†† 2020 Mathematical subject classification: 35K08, 35R11, 53C18

# A heat equation approach to intertwining

Nicola Garofalo  Dipartimento d’Ingegneria Civile e Ambientale (DICEA)
Università di Padova
Via Marzolo, 9 - 35131 Padova, Italy<EMAIL_ADDRESS>and  Giulio
Tralli  Dipartimento d’Ingegneria Civile e Ambientale (DICEA)
Università di Padova
Via Marzolo, 9 - 35131 Padova, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we present a new approach based on the heat equation and
extension problems to some intertwining formulas arising in conformal CR
geometry.

###### Key words and phrases:

Nonlocal intertwining. Parabolic extension problems. CR conformal invariance

Both authors are supported in part by a Progetto SID: “Non-local Sobolev and
isoperimetric inequalities”, University of Padova, 2019.

##  1\. Introduction

In this paper we present a heat semigroup approach to some intertwining
formulas which arise in conformal CR geometry. The prototypical example of the
questions we are interested in can be described as follows. In
$\mathbb{R}^{n}$ with $n\geq 2$, for $0<s<1$ consider the pseudodifferential
operator which in Fourier transform is given by
$\widehat{(-\Delta)^{s}u}(\xi)=(2\pi|\xi|)^{2s}\hat{u}(\xi)$, see [32, 35].
Then, for every $x\in\mathbb{R}^{n}$, and each fixed $y>0$, one has the
following intertwining relation

(1.1)
$\left(-\Delta\right)^{s}\left((|x|^{2}+y^{2})^{-\frac{n-2s}{2}}\right)=\frac{\Gamma\left(\frac{n}{2}+s\right)}{\Gamma\left(\frac{n}{2}-s\right)}\left(2y\right)^{2s}(|x|^{2}+y^{2})^{-\frac{n+2s}{2}}.$

The objective of this note is to use the heat equation to establish a suitable
variant of (1.1) in which $(-\Delta)^{s}$ is replaced by the nonlocal
conformal horizontal Laplacian $\mathscr{L}_{s}$ on a Lie group of Heisenberg
type $\mathbb{G}$. It is worth mentioning here that our approach is new even
in the classical case of (1.1).

To provide the reader with some historical perspective we recall that (1.1)
implicitly appeared in the celebrated 1983 work of Lieb concerning the best
constants in the Hardy-Littlewood-Sobolev inequalities, see (3.12), (3.13) in
the proof of [31, Theorem 3.1]. In this connection we observe that a notable
consequence of (1.1) is that it provides a family of positive solutions to the
nonlocal Yamabe equation in $\mathbb{R}^{n}$

(1.2) $\left(-\Delta\right)^{s}u=u^{\frac{n+2s}{n-2s}}.$

The uniqueness of such positive solutions was subsequently proved by Chen, Li,
and Ou in [8]. A different way of looking at (1.1), which also underscores its
geometric invariance, goes back to the ideas in [3, 4]. For a function $f$ on
$\mathbb{R}^{n}$ denote by $f^{\star}$ its push forward to
$\mathbb{S}^{n}\subset\mathbb{R}^{n+1}$ through the stereographic projection.
Branson characterized the conformal pseudodifferential operators $A_{s}$ of
order $2s$ on $\mathbb{S}^{n}$ satisfying the intertwining formula

(1.3)
$A_{s}\left(\left(\left(\frac{1+|\cdot|^{2}}{2}\right)^{\frac{n}{2}-s}f\right)^{\star}\right)=\left(\left(\frac{1+|\cdot|^{2}}{2}\right)^{\frac{n}{2}+s}(-\Delta)^{s}f\right)^{\star},$

and proved that they have the following spectral representation

(1.4)
$A_{s}=\frac{\Gamma(\sqrt{-\Delta_{\mathbb{S}^{n}}+\frac{(n-1)^{2}}{4}}+s+\frac{1}{2})}{\Gamma(\sqrt{-\Delta_{\mathbb{S}^{n}}+\frac{(n-1)^{2}}{4}}-s+\frac{1}{2})}.$

The number $A_{s}(1)=\frac{\Gamma(\frac{n}{2}+s)}{\Gamma(\frac{n}{2}-s)}$ is
called, up to a renormalising constant, the fractional $Q_{s}$-curvature of
$\mathbb{S}^{n}$. For more insights on these aspects we refer the reader to
[2, 7, 23] . We note that from (1.3) it is immediate to recognise the validity
of (1.1). It suffices to take
$f(x)=\big{(}\frac{1+|x|^{2}}{2}\big{)}^{s-\frac{n}{2}}$, and observe that

$(-\Delta)^{s}\big{(}\frac{2}{1+|x|^{2}}\big{)}^{\frac{n}{2}-s}=\big{(}\frac{2}{1+|x|^{2}}\big{)}^{\frac{n}{2}+s}\big{(}A_{s}(1^{\star})\big{)}^{\star}=\big{(}\frac{2}{1+|x|^{2}}\big{)}^{\frac{n}{2}+s}A_{s}(1)=\frac{\Gamma(\frac{n}{2}+s)}{\Gamma(\frac{n}{2}-s)}\big{(}\frac{2}{1+|x|^{2}}\big{)}^{\frac{n}{2}+s}.$

In our main result, Theorem 1.1 below, we present a new approach to
intertwining formulas such as (1.1) which is exclusively based on the heat
equation. To explain the main ideas without delving into technical aspects we
mention that, instead of looking at $(-\Delta)^{s}$, we consider the nonlocal
heat operator $(\partial_{t}-\Delta)^{s}$ and its _extension problem_ : given
a function $u\in C^{\infty}_{0}(\mathbb{R}^{n}_{x}\times\mathbb{R}_{t})$, find
$U\in
C^{\infty}(\mathbb{R}^{n}_{x}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y})$
such that

(1.5)
$\begin{cases}\mathfrak{P}^{(s)}U\overset{def}{=}\frac{\partial^{2}U}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial U}{\partial y}+\Delta_{x}U-\frac{\partial
U}{\partial t}=0,\\\ U(x,t,0)=u(x,t).\end{cases}$

We claim that the fundamental solution $q^{(s)}(x,y,t)$ of the operator
$\mathfrak{P}^{(s)}$ in (1.5) can be used to provide a simple proof of (1.1).
To see this we note that, if $w\in\mathbb{R}^{2(1-s)}$ and $y=|w|$, then
$\mathfrak{P}^{(s)}$ represents the action on functions
$U(x,w,t)=\overline{U}(x,y,t)$ of the heat operator
$\Delta_{x}+\Delta_{w}-\partial_{t}$ in the space with fractal dimension
$\mathbb{R}^{n+2(1-s)}\times\mathbb{R}^{+}_{t}$ whose fundamental solution
(with pole at the origin) is given by

(1.6) $q^{(s)}(x,y,t)=\frac{1}{(4\pi
t)^{\frac{n}{2}+1-s}}e^{-\frac{|x|^{2}+y^{2}}{4t}}.$

If we denote by $q^{(-s)}(x,y,t)$ the heat kernel obtained by replacing $s$
into $-s$ in (1.6), then by Bochner’s subordination the two functions

(1.7) $E^{(\pm s)}(x,y)\overset{def}{=}\int_{0}^{\infty}q^{(\pm s)}(x,y,t)dt$

are the fundamental solutions of the time-independent differential operators
$\frac{\partial^{2}}{\partial y^{2}}+\frac{1\mp 2s}{y}\frac{\partial}{\partial
y}+\Delta_{x}$. If we now use the elementary (but very important) consequence
of (1.6) that

(1.8) $\int_{0}^{\infty}q^{(\pm s)}(x,y,t)dt=\frac{\Gamma(\frac{n\mp
2s}{2})}{4\pi^{\frac{n}{2}+1\mp s}}(|x|^{2}+y^{2})^{-\frac{n\mp 2s}{2}},$

then in view of (1.7) the problem of proving (1.1) is reduced to that of
establishing the equivalent dimension-free relation

(1.9) $\left(-\Delta\right)^{s}\left(E^{(s)}(\cdot,y)\right)(x)=\left(2\pi
y\right)^{2s}E^{(-s)}(x,y).$

To prove (1.9) we once again bring to the center stage the heat semigroup
$P_{t}=e^{-t\Delta}$. If, in fact, instead of the Fourier transform definition
of $(-\Delta)^{s}$ we use the (equivalent) one given by the following formula
of Balakrishnan [1],

(1.10)
$(-\Delta)^{s}u(x)=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{t^{1+s}}(P_{t}u(x)-u(x))dt,$

then the proof of (1.9) hinges on the computation of
$P_{t}(E^{(s)}(\cdot,y))(x)$, with $E^{(s)}$ defined by (1.7). As we show in
(2.7) below and subsequent considerations, the Chapman-Kolmogorov identity and
an elementary change of variable allow to easily complete this computation and
establish (1.9). By comparison, a direct proof of (1.1) via Fourier transform
rests on elaborate computations involving special functions which tend to
overshadow the geometric content of the formula itself (for complete details
see [20, Lemma 8.6]).

