdot]$. We _define_
$H_{0}^{s}(Z)=W_{0}^{s,2}(Z)$ to be the closure on $O$ in this norm, and
consider the domain of $\mathcal{E}_{s}$ to be $H_{0}^{s}(Z)$ with core
$O$.101010During the process of writing this piece, we were informed by G.
Gerontogiannis (private communication) that in fact $H_{0}^{s}(Z)=H^{s}(Z)$.

######  Remark 10.1.

Assume for a moment that $\nu$ is the Hausdorff $D$-measure on $(Z,d)$. Then
since $|g^{\prime}|\in\operatorname{\mathrm{Lip}}(Z)$ for any
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $\pi_{s}^{(2)}$
preserves the core $O$, and thus extends to a uniformly bounded representation
of $\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$ on $H_{0}^{s}(Z)$ by
Theorem 2 (which we continue to denote $\pi_{s}^{(2)}$).

By general theory [30], $\mathcal{E}_{s}$ has an infinitesimal generator that
we will denote $-\Delta^{s}$, i.e. a non-positive self-adjoint operator with
domain $\mathcal{D}(\Delta^{s})\subset H^{s}_{0}(Z)$ such that
$\mathcal{E}_{s}[\phi]=\langle\sqrt{\Delta^{s}}\phi,\sqrt{\Delta^{s}}\phi\rangle$
for $\phi\in H^{s}_{0}(Z)$. Furthermore, there is a (strongly continuous) heat
semigroup $T_{t}=\exp(-t\Delta^{s})$ for $t>0$. Potential theory now tells us
that $T_{t}$ are kernel operators, and provides a _stable-like_ estimate on
the kernels:

$\displaystyle T_{t}\phi(\eta)$
$\displaystyle=\int_{Z}p_{t}(\xi,\eta)\phi(\xi)\mathrm{d}\nu(\xi),\quad\phi\in
L^{2}(Z),\,\eta\in Z,$ $\displaystyle p_{t}(\xi,\eta)$
$\displaystyle\asymp\frac{1}{t^{D/(2s)}}\left(1+\frac{d(\xi,\eta)}{t^{1/(2s)}}\right)^{-(D+2s)}\asymp\min\left(t^{-\frac{D}{2s}},\frac{t}{d^{D+2s}(\xi,\eta)}\right),$

for $t\in(0,\text{diam}(Z)^{2s})$ and $\nu$-almost all $\xi,\eta\in Z$, by
[34, Theorem 1.12] and the discussion afterwards; see also [20].

Next, we will use the heat kernel calculus from [39] to obtain an estimate for
the kernel of the analogue of the Bessel potential, $(1-\Delta^{s})^{-1}$.
More precisely: the heat kernel estimate means that $(p_{t})$ satisfies the
condition $(H_{\Phi})$ of [39, (4.5),(4.6)] (with $\Phi(t)=(1+t)^{-(D+2s)}$,
$\beta=2s$, $V(t)=t^{D}$). We aim to apply [39, Theorem 4.3] (with $\rho\equiv
1$, $\theta_{1}=D$). The assumption [39, (4.8)] requires verifying that

$\int_{0}^{\infty}t^{D-2s-1}\Phi(t)\mathrm{d}t=\int_{0}^{\infty}\left(\frac{t}{(1+t)}\right)^{D}\cdot\left(\frac{1}{t(1+t)}\right)^{2s}\frac{\mathrm{d}t}{t}<\infty,$

which holds as $D>2s$. Finally, the completely monotone function $f$ is
$f(t)=t^{-1}$. Applying [39, Theorem 4.3], we obtain that the positive
definite operator $J_{s}=(1-\Delta_{s})^{-1}$ (defined by functional calculus)
is a kernel operator with kernel $k_{s}$, satisfying

(10.1) $\displaystyle k_{s}(\xi,\eta)$
$\displaystyle\asymp\frac{1}{d^{D-2s}(\xi,\eta)},\quad\xi,\eta\in Z.$

Note that $J_{s}$ is injective by [39, Theorem 3.1].

