In this paper we construct the fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation approach. In addition, we characterize the powers of the fractional sub-Laplacian in the Heisenberg group, and as a byproduct we compute the $k$-th order momenta with respect to the heat kernel.

The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

Francesca Corni
Francesca Corni: Dipartimento di Matematica
Università di Bologna
Piazza di Porta S.Donato 5
40126, Bologna-Italy
Fausto Ferrari
Fausto Ferrari: Dipartimento di Matematica
Università di Bologna
Piazza di Porta S.Donato 5
40126, Bologna-Italy
F.C.  and F.F. are partially supported by INDAM-GNAMPA-2022 projet: Equazioni completamente non lineari locali e non locali
AMS-2020, 35R03, 35R11

§ INTRODUCTION

The fractional Laplacian may be considered as the simplest example of a non-local fractional integral operator. A great interest about this object arises, since it allows to model various phenomena related to different fields. It is well known that fractional operators are connected to potential theory, probability, harmonic and fractional analysis, pseudo-differential operators and partial differential equations, see for instance [13], [35], [10] and [19].

On the other side, the setting of stratified groups represents a generalization of the Euclidean one, endowed with a rich structure, both from the algebraic and geometrical point of view.

Actually, the goal of generalizing fractional operators to stratified groups gives rise to appealing questions and open problems.
The aim of this paper is to investigate the connection between the fractional sub-Laplacian and suitable generalized Riesz potentials in the context of Carnot groups.

In particular, following an idea recalled in the book  by Landkof, [27], based on the work of Marcel Riesz, [33], [34],  we propose a natural extension of the definition of the real powers of the sub-Laplacian on a stratified group based on a suitable analytic continuation approach. We would like to represent an alternative face to the problem already dealt with by Folland [14].

Indeed, we think that this approach is very interesting for its simplicity and it is worth to be focused.

We start by considering a stratified group $\G$, i.e. a nilpotent simply connected Lie group whose Lie algebra admits a stratification. It is a standard fact that $\G$ can be represented as a vector space endowed with both a structure of stratified Lie algebra, with grading
\begin{equation}\label{grading}
\G=V_1 \oplus \ldots V_{\iota},
\end{equation}
and Lie group.

We denote by $q$ and $Q=\sum_{i=1}^{\iota} i \dim(V_i)$ the topological and the homogeneous dimension of $\G$, respectively, and, following [14], we suppose that $Q \geq 3$. We assume that a homogeneous distance $d$ on $\G$ is fixed, i.e. a distance $d$ such that $d(zx,zy)=d(x,y)$ and $d(\delta_t(x), \delta_t(y))=td(x,y)$ for every $x,y,z \in \G$ and $t>0$, where $\delta_t$ is a semigroup anisotropic dilation  associated with the grading (<ref>) of $\G$.

We denote by $m_1$ the dimension of the first layer $V_1$ of the grading. Moreover, we assume that a scalar product on $\G$ is fixed and we consider an orthonormal basis $(e_1, \ldots, e_{m_1})$ of $V_1$. We set for every $i=1, \ldots, m_1$ the vector field $X_i \in \mathrm{Lie}(\G)$ such that $X_i(0)=e_i$. The sub-Laplacian associated with $(e_1, \ldots, e_{m_1})$ is the operator

It is known that $\mathcal{L}$ is hypoelliptic,[25],  left-invariant and $2$-homogeneous with respect to the anisotropic dilations $\{\delta_t \}_{t>0}$.

Fractional powers $\mathcal{L}^{-\frac{\alpha}{2}}$ of $\mathcal{L}$, for $\alpha \in (-2,0)$, have been introduced through the theory of semigroups, involving the fundamental solution $h$ of the heat operator $\mathcal{H}=\partial_t+\mathcal{L}$, that is the heat kernel on $\G$ of $\mathcal{H}$ (see Section <ref> for definitions).

The operator $\mathcal{L}^{-\frac{\alpha}{2}}$, $\alpha \in (-2,0)$ takes the name of fractional sub-Laplacian and it has been studied by many authors.

In [11] the authors established a pointwise integral representation of $\mathcal{L}^{-\frac{\alpha}{2}}\phi(x)$ through a family of suitable generalized Riesz $(\alpha-Q)$-homogeneous kernels $P_{\alpha} \in C^{\infty}(\G \setminus \{0\})$.
They showed that for every $\alpha  \in (-2,0)$ if $u \in \mathcal{S}(\G)$, then $\mathcal{L}^{-\frac{\alpha}{2}}u \in L^2(\G)$ and

\begin{equation}\label{onerep}\mathcal{L}^{-\frac{\alpha}{2}}u(x)= P.V. \int_{\G} (u(y)-u(x)) P_{\alpha}(x^{-1}y) dy.
\end{equation}

We start by considering the operator

\begin{equation}
\phi\star P_{\alpha}(x)= \int_{\G} \phi(xy) P_{\alpha}(y)dy,
\end{equation}

for $\phi \in \mathcal{S}(\G)$ and $x\in \G$, where the symbol $\star$ in (<ref>) denotes the  convolution in the group, and we notice that it is well defined for every $\alpha \in (0,Q)$.

The main result of our paper relies in the construction of an analytic continuation of the map

\begin{equation}\label{map}
(0, Q) \ni \alpha \to  \phi\star P_{\alpha}(x)= \int_{\G} \phi(xy) P_{\alpha}(y)dy,
\end{equation}

to the interval $(-2,Q)$ for any fixed $x \in \G$ and $\phi \in \mathcal{S}(\G)$. This allows to extend in a natural way the definition of the power $\mathcal{L}^{-\frac{\alpha}{2}}$ on the Schwartz class $\mathcal{S}(\G)$, for $\alpha \in (-2,Q)$.

In our setting, we need to rely on various tools of geometric measure theory. Among these, in particular, we exploit the coarea formula proved in [29, 30]

and the Eikonal equation for the Carnot-Carathéodory distance proved in [31].

On the other hand, we take into account various properties of the heat kernel $h$ associated with $\mathcal{H}$, developed in [14] and [12]. Actually, the main step in  our constuction is the estimate <cit.>, involving homogeneous Taylor polynomials on a stratified group. In fact, while in the Euclidean setting the main point of the continuation is the Pizzetti formula, in an arbitrary stratified group the Pizzetti formula is not available yet. More precisely, even if some results are available in the literature [4], [2], they don't seem directly exploitable in our approach.

In the second part of the paper, we restrict ourselves to the simplest non-trivial case, given by the Heisenberg group $\H^n$. We show by induction the construction of the analytic continuation of the map in (<ref>) up to

$$ (-\infty, Q) \ni \alpha \to \phi \star P_{\alpha}(x),$$

for any fixed $x \in \G$ and $\phi \in \mathcal{S}(\G)$. In a natural way, this leads to an extension of the definition of $\mathcal{L}^{-\frac{\alpha}{2}}$ on the Schwartz class $\mathcal{S}(\H^n)$, for $\alpha \in (-\infty,Q)$.

From our point of view, the Heisenberg group represents a privileged setting. In fact, in $\H^n$,  both the heat kernel $h$, the kernels $P_{\alpha}$

and the profile of the Carnot-Carathéodory ball $\partial \B_c(0,1)$

preserve various symmetries related to the structure of the group.
In an arbitrary stratified group it is not clear how to proceed.

These invariances allow to rely on the use of Taylor polynomials to extend the continuation of the map in (<ref>) to every strip $(-2m-2, -2m]$, with $m \in \N$, through an inductive procedure.

Eventually, we provide simplified representations of the extended map $\alpha \to \psi(x,\alpha)$  on the intervals $(-4,Q)$, $(-6,Q)$ and $(-8,Q)$. By combining these representations with <cit.>, we obtain several geometric consequences. In particular, we compute the value of the integrals

\begin{equation}\label{geocons}
\begin{split}
\int_{\H^n} y_i^2 h(1, x) dx &\quad \int_{\H^n} y_i^4 h(1, x) dx  \quad \int_{\H^n} y_{2n+1}^2 h(1, x) dx\\
\int_{\H^n} y_i^6 h(1, x) & dx \quad  \int_{\H^n} y_i^2 y_{2n+1}^2 h(1, x) dx,
\end{split}
\end{equation}

for $i \in \{1, \ldots, 2n\}$, that of course have an important probabilistic meaning.

Now, we briefly describe the content of the various sections. In Section <ref> we introduce stratified groups and we collect many related definitions and results available in literature. Moreover, we deal with the sub-Laplacian $\mathcal{L}$ and fractional sub-Laplacian $\mathcal{L}^s$, with $s \in (0,1)$ and we introduce the family of the generalized Riesz potentials $P_{\alpha}$.

In Section <ref> we study the map

$$ (0,Q) \ni \alpha \to \psi(x,\alpha)=\phi \star P_{\alpha}(x),$$

and we build up its analytic continuation to the strip $(-2,0]$. Hence, we are in position to extend the definition of the fractional sub-Laplacian $\mathcal{L}^{s}$ to $s \in (-\frac{Q}{2},1)$.

In Section <ref> we sketch the setting of the Heisenberg group $\H^n$ for fixing notation and we show some technical auxiliary results.

In Section <ref> we build up the analytic continuation of the function $\alpha \to \psi(x, \alpha)$ on the interval $(-\infty,Q)$. Then, we exploit the uniqueness of the analytic continuation to extend the notion of fractional sub-Laplacian $\mathcal{L}^s$ to every $s \in (-\frac{Q}{2}, \infty)$.

Finally, in Sections <ref>, <ref> and <ref> we obtain simplified representations of $\alpha \to \psi(x,\alpha)$ on the intervals $(-4,Q)$, $(-6,Q)$ and $(-8,Q)$, respectively. We apply these representations to establish the value of the integrals in (<ref>).

§ PRELIMINARIES

§.§ Stratified groups, Pansu differentiabily, coarea formula

In this section we collect some definitions and more or less known results about stratified groups, that we state for fixing the notation.

A stratified group or Carnot group $\G$ of step $\iota$ is a connected, simply connected and nilpotent Lie group whose Lie algebra is stratified of step $\iota$, i.e. there exists a sequence of subspaces $\cV_j$ with $j\in\N$, such that $\cV_j= \{ 0 \}$ if $j>\iota$, $[\cV_1, \cV_i] = \cV_{i+1}$ for every $i,j\ge1$, $\cV_\iota \neq \{ 0 \}$ and
$ \mathrm{Lie}(\G)=\cV_1 \oplus \dots \oplus \cV_\iota$,
where for $i,j \geq 1$ we have set

$$[\cV_i,\cV_j]=\text{span} \{ [X,Y]  :  X \in \cV_i, \ Y \in \cV_j\}.$$

The exponential map $\mathrm{exp}: \mathrm{Lie}(\G) \to \G$ is a global diffeomorphism which allows to identify in a standard way $\G$ with $\mathrm{Lie}(\G)$ (see for instance <cit.>).
As a consequence, we can model a stratified group as a vector space $\G$ endowed with both a Lie bracket $[ \cdot, \cdot]$ with respect to which $\G$ is a stratified Lie algebra

\begin{equation}\label{eq:cg}
\G=V_1 \oplus V_2 \ldots \oplus V_{\iota},
\end{equation}
and a Lie group structure, whose product is given by the Baker-Campbell-Hausdorff operation associated with $[ \cdot, \cdot]$ (see <cit.>).

For every $x \in \G$ we denote the corresponding left translation by
$$l_x:\G \to \G, \ l_x(y)=xy,$$
for every $y \in \G$. We denote by $H\G$ the distribution generated by $V_1$, it is called the horizontal distribution.
A basis $(e_1,\ldots,e_q)$ of $\G$ is called graded if
the ordered family $ (e_{h_{i-1}+1},\ldots,e_{h_i}) $ is a basis of $V_i$ for every $i=1,\ldots,\iota$.
We call $(Z_1, \cdots Z_q)$ a Jacobian basis of $\mathrm{Lie}(\G)$ if $(Z_1(0), \ldots Z_q(0))$ is a graded basis of $\G$.
A family of dilations is associated with the stratification of $\G$ in a natural way: for every $t>0$ we set
$$\delta_t \left(\sum_{i=1}^\iota v_i \right)=\sum_{i=1}^{\iota}t^i v_i \qquad \mathrm{with} \ v_i \in V_i.$$

Let $v \in \G$, we call $n \in \N$ the degree of $v$, and we write $\mathrm{deg}(v)=n$, if

$$ \delta_t(v)=t^n v,$$

hence if $v \in V_n$. Analogously, for a vector field $X \in \mathrm{Lie}(\G)$, we say that $\mathrm{deg}(X)=n$ if $X \in \mathrm{Lie}(V_n)$.