The pseudodifferential operators $(-\Delta)^{s}$ and (1.4) have a counterpart
in CR geometry. In this context the Heisenberg group
$\mathbb{H}^{n}\cong\mathbb{C}^{n}\times\mathbb{R}$ with coordinates
$(z,\sigma)$111we explicitly mention here that traditionally the letter $t$ is
reserved for the vertical variable in $\mathbb{H}^{n}$. However, since we want
to indicate the time variable with $t$, we have opted for the notation
$\sigma$. occupies a special position since, via the Cayley transform, such
group has a conformal identification with the boundary of the unit ball in
$\mathbb{C}^{n+1}$, the sphere $\mathbb{S}^{2n+1}$ with its standard CR
structure. Analogously to the term $(|x|^{2}+1)^{-1}$ in the stereographic
projection, the function
$|i+(4\sigma+i|z|^{2})|^{-2}=((|z|^{2}+1)^{2}+16\sigma^{2})^{-1}$ appears as
the conformal factor in the Cayley transform, see [26, Section 4], and in fact
appropriate powers of such factor played a central role in the works of
Jerison and Lee [27] and Frank and Lieb [15]. In $\mathbb{H}^{n}$, with
$T=\partial_{\sigma}$, the CR conformal nonlocal operator $\mathscr{L}_{s}$
was first introduced in the work by Branson et al. [5, see (1.33)] via the
spectral formula

(1.11)
$\mathscr{L}_{s}=2^{s}|T|^{s}\frac{\Gamma(-\frac{1}{2}\mathscr{L}|T|^{-1}+\frac{1+s}{2})}{\Gamma(-\frac{1}{2}\mathscr{L}|T|^{-1}+\frac{1-s}{2})},\
\ \ \ \ \ \ 0<s<1,$

where we have denoted by $\mathscr{L}$ the Kohn-Spencer horizontal Laplacian
in $\mathbb{H}^{n}$. We note that, when $0<s<1$ the pseudodifferential
operator $\mathscr{L}_{s}$ is dramatically different from
$\mathscr{L}^{s}=(-\mathscr{L})^{s}$ (for the definition of this operator see
(2.2) below), and these two operators coincide only in the local case $s=1$.
In fact, by formally letting $s\nearrow 1$ in (1.11), and using
$\Gamma(x+1)=x\Gamma(x)$, we obtain
$\mathscr{L}_{1}=-\mathscr{L}=\mathscr{L}^{1}$. In their work [16] Frank et
al. introduced a new extension problem for the nonlocal operator (1.11), very
different from that for its non-geometric counterpart $\mathscr{L}^{s}$ in
[13] (based on the work of Caffarelli and Silvestre [6]), and used scattering
theory to solve it. Subsequently, Roncal and Thangavelu employed a parabolic
version of the extension problem in [16] to establish some optimal Hardy
inequalities in $\mathbb{H}^{n}$ [33], or more in general in groups of
Heisenberg type [34]. Such time-dependent extension problem plays an important
role in our recent work [21], as well as in the present paper which can be
seen as a continuation of such work.

To introduce the relevant geometric framework we recall that Lie groups of
Heisenberg type were introduced by Kaplan [28] in connection with
hypoellipticity questions. They are geometrically interesting since on one
hand they retain most of the important symmetries of $\mathbb{H}^{n}$, on the
other they naturally arise as the nilpotent component $N$ in the Iwasawa
decomposition $\mathbb{G}=KAN$ of a simple group of rank one, see [11]. In a
group of Heisenberg type $\mathbb{G}$ (see Section 3 for the relevant
definitions) the following generalisation of (1.11) was introduced in [34].
Let $\mathfrak{g}=V_{1}\oplus V_{2}$ be the Lie algebra of $\mathbb{G}$, and
denote $m=\operatorname{dim}V_{1}$, $k=\operatorname{dim}V_{2}$ (we note that
the complex structure of $\mathbb{G}$ forces $m=2n$ for some
$n\in\mathbb{N}$). We routinely identify $\mathfrak{g}$ with
$\mathbb{R}^{m}\times\mathbb{R}^{k}$, and the generic point $g\in\mathbb{G}$
with its logarithmic coordinates
$(z,\sigma)\in\mathbb{R}^{m}\times\mathbb{R}^{k}$. Let $\mathscr{L}$ be a
given horizontal Laplacian in $\mathbb{G}$ associated with an orthonormal
basis of the horizontal layer $V_{1}$. Consider the pseudo-differential
operator of order $2s$ defined by

(1.12)
$\mathscr{L}_{s}=2^{s}(-\Delta_{\sigma})^{s/2}\frac{\Gamma(-\frac{1}{2}\mathscr{L}(-\Delta_{\sigma})^{-1/2}+\frac{1+s}{2})}{\Gamma(-\frac{1}{2}\mathscr{L}(-\Delta_{\sigma})^{-1/2}+\frac{1-s}{2})},\
\ \ \ \ \ 0<s<1.$

We note that for $\mathbb{H}^{n}$ the dimension of the vertical layer is $k=1$
and $(-\Delta_{\sigma})^{s/2}$ in (1.12) becomes $|T|^{s}$, thus giving back
(1.11). In a group of Heisenberg type $\mathbb{G}$ the counterpart of the
intertwining formula (1.1) is given by the following

(1.13)
$\displaystyle\mathscr{L}_{s}\left(((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k-2s}{4}}\right)$
$\displaystyle=\frac{\Gamma\left(\frac{m+2+2s}{4}\right)\Gamma\left(\frac{m+2k+2s}{4}\right)}{\Gamma\left(\frac{m+2-2s}{4}\right)\Gamma\left(\frac{m+2k-2s}{4}\right)}(4y)^{2s}((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k+2s}{4}},$

for $(z,\sigma)\in\mathbb{G}$, and $y>0$. We stress that in the particular
case of the Heisenberg group $\mathbb{H}^{n}$ (which corresponds to the case
$m=2n$ and $k=1$) the function appearing in the left-hand side of (1.13)
defines, up to group translations, the unique extremal of the Hardy-
Littlewood-Sobolev inequalities obtained by Frank and Lieb in [15]. We also
note that, similarly to (1.2) above, a remarkable by-product of (1.13) is that
up to a renormalising factor the function
$u(z,\sigma)=((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{Q-2s}{4}}$ (where
$Q=m+2k$ denotes the so-called homogeneous dimension of the group
$\mathbb{G}$) provides a positive solution to the following nonlocal Yamabe
equation

(1.14) $\mathscr{L}_{s}u=u^{\frac{Q+2s}{Q-2s}}.$

Following [16], one can consider semilinear equations as (1.14) in a suitable
class of CR manifolds having $\mathbb{H}^{n}$ as (flat-)model case: proving
existence of positive solutions in this framework would in fact resolve the
fractional CR Yamabe problem posed in [16, p. 103-104]. We also mention [24,
30] for multiplicity results for solutions of (1.14) with unrestricted sign in
$\mathbb{H}^{n}$.

In connection with (1.13) we recall that the existence of intertwining
operators in semisimple Lie groups is known since the pioneering work [29] by
Knapp and Stein. A systematic treatment of intertwining kernels in Lie groups
of Heisenberg type was developed by Cowling in [10] and subsequently by
Cowling and Haagerup in [12]. Formula (1.13) was recently proved by Roncal and
Thangavelu in [33, Theorem 3.1] and [34, Theorem 3.7], where they used it to
find a M. Riesz type inverse of the operator $\mathscr{L}_{s}$. Their approach
relies on non-commutative Fourier analysis and group representation theory,
and it is inspired to the results in [12, Section 3].

Our approach to the CR intertwining formula (1.13) is inspired to the above
described strategy leading to (1.9). While we refer the reader to Section 3
for the relevant details and a description of the background results from
[16], [33], [21] and [22], here we mention that the main step is to consider
the parabolic extension problem referred to above: given a function $u\in
C^{\infty}_{0}(\mathbb{G}\times\mathbb{R}_{t})$, find a function $U\in
C^{\infty}(\mathbb{G}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y})$ such that

(1.15)
$\begin{cases}\mathfrak{P}_{(s)}U\overset{def}{=}\frac{\partial^{2}U}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial U}{\partial
y}+\frac{y^{2}}{4}\Delta_{\sigma}U+\mathscr{L}U-\frac{\partial U}{\partial
t}=0,\ \ \ \ \ \text{in}\
\mathbb{G}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y},\\\
U(g,t,0)=u(g,t).\end{cases}$

In our recent work [21] we have proved that the fundamental solution (with
pole at the origin) of the operator $\mathfrak{P}_{(s)}$ in (1.15) is the
function in the thick space
$\mathbb{G}\times\mathbb{R}^{2(1-s)}\times\mathbb{R}^{+}$ given by

(1.16) $\displaystyle q_{(s)}((z,\sigma),t,y)$
$\displaystyle=\frac{2^{k}}{(4\pi
t)^{\frac{m}{2}+k+1-s}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\frac{m}{2}+1-s}e^{-\frac{|z|^{2}+y^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}d\lambda.$

Similarly to the case of $\mathbb{R}^{n}$, the conformal CR invariants of the
formula (1.13) are embedded in the fundamental solution
$q_{(s)}((z,\sigma),t,y)$ in (1.16). This claim will follow from two basic
results stated below: Theorem 1.1 and Theorem A. The former is the main result
in this paper, the latter represents a counterpart of (1.8) and was proved in
[21, Theorem 1.4]. From (1.16) and Bochner’s principle of subordination we
know that the distribution

(1.17)
$\mathfrak{e}_{(s)}((z,\sigma,y)\overset{def}{=}\int_{0}^{\infty}q_{(s)}((z,\sigma),t,y)dt$

is the fundamental solution with pole at the origin of the time-independent
part of $\mathfrak{P}_{(s)}$, i.e., the conformal extension operator
$\mathfrak{L}_{(s)}=\frac{\partial^{2}}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial}{\partial
y}+\frac{y^{2}}{4}\Delta_{\sigma}+\mathscr{L}.$ The reader should bear in mind
that $\mathfrak{e}_{(s)}((z,\sigma,y)$ is the CR counterpart of the function
$E^{(s)}(x,y)$ in (1.7) above. Along with $\mathfrak{e}_{(s)}((z,\sigma),y)$,
we consider the distribution $\mathfrak{e}_{(-s)}((z,\sigma),y)$ obtained by
changing $s$ into $-s$ in (1.16) and (1.17). We are finally ready to state our
main result.