As the norm on $H_{0}^{s}(Z)$ is equivalent to
$\langle(1-\Delta_{s})\phi,\phi\rangle_{L^{2}}^{1/2}$, the dual of
$H_{0}^{s}(Z)$, denoted $H_{0}^{-s}(Z)=H_{0}^{s}(Z)^{\prime}$, is the
completion of $L^{2}(Z)$ with respect to the (Hilbert space) norm
$\langle\phi,J_{s}\phi\rangle_{L^{2}}^{1/2}$. Together with (10.1), this
proves the first part of Theorem 4. The second part of the Theorem follows
from duality and Remark 10.1 (assuming that $\nu$ is the Hausdorff $D$-measure
of $(Z,\nu)$).

##  Appendix A A groupoid picture

Assume for simplicity that $G$ is a discrete group, acting on a locally
compact second countable space $Z$. We also make a blanket assumptions that
the representations considered are separable.

Denote $\mathcal{Z}=Z\times Z$ for the pair groupoid. The transformation
groupoid $\mathcal{G}=G\ltimes Z$ acts by automorphisms on $\mathcal{Z}$ by
$\mathcal{G}\to G\to\text{Aut}(\mathcal{Z})$. One can then define the _fibred
semi-direct product groupoid_ , $\mathcal{G}\rtimes\mathcal{Z}$, as follows.

As a topological space, $\mathcal{G}\ltimes\mathcal{Z}$ is the fibre product

$\mathcal{G}\ltimes\mathcal{Z}=_{\text{def.}}\\{(g,z):t_{\mathcal{Z}}(z)=s_{\mathcal{G}}(g)\\}\subset\mathcal{G}\times\mathcal{Z},$

with space of objects $Z$, and source and target map given respectively by
$t(g,z)=t_{\mathcal{G}}(g)$ and $s(g,z)=s_{\mathcal{Z}}(z)$. For $x\in Z$
denote $e_{x}=(x,x)\in\mathcal{Z}$, and $f_{x}=(\text{Id},x)\in\mathcal{G}$.
The maps $t$ and $s$ are continuous, and $\mathcal{G}$ and $\mathcal{Z}$
respectively identify with subsets
$\\{(g,e_{s_{\mathcal{G}}(g)}):g\in\mathcal{G}\\}$ and
$\\{(f_{t_{\mathcal{Z}}(z)},z):z\in\mathcal{Z}\\}$. The multiplication on
$\mathcal{G}\ltimes\mathcal{Z}$ is the unique one that makes $\mathcal{G}$ and
$\mathcal{Z}$ (identified as above) into subgroupoids, and satisfies
$(g,e_{s_{\mathcal{G}}(g)})(f_{t_{\mathcal{Z}}(z)},z)=(g,z)$ when
$t_{\mathcal{Z}}(z)=s_{\mathcal{G}}(g)$, and
$(f_{t_{\mathcal{Z}}(z)},z)(g,e_{s_{\mathcal{G}}(g)})=((\gamma,\gamma^{-1}t_{\mathcal{Z}}(z)),g^{-1}z)$
when $g=(\gamma,\gamma^{-1}s_{\mathcal{Z}}(z))\in\mathcal{G}$. Observe that
$\mathcal{G}\ltimes\mathcal{Z}$ has isotropy groups

$G_{z}=\left\\{\left((\gamma^{-1}z,\gamma),(\gamma^{-1}z,z)\right):\gamma\in
G\right\\}$

with $z\in Z$, all isomorphic to $G$. In particular every representation of
$\mathcal{G}\ltimes\mathcal{Z}$ induces isotropic representations of $G$.