We assume that a scalar product $ \langle \cdot, \cdot \rangle$ is fixed on $\G$
and we denote by $|\cdot|$ its associated norm. Moreover, we assume that $\langle \cdot, \cdot \rangle$ makes the $V_i$'s orthogonal. Taking into account the structure of vector space of $\G$, we may identify it with $T_0\G$. Thus, the scalar product $\langle \cdot, \cdot \rangle$ extends through the left translations to a left invariant Riemannian metric that we denote by $g$.

A distance $d$ on $\G$ is called homogeneous if $d(zx,zy)=d(x,y)$ and $d(\delta_t(x), \delta_t(y))=t d(x,y)$ for every $x,y,z \in \G$ and $t>0$.
Let us recall two concrete examples of homogeneous distances (see <cit.> and <cit.>).
First, we consider $x,y \in \G$ and we set $\Gamma_{x,y}$ the set of horizontal curves, i.e. absolutely continuous curves $\gamma:[0,T] \to \G$ with $\dot{\gamma(t)} \in H\G_{\gamma(t)}$ for a.e. $t \in [0,T]$ for some $T>0$, such that $\gamma(0)=x, \ \gamma(T)=y$.
The Carnot-Carathéodory distance or CC-distance between $x$ and $y$ is

$$ d_c(x,y)= \inf \left\{ \int^T_{0} \sqrt{g(\gamma(t))(\dot{\gamma}(t), \dot{\gamma}(t))} dt : \gamma \in \Gamma_{x,y} \right\}.$$

For every $x,y \in \G$, the Chow's theorem ensures that the set $\Gamma_{x,y}$ is non-empty hence $d_c$ is well defined. We denote by $\| x \|_c=d(x,0)$, for every $x \in \G$, the homogeneous norm associated with $d_c$.
The second homogeneous distance we introduce is the distance $d_{\infty}$, defined for all $x,y \in \G$ as

$$d_{\infty}(x,y)= \| y^{-1}  x \|_{\infty},$$
where the homogeneous norm $\| \cdot \|_{\infty}$ is

$$ \| x \|_{\infty}:= \max\{ \varepsilon_j | x^j|^{1/j}, \ j=1, \dots, \kappa  \},$$
for $x=\sum_{i=1}^\kappa x^j \in \G$, with $x^j \in V_j$, with $\varepsilon_1=1$, $\varepsilon_{j} \in (0,1]$ suitable positive constants depending on the group structure .
In this paper, we assume that a homogeneous distance $d$ is fixed on $\G$ and we denote by $\| x \|=d(x,0)$, for every $x \in \G$, the corresponding homogeneous norm. Moreover, we introduce the following closed balls

\begin{align*}
\B(x,r)&= \{ x \in \G: \|x\| \leq r \},\\
\B_E(x,r)&=\{ x \in \G: |x| \leq r \},
\end{align*}

for every $x \in \G$ and $r>0$.
The closed ball centered at $x$ of radius $r$ corresponding to $d_c$ will be denoted as $\B_c(x,r)$.
We recall that all homogeneous norms on $\G$ are equivalent (see for instance <cit.>), hence if $d_1$, $d_2$ are two homogeneous distances on $\G$, then there exist a positive constant $C>0$ such that

$$\frac{1}{C}d_1(x,y) \leq d_2(x,y) \leq C d_1(x,y),$$

for every $x,y \in \G$.

We denote by $q$ the topological (or linear) dimension of $\G$ and we call homogeneous dimension of $\G$ the integer
$$Q=\sum_{i=1}^\iota i \dim(V_i),$$
which is also the Hausdorff dimension of $\G$ with respect to a homogeneous distance $d$. We assume that $Q\geq 3$.
Let us recall also the notion of Hausdorff measure in this context. Let $\mathcal{F} \subset \mathcal{P}(\G) $ be a non-empty family of closed subsets of a Carnot group $\G$, endowed with a homogeneous distance $d$. Let $\zeta: \mathcal{F} \to \R^+$ be a function such that $0 \leq \zeta(S)<\infty$ for any $S \in \mathcal{F}$. If $\delta>0$, and $A \subset \G$, we define

\begin{equation}
\phi_{\delta, \zeta}(A)= \inf \left\{ \sum_{j=0}^{\infty} \zeta(B_j) \ : \ A \subset \bigcup_{j=0}^\infty B_j , \ \mathrm{diam}(B_j) \leq \delta, \ B_j \in \mathcal{F} \right\}.
\end{equation}

If $\mathcal{F}$ coincides with the family of closed balls with respect to the distance $d$ and $\zeta(\B(x,r))=r^{\alpha}$ we call

$$ \mathcal{S}^{\alpha}(A):= \sup_{\delta>0} \phi_{\delta, \zeta}(A)$$

the $\alpha$-spherical Hausdorff measure of $A$.
If $\mathcal{F}$ coincides with the family of all
closed sets, and for $ k \in \{1, \dots, q \}$ we set
$c_{k}=\omega_k$, the $k$-dimensional Euclidean measure of the unitary Euclidean ball in $\R^k$, and $\zeta(B)=c_{k} \left(\frac{\mathrm{diam}(B)}{2} \right)^{k},$ we call

$$ \mathcal{H}^{k}_E (A):= \sup_{\delta>0} \phi_{\delta, \zeta}(A)$$

the $k$-Euclidean Hausdorff measure.
In particular, we stress that $\mathcal{H}_E^k$ is the Hausdorff measure with respect to the distance generated by the Riemannian metric $g$ fixed on $\G$. Nevertheless, if we consider $\G$ in coordinates with respect to an orthonormal basis, then $\mathcal{H}_E^k$ coincides with the Euclidean Hausdorff measure and this justify our notation.
The $q$-dimensional Lebesgue measure on $\G$ is the Haar measure of $\G$. It is left invariant and $Q$-homogeneous.
On the other hand, for every $m>0$, for every measurable set $ A\subset \mathbb{G}$ and for all $x \in \G$ and $t>0$ we have

\begin{equation*}
\mathcal{S}^m(l_x( A))=\mathcal{S}^m(A)  \quad \mathrm{and} \quad
\mathcal{S}^m(\delta_t(A))=t^m \mathcal{S}^m( A).
\end{equation*}

We refer the reader for example to [36] for more details about an introduction to stratified groups.

§.§.§ Pansu differentiability

A map $L:\G \to \R$ is a h-homomorphism if it is a group homomorphism such that $L(\delta_t(x))=t(L(x))$, for every $x \in \G$ and $t>0$. It is not difficult to verify that $L$ is linear and $L|_{V_j}\equiv 0$ for every $j=2, \ldots, \iota$. We denote by $\mathcal{L}_h(\G, \R)$ the family of h-homomorphisms between $\G$ and $\R$.
Given $L, T \in \mathcal{L}_h(\G, \R)$, we define the distance
$$d_{\mathcal{L}_h(\G, \R)}(L,T):= \sup_{\|x\| \leq 1} |T(x)^{-1}L(x) |.$$
This class of maps is the starting point to state a notion of differentiability fitting the setting of stratified groups.
Let $\Omega$ be an open set in $\G$, let $f : \Omega \to \R$ be a function and let $x \in \Omega$.
The map $f$ is said  Pansu differentiable at $x$ if there exists a h-homomorphism $L:\G \to \R$ that satisfies

\begin{equation}
\label{pansudiff}
|  f(y) - f(x)-  L(x^{-1} y) |= o ( \|x^{-1} y \|) \quad  \text{as} \quad \|x^{-1} y \|  \to 0.
\end{equation}

When condition (<ref>) is verified, we call $L$ the Pansu differential of $f$ at $x$ and we denote it by $D_hf(x)$.
The unique vector $\nabla_{H}f(x)\in \G$ such that

\begin{equation}\label{def-grad}
\end{equation}
for every $z\in \G$ is called the horizontal gradient of $f$ at $x$.
Assuming that $(v_1, \dots, v_q)$ is a graded orthonormal basis of $\G$ and setting $X_j \in \mathrm{Lie}(\G)$ the left invariant vector field such that $X_j(0)=v_j$ for every $j=1, \dots, q$
\begin{equation}\label{forma-grad}
\nabla_Hf(x)=\sum_{i=1}^{m_1} X_if(x) v_i,
\end{equation}

where $m_1=\dim(V_1)$ (see <cit.> and the references therein).
We say that $f:\Omega\to\R$
is continuously Pansu differentiable on $\Omega$ if $f$ is Pansu differentiable at every point of $\Omega$ and the function
$D_hf: \Omega \to \mathcal{L}_h(\G, \R)$ is continuous with respect to the topology induced by $d_{\mathcal{L}_h(\G, \R)}$.
We denote by $C^1_h(\Omega)$ the family of continuously Pansu differentiable mappings from $\Omega$ to $\R$ and we observe that $C^1(\Omega) \subsetneq C^1_h(\Omega)$.

§.§.§ Coarea formula in stratified groups

In this subsection we recall a suitable coarea formula in Carnot groups. It is interesting to notice that the starting point of the proof of Theorem <ref> is a previous coarea formula for bounded variation maps ([16, 20, 31]).

Let $v \in V_1 \setminus \{ 0 \}$ be a vector and consider its orthogonal hyperplane $\mathcal{L}(v) \subset \G$. We define

$$ \theta^g_{Q-1}(v):= \max_{z \in \B(0,1)}\mathcal{H}_E^{q-1}(\mathcal{L}(v) \cap \B(z,1))$$

the metric factor with respect to the direction $v$.

We explicitly remark that $\theta^g_{Q-1}(v)$ depends both on the homogeneous distance $d$ and on the Riemannian metric $g$.

The following theorem is proved in <cit.> (see also [30]).

Let $A \subset \G$ be a measurable set and consider a Lipschitz function $u: A \to \R$. For any measurable function $h:A \to [0, \infty)$ the equality

$$ \int_A h(x) | \nabla_Hu(x)| dx = \int_\R \int_{u^{-1}(r) \cap A} h(x) \theta^g_{Q-1}( \nabla_H u(x)) d \mathcal{S}^{Q-1}(x) dr$$

holds. Notice that $\theta^g_{Q-1}( \nabla_H u(x))$ and $\mathcal{S}^{Q-1} $
are both considered with respect to $d$.

On the other hand, by <cit.> and <cit.> we know that the distance $d_{\infty}$ preserves enough symmetries to guarantee the constancy of the metric factor. This is the content of the following proposition.

If $d=d_{\infty}$, then there is a constant $\alpha_{Q-1}>0$ such that
$$\theta^g_{Q-1}(v)=\alpha_{Q-1}, \qquad \mathrm{ for \ every } \ v \in V_1 \setminus \{0\}.$$

Thus, in the setting of Proposition <ref>, we may set

$$\mathcal{S}^{Q-1}_{\infty}:= \alpha_{Q-1} \mathcal{S}^{Q-1}$$

to yield a simplified coarea formula, which follows from the combination of <cit.> and Proposition <ref>.

Let $d=d_{\infty}$, let $A \subset \G$ be a measurable set and let $u: A \to \R$
be a Lipschitz function. For any measurable function $h:A \to [0, \infty)$, the following equality holds

$$ \int_A h(x) | \nabla_Hu(x)| dx = \int_\R \int_{u^{-1}(r) \cap A} h(x) d \mathcal{S}_\infty^{Q-1} dr.$$

Now, in order to obtain a further simplified expression of the coarea formula, we recall the Eikonal equation associates with the CC-distance $d_c$, which is a consequence of <cit.>.

For a.e. $ x \in \G$ the equality
$ | \nabla_H d_c(x, 0)| =1$ holds.