######  Theorem 1.1 (Geometric intertwining).

Let $\mathbb{G}$ be a group of Heisenberg type and let $s\in(0,1)$. For every
$g\in\mathbb{G}$ and $y>0$ one has

(1.18) $\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)=(2\pi
y)^{2s}\mathfrak{e}_{(-s)}(g,y).$

Theorem 1.1 is the CR counterpart of the dimension-free identity (1.9). It is
worth mentioning here that in our proof of (1.18) we do not use the definition
(1.12) of $\mathscr{L}_{s}$ since the latter would immediately lead into the
elaborate machinery of non-commutative Fourier analysis and group
representation theory. Instead, we base our analysis on the equivalent
representation (3.8) below which is only formally similar to (1.10).

With Theorem 1.1 in hands, we combine it with the following result, which is
[21, Theorem 1.4], that further underscores the geometric relevance of the
functions $\mathfrak{e}_{(\pm s)}((z,\sigma),y)$.

Theorem A.  _Let $0<s\leq 1$. In any group of Heisenberg type $\mathbb{G}$ the
distribution in the thick space $\mathbb{G}\times\mathbb{R}^{+}_{y}$ defined
by (1.17) is given by_

(1.19)
$\mathfrak{e}_{(s)}((z,\sigma),y)=\frac{\Gamma(s)}{(4\pi)^{1-s}}C_{(s)}(m,k)\
((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{1}{2}(\frac{m}{2}+k-s)},$

_where we have let_

(1.20)
$C_{(s)}(m,k)=\frac{2^{\frac{m}{2}+2k-3s-1}\Gamma(\frac{1}{2}(\frac{m}{2}+1-s))\Gamma(\frac{1}{2}(\frac{m}{2}+k-s))}{\pi^{\frac{m+k+1}{2}}\Gamma(s)}.$

_An equation similar to ( 1.19) holds if we replace $s$ with $-s$, provided
that $\Gamma(s)$ is replaced by $|\Gamma(-s)|$_.

It should now be clear to the reader that the combination of Theorem 1.1 and
Theorem A immediately implies the intertwining formula (1.13), see Corollary
3.3 below.

In closing, we provide a brief description of the paper. In Section 2 we
present a fairly general non-geometric version of the intertwining phenomenon
that goes well beyond the setting of Lie groups of Heisenberg type. We warn
the reader that, although on a formal level the statement of Theorem 2.1 seems
similar that of Theorem 1.1, the geometric content of the latter result is
lost in the former. Our intent is to illustrate the flexibility of the heat
equation approach which allows to treat situations in which the arsenal of
Fourier analysis is not readily available. For this we have chosen the
framework of stratified nilpotent Lie groups (aka Carnot groups) of arbitrary
step, but the proof of Theorem 2.1 holds with no changes for the fractional
powers of the infinitesimal generator of a Dirichlet form under by now
standard assumptions which guarantee Gaussian estimates for the relevant heat
semigroup. Section 3 is devoted to proving Theorem 1.1. We begin by explaining
in some detail the geometric extension problem (1.15), and we introduce the
two operators $P_{(\pm s),t}$. We highlight the new difficulties that one
encounters with respect to the non-geometric case and we establish two
preliminary results of an elementary character, Lemma 3.1 and Lemma 3.2, that
serve to overcome such obstructions. Finally, we prove Theorem 1.1. In
Corollary 3.3, which closes the paper, we combine this result with Theorem A
to establish (1.13).

##  2\. Non-conformal intertwining formula

In this section we establish in the setting of stratified nilpotent Lie groups
of arbitrary steps (aka Carnot groups) a general version of the intertwining
formula (1.9), see Theorem 2.1 below. On one hand, our purpose is to show that
the heat equation approach outlined in the case of flat $\mathbb{R}^{n}$ in
the introduction works in great generality. On the other hand, with the
present section we intend to highlight the dramatic differences between the
geometric and the non-geometric operators $\mathscr{L}_{s}$ and
$\mathscr{L}^{s}$. To introduce our discussion we recall that a Carnot group
of step $r$ is a simply-connected Lie group $\mathbb{G}$ whose Lie algebra
$\mathfrak{g}$ is stratified and $r$-nilpotent. This means that
$\mathfrak{g}=V_{1}\oplus...\oplus V_{r}$, with $[V_{1},V_{i}]=V_{i+1}$,
$i=1,...,r-1$, and $[V_{1},V_{r}]=\\{0\\}$. The bracket generating layer
$V_{1}$ of $\mathfrak{g}$ is called the horizontal layer. If we denote by
$L_{g}(g^{\prime})=g\circ g^{\prime}$ the operator of left-translation on
$\mathbb{G}$, and define left-invariant vector fields
$X_{j}(g)=dL_{g}(e_{j})$, where $\\{e_{1},...,e_{m}\\}$ is an orthonormal
basis of $V_{1}$, then a horizontal Laplacian in $\mathbb{G}$ (associated with
the basis $\\{e_{1},...,e_{m}\\}$) is defined by
$\mathscr{L}=\sum_{j=1}^{m}X_{j}^{2}$. Such differential operator fails to be
elliptic at every point $g\in\mathbb{G}$, but according to Hörmander’s theorem
in [25] it is hypoelliptic in $\mathbb{G}$. In [14] Folland proved the
existence of a strictly positive kernel $p(g,g^{\prime},t)$, $C^{\infty}$ off
the diagonal, such that the heat semigroup $P_{t}=e^{-t\mathscr{L}}$ is
represented by
$P_{t}f(g)=\int_{\mathbb{G}}p(g,g^{\prime},t)f(g^{\prime})dg^{\prime}$. Such
semigroup is stochastically complete ($P_{1}1=1$) and a contraction on
$L^{p}(\mathbb{G})$ for every $1\leq p\leq\infty$. Furthermore, the kernel
$p(g,g^{\prime},t)$ satisfies the Gaussian estimates

(2.1) $Ct^{-Q/2}e^{-\alpha\frac{d(g,g^{\prime})^{2}}{t}}\leq
p(g,g^{\prime},t)\leq C^{-1}t^{-Q/2}e^{-\beta\frac{d(g,g^{\prime})^{2}}{t}},$

for appropriate universal constants $C,\alpha,\beta>0$. In (2.1) we have
indicated with $Q=\sum_{j=1}^{r}j\operatorname{dim}V_{j}$ the homogeneous
dimension of $\mathbb{G}$ with respect to the anisotropic dilations associated
with the grading of the Lie algebra, whereas $d(g,g^{\prime})$ denotes the
control distance, see [14, 36]. The semigroup $P_{t}$ is all that is needed to
introduce the fractional powers $\mathscr{L}^{s}=(-\mathscr{L})^{s}$ by means
of the following formula due to Balakrishnan

(2.2)
$\mathscr{L}^{s}u(g)=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{t^{1+s}}(P_{t}u(g)-u(g))dt.$

Next, we consider the parabolic extension problem for the pseudodifferential
operator $(\partial_{t}-\mathscr{L})^{s}$, see [19]: given a function $u\in
C^{\infty}_{0}(\mathbb{G}\times\mathbb{R}_{t})$, find $U\in
C^{\infty}(\mathbb{G}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y})$ such that

$\begin{cases}\mathfrak{P}^{(s)}U\overset{def}{=}\frac{\partial^{2}U}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial U}{\partial y}+\mathscr{L}U-\frac{\partial
U}{\partial t}=0,\\\ U(g,t,0)=u(g,t).\end{cases}$

Since we can think of the differential operator $\mathfrak{P}^{(s)}$ as a heat
operator on the group
$\mathbb{G}\times\mathbb{R}^{2(1-s)}_{w}\times\mathbb{R}^{+}_{t}$ acting on
functions $U(g,w,t)=\overline{U}(g,y,t)$, with $y=|w|$, its fundamental
solution with pole at the identity is given by

(2.3) $q^{(s)}(g,t,y)=\frac{1}{(4\pi t)^{1-s}}e^{-\frac{y^{2}}{4t}}p(g,e,t),$

where we have denote by $e\in\mathbb{G}$ the identity element. We stress that
the function $q^{(s)}(g,t,y)$ in (2.3) is dramatically different from its
geometric counterpart $q_{(s)}((z,\sigma),t,y)$ in (1.16) above which is not a
product. Hereafter, we indicate with the $q^{(-s)}(g,t,y)$ the function
obtained by changing $s$ into $-s$ in (2.3) (see (2.9) below), and we define

(2.4) $\mathfrak{e}^{(\pm s)}(g,y)\overset{def}{=}\int_{0}^{\infty}q^{(\pm
s)}(g,t,y)dt.$

It should be clear to the reader that the function $\mathfrak{e}^{(s)}$ in
(2.4) is the fundamental solution (with pole at the identity) of the time-
independent extension operator $\frac{\partial^{2}}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial}{\partial y}+\mathscr{L}$. We have the
following result.