Concretely, let $(\Pi,\mathcal{V}_{Z})$ be a unitary representation of the
transformation groupoid $\mathcal{G}=G\ltimes Z$. A cocycle,$c$ on
$\mathcal{V}_{Y}$, i.e. a representation of the pair groupoid
$\mathcal{Z}\simeq Z\times Z$ is $\mathcal{G}$-equivariant if it satisfies an
extra condition

$\Pi(b;\gamma)^{-1}c(\gamma b,\gamma a)\Pi(a;\gamma)=c(b,a)$

for $a,b\in Z$ and $\gamma\in G$. Cocycles of this type correspond to
representations of $\mathcal{G}\ltimes\mathcal{Z}$, and induce a class of
representations of $G$ by the formula

$\pi_{a}(\gamma)=_{\text{def.}}c(a,\gamma
a)\Pi(a;\gamma)=\Pi(\gamma^{-1}a;\gamma)c(\gamma^{-1}a,a)$

for $a\in Z$ and $g\in G$. Observe that $c$ is unitary (respectively uniformly
bounded) if and only if $\pi_{a}$ is unitary (respectively uniformly bounded).
Moreover $c(b,a)$ intertwines the representation $(\pi_{a},\mathcal{H}_{a})$
with $(\pi_{b},\mathcal{H}_{b})$ for all $a,b\in Z$.

On the other hand every uniformly bounded representation of $G$ on a Hilbert
space (or even a Banach space) induces an uniformly bounded representation of
$\mathcal{G}\ltimes\mathcal{Z}$:

######  Proposition A.1.

Assuming that the action of $G$ on $Z$ is amenable, every uniformly bounded
representation of $G$ can be seen as the isotropy representation obtained from
a unitary representation of $\mathcal{G}$ and a $G$-equivariant uniformly
bounded cocycle over $\mathcal{Z}$.

###### Proof.

Given a uniformly bounded representation $\pi$ of $G$ on a Hilbert space
$\mathcal{H}$, let

$T_{\pi}:\mathcal{H}\otimes_{\text{proj}}\mathcal{H}\rightarrow
L^{\infty}(G\ltimes Z);\quad(v\otimes w)\mapsto T_{\pi}(v\otimes w)$

be defined by $T_{\pi}(v\otimes w)(g,z)=(\pi(g)v,\pi(g)w)$. Observe that
$T_{\pi}$ is $G$-equivariant in the following sense:
$T_{\rho}\pi(g)\otimes\pi(g)=\lambda(g)T_{\rho}$, where $\lambda$ stands for
the left translation. If $E:\mathcal{L}^{\infty}(G\ltimes
Z)\rightarrow\mathcal{L}^{\infty}(Z)$ denotes the $G$-equivariant conditional
expectation — which exists by amenability of the action of $G$ on $Z$ — then
$E\circ
T_{\pi}:\mathcal{H}\otimes_{\text{proj}}\mathcal{H}\rightarrow\mathcal{L}^{\infty}(Z)$
is non-zero, $G$-equivariant and $B_{z}(v,w)=_{\text{def.}}E\circ
T_{\pi}(v\otimes w)(z)$ defines a scalar product on $\mathcal{H}$ for all
$z\in Z$. Let $\mathcal{H}_{z}$ be the completion of $\mathcal{H}$ with
respect to the scalar product $B_{z}$. Then
$\pi(g,z)=\pi(g):\mathcal{H}_{z}\rightarrow\mathcal{H}_{gz}$ is isometric and
thus $(\pi|_{G\ltimes Z},(\mathcal{H}_{z})_{z})$ is an isometric
representation of $G\ltimes Z$.

As $\|v\|_{z}\asymp_{\|\pi\|}\|v\|$ for all $z\in Z$ and $v\in\mathcal{H}$,
there exists a bounded invertible operator
$S_{z}:\mathcal{H}_{z}\rightarrow\mathcal{H}$ that satisfies
$B_{z}(v,w)=(S_{z}v,S_{z}w)$ for all $z\in Z$ and $v,w\in\mathcal{H}$ with
$\|S_{z}^{\pm 1}\|\asymp\|\pi\|^{\pm 1}$. It is an exercise to prove that
$c(a,b)=_{\text{def.}}S_{a}^{-1}S_{b}$ is $\mathcal{G}$-equivariant cocycle,
and thus $(\pi|_{G\ltimes Z},(\mathcal{H}_{z})_{z},c)$ defines a
representation of $\mathcal{G}\ltimes\mathcal{Z}$. Observe that the isotropic
representations $(\pi_{z})_{z}$ satisfy $\pi_{z}(g)=S_{gz}^{-1}\pi(g)S_{z}$
for all $z\in Z$. ∎

##  Appendix B Some calculations

In this Appendix, we spell out some calculations postponed from Section 2.
Here $(Z,d)$ is a complete metric space without isolated points, and $\nu$ is
an Ahlfors $D$-regular Hausdorff $D$-measure on $(Z,d)$.