By combining Theorems <ref> and Theorem <ref>, setting $u(x):\G \to \R, \ u(x)=d_c(x,0)$ as the Lipschitz function in Theorem <ref>, we get the following formula, which retraces the Euclidean coarea formula.
(Coarea formula)

If $d=d_{\infty}$ and $A \subset \G$ is a measurable set,
then for any measurable function $h:A \to [0, \infty)$ we have

$$ \int_A h(x) dx = \int_\R \int_{\partial \B_c(0,r) \cap A} h(x) d \mathcal{S}_\infty^{Q-1} dr.$$

§.§ Taylor polynomial on stratified groups

In this subsection we introduce homogeneous Taylor polinomials on $\G$. We mainly follow <cit.>.
We choose $m \in \N$ and for every $m$-multi-index $\gamma =(\gamma_1, \ldots, \gamma_m)\in (\N \cup \{ 0 \})^m$ we set

$$|\gamma|=\gamma_1+\gamma_2+\dots \gamma_m \qquad \gamma!=\gamma_1! \gamma_2! \dots \gamma_m! \qquad \mathrm{and} \qquad x^{\gamma}=x_1^{\gamma_1}x_2^{\gamma_2}\dots x_m^{\gamma_m} \quad \mathrm{for \ every \ } x \in \R^m.$$

Moreover, we introduce

$$(Z_1, \cdots, Z_m)^{\gamma}=Z_1^{\gamma_1} \cdots Z_m^{\gamma_m}, \quad \mathrm{ for \ every \ } Z_1, \ldots Z_m \in \Lie(\G).$$

We say that an $m$-multi-index $\gamma =(\gamma_1, \ldots, \gamma_m)\in (\N \cup \{ 0 \})^m$ is even if $\gamma_j$ is even (0 being even) for every $j \in \{1, \ldots, m \}$.

If $f \in C^{\infty}(\G,\R)$ and $(Z_1, \ldots, Z_q)$ is a Jacobian basis of $\mathrm{Lie}(\G)$, then for every $n \in \N \cup \{0 \}$ there exists one and only one polynomial $p$ homogeneous of degree at most $n$ such that

\begin{equation}\label{taipol}
(Z_1 \cdots Z_q)^{\beta}(p)(0)= (Z_1 \cdots Z_q)^{\beta}(f)(0),
\end{equation}

for every multi-index $\beta \in ( \N \cup \{0\})^q$ such that $|\beta|\leq n$.
The unique polynomial satisfying (<ref>) is denoted by $\mathcal{T}_n(f,0)(x)$ and it is called the $Z$-Mac Laurin polynomial homogeneous of degree $n$ related to $f$.
Then, for every $x,x_0 \in \G$ we introduce the so-called $Z$-Taylor polynomial of $\G$-degree $n$ related to $f$ centered at $x_0$ as

\begin{equation}\label{transla}
\mathcal{T}_n(f,x_0)(x):=\mathcal{T}_n(f \circ l_{x_0},0)(x_0^{-1}x).
\end{equation}

Let us recall an important estimate involving $Z$-Taylor polynomials, obtained by <cit.>, setting $d=d_c$ (see also <cit.>).
(Stratified Taylor formula)

Let $\G$ be a stratified group. There is a constant $b=b(\G)$ such that for every $n \in \N \cup \{ 0 \}$, there exists a constant $c=c(\G,n)>0$ such that

\begin{equation} \label{Taylorn}
|f(xy)-\mathcal{T}_n(f,x)(xy)| \leq c \|y\|_c^{n+1} \sup_{\{\|z\|_c \leq b^{n+1} \|y\|_c\}} \sup_{\{|\beta|=n+1\}} |(Z_1 \cdots Z_{m_1})^{\beta} f(xz)|,
\end{equation}

for all $f \in C^{n+1}(\G,\R)$, for every $x,y \in \G$. Here $\beta$ is a $m_1$-multi-index and $m_1=\dim(V_1)$.

An explicit form of the $Z$-Taylor polynomial homogeneous of $\G$-degree $n$ is known. We refer the reader to <cit.> (see also <cit.> formula (43)).

Let $n \in \N \cup \{0\}$, let $f \in C^{n+1}(\G,\R)$ and choose $x_0 \in \G$. Let $(Z_1, \dots, Z_q)$ be a Jacobian basis of $\mathrm{Lie}(\G)$, then

$$ \mathcal{T}_n(f,x_0)(x)=f(x_0)+ \sum_{h=1}^n \sum_{\substack{k=1, \ldots, n\\ i_1, \ldots,i_k \leq q, \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=h}} \frac{(x_0^{-1}x)_{i_1} \ldots (x_0^{-1}x)_{i_k}}{k!} Z_{i_1} \ldots Z_{i_k} f(x_0),$$

for every $x \in \G$.

§.§ Fractional sub-Laplacian on stratified groups

In this subsection, we introduce the sub-Laplacian on $\G$ and its powers. We choose a graded orthonormal basis $(v_1 , \dots, v_{m_1})$ of $V_1$ and we let $X_i \in \mathrm{Lie}(\G)$ be the left invariant vector field such that $X_i(0)=v_i$ for every $i=1, \dots, m_1$.
We call sub-Laplacian of $\G$ the second-order differential operator

\begin{equation}\label{sublaplacian}
\mathcal{L}:= - \sum_{i=1}^{m_1} X_i^2.
\end{equation}

It is a positive operator associated with the choice of the basis $(v_1, \ldots, v_{m_1})$.

Since $\G$ is a stratified group, by Hormander's theorem the operator $\mathcal{L}$ is hypoelliptic. By <cit.> there exists a unique fundamental solution for $\mathcal{ L}$.

Let $\widehat{\G}$ be the homogeneous group
$$\widehat{\G}=\R \times \G,$$
endowed with the product

$$(t,x) \cdot (t'.y)=(t+t', xy)$$
and the following stratification

$$ \widehat{\G}= V_1 \oplus \widehat{V_2}  \oplus \dots \oplus V_{\iota}, \quad \mathrm{ with  }\quad\mathrm{Lie}(\widehat{V_2})=\mathrm{span} \left\{ \frac{\partial}{\partial t}, \mathrm{Lie}(V_2) \right\}.$$
The group $\widehat{G}$ may be equipped with the dilations

\begin{equation}\label{dil}
\widehat{\delta}_s(t,x)=(s^2t, \delta_s(x)),
\end{equation}

for every $s>0$. We consider on $\widehat{\G}$ the heat operator
$$\mathcal{H}= \mathcal{L} + \frac{\partial}{\partial t},$$
$\mathcal{L}$ is the sub-Laplacian in (<ref>).
The operator $\mathcal{H}$ is invariant with respect to the left translations of $\widehat{\G}$ and homogeneous of degree 2 with respect to the dilations in (<ref>), i.e. $\mathcal{H}(u \circ \widehat{\delta}_s)=s^2 \mathcal{H}u$ for $s>0$. Moreover, $\mathcal{H}$ is hypoelliptic and it has a unique fundamental solution $h$ called the heat kernel on $\G$ (see [14]). We collect some properties of $h$ addressing the reader to <cit.>, <cit.> and <cit.> and to the references therein.

Let $h:\R \times \G \to \R$ be the heat kernel associated with the sub-Laplacian $\mathcal{L}$ on $\G$, then the following conditions hold

(i) $h \in C^{\infty}(\widehat{\G} \setminus \{(0,0)\})$.

(ii) $ \mathcal{H} h=0$ on $(0, \infty) \times \G $.

(iii) $\int_{\G} h(t,x) dx=1$, for every $t>0$; $h(t,x)=h(t,x^{-1})$, $h(t,x)\geq 0$, for every $x \in \G$ and $t \in \R$ and $h(t,x)=0$, when $t\leq 0$.

(iv) $h(t, \cdot ) \star h(\tau, \cdot)= h(t+\tau, \cdot)$, for every $t, \tau >0$.

(v) $h(\widehat{\delta}_s(t,x))=s^{-Q} h(t,x)$, for every $s, t>0$ and $x \in \G$.

(vi) There exists a constant $c>0$ such that

$$ c^{-1} t^{-Q/2} \mathrm{exp}\Big(-\frac{ \| x \|^2}{c^{-1}t}\Big) \leq h(t,x) \leq c t^{-Q/2} \mathrm{exp}\Big(-\frac{\| x \|^2}{ct}\Big),$$

for every $x \in \G$ and $t>0$.

$h(t, \cdot) \in \mathcal{S}(\G)$, for every $t>0$.

Now, we introduce the heat semigroup:

\begin{equation}
e^{-t \mathcal{L}}f(x):=(f \star h(t, \cdot))(x)=\int_\G h(t,y^{-1}x)f(y)dy,
\end{equation}

for every $f \in L^1(\G)$, where $\star$ denotes the group convolution.
We define the fractional sub-Laplacian $\mathcal{L}^s$, $0<s<1$ as follows (see [37]). For every map $u \in W^{2s,2}(\G)$ and for every $x \in \G$

\begin{equation}\label{frac-lap}
\mathcal{L}^s u(x):= \frac{1}{\Gamma(-s)} \int_0^{\infty}t^{-s-1} (e^{-t\mathcal{L}} u(x)-u(x)) dt.
\end{equation}

See <cit.> and the references therein for the definition and an introduction to fractional Sobolev space on a stratified group. The operator $\mathcal{L}^s$ is non-local.

We recall the following result about the behaviour of $\mathcal{L}^s$, when $s$ is close to one, see also <cit.>. We remind the proof of the theorem to point out some technical aspects of the result. By $\mathcal{S}(\G)$ we denote the Schwartz class on $\G$ (see <cit.>).

For every $\phi \in \mathcal{S}(\G)$ and for every $x \in \G$

$$\lim_{s \to 1^-} \mathcal{L}^{s}\phi(x)=\mathcal{L}\phi(x).$$

Using the change of variables $\xi=t^{1-s}$ on (<ref>), we get

\begin{equation}\mathcal{L}^s\phi(x)=\frac{1}{(1-s)\Gamma(-s)} \int_0^{\infty} (e^{-\xi^{\frac{1}{1-s}\mathcal{L}}}\phi(x)-\phi(x))\xi^{-\frac{1}{1-s}} d \xi.
\end{equation}

Clearly, if $0\leq \xi <1$ then $\lim_{s \to 1^-}\xi^{\frac{1}{1-s}}=0$. Since $\phi \in \mathcal{S}(\G)$, by <cit.> (notice that $\mathcal{D(\G)}$ is $L^{\infty}$-dense in $\mathcal{S}(\G)$) we have

\begin{equation}\label{1.17}
\left\| \frac{e^{-h\mathcal{L}}\phi-\phi}{h}+\mathcal{L}\phi \right\|_{L^{\infty}(\G)} \to 0 \ \mathrm{as} \ h \to 0.
\end{equation}

Set $c(s)=\frac{1}{(1-s)\Gamma(-s)}=\frac{-s}{(1-s)\Gamma(1-s)}$. We claim that for every $x \in \G$

\begin{equation}\label{1.18}
\lim_{s \to 1^-} c(s) \int_0^1 (e^{-\xi^{\frac{1}{1-s}\mathcal{L}}}\phi(x)-\phi(x))\xi^{-\frac{1}{1-s}} d \xi =\frac{1}{2}\mathcal{L}(x).
\end{equation}