######  Theorem 2.1 (Non-geometric intertwining).

Let $\mathbb{G}$ be a Carnot group and $s\in(0,1)$. For any $g\in\mathbb{G}$
and $y>0$ one has

(2.5) $\mathscr{L}^{s}(\mathfrak{e}^{(s)}(\cdot,y))(g)=(2\pi
y)^{2s}\mathfrak{e}^{(-s)}(g,y).$

###### Proof.

First, we observe that if $u$ is a $C^{\infty}(\mathbb{G})$ function which
suitably decays at infinity, then the estimates (2.1) allow to integrate by
parts in (2.2) obtaining

(2.6)
$\mathscr{L}^{s}u(g)=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{\tau^{1+s}}\int_{0}^{\tau}\partial_{t}P_{t}u(g)dtd\tau=-\frac{1}{\Gamma(1-s)}\int_{0}^{\infty}t^{-s}\partial_{t}P_{t}u(g)dt.$

If we apply $P_{t}$ to $u(g)=\mathfrak{e}^{(s)}(g,y)$, and we use (2.4), we
find

(2.7) $\displaystyle P_{t}\mathfrak{e}^{(s)}(g,y)$
$\displaystyle=\int_{\mathbb{G}}p(g,g^{\prime},t)\int_{0}^{\infty}\frac{1}{(4\pi\tau)^{1-s}}e^{-\frac{y^{2}}{4\tau}}p(g^{\prime},0,\tau)d\tau
dg^{\prime}$
$\displaystyle=\int_{0}^{\infty}\frac{1}{(4\pi\tau)^{1-s}}e^{-\frac{y^{2}}{4\tau}}\int_{\mathbb{G}}p(g,g^{\prime},t)p(g^{\prime},e,\tau)dg^{\prime}d\tau$
$\displaystyle=\int_{0}^{\infty}\frac{1}{(4\pi\tau)^{1-s}}e^{-\frac{y^{2}}{4\tau}}p(g,e,t+\tau)d\tau,$

where in the last equality we have used the Chapman-Kolmogorov equation
(semigroup property for $P_{t}$)

$p(g,g^{\prime\prime},t+\tau)=\int_{\mathbb{G}}p(g,g^{\prime},t)p(g^{\prime},g^{\prime\prime},\tau)dg^{\prime}\quad\mbox{
for }g^{\prime\prime}\in\mathbb{G}.$

Substituting this formula in (2.6), we find

$\displaystyle\mathscr{L}^{s}(\mathfrak{e}^{(s)}(\cdot,y))(g)$
$\displaystyle=-\frac{(4\pi)^{(s-1)}}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}\left(\frac{\tau}{t}\right)^{s}\partial_{t}\left(e^{-\frac{y^{2}}{4\tau}}p(g,e,t+\tau)\right)\frac{d\tau}{\tau}dt.$

The change of variable

(2.8)
$(v,\rho)=\left(t+\tau,\frac{\tau}{t}\right),\qquad\qquad\text{or}\qquad\qquad(t,\tau)=\left(\frac{v}{1+\rho},\frac{v\rho}{1+\rho}\right),$

for which $dtd\tau=\frac{v}{(1+\rho)^{2}}dvd\rho$ and
$\partial_{t}=\partial_{v}-\frac{\rho(1+\rho)}{v}\partial_{\rho}$, now gives

$\mathscr{L}^{s}(\mathfrak{e}^{(s)}(\cdot,y))(g)=\frac{-(4\pi)^{s-1}}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}\frac{\rho^{s-1}}{1+\rho}\left(\frac{\partial}{\partial
v}-\frac{\rho(1+\rho)}{v}\frac{\partial}{\partial\rho}\right)\left(e^{-\frac{y^{2}}{4v}\frac{1+\rho}{\rho}}p(g,0,v)\right)dvd\rho.$

Observing that, as a function of $v$,
$e^{-\frac{y^{2}}{4v}\frac{1+\rho}{\rho}}p(g,0,v)$ tends to $0$ both as $v\to
0^{+}$ and $v\to\infty$ thanks to (2.1), we deduce that

$\displaystyle\mathscr{L}^{s}(\mathfrak{e}^{(s)}(\cdot,y))(g)=\frac{(4\pi)^{s-1}}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}\frac{\rho^{s}}{v}\frac{\partial}{\partial\rho}\left(e^{-\frac{y^{2}}{4\rho
v}}e^{-\frac{y^{2}}{4v}}p(g,0,v)\right)dvd\rho$
$\displaystyle=\frac{(4\pi)^{s}}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{4\pi
v}e^{-\frac{y^{2}}{4v}}p(g,0,v)\left(\int_{0}^{\infty}\rho^{s}\frac{\partial}{\partial\rho}\left(e^{-\frac{y^{2}}{4\rho
v}}\right)d\rho\right)dv$
$\displaystyle=\frac{(4\pi)^{s}}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{4\pi
v}e^{-\frac{y^{2}}{4v}}p(g,0,v)\left(\frac{y^{2}}{4v}\int_{0}^{\infty}\rho^{s-1}e^{-\frac{y^{2}}{4\rho
v}}\frac{d\rho}{\rho}\right)dv.$

Finally, keeping in mind that

$\frac{y^{2}}{4v}\int_{0}^{\infty}\rho^{s-1}e^{-\frac{y^{2}}{4\rho
v}}\frac{d\rho}{\rho}=\frac{\Gamma(1-s)}{(4v)^{s}}y^{2s},$

and recalling

(2.9) $q^{(-s)}(g,v,y)=\frac{1}{(4\pi v)^{1+s}}e^{-\frac{y^{2}}{4v}}p(g,e,v),$

we reach the desired conclusion (2.5). ∎

In the Abelian case when $\mathbb{G}=\mathbb{R}^{n}$, the functions
$\mathfrak{e}^{(\pm s)}(g,y)$ coincide with $E^{(\pm s)}(x,y)$ in (1.7).
However, we warn the reader that for a general non-Abelian group $\mathbb{G}$
the conformal significance of formulas (1.6) and (1.8) is lost. We close this
section by recording that, as an immediate consequence of Theorem 2.1 and
formulas (1.6)-(1.8), we have the following.

######  Corollary 2.2.

In $\mathbb{R}^{n}$, with $n\geq 2$, we have for $0<s<1$,

$\left(-\Delta\right)^{s}\left((|x|^{2}+y^{2})^{-\frac{n-2s}{2}}\right)=\frac{\Gamma\left(\frac{n}{2}+s\right)}{\Gamma\left(\frac{n}{2}-s\right)}\left(2y\right)^{2s}(|x|^{2}+y^{2})^{-\frac{n+2s}{2}},\qquad
x\in\mathbb{R}^{n},\ y>0.$

###### Proof.

We observe that (2.3) and its counterpart obtained by changing $s$ into $-s$
presently give

(2.10) $q^{(\pm s)}(g,t,y)=\frac{1}{(4\pi t)^{1\mp
s}}e^{-\frac{y^{2}}{4t}}\frac{1}{(4\pi
t)^{\frac{n}{2}}}e^{-\frac{|x|^{2}}{4t}}=\frac{1}{(4\pi t)^{\frac{n}{2}+1\mp
s}}e^{-\frac{|x|^{2}+y^{2}}{4t}}.$

From (2.10) and the definition (2.4) we find

(2.11) $\mathfrak{e}^{(\pm
s)}(g,y)\overset{def}{=}\int_{0}^{\infty}\frac{1}{(4\pi t)^{\frac{n}{2}+1\mp
s}}e^{-\frac{|x|^{2}+y^{2}}{4t}}dt=\frac{\Gamma(\frac{n\mp
2s}{2})}{4\pi^{\frac{n}{2}+1\mp s}}(|x|^{2}+y^{2})^{-\frac{n\mp 2s}{2}}.$

Substituting the right-hand side of (2.11) in (2.5) of Theorem 2.1 we reach
the desired conclusion.

∎

##  3\. Conformal intertwining formula

The objective of this section is to prove Theorem 1.1 following the ideas
outlined in Sections 1 and 2. We begin by introducing the relevant geometric
framework. Let $\mathbb{G}$ be a Carnot group of step $r=2$ with Lie algebra
$\mathfrak{g}=V_{1}\oplus V_{2}$, which we assume endowed with an inner
product $\langle\cdot,\cdot\rangle$ and induced norm $|\cdot|$. We fix
orthonormal basis $\\{e_{1},...,e_{m}\\}$ and
$\\{\varepsilon_{1},...,\varepsilon_{k}\\}$ for $V_{1}$ and $V_{2}$
respectively. Points $z\in V_{1}$ and $\sigma\in V_{2}$ will be identified
with either one of the representations $z=\sum_{j=1}^{m}z_{j}e_{j}$,
$\sigma=\sum_{\ell=1}^{k}\sigma_{\ell}\varepsilon_{\ell}$, or also
$z=(z_{1},...,z_{m})$, $\sigma=(\sigma_{1},...,\sigma_{k})$. Accordingly,
whenever convenient we will identify the point $g=\exp(z+\sigma)\in\mathbb{G}$
with its logarithmic coordinates
$(z,\sigma)\in\mathbb{R}^{m}\times\mathbb{R}^{k}$. The Kaplan mapping
$J:V_{2}\to\operatorname{End}(V_{1})$ is defined by

(3.1) $\langle J(\sigma)z,\zeta\rangle=\langle[z,\zeta],\sigma\rangle=-\langle
J(\sigma)\zeta,z\rangle,$

for $z,\zeta\in V_{1}$ and $\sigma\in V_{2}$. Clearly,
$J(\sigma)^{\star}=-J(\sigma)$, and one has $<J(\sigma)z,z>=0$. By (3.1) and
the Baker-Campbell-Hausdorff formula, see e.g. p. 12 of [9],

$\exp(z+\sigma)\exp(\zeta+\tau)=\exp{\left(z+\zeta+\sigma+\tau+\frac{1}{2}[z,\zeta]\right)},$

we obtain the non-Abelian multiplication in $\mathbb{G}$

(3.2) $g\circ
g^{\prime}=(z,\sigma)\circ(\zeta,\tau)=\big{(}z+\zeta,\sigma+\tau+\frac{1}{2}\sum_{\ell=1}^{k}<J(\varepsilon_{\ell})z,\zeta>\varepsilon_{\ell}\big{)}.$

From (3.2) it is easy to recognise that $e=(0,0)$ and
$(z,\sigma)^{-1}=(-z,-\sigma)$.