######  Calculation B.1.

We verify the conformal identity for $\Delta_{a}^{s}$ from (2.1), i.e. that
$\Delta_{ga}^{s}(g^{*}\phi)=|g^{\prime}|^{2s}(a)\cdot
g^{*}(\Delta_{a}^{s}\phi)$. Here
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $a\in Z$,
$\phi\in\mathcal{D}_{a}$, and $\xi\in Z_{a}$:

$\displaystyle\Delta_{ga}^{s}(g^{*}\phi)(\xi)$
$\displaystyle=-\int_{Z_{ga}}\frac{g^{*}\phi(\eta)-g^{*}\phi(\xi)}{d^{D+2s}_{ga}(\eta,\xi)}\,\mathrm{d}\nu_{ga}(\eta)$
$\displaystyle=-\int_{Z_{ga}}\frac{\phi(\omega)-g^{*}\phi(\xi)}{d_{ga}^{D+2s}(g\omega,\xi)}\,\mathrm{d}\nu_{ga}(g\omega)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:change-of-var-
nu_a}\eqref{eq:GMV-for-
d_a}}}{{=}}-\int_{Z_{a}}\frac{\phi(\omega)-\phi(g^{-1}\xi)}{d_{a}^{D+2s}(\omega,g^{-1}\xi)}\cdot|g^{\prime}|^{D+2s}(a)\cdot\frac{\mathrm{d}\nu_{a}(\omega)}{|g^{\prime}|^{D}(a)}$
$\displaystyle=|g^{\prime}|^{2s}(a)\cdot\Delta_{a}^{s}\phi(g^{-1}\xi)=|g^{\prime}|^{2s}(a)\cdot
g^{*}(\Delta_{a}^{s}\phi)(\xi).$

######  Calculation B.2.

We verify that for an $s$-conformal family $(\mathcal{D}_{a},T_{a})_{a\in Z}$,
$g\in\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$, $a\in Z$, $p\geq 2$,
and $z=s+it\in\mathbb{C}$, the operator
$\Pi_{z}(g;a):\mathcal{H}_{T_{a}}\to\mathcal{H}_{T_{ga}}$ is unitary. Recall
that $\Pi_{z}(g;a)\phi=|g^{\prime}|^{\frac{D}{2}-z}(a)\,g^{*}\phi$. Here
$\phi,\psi\in\mathcal{D}_{a}$:

$\displaystyle\langle
T_{ga}\Pi_{z}(g;a)\phi,\Pi_{z}(g;a)\psi\rangle_{L^{2}(Z_{ga})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Pi_s_g_a-
defn}}}{{=}}|g^{\prime}|^{D-z-\overline{z}}(a)\langle
T_{ga}(g^{*}\phi),g^{*}\psi\rangle_{L^{2}_{ga}}$
$\displaystyle=|g^{\prime}|^{D-2s}(a)\langle
T_{ga}(g^{*}\phi),g^{*}\psi\rangle_{L^{2}_{ga}}$
$\displaystyle=|g^{\prime}|^{D}(a)\langle
g^{*}(T_{a}\phi),g^{*}\psi\rangle_{L^{2}_{ga}}$
$\displaystyle=|g^{\prime}|^{D}(a)\int_{Z_{g}a}T_{a}\phi(g^{-1}\xi)\overline{\psi(g^{-1}\xi)}\,\mathrm{d}\nu_{ga}(\xi)$
$\displaystyle=\int_{Z_{ga}}T_{a}\phi(\omega)\overline{\psi(\omega)}\,|g^{\prime}|^{D}(a)\,\mathrm{d}\nu_{ga}(g\omega)$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:change-of-var-
nu_a}}}{{=}}\int_{Z_{a}}T_{a}\phi(\omega)\overline{\psi(\omega)}\,\mathrm{d}\nu_{a}(\omega)$
$\displaystyle=\langle T_{a}\phi,\psi\rangle_{L^{2}(Z_{a})}.$

##  Appendix C A remark on tangent spaces and conformal rescaling of metric
operators

In this Appendix, we offer a heuristic for thinking about $(Z_{a},d_{a})$ as
tangent spaces of $(Z,d)$, and how to interpret the conformal operators on the
bundle $(Z_{a},d_{a})_{a\in Z}$ as a rescaling of operators on $Z$.