Indeed, denoting by $f_{s,x}(\xi):=((e^{-\xi^{\frac{1}{1-s}\mathcal{L}}}\phi(x)-\phi(x))\xi^{-\frac{1}{1-s}}$ and using (<ref>) we get

\begin{equation}\label{1.19}
f_{s,x}(\xi) \to -\mathcal{L}(x)
\end{equation}

for every $x \in \G$. Moreover,

\begin{equation}\label{1.20}
\begin{split}
|f_{s,x}(\xi)|&=\left| ((e^{-\xi^{\frac{1}{1-s}\mathcal{L}}}\phi(x)-\phi(x))\xi^{-\frac{1}{1-s}} \right|=\left| \frac{1}{\xi^{\frac{1}{1-s}}} \int_0^{\xi^{\frac{1}{1-s}}} \frac{d}{d\tau} e^{-\tau \mathcal{L}}\phi(x) d \tau \right|\\
&= \left| \frac{1}{\xi^{\frac{1}{1-s}}} \int_0^{\xi^{\frac{1}{1-s}}}  e^{-\tau \mathcal{L}} \mathcal{L}\phi(x) d \tau \right| \leq \| \mathcal{L}\phi\|_{L^{\infty}(\G)},
\end{split}
\end{equation}

where in the last inequality we used that $\phi \in \mathcal{S}(\G)$ and the fact, proved in <cit.>, that the heat semigroup preserves one-sided bounds, namely

\begin{equation}\label{1.6}
f \leq C \mathrm{ \ a.e. \ in \ }\G \Rightarrow e^{-t\mathcal{L}}f \leq C \mathrm{ \ a.e. \ in \ } \G \qquad \forall t \geq 0.
\end{equation}

Thus, our claim (<ref>) follows using the definition of $c(s)$, hence from the fact that $\lim_{s\to 1^-}c(s)=-1$, and from (<ref>) and (<ref>). Furthermore, using (<ref>), we obtain

\begin{align*}
c(s) \int_{1}^{\infty} |e^{-\xi^\frac{1}{1-s}\mathcal{L}}\phi(x)-\phi(x)| \xi^{-\frac{1}{1-s}} d \xi &\leq 2c(s) \|\phi\|_{L^{\infty}(\G)} \int_1^{\infty}\xi^{-\frac{1}{1-s}}d \xi\\
&= - 2  \|\phi\|_{L^{\infty}(\G)}\frac{-s}{(1-s)\Gamma(1-s)} \frac{1-s}{-s}\\
&= - 2 \frac{1}{\Gamma(1-s)}  \|\phi\|_{L^{\infty}(\G)} \to 0
\end{align*}

as $s \to 1^-$. Thus, letting $s \to 1^-$ we get the thesis.

§.§ Generalized Riesz potentials

In this section we introduce a family of generalized Riesz potentials.
For every $ \alpha< Q$, $\alpha \neq 0, -2, -4, \cdots$ we set $P_{\alpha}:\G \to \R$ as

$$ P_{\alpha}(x)=\frac{1}{\Gamma \left( \frac{\alpha}{2} \right)}\int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt.$$

We observe that $P_{\alpha}$ coincides with the map $R_{\alpha}$ defined in [14, 11, 12] for $0<\alpha<Q$.
We collect below some properties of the $P_{\alpha}$ (see <cit.>).

*  By Theorem <ref>(vi), for every $ x \neq 0$
$$h(t,x)=o(t^N) \ \mathrm{as} \ t \to 0^+$$
for every $N \in \N$. For every $x \in \G$,
$$h(t,x) \leq c t^{-\frac{Q}{2}}$$
for some constant $c\geq 0$ and for every $t>0$.

Thus, $P_{\alpha}(x)$ converges absolutely if $\alpha<Q$ for every $x \neq 0$.

*  For every $\alpha<Q$, $P_{\alpha} \in C^{\infty}(\G \setminus \{0\})$, since $h(t,x) \in C^{\infty}(\widehat{\G }\setminus \{(0,0)\})$.

*  For every $0<\alpha<Q$, $P_{\alpha}$ is non-negative. Moreover, for every $\alpha<Q$ it is homogeneous of degree $\alpha-Q$ with respect to the intrinsic dilations $\delta_s$, $s>0$. Let us show the homogeneity, for $s>0$ we have

\begin{align*}
P_{\alpha}(\delta_s(x))&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,\delta_s(x)) dt\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h\left(\frac{1}{s^2}s^2t,\delta_s(x)\right) dt\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_0^{\infty} t^{\frac{\alpha}{2}-1} s^{-Q}h\left(\frac{1}{s^2}t,x \right) dt,
\end{align*}

where we exploited Theorem <ref>(v). Let us now perform the change of variables $t=s^2\tau$ so that we get

\begin{align*}
P_{\alpha}(\delta_s(x))&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_0^{\infty} (s^2 \tau)^{\frac{\alpha}{2}-1} s^{-Q}h(\tau,x) s^2 d\tau\\
&=s^{\alpha-2-Q+2}\frac{1}{\Gamma(\frac{\alpha}{2})} \int_0^{\infty}  \tau^{\frac{\alpha}{2}-1} h(\tau,x) d\tau\\
\end{align*}

We introduce for every $ \alpha<Q$ and for every $x \in \G$, $x \neq 0$, the function

\begin{equation}
\| x \|_{\alpha}:=\left( \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt \right)^{\frac{1}{\alpha-Q}}.
\end{equation}

Notice that for every $\alpha<Q$, $\alpha \neq 0,-2,-4, \ldots$ and $x \in \G$, $x \neq 0$, the equality
$$P_{\alpha}(x)= \frac{1}{\Gamma(\frac{\alpha}{2})} \|x\|_{\alpha}^{\alpha-Q}$$

The map $\| \cdot \|_{\alpha}$ is a homogeneous norm on $\G$ smooth outside the identity element (see <cit.> and Item <ref> in Section <ref>). Since on a Carnot group every homogeneous norms are equivalent, for every $\alpha<Q$ there exists a constant $c_{\alpha}>0$ such that

\begin{equation}\label{relcc-alfa}
\frac{1}{c_{\alpha}} \| x \|_c \leq \|x \|_{\alpha} \leq c_{\alpha} \|x\|_c,
\end{equation}

for every $x \in \G$.

Let us verify that if $0<\alpha<Q$, then $P_{\alpha} \in L^1_{loc}(\G)$. In fact, by combining the coarea formula of Corollary <ref>, the homogeneity of the measure $\mathcal{S}_{\infty}^{Q-1}$ and the homogeneity of the norm $\| \cdot \|_{\alpha}$, for every $R>0$ we get

\begin{align*}
\frac{1}{\Gamma(\frac{\alpha}{2}) }\int_{\B_c(0,R)} \|x \|_{\alpha}^{\alpha-Q} dx &= \frac{1}{\Gamma(\frac{\alpha}{2}) }\int_0^R \int_{\partial \B_c(0,r)} \|x \|_{\alpha}^{\alpha-Q} d\mathcal{S}^{Q-1}_{\infty}(x) dt\\
&= \frac{1}{\Gamma(\frac{\alpha}{2}) }\int_0^R \int_{\partial \B_c(0,1)} r^{\alpha-Q} \|z \|_{\alpha}^{\alpha-Q} r^{Q-1} d\mathcal{S}^{Q-1}_{\infty}(z) dt \\
&\leq \frac{1}{\Gamma(\frac{\alpha}{2}) } \int_{0}^{R} r^{\alpha-1} dr \  \mathcal{S}_{\infty}^{Q-1}( \partial \B_c(0,1)) \max_{\partial \B_c(0,1)} \|z\|_{\alpha}^{\alpha-Q},
\end{align*}

which is finite since $\alpha>0$.

Taking into account the previous Items <ref>, <ref> and Remark <ref>, it holds that if $0<\alpha<Q$, then $P_{\alpha}$ is a kernel of type $\alpha$, according to the definition in [14]. We list below various results known in the literature that relate the maps $P_{\alpha}$ with $\mathcal{L}$ and $\mathcal{L}^s$, for $0<s<1$. We give the details of their own proofs only in those cases that are useful for the sequel of the paper.

(i) The map $P_2$ is the fundamental solution of $\mathcal{L}$. Indeed for $x \in \G$
$$\mathcal{L} P_2(x)= \int_0^{\infty}  \mathcal{L}h(t,x) dt= -\int_0^{\infty} \frac{\partial}{\partial t}h(t,x) dt =- \lim_{t \to \infty} h(t,x)+h(0,x)=-\lim_{t \to \infty} h(t,x)=0, $$

where we exploited Theorem <ref>(iii),(vi) and the known estimates about the derivatives of the heat kernel (see for example the proof of <cit.>)

(ii) By <cit.>, for every $s \in (0,1)$ the kernel $P_{2s}$ is the fundamental solution of $\mathcal{L}^s$.

(iii)  By <cit.> and <cit.> we can deduce that for every $s \in (0,1)$ and $u \in C^{\infty}_0(\G)$ the equality
$$ \mathcal{L}^su= \mathcal{L} u \star P_{2-2s}$$

(iv) For every $s \in (0,1)$ the following relation holds outside the origin

$$ \mathcal{L} P_{2-2s}=P_{-2s}.$$

In fact, let $x \in \G$, $x \neq 0$

\begin{align*}
\mathcal{L}P_{2-2s}(x)&= \mathcal{L} \left( \frac{1}{\Gamma \left( 1-s \right)} \int_0^{\infty} t^{\frac{2-2s}{2}-1} h(t,x) dt \right)\\
& =  \frac{1}{-s \Gamma(-s)} \int_0^{\infty} t^{-s} \mathcal{L} h(t,x) dt \\
& = - \left( \frac{1}{-s \Gamma(-s)} \int_0^{\infty} t^{-s} \frac{\partial}{\partial t} h(t,x) dt \right)\\
&=  \frac{1}{-s \Gamma(-s)} \int_0^{\infty}(-s) t^{-s-1}  h(t,x) dt \\
&=   \frac{1}{ \Gamma(-s)} \int_0^{\infty} t^{-s-1}  h(t,x) dt =P_{-2s}(x),
\end{align*}

where we have considered as well the known estimates about the derivatives of the heat kernel (see <cit.>) and the integration by parts.

(v) Reading <cit.> in light of item (iv), for every $s \in (0,1)$ if $u \in \mathcal{S}(\G)$, then $\mathcal{L}^su \in L^2(\G)$ and

\begin{equation}\label{3.11} \mathcal{L}^su(x)= P.V. \int_{\G} (u(y)-u(x)) P_{-2s}(x^{-1}y) dy.
\end{equation}

We have taken into account the equivalence proved in <cit.>.

(vi)For every $ 0< \alpha, \beta<Q$ such that $\alpha+\beta<Q$ and for every $x \in \G$, $x \neq 0$ the following relation holds

\begin{equation}
\label{convolution}
P_{\alpha+ \beta}(x)= P_{\alpha} \star P_{\beta}(x).
\end{equation}

Let us show (<ref>):

\begin{align*}
P_{\alpha}\star P_{\beta}(x)&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_{\G} \int_0^{\infty} t^{\frac{\alpha}{2}-1}h(t, y)dt \int_0^{\infty} \tau^{\frac{\beta}{2}-1} h(\tau, y^{-1}x) d \tau  dy\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_{\G} \int_{(0, \infty)\times(0, \infty)} t^{\frac{\alpha}{2}-1}\tau^{\frac{\beta}{2}-1} h(t,y) h(\tau, y^{-1}x) dt d \tau  dy\\
&=\frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})}  \int_{(0, \infty)\times(0, \infty)} t^{\frac{\alpha}{2}-1}\tau^{\frac{\beta}{2}-1} \int_{\G} h(t, y) h(\tau, y^{-1}x) dy dt d \tau  \\
&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})}  \int_{(0, \infty)\times(0, \infty)} t^{\frac{\alpha}{2}-1}\tau^{\frac{\beta}{2}-1} h(t + \tau, x) dt d \tau ,
\end{align*}

where we used Fubini's theorem and Theorem <ref>(iv). Then, we consider the change of variables
$$ \begin{cases} t=s(1-u)\\
\tau=us, \end{cases}$$

\begin{align*}
P_{\alpha} \star P_{\beta}(x)&=\frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_{(0, \infty) \times (0,1)} (s(1-u))^{\frac{\alpha}{2}-1}(us)^{\frac{\beta}{2}-1} h(s, x) s  \ dsdu\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_0^{\infty} s^{\frac{\alpha}{2}-1}s^{\frac{\beta}{2}-1} s h(s, x)  ds \int_0^1(1-u)^{\frac{\alpha}{2}-1} u^{\frac{\beta}{2}-1} du\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_0^{\infty} s^{\frac{\alpha+ \beta}{2}-1} h(s, x)  ds \ B\left( \frac{\alpha}{2}, \frac{\beta}{2}\right),
\end{align*}

where $B(\cdot, \cdot )$ denotes the Beta Euler function. Exploiting the relation between the Beta and the Gamma Euler functions, we have

\begin{align*}
P_{\alpha} \star P_{\beta}(x)&= \frac{1}{\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})} \int_0^{\infty} s^{\frac{\alpha+ \beta}{2}-1} h(s, x)  ds \frac{\Gamma(\frac{\alpha}{2}) \Gamma(\frac{\beta}{2})}{\Gamma(\frac{\alpha+\beta}{2})}\\
&= \frac{1}{\Gamma(\frac{\alpha+\beta}{2})}\int_0^{\infty} s^{\frac{\alpha+ \beta}{2}-1} h(s, x)  ds= P_{\alpha+\beta}(x),
\end{align*}

and (<ref>) is proved.