Henceforth in this section we assume that $\mathbb{G}$ is of Heisenberg type.
By this we mean that for every $\sigma\in V_{2}$ one has

$J(\sigma)^{2}=-|\sigma|^{2}\ {\mathbb{I}}_{V_{1}}.$

We now denote by $\mathscr{L}$ the horizontal Laplacian associated with the
orthonormal basis $\\{e_{1},\ldots,e_{m}\\}$ of the horizontal layer $V_{1}$,
see the opening of Section 2.

We consider the extension problem (1.15) above: given a function $u\in
C^{\infty}_{0}(\mathbb{G}\times\mathbb{R}_{t})$, find a function $U\in
C^{\infty}(\mathbb{G}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y})$ such that

(3.3)
$\begin{cases}\mathfrak{P}_{(s)}U\overset{def}{=}\frac{\partial^{2}U}{\partial
y^{2}}+\frac{1-2s}{y}\frac{\partial U}{\partial
y}+\frac{y^{2}}{4}\Delta_{\sigma}U+\mathscr{L}U-\frac{\partial U}{\partial
t}=0,\ \ \ \ \ \text{in}\
\mathbb{G}\times\mathbb{R}_{t}\times\mathbb{R}^{+}_{y},\\\
U(g,t,0)=u(g,t).\end{cases}$

For the Heisenberg group $\mathbb{H}^{n}$ the time-independent version of
(3.3) was first introduced in the cited work of Frank et al. [16].
Subsequently, Roncal and Thangavelu considered the time-dependent problem
(3.3) in their cited papers [33, 34]. Our approach is quite different from
that in these works since, as we have mentioned above, these authors do not
rely on (3.3) to establish (1.13), but instead use non-commutative Fourier
analysis and group representation theory.

We continue to denote by $q_{(\pm s)}((z,\sigma),t,y)$ respectively the
fundamental solution of $\mathfrak{P}_{(s)}$ introduced in (1.16), and the
function obtained from it by changing $s$ into $-s$. We now consider the
kernels in the thin space $\mathbb{G}\times(0,\infty)$ defined by

(3.4) $\mathscr{K}_{(\pm s)}((z,\sigma),t)=(4\pi t)^{1\mp s}q_{(\pm
s)}((z,\sigma),t,0).$

From (1.16) and the definition (3.4) it is easy to see that the kernels
$\mathscr{K}_{(\pm s)}$ have the following explicit expression

(3.5) $\mathscr{K}_{(\pm s)}((z,\sigma),t)=\frac{2^{k}}{\left(4\pi
t\right)^{\frac{m}{2}+k}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\frac{m}{2}+1\mp
s}e^{-\frac{|z|^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}d\lambda,\ \ \ \
0<s\leq 1.$

It is worth noting here that if we take $s=1$ in (3.5) we obtain the Gaveau-
Hulanicki-Cygan heat kernel in $\mathbb{G}$ (we refer the reader to [22] for a
recent pde-based derivation of such kernel). With (3.5) in hands, and by
slightly abusing the notation, we define

$\mathscr{K}_{(\pm s)}(g,g^{\prime},t)=\mathscr{K}_{(\pm s)}(g^{-1}\circ
g^{\prime},t)=\mathscr{K}_{(\pm s)}((g^{\prime})^{-1}\circ g,t).$

With such functions in hands, we next introduce two operators on
$L^{p}(\mathbb{G})$ by the formula

(3.6) $P_{(\pm s),t}u(g)=\int_{\mathbb{G}}\mathscr{K}_{(\pm
s)}(g,g^{\prime},t)u(g^{\prime})dg^{\prime}.$

As for $P_{(s),t}$, we mention that its _raison d’être_ is in the fact, which
is one of the main results in [21], that the operator

(3.7)
$\mathscr{I}_{(2s)}u(g)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}P_{(s),t}u(g)dt$

provides the inverse of $\mathscr{L}_{s}$. The relevance of the operator
$P_{(-s),t}$, instead, is underscored by the following formula which was
proved in [34]

(3.8)
$\mathscr{L}_{s}u(g)=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\frac{1}{t^{1+s}}\left[P_{(-s),t}u(g)-u(g)\right]dt.$

This preliminary discussion brings us to the heart of the present section, the
proof of Theorem 1.1. Our plan is to proceed as closely as possible to the
proof of Theorem 2.1, but we immediately encounter some difficulties. To
explain this point we mention that in the non-geometric setting of Section 2
there are two aspects that play a crucial role: (i) the same heat kernel
$p(\cdot,t)$ occurs both in the expression (2.6) of $\mathscr{L}^{s}$ and in
that of the function $\mathfrak{e}^{(s)}$; (ii) the Chapman-Kolmogorov
identity enters crucially in the final equality of the key identity (2.7).
Both facts fail to hold in the present conformal setting. A third more
pervasive complication is represented by the very different nature of the
fundamental solutions $q^{(\pm s)}$ and $q_{(\pm s)}$ of the parabolic
extension problems for $\mathscr{L}^{s}$ and $\mathscr{L}_{s}$.

The proof of Theorem 1.1 rests on two preliminary lemmas in which we
circumvent the difficulties listed above. In the first one we establish a
replacement of the semigroup property for the group convolution of the
different kernels $q_{(-s)}(\cdot,t,0)$ and $q_{(s)}(\cdot,\tau,y)$ which
respectively appear in $P_{(-s),t}$ and $\mathfrak{e}_{(s)}(\cdot,y)$). We
mention that the case $y=0$ of the following lemma was proved in [21, Lemma
4.2].

######  Lemma 3.1.

Fix $s\in(0,1)$, $g\in\mathbb{G}$, $t,\tau>0$, and $y\geq 0$. Then, we have

(3.9) $\displaystyle\int_{\mathbb{G}}$ $\displaystyle
q_{(-s)}((g^{\prime})^{-1}\circ
g,t,0)q_{(s)}(g^{\prime},\tau,y)\,dg^{\prime}=\int_{\mathbb{R}^{k}}e^{2\pi
i\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{2\sinh
2\pi\tau|\lambda|}\right)^{1-s}\times$
$\displaystyle\times\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{1+s}\left(\frac{|\lambda|}{2\sinh
2\pi(t+\tau)|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi(t+\tau)|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}d\lambda.$

###### Proof.

To establish (3.9), we fix $t,\tau>0$, $y\geq 0$, and $z\in\mathbb{R}^{m}$. By
partial Fourier transform with respect to the vertical variable
$\sigma\in\mathbb{R}^{k}$, we observe that the desired conclusion (3.9) will
be true if we can prove that for all $\lambda\in\mathbb{R}^{k}$ the following
holds

(3.10) $\displaystyle\int_{\mathbb{R}^{k}}e^{-2\pi
i\langle\sigma,\lambda\rangle}\int_{\mathbb{G}}q_{(-s)}((g^{\prime})^{-1}\circ
g,t,0)q_{(s)}(g^{\prime},\tau,y)\,dg^{\prime}d\sigma=\left(\frac{|\lambda|}{2\sinh
2\pi\tau|\lambda|}\right)^{1-s}\times$
$\displaystyle\times\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{1+s}\left(\frac{|\lambda|}{2\sinh
2\pi(t+\tau)|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi(t+\tau)|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}.$

The rest of the proof of the lemma will thus be devoted to establishing
(3.10). Recalling the definition of $q_{(\pm s)}$ in (1.16), by a simple
change of variable we see that

(3.11) $q_{(\pm s)}(g^{\prime},\tau,y)=\int_{\mathbb{R}^{k}}e^{2\pi
i\langle\sigma^{\prime},\lambda\rangle}\left(\frac{|\lambda|}{2\sinh
2\pi\tau|\lambda|}\right)^{\frac{m}{2}+1\mp
s}e^{-\frac{\pi}{2}(|z^{\prime}|^{2}+y^{2})\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}d\lambda.$

From the expression (3.2) of the group law, we also have

(3.12) $\displaystyle q_{(\pm s)}((g^{\prime})^{-1}\circ g,t,0)$
$\displaystyle=\int_{\mathbb{R}^{k}}e^{2\pi
i\left(\langle\sigma-\sigma^{\prime},\lambda\rangle+\frac{1}{2}\left\langle
J(\lambda)z,z^{\prime}\right\rangle\right)}\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{\frac{m}{2}+1\mp
s}e^{-\frac{\pi}{2}|z-z^{\prime}|^{2}\frac{|\lambda|}{\tanh 2\pi
t|\lambda|}}d\lambda.$