Throughout this section, we suppose that $(X,\rho)$ is a roughly geodesic,
strongly hyperbolic metric space (cf. [56, 59] for background). Fix a
basepoint $o\in X$, and let $Z=\partial X$. This setup has been explained
before, cf. Theorem 2 and Section 9: denote by $\langle\cdot,\cdot\rangle_{o}$
the Gromov product (which continuously extends to the compactification $X\cup
Z$), by $d(\xi,\eta)=\exp(-\varepsilon_{0}\langle\xi,\eta\rangle_{o})$ (for a
suitable $\varepsilon_{0}>0$) a metric on $Z$, by $D$ the Hausdorff dimension
of $(Z,d)$, and by $\nu$ the (Ahlfors $D$-regular) measure on $Z$.

Suppose that a group $\Gamma$ acts properly discontinuously and cocompactly by
isometries on $(X,\rho)$ (and hence
$\Gamma<\operatorname{M\mathnormal{\ddot{\mathrm{o}}}b}(Z)$). We shall also
need that in this situation, for $g\in\Gamma$ and $\xi\in Z$, we have
$|g^{\prime}|(\xi)=\exp(-\varepsilon_{0}\beta(\xi,go,o))$, where for $x\in X$
we denote $\beta(\xi,x,o)=-2\langle\xi,x\rangle_{o}+\rho(x,o)$ the Busemann
function [59, Section 3].

###  C.1. Metric tangent spaces

Fix $a\in Z$. Roughly speaking, we explain below that any fixed ball
$B(z_{0},R)$ in $(Z_{a},d_{a})$ is a (Gromov–Hausdorff) limit of rescaled
small balls in $Z$ “in the direction of $a$”.

In this subsection, let us use the notation $B_{d}(z,R)$ to denote (open)
balls centred at $z$, with radius $R$, with respect to the metric $d$.

Suppose that we fix a sequence $(g_{n})_{n\in\mathbb{N}}\subset\Gamma$ such
that $g_{n}o\to a$ in the compactification $X\cup Z$, as $n\to\infty$. Denote
$\lambda_{n}=\exp(\varepsilon_{0}\rho(g_{n}o,o))$; then necessarily
$\lambda_{n}\to\infty$.

Fix a basepoint $z_{0}\in Z_{a}$, and $R>0$. We now show that the balls
$B_{\lambda_{n}d}(g_{n}z_{0},R)$ “converge” to the ball $B_{d_{a}}(z_{0},R)$
in $(Z_{a},d_{a})$ in the following sense: there exist functions
$f_{n}:B_{\lambda_{n}d}(g_{n}z_{0},R)\to Z_{a}$, such that

  * •

$f_{n}(g_{n}z_{0})=z_{0}$ for all $n\in\mathbb{N}$,

  * •

for any $\delta>0$, $f_{n}$ are eventually surjective onto
$B_{d_{a}}(z_{0},R-\delta)$, and

  * •

the restrictions of the metrics to the balls converge uniformly to the metric
on $B_{d_{a}}(z_{0},R)$, i.e.