§.§.§ Regularity

Here we deal with the regularity of the map $\alpha \to P_{\alpha}(x)$, for $ x \in \G \setminus \{0\}$.

For every $x \in \G, \ x \neq 0$, the map

\begin{align*}
(-\infty, Q) \ni \alpha \to \| x \|_{\alpha}^{\alpha-Q}&= \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt
\end{align*}
is analytic.

We fix $x \in \G$, $x \neq 0$, $t>0$ and $k \in \N$ and we compute the $k$-th derivative of the map $\alpha \to t^{\frac{\alpha}{2}-1} h(t,x)$ that is

$$ \frac{\partial^k}{\partial \alpha^k} \Big( t^{\frac{\alpha}{2}-1} h(t,x) \Big)= t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x).$$

We notice that from the estimates in Item <ref> of Section <ref> we have
$$\int_0^{\infty}t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt < \infty,$$
for every $\alpha<Q$, then we can deduce that for every $\alpha<Q$ we have

$$ \frac{\partial^k}{\partial \alpha^k} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x)dt=  \int_0^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt.$$

Now, we claim that for every closed interval $[a,b] \subset (-\infty, Q)$ there are two positive constants $C$ and $M$ such that

\begin{equation}
\label{claim}
\left| \frac{\partial^k}{\partial \alpha^k} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x)dt  \right| \leq C M^{k+1} k!,
\end{equation}

for every $\alpha \in [a,b]$.
This would conclude the proof. Hence, we need to prove (<ref>). First, we consider

\begin{equation}
\label{stima1}
\left| \frac{\partial^k}{\partial \alpha^k} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x)dt  \right| \leq \left| \int_0^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt \right| + \left| \int_1^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt \right|.
\end{equation}

By combining the homogeneity of $h$ and Theorem <ref>(vii), which ensures that $h(1, \cdot ) \in \mathcal{S}(\G)$, we can estimate the second term of the right-hand side of (<ref>) getting

\begin{align*}
\left|\int_1^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt \right| &=\int_1^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt\\ &= \int_1^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) h(1,\delta_{\frac{1}{\sqrt{t}}}(x))dt\\
& \leq \|h(1, \cdot) \|_{L^{\infty}(\G)}\int_1^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t)dt.
\end{align*}

Now, for $k \in \N$ we set

\begin{align}
\label{ak}
a_k:&= \int_1^{\infty} t^{\frac{\alpha-Q}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) dt,
\end{align}

and we notice that

\begin{align*}
a_k&=\lim_{a \to \infty} \left[  \frac{2}{\alpha-Q} t^{\frac{\alpha-Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \right]_{t=1}^{a}+ \int_{1}^{\infty} \frac{2}{Q-\alpha} t^{\frac{\alpha}{2}-1-\frac{Q}{2}}  \left( \frac{1}{2} \right)^k k \ln^{k-1}(t)dt\\
&= \frac{1}{Q-\alpha} k a_{k-1},  \notag
\end{align*}

where we have exploited the integration by parts.
Then, recursively we get that

\begin{equation}
\label{ric1}
a_k=k \frac{1}{Q-\alpha} a_{k-1}= k(k-1)\frac{1}{(Q-\alpha)^2} a_{k-2} = \ldots = k! \frac{1}{(Q-\alpha)^{k}}a_0=k! \frac{2}{(Q-\alpha)^{k+1}}.
\end{equation}

Let us now focus on the first contribute of the right-hand side of (<ref>).
We exploit the fact that $h(t,x)=o(t^N)$ as $t \to 0$ for every $x \neq 0$ and for every $N \in \N$ (hence for every $N \in \N$, there is $c>0$ such that $h(t,x)\leq ct^N$ as $t<1$) and we can state that

$$\left| \int_0^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x)dt \right| \leq c \left| \int_0^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k \ln^k(t) dt  \right| \leq c \int_0^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt. $$

Let us now set for $k \in \N$ and $N \in \N$ the term

\begin{align}
\label{bk}
b_k:&= \int_0^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt.
\end{align}

We have that

\begin{align*}
b_k&=\lim_{a \to 0} \left[ t^{\frac{\alpha}{2}+N} \frac{2}{\alpha+2N} \left( \frac{1}{2} \right)^k |\ln^k (t)| \right]_{t=a}^1 - \int_0^1 \frac{1}{\alpha+2N}  t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^{k-1} k |\ln (t)|^{k-1} \sgn(\ln(t))dt. \notag
\end{align*}

If $N>-\frac{\alpha}{2}$, we can go on getting

\begin{equation}
\label{ric2}
\begin{split}
b_k &=  \int_0^1 \frac{1}{\alpha+2N}  t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^{k-1} k |\ln (t)|^{k-1} dt\\
&=\frac{1}{\alpha+2N} k b_{k-1}= \frac{1}{(\alpha+2N)^2}k(k-1)b_{k-2}= \ldots = \frac{1}{(\alpha+2N)^k} k! b_0=\frac{2}{(\alpha+2N)^{k+1}} k!. \\
\end{split}
\end{equation}

By combining (<ref>), (<ref>) and (<ref>), we get that

\begin{equation}
\label{stima-ana}
\begin{split}
\left| \frac{\partial^k}{\partial \alpha^k} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x)dt  \right| \leq  k! \| h(1, \cdot)\|_{L^{\infty}(\G)}\frac{2}{(Q-\alpha)^{k+1}} +  \frac{2c}{(\alpha+2N)^{k+1}} k!\\
=2(1+c)(1+\|h(1, \cdot)\|_{L^{\infty}(\G)}) k! \max \left\{ \frac{1}{Q-\alpha}, \frac{1}{\alpha+2N} \right\}^{k+1}.
\end{split}
\end{equation}

Since $N$ can be chosen arbitrarily large (once fixed $\alpha$), our claim (<ref>) is proved and this completes the proof.

As a consequence of Proposition <ref>, the map $\alpha \to P_{\alpha}(x)$ is analytic for $\alpha<Q$, for every $x \in \G, \ x \neq 0$, since it is a product of analytic functions.

Now we are in position to introduce the following map

\begin{equation}\label{sigmaalfa}
(-\infty,Q) \ni \alpha \to \sigma(\alpha):= \int_{\partial \B_c(0,1)} \| z \|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z).
\end{equation}

§.§ Main properties of the map $\sigma(\alpha)$

In this subsection we collect and prove some properties of the map $\sigma$ defined in (<ref>).

* The map $\sigma(\alpha)$ is well defined for every $\alpha<Q$. In fact, if we exploit the estimates in Item <ref> of Section <ref>, we have

\begin{align*}
\sigma(\alpha)&= \int_{\partial \B_c(0,1)} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt d \mathcal{S}^{Q-1}_{\infty}(x)\\
&=  \int_{\partial \B_c(0,1)} \int_0^1  t^{\frac{\alpha}{2}-1} h(t,x) dt d \mathcal{S}^{Q-1}_{\infty}(x) +   \int_{\partial \B_c(0,1)} \int_1^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt d \mathcal{S}^{Q-1}_{\infty}(x)  \\
& \leq c_1  \int_{\partial \B_c(0,1)} \int_0^1 t^{\frac{\alpha}{2}-1} t^Nd \mathcal{S}^{Q-1}_{\infty}(x) dt + c_2 \int_{\partial \B_c(0,1)} \int_1^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}}  dt d \mathcal{S}^{Q-1}_{\infty}(x) \\
&= \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \left( c_1\int_0^1   t^{\frac{\alpha}{2}-1+N} dt + c_2 \int_1^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} dt\right),
\end{align*}

for some positive constants $c_1$ and $c_2$ for every arbitrarily fixed $N \in \N$. Therefore, $\sigma(\alpha)$ converges if $\alpha <Q$.

*  We show that $\alpha \to \sigma(\alpha)$ is analytic if $\alpha<Q$. For every $k \in \N$ we have

\begin{equation}
\label{der-c-alfa}
\begin{split}
\left| \frac{\partial}{\partial \alpha^k} \sigma(\alpha) \right|&=\left| \frac{\partial^k}{\partial \alpha^k} \int_{\partial \B_c(0,1)} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x) dt d\mathcal{S}^{Q-1}_{\infty}(x)\right| \\
&=\left| \int_{\partial \B_c(0,1)} \int_0^{\infty}\frac{\partial^k}{\partial \alpha^k} \left( t^{\frac{\alpha}{2}-1} h(t,x) \right) dt d\mathcal{S}^{Q-1}_{\infty}(x)\right|\\
&=\left| \int_{\partial \B_c(0,1)} \int_0^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2}\right)^k \ln^k(t) h(t,x)  dt d\mathcal{S}^{Q-1}_{\infty}(x)\right|,
\end{split}
\end{equation}

since for every $\alpha<Q$ and $k=0, 1, \dots$ we have that

\begin{align}\label{servez}
&\int_{\partial \B_c(0,1)} \int_0^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2}\right)^k \ln^k(t) h(t,x)  dt d\mathcal{S}^{Q-1}_{\infty}(x) \leq \notag \\
& \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \int_0^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2}\right)^k \ln^k(t) h(t,x_M)  dt < \infty,
\end{align}

where $x_M \in \partial \B_c(0,1)$ is such that $h(t,x_M)=\max_{\partial \B_c(0,1)} \left( \int_0^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2}\right)^k \ln^k(t) h(t,x)  dt\right)$. By combining (<ref>), (<ref>) and (<ref>), we get that

\begin{align*}
\left| \frac{\partial^k}{\partial \alpha^k} \sigma(\alpha) \right| &\leq \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \left| \frac{\partial^k}{\partial \alpha^k} \int_0^{\infty} t^{\frac{\alpha}{2}-1} h(t,x_M)dt  \right| \\
&= \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1))  2(1+c)(1+\|h(1, \cdot)\|_{L^{\infty}(\G)}) k! \max \left\{ \frac{1}{Q-\alpha}, \frac{1}{\alpha+2N} \right\}^{k+1},
\end{align*}

so that $\sigma(\alpha)$ is analytic with respect to $\alpha$ for $\alpha<Q$.