Next, we note that by (3.11), (3.12), and applying twice the Fourier inversion
formula, for every fixed $\lambda\in\mathbb{R}^{k}$ we can rewrite the left-
hand side of (3.10) in the following way

(3.13) $\displaystyle\int_{\mathbb{R}^{k}}e^{-2\pi
i\langle\sigma,\lambda\rangle}\int_{\mathbb{G}}q_{(-s)}((g^{\prime})^{-1}\circ
g,t,0)q_{(s)}(g^{\prime},\tau,y)\,dg^{\prime}d\sigma$
$\displaystyle=\int_{\mathbb{R}^{k}}\int_{\mathbb{R}^{k}}\int_{\mathbb{R}^{k}}\int_{\mathbb{G}}e^{-2\pi
i\langle\sigma,\lambda\rangle}e^{2\pi
i\langle\sigma-\sigma^{\prime},\mu\rangle}e^{\pi i\langle
J(\mu)z,z^{\prime}\rangle}e^{2\pi
i\langle\sigma^{\prime},\omega\rangle}\left(\frac{|\mu|}{2\sinh 2\pi
t|\mu|}\right)^{\frac{m}{2}+1+s}\times$
$\displaystyle\times\left(\frac{|\omega|}{2\sinh
2\pi\tau|\omega|}\right)^{\frac{m}{2}+1-s}\times
e^{-\frac{\pi}{2}|z-z^{\prime}|^{2}\frac{|\mu|}{\tanh 2\pi
t|\mu|}}e^{-\frac{\pi}{2}(|z^{\prime}|^{2}+y^{2})\frac{|\omega|}{\tanh
2\pi\tau|\omega|}}dg^{\prime}d\omega d\mu d\sigma$
$\displaystyle=\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{\frac{m}{2}+1+s}\int_{\mathbb{R}^{k}}\int_{\mathbb{G}}e^{-2\pi
i\langle\sigma^{\prime},\lambda\rangle}e^{\pi i\langle
J(\lambda)z,z^{\prime}\rangle}e^{2\pi
i\langle\sigma^{\prime},\omega\rangle}\left(\frac{|\omega|}{2\sinh
2\pi\tau|\omega|}\right)^{\frac{m}{2}+1-s}\times$ $\displaystyle\times
e^{-\frac{\pi}{2}|z-z^{\prime}|^{2}\frac{|\lambda|}{\tanh 2\pi
t|\lambda|}}e^{-\frac{\pi}{2}(|z^{\prime}|^{2}+y^{2})\frac{|\omega|}{\tanh
2\pi\tau|\omega|}}dg^{\prime}d\omega$
$\displaystyle=\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{\frac{m}{2}+1+s}\left(\frac{|\lambda|}{2\sinh
2\pi\tau|\lambda|}\right)^{\frac{m}{2}+1-s}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}\times$ $\displaystyle\times\int_{\mathbb{R}^{m}}e^{\pi
i\langle
J(\lambda)z,z^{\prime}\rangle}e^{-\frac{\pi}{2}|z-z^{\prime}|^{2}\frac{|\lambda|}{\tanh
2\pi t|\lambda|}}e^{-\frac{\pi}{2}|z^{\prime}|^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}dz^{\prime}.$

We now notice that the last integral is the same appearing in proof of [21,
Lemma 4.2]: in fact, exploiting the properties of the $J$ map (3.1), and
elementary manipulations of hyperbolic functions, in the final equality of
formula (4.6) in [21] we showed that

(3.14) $\displaystyle\int_{\mathbb{R}^{m}}e^{\pi i\langle
J(\lambda)z,z^{\prime}\rangle}e^{-\frac{\pi}{2}|z-z^{\prime}|^{2}\frac{|\lambda|}{\tanh
2\pi t|\lambda|}}e^{-\frac{\pi}{2}|z^{\prime}|^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}dz^{\prime}$ $\displaystyle=\left(\frac{|\lambda|}{2\sinh
2\pi(t+\tau)|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi(t+\tau)|\lambda|}}.$

Inserting (3.14) in (3.13) we obtain the desired conclusion (3.10). This
completes the proof of the lemma.

∎

The next lemma is purely technical and its significance will be clear in the
subsequent discussion.

######  Lemma 3.2.

Let $0<s<1$, $B,\mu>0$. We have

(3.15)
$\displaystyle\int_{0}^{\infty}\left(\frac{\mu}{(1+\rho)\sinh\frac{\rho}{1+\rho}\mu}\right)^{2}\left(\frac{\sinh\frac{\rho}{1+\rho}\mu}{\sinh\frac{\mu}{1+\rho}}\right)^{s}e^{-\frac{B\mu}{\tanh\frac{\rho}{1+\rho}\mu}}d\rho$
$\displaystyle=B^{s-1}\left(\frac{\mu}{\sinh\mu}\right)^{s}\Gamma(1-s)e^{-\frac{B\mu}{\tanh\mu}}.$

###### Proof.

If we keep in mind the following identity

$\frac{1}{\tanh\mu}-\frac{1}{\tanh\frac{\rho}{1+\rho}\mu}=-\frac{1}{\sinh\mu}\frac{\sinh\frac{\mu}{1+\rho}}{\sinh\frac{\rho}{1+\rho}\mu},$

we have

$\displaystyle
e^{\frac{B\mu}{\tanh\mu}}\int_{0}^{\infty}\left(\frac{\mu}{(1+\rho)\sinh\frac{\rho}{1+\rho}\mu}\right)^{2}\left(\frac{\sinh\frac{\rho}{1+\rho}\mu}{\sinh\frac{\mu}{1+\rho}}\right)^{s}e^{-\frac{B\mu}{\tanh\frac{\rho}{1+\rho}\mu}}d\rho$
$\displaystyle=\int_{0}^{\infty}\left(\frac{\mu}{(1+\rho)\sinh\frac{\rho}{1+\rho}\mu}\right)^{2}\left(\frac{\sinh\frac{\rho}{1+\rho}\mu}{\sinh\frac{\mu}{1+\rho}}\right)^{s}e^{-\frac{B\mu}{\sinh\mu}\frac{\sinh\frac{\mu}{1+\rho}}{\sinh\frac{\rho}{1+\rho}\mu}}d\rho.$

In the last integral we now make the change of variable
$\rho\to\tau=\frac{B\mu}{\sinh\mu}\frac{\sinh\frac{\mu}{1+\rho}}{\sinh\frac{\rho}{1+\rho}\mu}$,
which yields
$d\tau=-\frac{B\mu^{2}}{(1+\rho)^{2}\sinh^{2}\frac{\rho}{1+\rho}\mu}d\rho$. We
thus have

$\displaystyle
e^{\frac{B\mu}{\tanh\mu}}\int_{0}^{\infty}\left(\frac{\mu}{(1+\rho)\sinh\frac{\rho}{1+\rho}\mu}\right)^{2}\left(\frac{\sinh\frac{\rho}{1+\rho}\mu}{\sinh\frac{\mu}{1+\rho}}\right)^{s}e^{-\frac{B\mu}{\tanh\frac{\rho}{1+\rho}\mu}}d\rho$
$\displaystyle=\frac{1}{B}\int_{0}^{\infty}\left(\frac{B\mu}{\tau\sinh\mu}\right)^{s}e^{-\tau}d\tau=B^{s-1}\left(\frac{\mu}{\sinh\mu}\right)^{s}\Gamma(1-s).$

This proves (3.15).

∎

We are now ready to present the proof of the main result in this note.

###### Proof of Theorem 1.1.

For $u\in C^{\infty}(\mathbb{G})$ and suitably decay at infinity, and
$g\in\mathbb{G}$, one has

(3.16)
$\displaystyle\mathscr{L}_{s}u(g)=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\tau^{-s-1}\int_{0}^{\tau}\frac{\partial}{\partial
t}\left(P_{(-s),t}u(g)\right)dtd\tau$
$\displaystyle=-\frac{s}{\Gamma(1-s)}\int_{0}^{\infty}\int_{t}^{\infty}\tau^{-s-1}\frac{\partial}{\partial
t}\left(P_{(-s),t}u(g)\right)d\tau dt$
$\displaystyle=-\frac{1}{\Gamma(1-s)}\int_{0}^{\infty}t^{-s}\frac{\partial}{\partial
t}\left(P_{(-s),t}u(g)\right)dt,$

which is the conformal counterpart of (2.6). For $y>0$ we want to apply (3.16)
with the choice

$u(\cdot)=\mathfrak{e}_{(s)}(\cdot,y).$

By (3.4), (3.6) and (1.17), we have

$\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)=-\frac{1}{\Gamma(1-s)}\int_{0}^{\infty}t^{-s}\frac{\partial}{\partial
t}\left(P_{(-s),t}(\mathfrak{e}_{(s)}(\cdot,y))(g)\right)dt$
$\displaystyle=-\frac{1}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}t^{-s}\frac{\partial}{\partial
t}\left(\int_{\mathbb{G}}(4\pi t)^{1+s}q_{(-s)}((g^{\prime})^{-1}\circ
g,t,0)q_{(s)}(g^{\prime},\tau,y)dg^{\prime}\right)d\tau dt.$