$\lim_{n\to\infty}\sup_{\xi,\eta\in
B_{d_{a}}(z_{0},R)}\\{|\lambda_{n}d(\xi_{n},\eta_{n})-d_{a}(\xi,\eta)|:f_{n}(\xi_{n})=\xi,f_{n}(\eta_{n})=\eta\\}=0.$

For $n\in\mathbb{N}$ and $\xi\in B_{\lambda_{n}d}(g_{n}z_{0},R)$, we set
$f_{n}(\xi)=g_{n}^{-1}\xi$. Then the above properties follow from the
calculation below. Let $\xi_{n},\eta_{n}\in B_{\lambda_{n}d}(g_{n}z_{0},R)$,
and denote $\xi=f_{n}(\xi_{n})$, $\eta=f_{n}(\eta_{n})$. Then

$\displaystyle\lambda_{n}d(\xi_{n},\eta_{n})$
$\displaystyle=\lambda_{n}d(g_{n}\xi,g_{n}\eta)\stackrel{{\scriptstyle\eqref{eq:GMV}}}{{=}}\lambda_{n}\cdot|g_{n}^{\prime}|^{\frac{1}{2}}(\xi)\cdot|g_{n}^{\prime}|^{\frac{1}{2}}(\eta)\cdot
d(\xi,\eta)$
$\displaystyle=\exp(\varepsilon_{0}\rho(g_{n}o,o))\cdot\exp(\varepsilon_{0}(\langle\xi,g_{n}o\rangle_{o}-\rho(g_{n}o,o)/2))\cdot$
$\displaystyle\quad\cdot\exp(\varepsilon_{0}(\langle\eta,g_{n}o\rangle_{o}-\rho(g_{n}o,o)/2))\cdot
d(\xi,\eta)$
$\displaystyle=\frac{d(\xi,\eta)}{\exp(-\varepsilon_{0}\langle\xi,g_{n}o\rangle_{o})\cdot\exp(-\varepsilon_{0}\langle\xi,g_{n}o\rangle_{o})}$
$\displaystyle\longrightarrow_{n\to\infty}\frac{d(\xi,\eta)}{d(\xi,a)d(\eta,a)}=d_{a}(\xi,\eta).$

This fits into the framework of weak tangent spaces from [19] (see also [24]).
One may also extend this to the framework of measured Gromov–Hausdorff
convergence to include the measures $\nu$ and $\nu_{a}$.

###  C.2. Rescaling of operators

We continue in the setup above. Suppose we have an operator $T$ on $L^{2}(Z)$,
with domain $\mathcal{D}(T)\subset L^{2}(Z)$ stable under the action of
$\Gamma$. We say that an operator $(\mathcal{D}(T_{a}),T_{a})$ on
$L^{2}(Z_{a})$ is a _rescale_ of $T$, if there is a sequence
$(\lambda_{n})_{n\in\mathbb{N}}\subset(0,\infty)$ with $\lambda_{n}\to\infty$
as $n\to\infty$, and a sequence $(g_{n})_{n\in\mathbb{N}}\subset\Gamma$ with
$g_{n}o\to a$ in $X\cup Z$, such that

$\displaystyle\phi\in\mathcal{D}(T_{a})\implies
g_{n}^{*}\phi\in\mathcal{D}(T)\text{ for all }n\in\mathbb{N},\text{ and}$
$\displaystyle\langle
T_{a}\phi,\psi\rangle_{L^{2}(Z_{a})}=\lim_{n\to\infty}\left\langle
T(\lambda_{n}g_{n}^{*}\phi),\lambda_{n}g_{n}^{*}\psi\right\rangle_{L^{2}(Z)},$

for $\phi\in\mathcal{D}(T_{a})$, $\psi\in L^{2}(Z_{a})$.

We shall check that for $s\in(0,1)$, the fractional Laplacian (2.1) is a
rescale of

$\Delta^{s}\phi(\xi)=-\int_{Z}\frac{\phi(\eta)-\phi(\xi)}{d^{D+2s}(\xi,\eta)}\,\mathrm{d}\nu(\eta),\quad\xi\in
Z,$

with the domain $\mathcal{D}(\Delta^{s})$ consisting of compactly supported
Lipschitz functions on $Z$. We fix any $(g_{n})_{n\in\mathbb{N}}$ as above
(i.e. such that $g_{n}o\to a$), denote
$\lambda_{n}=\exp(\varepsilon_{0}(D-sp)\rho(g_{n}o,o)/2)$.