*  Let us prove that

$$\lim_{\alpha \to 0} \sigma(\alpha)=2,$$

hence that

$$ \lim_{\alpha \to 0} \int_{\partial \B_c(0,1)} \| z \|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z) =2.$$

Let us start by considering Theorem <ref>(iii). By exploiting the coarea formula and the homogeneity of $\mathcal{S}^{Q-1}_{\infty}$, for every $t>0$ we get

\begin{equation}
\label{fa1}
\begin{split}
1=\int_\G h(t,x) dx&= \int_0^{\infty} \int_{\partial \B_c(0,r)} h(t,x) d \mathcal{S}^{Q-1}_{\infty}(x) dt\\
&=\int_0^{\infty}r^{Q-1} \int_{\partial \B_c(0,1)} h(t, \delta_r(z)) d \mathcal{S}^{Q-1}_{\infty}(z) dr.
\end{split}
\end{equation}

Now, let us consider

\begin{align*}
\sigma(\alpha) &= \int_{\partial \B_c(0,1) } \| z \|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z)\\
&= \int_{\partial \B_c(0,1)} \int_0^{\infty} t^{\frac{\alpha}{2}-1}h(t,z) dt d \mathcal{S}^{Q-1}_{\infty}(z)\\
&= \int_0^{\infty}t^{\frac{\alpha}{2}-1} \int_{\partial \B_c(0,1)} h(t,z) d \mathcal{S}^{Q-1}_{\infty}(z) dt\\
&= \int_0^{\infty}t^{\frac{\alpha}{2}-1} \int_{\partial \B_c(0,1)} (\sqrt{t})^{-Q} h(1,\delta_{\frac{1}{\sqrt{t}}}(z)) d \mathcal{S}^{Q-1}_{\infty}(z) dt\\
&= \int_0^{\infty}t^{\frac{\alpha}{2}-1- \frac{Q}{2}} \int_{\partial \B_c(0,1)} h(1,\delta_{\frac{1}{\sqrt{t}}}(z)) d \mathcal{S}^{Q-1}_{\infty}(z) dt,
\end{align*}

where we used the homogeneity of $h$. Let us change variable setting $r= \frac{1}{\sqrt{t}}$, hence $t=r^{-2}$ ($dt=-2r^{-3}$) and we get

\begin{align*}
\sigma(\alpha) &= -2 \int_{\infty}^{0}r^{-\alpha+2+Q} \int_{\partial \B_c(0,1)} h(1,\delta_{r}(z)) d \mathcal{S}^{Q-1}_{\infty}(z) r^{-3} dr\\
&= 2 \int_{0}^{\infty}r^{-\alpha-1+Q} \int_{\partial \B_c(0,1)} h(1,\delta_{r}(z)) d \mathcal{S}^{Q-1}_{\infty}(z) dr,
\end{align*}

which converges to

\begin{align*}
2 \int_{0}^{\infty}r^{Q-1} \int_{\partial \B_c(0,1)} h(1,\delta_{r}(z)) d \mathcal{S}^{Q-1}_{\infty}(z) dr,
\end{align*}

as $\alpha \to 0$. Hence the proof of Item <ref> is now complete taking into account (<ref>).

§ ANALYTIC CONTINUATION IN $\G$

Let $\phi \in \mathcal{S}(\G)$ and $x \in \G$.
For every $r>0$ and $0<\alpha<Q$ we denote the closed ball
$$\B_{\alpha}(x,r):= \{ y \in \G : \| x^{-1} y \|_{\alpha} \leq r \}.$$

For every $ \alpha>0 $ we introduce the map

\begin{equation}\label{psi}\psi(x,\alpha)= \int_{\G} \phi(xy)P_{\alpha}(y) dy= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_\G \phi(xy) \|y \|_{\alpha}^{\alpha-Q} dy.
\end{equation}

For every $x \in \G$ the map $(0, Q) \ni \alpha \to \psi(x, \alpha)$ is well defined.

By exploiting the estimate in (<ref>), the fact that $\phi$ is in the Schwartz class $ \mathcal{S}(\G)$ on $\G$ and the coarea formula of Corollary <ref>, we have

\begin{equation}
\label{converge}
\begin{split}
\int_{\G} \phi(xy) P_{\alpha}(y) dy & \leq  \int_{\G} |\phi(xy)| |P_{\alpha}(y) |dy = \frac{1}{  \Gamma(\frac{\alpha}{2})} \int_{\G} |\phi(z)| \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz \\
&= \frac{1}{ \Gamma(\frac{\alpha}{2})} \int_{\B_{\alpha}(x,1)} |\phi(z)| \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz  +  \frac{1}{ \Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_{\alpha}(x,1)} |\phi(z)| \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz\\
& \leq  \frac{1}{ \Gamma(\frac{\alpha}{2})} \|\phi \|_{L^{\infty}(\G)} \int_{\B_c(x,c_{\alpha})} \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz  +  \frac{1}{ \Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_{\alpha}(x,1)}|\phi(z)| \frac{(1+\| z \|_c^k)}{(1+\| z \|_c^k)}  dz,
\end{split}
\end{equation}

thus, taking into account the homogeneity of the norm $\| \cdot \|_{\alpha}$ and of the measure $\mathcal{S}^{Q-1}_{\infty}$, and the definition of $\sigma(\alpha)$ we can continue from (<ref>) getting

\begin{align*}
\int_{\G} \phi(xy) P_{\alpha}(y) dy &\leq  \frac{1}{\Gamma(\frac{\alpha}{2})} \sigma ( \alpha) \|\phi\|_{L^{\infty}(\G)} \int_0^{c_{\alpha}} r^{\alpha-1} dr   +  \frac{1}{\Gamma(\frac{\alpha}{2})} c_k \int_{\G } \frac{1}{(1+\| z \|_c^k)} dz < \infty \\
&\leq  \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{c_{\alpha}}{\alpha} \sigma ( \alpha) \|\phi  \|_{L^{\infty}(\G)}+  \frac{1}{\Gamma(\frac{\alpha}{2})} c_k \int_{\G } \frac{1}{(1+\| z \|_c^k)} dz,
\end{align*}

where, choosing $k>Q$, the last integral converges.

The map $\alpha \to \psi(x,\alpha)$ is well defined also if we weaken the hypothesis on $\phi$.

If $\phi \in C^{\infty}(\G)$ and $|\phi(z)| \leq K\|z\|_c^{-L}$ as $\|z\|_c \geq S$ for some $L, K, S>0$ then the map
$$(0,\min \{L,Q \}) \ni \alpha \to \psi(x, \alpha)$$
is well defined.

Keeping in mind that $x$ is fixed, we consider $R(x):= \max \{1+ \|x\|_c, S \}$ and we set $\widetilde{R}(x)>0$ such that $\B_c(0,R(x)) \subset \B_c(x,\widetilde{R}(x))$. For $\alpha>0$, by exploiting the hypotheses on $\phi$ and the coarea formula in Corollary <ref>, we get
\begin{equation}
\label{continue4}
\begin{split}
\int_{\G} \phi(xy) P_{\alpha}(y) dy &\leq  \int_{\G}| \phi(xy) P_{\alpha}(y)| dy= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G}| \phi(z)| \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz \\
= \frac{1}{\Gamma(\frac{\alpha}{2})} &\int_{\B_{c}(0,R(x))}| \phi(z) |\| x^{-1} z \|_{\alpha}^{\alpha-Q} dz  +  \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(0,R(x))} |\phi(z)| \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz\\
\leq  \frac{1}{\Gamma(\frac{\alpha}{2})} &  \|\phi\|_{L^{\infty}(\G)} \int_{\B_c(0,R(x))} \| x^{-1} z \|_{\alpha}^{\alpha-Q} dz  +  K\frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(0,R(x))}\frac{1}{\|z\|_c^{L}}  \| x^{-1} z \|_{\alpha}^{\alpha-Q}  dz.
\end{split}
\end{equation}

Thus, taking into account the homogeneity of the norm $\| \cdot \|_{\alpha}$ and of the measure $\mathcal{S}^{Q-1}_{\infty}$, and the definition of $\sigma(\alpha)$, we can continue from (<ref>) getting

\begin{align}\label{serve2}
\int_{\G} \phi(xy) P_{\alpha}(y) dy &\leq  \frac{1}{\Gamma(\frac{\alpha}{2})} \sigma(\alpha)  \|\phi\|_{L^{\infty}(\G)} \int_0^{\widetilde{R}(x)} r^{\alpha-1} dr   + K \frac{1}{ \Gamma(\frac{\alpha}{2})}  \int_{\G \setminus \B_c(0,R(x))}\frac{1}{\|z\|_c^{L+Q-\alpha}}  \frac{\|z\|_c^{Q-\alpha}}{\| x^{-1} z \|_{\alpha}^{Q-\alpha}}   dz  \notag \\
&\leq  \frac{1}{\Gamma(\frac{\alpha}{2})} \sigma(\alpha)  \|\phi\|_{L^{\infty}(\G)} \int_0^{\widetilde{R}(x)} r^{\alpha-1} dr   +  K\frac{1}{\Gamma(\frac{\alpha}{2})} (c_{\alpha}(\|x\|_c+1))^{Q-\alpha}\int_{\G \setminus \B_c(0,R(x))}\frac{1}{\|z\|_c^{L+Q-\alpha}}  dz,
\end{align}

since for every $z \notin \B_c(0,R(x))$, combining the definition of $R(x)$, the estimate (<ref>) and some properties of homogeneous norms we have

\begin{align}\label{serve3}
& \frac{\| x^{-1}z \|_{\alpha}}{\| z \|_c} \geq \frac{1}{c_{\alpha}} \frac{\| x^{-1}z \|_c}{\| z \|_c} = \frac{1}{c_{\alpha}}  \| \delta_{\frac{1}{\| z \|_c}}(x)^{-1}\delta_{\frac{1}{\| z \|_c}}(z) \|_c \notag \\
& \geq \frac{1}{c_{\alpha}} | \| \delta_{\frac{1}{\| z \|_c}}(x)\|_c - \| \delta_{\frac{1}{\| z \|_c}}(z)  \|_c |\\
& \geq \frac{1}{c_{\alpha}}\bigg(1-\frac{\|x\|_c}{\|x\|_c+1} \bigg) =\frac{1}{c_{\alpha}}\frac{1}{\|x\|_c+1}. \notag
\end{align}

Therefore, we can continue from (<ref>) getting

\begin{align*}
\int_{\G} \phi(xy) P_{\alpha}(y) dy  \leq \  & \frac{1}{\Gamma(\frac{\alpha}{2})}  \sigma(\alpha)  \|\phi\|_{L^{\infty}(\G)} \int_0^{\widetilde{R}(x)} r^{\alpha-1} dr\\
&   + K \frac{1}{\Gamma(\frac{\alpha}{2})} (c_{\alpha}(\|x\|_c+1))^{Q-\alpha}\mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \int_{R(x)}^{\infty} r^{\alpha-L-1} dr.
\end{align*}

The last term of previous equation converges for every $\alpha>0$ such that $L+1-\alpha>1$, hence when $\alpha<L$. This concludes the proof.

From now on, in the current section we show that the map $\alpha \to \psi(x, \alpha)$ is analytic for $0 < \alpha <Q$ and we construct an analytic continuation of $\alpha \to \psi(x, \alpha)$ on $-2<\alpha\leq 0 $. This construction allows to introduce a consistent suitable definition of negative powers of the sub-Laplacian $\mathcal{L}$. We mainly follow the structure of <cit.>, where an analogous construction is carried out in Euclidean spaces.

§.§ The map $\alpha \to \psi(x, \alpha)$ is analytic on $(0,Q)$

In this subsection, we show that the map $\alpha \to \psi(x, \alpha)$ is analytic on $(0,Q)$. First, we consider the following remark.

For every $x \neq 0$, we know that $h(t,x)=o(t^N)$ for every $N \in \N$. This property is uniform for $x \in \G \setminus \B(0,r)$, for every $r>0$. In fact, by Theorem <ref>(vi), there exists a constant $c>0$ such that for every $x \in \G$ and $t>0$

$$ c^{-1} t^{-Q/2} \exp\left(-\frac{ \| x \|^2}{c^{-1}t}\right) \leq h(t,x) \leq c t^{-Q/2} \exp\left(-\frac{\| x \|^2}{ct}\right),$$

hence for every $y \in \G \setminus \B_c(0,r)$ and $t>0$ the estimate

$$h(t,y) \leq c t^{-Q/2} e^{-\frac{r}{ct}}$$

holds, so that for every $N \in \N$

$$t^{-N} h(t,y) \leq c t^{-Q/2-N} e^{-\frac{r}{ct}} \to 0$$

as $t \to 0$ independently of $y$. We are ready to treat the regularity of $\psi(x,\alpha)$.