In view of Lemma 3.1 we thus infer

$\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=\frac{-1}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{k}}t^{-s}\frac{\partial}{\partial
t}\left((4\pi t)^{1+s}e^{2\pi
i\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{2\sinh
2\pi\tau|\lambda|}\right)^{1-s}\left(\frac{|\lambda|}{2\sinh 2\pi
t|\lambda|}\right)^{1+s}\times\right.$
$\displaystyle\times\left.\left(\frac{|\lambda|}{2\sinh
2\pi(t+\tau)|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi(t+\tau)|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}\right)d\lambda d\tau dt$
$\displaystyle=-\frac{(4\pi)^{s-1}}{\Gamma(1-s)}\int_{0}^{\infty}\int_{0}^{\infty}\int_{\mathbb{R}^{k}}\frac{e^{2\pi
i\langle\sigma,\lambda\rangle}}{\tau}\left(\frac{\tau}{t}\right)^{s}\frac{\partial}{\partial
t}\left(\left(\frac{2\pi\tau|\lambda|}{\sinh
2\pi\tau|\lambda|}\right)^{1-s}\left(\frac{2\pi t|\lambda|}{\sinh 2\pi
t|\lambda|}\right)^{1+s}\times\right.$
$\displaystyle\times\left.\left(\frac{|\lambda|}{2\sinh
2\pi(t+\tau)|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi(t+\tau)|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\tau|\lambda|}}\right)d\lambda d\tau dt.$

Hence, the same change of variable as in (2.8) leads to the following identity

(3.17) $\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=\frac{-(4\pi)^{s-1}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{\rho^{s-1}}{1+\rho}\left(\frac{\partial}{\partial
v}-\frac{\rho(1+\rho)}{v}\frac{\partial}{\partial\rho}\right)\left(\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\times\right.$
$\displaystyle\times\left.\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}\left(\frac{|\lambda|}{2\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\right)d\rho dvd\lambda.$

We notice that, for any $\lambda\in\mathbb{R}^{k}$ and $\rho>0$, the function

$\displaystyle\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}\left(\frac{|\lambda|}{2\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}$

converges to $0$ both as $v\to 0^{+}$ and $v\to\infty$ (this holds for any
$z\in\mathbb{R}^{m}$ and $y>0$). Inserting this information in (3.17) we
deduce

(3.18) $\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=\frac{(4\pi)^{s-1}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{\rho^{s}}{v}\frac{\partial}{\partial\rho}\left(\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}\times\right.$
$\displaystyle\times\left.\left(\frac{|\lambda|}{2\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\right)d\rho dvd\lambda$
$\displaystyle=\frac{(4\pi)^{s}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+1}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}\times$
$\displaystyle\times\left(\int_{0}^{\infty}\rho^{s}\frac{\partial}{\partial\rho}\left(\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\right)d\rho\right)dvd\lambda.$

To deal with the integral in the last equality in (3.18), for $0<s<1$ and
$\mu>0$ we now consider the function $h_{s,\mu}:(0,\infty)\to\mathbb{R}$
defined by

(3.19)
$h_{s,\mu}(\rho)=\frac{\mu}{\sinh\mu}\left(\frac{\sinh\frac{\rho\mu}{1+\rho}}{\sinh\frac{\mu}{1+\rho}}\right)^{s}\left[\frac{\mu}{\sinh\mu}\frac{\rho}{(1+\rho)^{2}}\left(\frac{\sinh\frac{\rho\mu}{1+\rho}}{\sinh\frac{\mu}{1+\rho}}+2\cosh\mu+\frac{\sinh\frac{\mu}{1+\rho}}{\sinh\frac{\rho\mu}{1+\rho}}\right)-1\right].$

This function was introduced in [21, Section 4], and the main motivation
behind it is in the formula

(3.20)
$h^{\prime}_{s,\mu}(\rho)=\rho^{s}\frac{\partial}{\partial\rho}\left(\left(\frac{\frac{\rho}{1+\rho}\mu}{\sinh\frac{\rho}{1+\rho}\mu}\right)^{1-s}\left(\frac{\frac{\mu}{1+\rho}}{\sinh\frac{\mu}{1+\rho}}\right)^{1+s}\right),$

see [21, equation (4.19)]. This property was critical in showing that the
operator $\mathscr{I}_{(2s)}$ in (3.7) inverts $\mathscr{L}_{s}$. If in (3.19)
we now take $\mu=2\pi v|\lambda|$, we obtain from (3.18)

(3.21)
$\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)=\frac{(4\pi)^{s}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+1}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}\times$ $\displaystyle\times
e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh 2\pi
v|\lambda|}}\int_{0}^{\infty}\left(h^{\prime}_{s,2\pi
v|\lambda|}(\rho)e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}+\right.$
$\displaystyle+\left.\rho^{s}\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}\frac{\partial}{\partial\rho}\left(e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\right)\right)d\rho dvd\lambda,$

where in the last identity we have used (3.20). We also observe that from
(3.19) we have $h_{s,\mu}(\rho)\to 0$ as $\rho\to 0$ and $\rho\to\infty$ (see
[21, (4.18)]). From (3.21) we thus find

(3.22) $\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=\frac{(4\pi)^{s}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+1}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}\times$
$\displaystyle\times\int_{0}^{\infty}\frac{\partial}{\partial\rho}\left(e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\right)\times$
$\displaystyle\times\left(\rho^{s}\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}-h_{s,2\pi
v|\lambda|}(\rho)\right)d\rho dvd\lambda$
$\displaystyle=\frac{4^{s}\pi^{s+1}y^{2}}{\Gamma(1-s)}\int_{\mathbb{R}^{k}}\int_{0}^{\infty}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+2}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}\times$
$\displaystyle\times\int_{0}^{\infty}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}\left(\frac{2\pi v|\lambda|}{(1+\rho)\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{2}\times$
$\displaystyle\times\left(\rho^{s}\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}-h_{s,2\pi
v|\lambda|}(\rho)\right)d\rho dvd\lambda.$

From the identity

$\displaystyle\sinh^{2}2\pi v|\lambda|=$
$\displaystyle\sinh^{2}2\pi\frac{\rho}{1+\rho}v|\lambda|+\sinh^{2}2\pi\frac{v}{1+\rho}|\lambda|+$
$\displaystyle+2(\cosh 2\pi v|\lambda|)(\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|)(\sinh 2\pi\frac{v}{1+\rho}|\lambda|)$

and the definition (3.19), a straightforward computation shows that

$\rho^{s}\left(\frac{2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{1-s}\left(\frac{2\pi\frac{v}{1+\rho}|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{1+s}-h_{s,2\pi
v|\lambda|}(\rho)=\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\left(\frac{\sinh 2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{s}.$

From (3.22) we thus infer

(3.23) $\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=\frac{4^{s}\pi^{s+1}y^{2}}{\Gamma(1-s)}\int_{0}^{\infty}\int_{\mathbb{R}^{k}}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+2}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}+1}e^{-\frac{\pi}{2}|z|^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}\times$
$\displaystyle\times\left(\int_{0}^{\infty}\left(\frac{2\pi
v|\lambda|}{(1+\rho)\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{2}\left(\frac{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{s}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}d\rho\right)d\lambda dv.$

For any $\lambda\neq 0$ and $v>0$ we now apply Lemma 3.2 with the choices
$\mu=2\pi v|\lambda|$ and $B=\frac{y^{2}}{4v}$, and find

(3.24) $\displaystyle\int_{0}^{\infty}\left(\frac{2\pi
v|\lambda|}{(1+\rho)\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}\right)^{2}\left(\frac{\sinh
2\pi\frac{\rho}{1+\rho}v|\lambda|}{\sinh
2\pi\frac{v}{1+\rho}|\lambda|}\right)^{s}e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi\frac{\rho}{1+\rho}v|\lambda|}}d\rho$
$\displaystyle=\frac{y^{2s-2}}{(4v)^{s-1}}\left(\frac{2\pi v|\lambda|}{\sinh
2\pi
v|\lambda|}\right)^{s}\Gamma(1-s)e^{-\frac{\pi}{2}y^{2}\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}.$

Inserting (3.24) in (3.23), we finally obtain

$\displaystyle\mathscr{L}_{s}(\mathfrak{e}_{(s)}(\cdot,y))(g)$
$\displaystyle=(2\pi)^{2s}y^{2s}\int_{0}^{\infty}\int_{\mathbb{R}^{k}}e^{2\pi
i\langle\sigma,\lambda\rangle}\frac{1}{(4\pi
v)^{\frac{m}{2}+1+s}}\left(\frac{2\pi v|\lambda|}{\sinh 2\pi
v|\lambda|}\right)^{\frac{m}{2}+1+s}e^{-\frac{\pi}{2}(|z|^{2}+y^{2})\frac{|\lambda|}{\tanh
2\pi v|\lambda|}}d\lambda dv$ $\displaystyle=(2\pi
y)^{2s}\int_{0}^{\infty}q_{(-s)}((z,\sigma),v,y)dv=(2\pi
y)^{2s}\mathfrak{e}_{(-s)}((z,\sigma),y),$

which gives the desired conclusion (1.18).

∎

Although it should at this point be self-evident, for the sake of completeness
we show how (1.13) can be derived from Theorem 1.1 and Theorem A.