For notational convenience, we shall write
$d(\xi,x)=\exp(-\varepsilon_{0}\langle\xi,x\rangle_{o})$ for $\xi\in Z$ and
$x\in X$. Although this is not a metric, we have that $d(\xi,x)\to d(\xi,a)$
for $x\to a$ in $X\cup Z$. Similarly, we write
$\mathrm{d}\nu_{x}(\xi)=d^{-2D}(x,\xi)\mathrm{d}\nu(\xi)$; these are measures
on $Z$ such that $\nu_{x}\to\nu_{a}$ as $x\to a$. Finally, we write
$d_{x}(\xi,\eta)=d(\xi,\eta)/(d(\xi,x)d(\eta,x))$.

As a preparatory step, we calculate for $n\in\mathbb{N}$ and $\xi,\eta\in Z$:

(C.1) $\displaystyle\begin{aligned}
\lambda_{n}^{2}|g_{n}^{\prime}|^{\frac{D}{2}-s}(\xi)|g_{n}^{\prime}|^{\frac{D}{2}-s}(\eta)&=\exp(\varepsilon_{0}(D-2s)\rho(g_{n}o,o))\cdot\\\
&\qquad\cdot\exp\\!\left(\varepsilon_{0}\\!\left(\tfrac{D}{2}-s\right)\\!\left(2\langle\xi,g_{n}o\rangle-\rho(g_{n}o,o)\right)\right)\cdot\\\
&\qquad\cdot\exp\\!\left(\varepsilon_{0}\\!\left(\tfrac{D}{2}-s\right)\\!\left(2\langle\eta,g_{n}o\rangle-\rho(g_{n}o,o)\right)\right)\\\
&=d^{-D+2s}(\xi,g_{n}o)\,d^{-D+2s}(\eta,g_{n}o).\end{aligned}$

Now we calculate

$\left\langle\Delta^{s}(\lambda_{n}g_{n}^{*}\phi),\lambda_{n}g_{n}^{*}\psi\right\rangle_{L^{2}(Z)}\\\
\begin{aligned}
&=\frac{\lambda_{n}^{2}}{2}\iint_{Z^{2}}\frac{(\phi(g_{n}^{-1}\xi)-\phi(g_{n}^{-1}\eta))\overline{(\psi(g_{n}^{-1}\xi)-\psi(g_{n}^{-1}\eta))}}{d^{D+2s}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)\\\
&\stackrel{{\scriptstyle\eqref{eq:change-of-
var},\eqref{eq:GMV}}}{{=}}\frac{\lambda_{n}^{2}}{2}\iint_{Z^{2}}\frac{(\phi(\xi)-\phi(\eta))\overline{(\psi(\xi)-\psi(\eta))}}{d^{D+2s}(\xi,\eta)}|g_{n}^{\prime}|^{\frac{D}{2}-s}(\xi)|g_{n}^{\prime}|^{\frac{D}{2}-s}(\eta)\,\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta)\\\
&\stackrel{{\scriptstyle\eqref{localeq:prep-for-rescale-
calculation}}}{{=}}\frac{1}{2}\iint_{Z^{2}}\frac{(\phi(\xi)-\phi(\eta))\overline{(\psi(\xi)-\psi(\eta))}}{d_{g_{n}o}^{D+2s}(\xi,\eta)}\mathrm{d}\nu_{g_{n}o}(\xi)\mathrm{d}\nu_{g_{n}o}(\eta)\\\
&\longrightarrow\langle\Delta_{a}^{s}\phi,\psi\rangle_{L^{2}(Z_{a})},\end{aligned}$

Note that a similar calculation works more generally for the fractional
$p$-Laplacian ($p\geq 1$), which is a $(p-1)$-homogeneous nonlinear operator
$\Delta_{p}^{2}:W^{s,p}(Z)\to W^{s,p}(Z)^{\prime}$, defined by

$\langle\Delta_{p}^{s}(u),v\rangle_{(W^{s,p})^{\prime},W^{s,p}}=\frac{1}{2}\iint\frac{|u(\xi)-u(\eta)|^{p-2}(u(\xi)-u(\eta))\overline{(v(\xi)-v(\eta))}}{d^{D+sp}(\xi,\eta)}\mathrm{d}\nu(\xi)\mathrm{d}\nu(\eta),$

for $u,v\in W^{s,p}(Z)$.

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