For every $\phi \in \mathcal{S}(\G)$ and $x \in \G$ the function

$$ \alpha \to  \psi(x, \alpha)=\int_{\G} \phi(xy) P_{\alpha}(y) dy$$
is analytic on $(0,Q).$

Since the map $\alpha \to \frac{1}{\Gamma(\frac{\alpha}{2})}$ is analytic on $\R$ and analytic functions well behave with respect to the product, it is enough to prove that

\begin{equation*}
\alpha \to \int_{\G} \phi(xy) \| y \|_{\alpha}^{\alpha-Q}dy= \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} h(t,y)dt dy
\end{equation*}

is analytic on $(0,Q).$ Let us consider that for every $k \in \N$ for every $t>0$ and $x \in \G$

$$ \frac{\partial^k}{\partial \alpha^k} t^{\frac{\alpha}{2}-1} h(t,x)= t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,x).$$

We obtain the needed estimate on the $k$-th derivative of the map $\psi(x, \alpha)$, showing at the same time that it is summable on $ (0, \infty) \times \G $

\begin{align}
\label{zeropezzo}
\left| \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy \right| & \leq  \int_{\G} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \notag \\
=  \int_{\G \setminus \B_c(0,1)} |\phi(xy)| & \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \\
+   \int_{\B_c(0,1)} |\phi(xy)| &  \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy . \notag
\end{align}

Let us split the second term of (<ref>) as follows

\begin{align}
\label{zeropezzo1}
&\int_{\B_c(0,1)} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \notag \\
=&  \int_{\B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \\
&+  \int_{\B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y)dt dy. \notag
\end{align}

Let us then consider the first term of the sum in (<ref>), getting the estimate

\begin{align}
\label{primopezzo}
& \int_{\B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dy  \notag \\
\leq & \sup_{\B_c(0,1)} | \phi(x \cdot) | \int_0^1  t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\B(0,1)} h(t,y) dy dt  \notag \\
\leq & \sup_{\B_c(0,1)} | \phi(x \cdot) | \int_0^1  t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G} h(t,y) dy dt\\
\leq & \sup_{\B_c(0,1)}| \phi(x \cdot) | \int_0^1  t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt.  \notag
\end{align}

Now, consider that $\int_0^1  t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt$ equals the $b_k$ defined in (<ref>) for $N=0>- \frac{\alpha}{2}$, hence we can continue (<ref>) exploiting the estimate in (<ref>) and we get that

\begin{equation}
\label{first}
\begin{split}
\int_{\B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dy \leq \sup_{\B_c(0,1)} | \phi(x \cdot) |  \ \frac{2}{\alpha^{k+1}} k!.
\end{split}
\end{equation}

Let us bound the second term of the sum in (<ref>)

\begin{align}
\label{secondopezzo}
&\int_{\B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y)dt dy \notag \\
& = \int_{\B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) h(1,\delta_{\frac{1}{\sqrt{t}}}(y))dt dy  \\
& \leq \sup_{\B_c(0,1)}| \phi(x \cdot) | \sup_{\B_c(0,1)} h(1, \cdot) \mu(\B_c(0,1))  \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) dt. \notag
\end{align}

Now, consider that $\int_1^\infty  t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) dt$ equals the $a_k$ introduced in (<ref>), hence we can continue from (<ref>) exploiting the estimate in (<ref>) and we get that

\begin{equation}
\label{second}
\begin{split}
&\int_{\B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y)dt dy \\
&\leq \sup_{\B_c(0,1)}| \phi(x \cdot) | \sup_{\B_c(0,1)} h(1, \cdot) \mu(\B_c(0,1)) \frac{2}{(Q-\alpha)^{k+1}}   k!.
\end{split}
\end{equation}

Let us split the first term of the sum in (<ref>) as follows

\begin{align}
\label{zeropezzo2}
&\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
=&\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \\
&+ \int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy.  \notag
\end{align}

Keeping in mind Remark <ref>, we can estimate the first term of the sum in (<ref>) as follows for $N \in \N$

\begin{align}
\label{terzopezzo}
&\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
\leq \ & c  \int_{0}^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(0,1)} | \phi(xy)| dy dt  \notag \\
\leq \ & c \int_{0}^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G } |\phi(z)| \frac{(1+\|z\|_c^\ell)}{(1+\|z\|_c^\ell)} dz dt \\
\leq \ & c c_{\ell} \int_{\G} \frac{1}{1+\|z\|_c^\ell} dz  \int_{0}^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt   \notag \\
\leq \ & c \widetilde{c}_{\ell} \int_{0}^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k dt,  \notag
\end{align}

where $c$, $c_{\ell}$ and $\widetilde{c}_{\ell}$ are suitable positive constants and $\ell$ is understood such that $\ell>Q$.
Analogously to (<ref>), when $N> -\frac{\alpha}{2}$ we can bound (<ref>) as follows

\begin{equation}
\label{third}
\begin{split}
&\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1+N} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \leq \tilde{c}_\ell  \frac{2}{(\alpha+2N)^{k+1}} k!.
\end{split}
\end{equation}

Let us finally consider the second term of the sum in (<ref>)

\begin{align}
\label{quartopezzo}
&\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln(t)^k h(t,y) dt dy  \notag \\
&= \int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln(t)^k h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dt dy  \notag \\
&=   \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln(t)^k \int_{\G \setminus \B_c(0,1)} |\phi(xy)|h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dy dt  \notag \\
&\leq  \|h(1,\cdot)\|_{L^{\infty}(\G)} \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) dt \int_{\G } |\phi(h)| dh  \\
& =  \|h(1,\cdot)\|_{L^{\infty}(\G)} \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) dt \int_{\G } |\phi(h)| \frac{(1+ \|h \|_c^\ell)}{(1+ \|h \|_c^\ell)}dh  \notag \\
& \leq  c_\ell  \|h(1,\cdot)\|_{L^{\infty}(\G)} \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G } \frac{1}{(1+ \|h \|_c^\ell)}dh  dt   \notag \\
& \leq  \widetilde{c}_\ell  \|h(1,\cdot)\|_{L^{\infty}(\G)} \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t)  dt,  \notag
\end{align}

where $c_{\ell}$ and $\widetilde{c}_\ell$ are suitable positive constants depending on $\ell$, and $\ell$ is understood to be strictly larger than $Q$. Now, analogously to the reasoning used in (<ref>), exploiting the estimate in (<ref>), we get

\begin{equation}
\label{fourth}
\int_{\G \setminus \B_c(0,1)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \leq c_\ell  \|h(1,\cdot)\|_{L^{\infty}(\G)} \frac{2}{(Q-\alpha)^{k+1}}   k!.
\end{equation}

Now, by combining (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we get that for every $k \in \N$

\begin{align}
\label{conclusione}
&\left| \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy \right|  \notag \\
\leq & \sup_{\G} | \phi(x \cdot) |  \ \frac{2}{\alpha^{k+1}} k! + \sup_{\G}| \phi(x \cdot) | \sup_{\B_c(0,1)} h(1, \cdot) \mu(\B_c(0,1)) \frac{2}{(Q-\alpha)^{k+1}}   k!  \notag \\
& + \widetilde{c}_\ell  \frac{2}{(\alpha+2N)^{k+1}} k!+ \widetilde{c}_\ell   \|h(1,\cdot)\|_{L^{\infty}(\G)} \frac{2}{(Q-\alpha)^{k+1}}   k!\\
\leq & C(\phi, \ell)   \max \left\{\frac{1}{\alpha+2N}, \frac{1}{Q-\alpha}, \frac{1}{\alpha} \right\}^{k+1} k!,  \notag
\end{align}

where $c(\phi,\ell)$ is a suitable positive constant.
Thus, $\psi(x, \alpha)$ is analytic on $0 < \alpha < Q$ with respect to $\alpha$.

Following the scheme of the previous proof, exploiting the estimates in (<ref>) and (<ref>), it holds that the map
$$\alpha \to \int_{\G \setminus \B_c(0,1)} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dt$$
is analytic for $\alpha<Q$ (where it surely converges by direct computations similar to the ones in Proposition <ref>).

§.§ Further properties of the map $\alpha \to \psi(x,\alpha)$

Here we generalize Proposition <ref>. We assume that $\phi \in C^{\infty}(\G)$ and $|\phi(z)| \leq K \|z\|_c^{-L}$ when $\|z\|_c \geq S$ for some $L, S,K>0$.

For any $\phi \in C^{\infty}(\G)$ such that $|\phi(z)| \leq K \|z\|_c^{-L}$ when $\|z\|_c \geq S$ for some $L, S, K>0$, for every $x \in \G$ the map

$$ \alpha \to \int_{\G} \phi(xy) P_{\alpha}(y) dy$$
is analytic on $(0,\min\{L,Q\})$.

It is enough to prove that

\begin{equation*}
\alpha \to \int_{\G} \phi(xy) \| y \|_{\alpha}^{\alpha-Q}dy= \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} h(t,y)dt dy
\end{equation*}

is analytic on $(0,\min\{L,Q\})$.
Let $x \in \G$ and set $R(x)= \max \{ 1+\|x\|_c, S \}$. For $k \in \N$ we consider

\begin{align}
\label{zeropezzo23}
&\left| \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy \right|  \notag \\
\leq & \int_{\G} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
= & \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \\
&+   \int_{\B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy.  \notag
\end{align}

By computations analogous to the ones  used in the previous Proposition <ref>, one verifies that

\begin{align}
\label{zeropezzo3}
& \int_{\B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
= & \int_{\B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
&+  \int_{\B_c(x^{-1},R(x))} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y)dt dy\\
\leq & \sup_{\B_c(x^{-1},R(x))} | \phi(x \cdot) | \sup_{\B_c(x^{-1},R(x))}|h(1,\cdot)|  2 \max \left\{\frac{1}{\alpha}, \frac{1}{Q-\alpha} \right\}^{k+1}k!.  \notag
\end{align}

Let us split the first term of the sum in (<ref>)

\begin{align}
\label{zeropezzo21}
&\int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
=&\int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{0}^{1} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy \\
&+ \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy.  \notag
\end{align}

Let us consider the first term of the sum in (<ref>) as follows

\begin{align}
\label{terzopezzo1}
&\int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \notag \\
&= \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dt dy \\
&=   \int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)|h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dy dt . \notag
\end{align}

Let us change the variables as $z= \delta_{\frac{1}{\sqrt{t}}}(y)$, so that, since the Lebesgue measure on $\G$ is $Q$-homogeneous, we get the estimate

\begin{align*}
&\int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)|h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dy dt \\
&=  \int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} |\phi(x\delta_{\sqrt{t}}(z))|h(1,z) dz  \ t^{\frac{Q}{2}} dt \\
&=  \int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} |\phi(x\delta_{\sqrt{t}}(z))|h(1,z) dz   dt \\
& \leq K \int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L} h(1,z) \frac{\|z\|_c^\ell}{\|z\|_c^{\ell}}dz  dt \\
&  \leq K c_\ell \int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L}  \frac{1}{\|z\|_c^{\ell}}dz  dt,
\end{align*}

where $c_{\ell}$ is a suitable positive constant obtained exploiting the fact that the map $h(1, \cdot) \in \mathcal{S}(\G)$, for an arbitrary $\ell \in \N$. Let us change again variables, setting $h= x\delta_{\sqrt{t}}(z)$ and considering that the Lebesgue measure is left-invariant and $Q$-homogeneous, so that we get the estimate

\begin{align}\label{serve4}
& K c_\ell \int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L}  \frac{1}{\|z\|_c^{\ell}}dz  dt \notag \\
&  \leq K c_\ell \int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{1}{\|\delta_{\frac{1}{\sqrt{t}}}(x^{-1}h)\|_c^{\ell}}dh  dt \notag \\
&  \leq K c_\ell \int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{t^{\frac{\ell}{2}}}{\|x^{-1}h\|_c^{\ell}}dh  dt \\
&  \leq K c_\ell\int_0^1 t^{\frac{\alpha}{2}-1-\frac{Q}{2}+\frac{\ell}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{\|h\|^\ell}{\|x^{-1}h\|_c^{\ell}} \frac{1}{\|h\|_c^{\ell} }dh  dt \notag \\
&  \leq K c_\ell \int_0^1t^{\frac{\alpha}{2}-1-\frac{Q}{2}+\frac{\ell}{2}} \left( \frac{1}{2} \right)^k |\ln(t)|^k (1+\|x\|_c)^{\ell}\int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^{L+\ell}}  dh  dt. \notag
\end{align}