######  Corollary 3.3.

Let $\mathbb{G}$ be of Heisenberg type, and let $0<s<1$. For every
$(z,\sigma)\in\mathbb{G}$, and $y>0$ one has

$\displaystyle\mathscr{L}_{s}\left(((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k-2s}{4}}\right)$
$\displaystyle=\frac{\Gamma\left(\frac{m+2+2s}{4}\right)\Gamma\left(\frac{m+2k+2s}{4}\right)}{\Gamma\left(\frac{m+2-2s}{4}\right)\Gamma\left(\frac{m+2k-2s}{4}\right)}(4y)^{2s}((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k+2s}{4}}.$

###### Proof.

Combining (1.19) and (1.18), we obtain

(3.25)
$\displaystyle\mathscr{L}_{s}\left(((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k-2s}{4}}\right)$
$\displaystyle=\frac{(4\pi)^{1-s}}{\Gamma(s)C_{(s)}(m,k)}\mathscr{L}_{s}\left(\mathfrak{e}_{(s)}((z,\sigma),y)\right)=\frac{(4\pi)^{1-s}(2\pi
y)^{2s}}{\Gamma(s)C_{(s)}(m,k)}\mathfrak{e}_{(-s)}((z,\sigma),y)$
$\displaystyle=\frac{y^{2s}}{4^{s}}\frac{|\Gamma(-s)|C_{(-s)}(m,k)}{\Gamma(s)C_{(s)}(m,k)}((|z|^{2}+y^{2})^{2}+16|\sigma|^{2})^{-\frac{m+2k+2s}{4}}.$

On the other hand, by (1.20) we have

(3.26)
$\frac{|\Gamma(-s)|C_{(-s)}(m,k)}{\Gamma(s)C_{(s)}(m,k)}=4^{3s}\frac{\Gamma\left(\frac{m+2+2s}{4}\right)\Gamma\left(\frac{m+2k+2s}{4}\right)}{\Gamma\left(\frac{m+2-2s}{4}\right)\Gamma\left(\frac{m+2k-2s}{4}\right)}.$

Substituting (3.26) in (3.25) we reach the sought for conclusion.

∎

In closing, we observe that it is clear from Corollary 3.3 that, for any
$y>0$, the function

(3.27)
$u_{y}(z,\sigma)=\left(\frac{\Gamma\left(\frac{m+2+2s}{4}\right)\Gamma\left(\frac{m+2k+2s}{4}\right)}{\Gamma\left(\frac{m+2-2s}{4}\right)\Gamma\left(\frac{m+2k-2s}{4}\right)}\right)^{\frac{m+2k-2s}{4s}}\left(\frac{16y^{2}}{(|z|^{2}+y^{2})^{2}+16|\sigma|^{2}}\right)^{\frac{m+2k-2s}{4}}$

satisfies $\mathscr{L}_{s}u_{y}=u_{y}^{\frac{m+2k+2s}{m+2k-2s}}$. It is worth
emphasising here that Corollary 3.3 is stable under the convergence of
$s\nearrow 1$. In particular, we recover from (3.27) the functions that, in
the local case $s=1$, were found in [26] in the Heisenberg group
$\mathbb{H}^{n}$, see also [17, Theor. 1.1] and [18] for partial results in
groups of Heisenberg type.

## References

  * [1] A. V. Balakrishnan, _An operational calculus for infinitesimal generators of semigroups_. Trans. Amer. Math. Soc. 91 (1959), 330-353.
  * [2] H. Baum & A. Juhl, _Conformal differential geometry. $Q$-curvature and conformal holonomy_. Oberwolfach Seminars, 40, Birkhäuser Verlag, Basel, 2010.
  * [3] T. P. Branson, _Sharp inequalities, the functional determinant, and the complementary series_. Trans. Amer. Math. Soc. 347 (1995), no. 10, 3671-3742.
  * [4] T. P. Branson, _Spectral theory of invariant operators, sharp inequalities, and representation theory_. The Proceedings of the 16th Winter School “Geometry and Physics” (Srní, 1996). Rend. Circ. Mat. Palermo (2) Suppl. (1997), no. 46, 29-54.
  * [5] T. P. Branson, L. Fontana & C. Morpurgo, _Moser-Trudinger and Beckner-Onofri’s inequalities on the $\operatorname{CR}$ sphere_. Ann. of Math. (2) 177 (2013), no. 1, 1-52.
  * [6] L. Caffarelli & L. Silvestre, _An extension problem related to the fractional Laplacian_. Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260.
  * [7] S.-Y. A. Chang & M. d. M. González, _Fractional Laplacian in conformal geometry_. Adv. Math. 226 (2011), no. 2, 1410-1432.
  * [8] W. Chen, C. Li & B. Ou, _Classification of solutions for an integral equation_. Comm. Pure Appl. Math. 59 (2006), no. 3, 330-343.
  * [9] L. Corwin & F. P. Greenleaf, _Representations of nilpotent Lie groups and their applications, Part I: basic theory and examples_. Cambridge Studies in Advanced Mathematics 18, Cambridge University Press, Cambridge (1990).
  * [10] M. Cowling, _Unitary and uniformly bounded representations of some simple Lie groups_. In “Harmonic analysis and group representations”, A. Figà Talamanca (Ed.), C.I.M.E. Summer Schools, Cortona, Italy, 1980.
  * [11] M. Cowling, A. H. Dooley, A. Korányi & F. Ricci, _$H$ -type groups and Iwasawa decompositions_. Adv. Math. 87 (1991), no. 1, 1-41.
  * [12] M. Cowling & U. Haagerup, _Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one_. Invent. Math. 96 (1989), no. 3, 507-549.
  * [13] F. Ferrari & B. Franchi, _Harnack inequality for fractional sub-Laplacians in Carnot groups_. Math. Z. 279 (2015), no. 1-2, 435-458.
  * [14] G. B. Folland, _Subelliptic estimates and function spaces on nilpotent Lie groups_. Ark. Mat. 13 (1975), no. 2, 161-207.
  * [15] R. L. Frank & E. H. Lieb, _Sharp constants in several inequalities on the Heisenberg group_. Ann. of Math. (2) 176 (2012), no. 1, 349-381.
  * [16] R. L. Frank, M. d. M. González, D. Monticelli & J. Tan, _An extension problem for the $CR$ fractional Laplacian_. Adv. Math. 270 (2015), 97-137.
  * [17] N. Garofalo & D. Vassilev, _Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups_. Math. Ann. 318 (2000), no. 3, 453-516.
  * [18] N. Garofalo & D. Vassilev, _Symmetry properties of positive entire solutions of Yamabe-type equations on groups of Heisenberg type_. Duke Math. J. 106 (2001), no. 3, 411-448.
  * [19] N. Garofalo, _Some properties of sub-Laplaceans_. Proceedings of the International Conference ”Two nonlinear days in Urbino 2017”, 103-131, Electron. J. Differ. Equ. Conf., 25, Texas State Univ.-San Marcos, Dept. Math., San Marcos, TX, 2018.
  * [20] N. Garofalo, _Fractional thoughts_. New developments in the analysis of nonlocal operators, 1-135, Contemp. Math., 723, Amer. Math. Soc., Providence, RI, 2019.
  * [21] N. Garofalo & G. Tralli, _Feeling the heat in a group of Heisenberg type_. Adv. Math. 381 (2021), 107635.
  * [22] N. Garofalo & G. Tralli, _Mehler met Ornstein and Uhlenbeck: the geometry of Carnot groups of step two and their heat kernels_. ArXiv preprint 2007.10862.
  * [23] M. d. M. González, _Recent progress on the fractional Laplacian in conformal geometry_. Recent developments in nonlocal theory, 236-273, De Gruyter, Berlin, 2018.
  * [24] C. Guidi, A. Maalaoui & V. Martino, _Palais-Smale sequences for the fractional CR Yamabe functional and multiplicity results_. Calc. Var. Partial Differential Equations 57 (2018), article no. 152.
  * [25] L. Hörmander, Hypoelliptic second order differential equations. Acta Math. 119 (1967), 147-171.
  * [26] D. Jerison & J. M. Lee, _The Yamabe problem on CR manifolds_. J. Differential Geom. 25 (1987), no. 2, 167-197.
  * [27] D. Jerison & J. M. Lee, _Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem_. J. Amer. Math. Soc. 1 (1988), no. 1, 1-13.
  * [28] A. Kaplan, _Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms_. Trans. Amer. Math. Soc. 258 (1980), no. 1, 147-153.
  * [29] A. W. Knapp & E. M. Stein, _Intertwining operators for semisimple groups_. Ann. of Math. (2) 93 (1971), 489-578.
  * [30] A. Kristály, _Nodal solutions for the fractional Yamabe problem on Heisenberg groups_. Proc. Roy. Soc. Edinburgh Sect. A 150 (2020), 771-788.
  * [31] E. H. Lieb, _Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities_. Ann. of Math. (2) 118 (1983), no. 2, 349-374.
  * [32] M. Riesz, _Intégrales de Riemann-Liouville et potentiels_. Acta Sci. Math. Szeged, 9 (1938), 1-42.
  * [33] L. Roncal & S. Thangavelu, _Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group_. Adv. Math. 302 (2016), 106-158.
  * [34] L. Roncal & S. Thangavelu, _An extension problem and trace Hardy inequality for the sublaplacian on $H$-type groups_. Int. Math. Res. Not. IMRN (2020), no. 14, 4238-4294.
  * [35] E. M. Stein, _Singular integrals and differentiability properties of functions_. Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970 xiv+290 pp.
  * [36] N. Th. Varopoulos, L. Saloff-Coste & T. Coulhon, _Analysis and geometry on groups_. Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992. xii+156 pp.