We have used an argument analogous to the one used in (<ref>).
The integral in the last line of (<ref>) converges for every $\ell>Q-L$. If we choose some $\ell>Q-L$ large enough that also $\ell-Q > -\alpha$ holds, then we can choose also $\ell>Q$ and we can introduce $N=[\frac{\ell-Q}{2}]+1$ and we can repeat the computations leading to (<ref>) getting

\begin{equation}
\label{third1}
\int_{\G \setminus \B_c(x^{-1},1)} |\phi(xy)|\int_0^1 t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k h(t,y) dt dy  \leq K \widetilde{c}_\ell \frac{2}{(\alpha+(\ell-Q))^{k+1}} k! .
\end{equation}

Let us then consider the second term of the sum in (<ref>)

\begin{align}
\label{quartopezzo1}
&\int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy  \notag \\
&= \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dt dy \\
&=   \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)|h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dy dt .  \notag
\end{align}

Let us change the variables as $z= \delta_{\frac{1}{\sqrt{t}}}(y)$, so that, since the Lebesgue measure is $Q$-homogeneous we get

\begin{align*}
& \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \B_c(x^{-1},R(x))} |\phi(xy)|h(1,\delta_{\frac{1}{\sqrt{t}}}(y)) dy dt \\
&= \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} |\phi(x\delta_{\sqrt{t}}(z))|h(1,z) dz  \ t^{\frac{Q}{2}} dt \\
&= \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} |\phi(x\delta_{\sqrt{t}}(z))|h(1,z) dz   dt \\
&  \leq K \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L} h(1,z) \frac{\|z\|_c^\ell}{\|z\|^{\ell}}dz  dt \\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L}  \frac{1}{\|z\|^{\ell}}dz  dt,
\end{align*}

where $c_{\ell}$ denotes a suitable positive constant depending on $\ell$, obtained considering that $h(1, \cdot) \in \mathcal{S}(\G)$.
Let us change again variables, setting $h= x\delta_{\sqrt{t}}(z)$ and recalling that the Lebesgue measure on $\G$ is left-invariant and $Q$-homogeneous, so that we get

\begin{align*}
& K  c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k |\ln(t)|^k \int_{\G \setminus \delta_{\frac{1}{\sqrt{t}}}(\B_c(x^{-1},R(x)))} \frac{1}{\|x\delta_{\sqrt{t}}(z)\|_c^L}  \frac{1}{\|z\|^{\ell}}dz  dt \\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{1}{\|\delta_{\frac{1}{\sqrt{t}}}(x^{-1}h)\|^{\ell}}dz  dt \\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{t^{\frac{\ell}{2}}}{\|x^{-1}h\|^{\ell}}dz  dt \\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}+\frac{\ell}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) dt\int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^L}  \frac{\|h\|^\ell}{\|x^{-1}h\|^{\ell}} \frac{1}{\|h\|^{\ell} }dz   \\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{Q}{2}+\frac{\ell}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) (1+\|x\|_c)^{\ell} dt \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^{L+\ell}}  dz\\
&  \leq K c_\ell \int_{1}^{\infty} t^{\frac{\alpha}{2}-1-\frac{L}{2}} \left( \frac{1}{2} \right)^k \ln^k(t) (1+\|x\|_c)^{\ell} dt \int_{\G \setminus \B_c(0,R(x))} \frac{1}{\|h \|_c^{L+\ell}}  dz.
\end{align*}

The integral in the last line is justified if we consider $\ell>Q-L$, hence when $L>Q-\ell$. Now, similarly to the procedure applied to deal with (<ref>), exploiting an estimate analogous to (<ref>), we can conclude that

\begin{equation}
\label{fourth1}
\begin{split}
\int_{\G \setminus \B_c(x^{-1},R)} |\phi(xy)| \int_{1}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy  &\leq K c_\ell \frac{2}{(L-\alpha)^{k+1}}   k!.
\end{split}
\end{equation}

Now, by combining (<ref>), (<ref>),  (<ref>) and (<ref>), we get that for every $k \in \N$, and for every $\ell>\max \{Q-L, Q-\alpha, 2Q \}$

\begin{equation}
\label{conclusione1}
\begin{split}
&\left| \int_{\G} \phi(xy) \int_{0}^{\infty} t^{\frac{\alpha}{2}-1} \left( \frac{1}{2} \right)^k \ln^k(t) h(t,y) dt dy \right| \\
& \leq C \max \left\{ \frac{1}{Q-\alpha},\frac{1}{L-\alpha} ,\frac{1}{\alpha}, \frac{1}{\alpha+(\ell-Q)} \right\}^{k+1} k!,
\end{split}
\end{equation}

for some suitable constant $C>0$. Therefore, $\psi(x, \alpha)$ is analytic on $0 < \alpha < \min\{Q,L\}$.

Following the previous proof, in the same hypotheses of Proposition <ref>, we obtain that the map
$$\alpha \to \int_{\G \setminus \B_c(x^{-1},R(x))} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dt$$
is analytic on $(-\infty, \min\{Q,L\})$. It is enough to consider (<ref>) and the estimates in (<ref>) and (<ref>) that are valid for every $k \in \N$, and for every $\ell>\max\{ Q-L, Q-\alpha, 2Q\}$. The analyticity is guaranteed on $(-\infty, \min\{Q,L\})$ also for the map
$$\alpha \to \int_{\G \setminus \B_c(0,1)} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dt,$$
since, in order to estimate the $k$-th derivative of
$$ \alpha \to \int_{\B_c(x^{-1}, R) \setminus \B_c(0,1)} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dt,$$
one can simplify the arguments carried out in the proof of Proposition <ref>.

§.§ The map $\alpha \to \psi(x, \alpha)$ is analytically continued on (-2,0]

In this subsection, we build up the analytic continuation of $\alpha \to \psi(x,\alpha)$ on the strip $(-2,0]$.

For every $\phi \in \mathcal{S}(\G)$ and $x \in \G$, the map

$$(0,Q) \to \psi(x,\alpha)$$

can be analytically continued to the interval $(-2,Q)$. Moreover, the representation

\begin{align}\label{peralfamag-2}
\psi(x,\alpha)= \int_{\G \setminus \B_c(0,1)} \phi(xy)P_{\alpha}(y) dy + \int_{\B_c(0,1)} (\phi(xy)-\phi(x))P_{\alpha}(y) dy + \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha)
\end{align}

holds for $\alpha \in (-2,Q)$.

Before proving Theorem <ref>, we prove two technical results concerning the convergence and the regularity of the second term of the sum in (<ref>), respectively.

For every $\phi \in \mathcal{S}(\G)$ and for every $x \in \G$ the integral

$$\int_{\B_c(0,1)}(\phi(xy)-\phi(x))P_{\alpha}(y) dy $$
converges for every $-2<\alpha<Q$.

By the coarea formula of Corollary <ref> and the homogeneity of $P_{\alpha}$ and of $\mathcal{S}^{Q-1}_{\infty}$ we get

\begin{align*}
\int_{\B_c(0,1)}(\phi(xy)-\phi(x))P_{\alpha}(y) dy &= \int_0^1 \int_{\partial \B_c(0,r)} (\phi(xy)-\phi(x))P_{\alpha}(y) d \mathcal{S}^{Q-1}_{\infty}(y) dr\\
&= \int_0^1 r^{\alpha-1} \int_{\partial \B_c(0,1)} (\phi(x \delta_r(z))-\phi(x))P_{\alpha}(z) d \mathcal{S}^{Q-1}_{\infty}(z) dr\\
&= \int_0^1 r^{\alpha-1} \int_{\partial \B_c(0,1)} (\phi(x \delta_r(z))-\phi(x)-D_h \phi(x)(\delta_r(z)))P_{\alpha}(z) d \mathcal{S}^{Q-1}_{\infty}(z) dr.
\end{align*}

In order to obtain the last equality, we have exploited the equality

\begin{equation}\label{rel-palla-diffe}
\int_{\B_c(0,1)} D_h\phi(x)(y) P_{\alpha}(y) dy=0,
\end{equation}

that is justified by the combination of few results. In fact, first we know that $\phi \in \mathcal{S}(\G) \subset C^{\infty}(\G) \subset C^1(\G) \subset C^1_h(\G)$, then by the definition of h-homomorphism we know that for every $x, y \in \G$ it holds that

$$D_h\phi(x)(y)=\langle \nabla_H\phi(x) ,y\rangle = -\langle \nabla_H\phi(x) ,-y\rangle =- \langle \nabla_H\phi(x) ,y^{-1}\rangle=-D_h\phi(x)(y^{-1}). $$

Thus (<ref>) follows by the following remarks

* by the homogeneity of $d_c$, if $y \in \B_c(0,1)$, then $y^{-1} \in \B_c(0,1)$.

* by the property of the heat kernel $h$ stated in Theorem <ref>(iii), it can be verified that
for every $y \in \G$, $y \neq 0$.
Hence, also $P_{\alpha}(y)=P_{\alpha}(y^{-1})$ holds.

Now, by Theorem <ref> (in light of (<ref>) and (<ref>)), we know that
$$|\phi(x \delta_r(z))-\phi(x)-D_h \phi(x)(\delta_r(z)))| =O(r^2) \quad \mathrm{ \ as  \ } r \to 0,$$
as a consequence we know that

\begin{align*}
\int_{\B_c(0,1)}|\phi(xy)-\phi(x)||P_{\alpha}(y)| dy & \leq  c \int_0^1 r^{\alpha+1} dr \int_{\partial \B_c(0,1)} |P_{\alpha}(z)| d \mathcal{S}^{Q-1}_{\infty}(z) \\
& = c \ \frac{1}{| \Gamma(\frac{\alpha}{2})|} \frac{1}{\alpha+2}\sigma(\alpha),
\end{align*}

for some suitable constant $c>0$. The integral converges for every $\alpha$ such that $\alpha >-2$, since

$$\frac{1}{| \Gamma(\frac{\alpha}{2})|} \frac{1}{\alpha+2}\sigma(\alpha)=\frac{1}{| \Gamma(\frac{\alpha}{2})|\alpha} \frac{\alpha}{\alpha+2}\sigma(\alpha)$$
$\lim_{\alpha \to 0} \Gamma(\frac{\alpha}{2}) \alpha=2$.

For every $\phi \in \mathcal{S}(\G)$ and for every $x \in \G$ the map

\begin{equation}
\label{partedentro}
\alpha \to \int_{\B_c(0,1)} (\phi(xy)-\phi(x))\|y \|_{\alpha}^{\alpha-Q}dy
\end{equation}

is analytic on $(-2,Q)$.

Let us consider the $k$-th derivative of (<ref>) with respect to $\alpha$, for $k \in \N$. In particular, we will show that derivative and integral can be exchanged by estimating the term

\begin{equation}
\left| \int_{\B_c(0,1)} (\phi(xy)-\phi(x))\int_0^{\infty} t^{\frac{\alpha}{2}-1} (\ln(t))^k \left( \frac{1}{2} \right)^k h(t,y)dt dy \right|.
\end{equation}

In fact, by exploiting the equality (<ref>) justified in Lemma <ref>, we get

\begin{align}
\label{un}
&\left| \int_{\B_c(0,1)} (\phi(xy)-\phi(x))\int_0^{\infty} t^{\frac{\alpha}{2}-1} (\ln(t))^k \left( \frac{1}{2} \right)^k h(t,y)dt dy \right|  \notag \\
=&\left| \int_{\B_c(0,1)} (\phi(xy)-\phi(x)-D_h\phi(x)(y) )\int_0^{\infty} t^{\frac{\alpha}{2}-1} (\ln(t))^k \left( \frac{1}{2} \right)^k h(t,y)dt dy \right|  \notag \\
\leq & \int_{\B_c(0,1)} |\phi(xy)-\phi(x)-D_h\phi(x)(y) |\int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k h(t,y)dt dy \\
=  & \int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k  \int_{\B_c(0,1)} |\phi(xy)-\phi(x)-D_h\phi(x)(y) | h(t,y)dy dt. \notag
\end{align}

Now, by Theorem <ref> (in light of (<ref>) and (<ref>)), we know that
$$|\phi(x y)-\phi(x)-D_h \phi(x)(y))| =O(\|y \|_c^2) \quad \mathrm{ \  as \ } \|y \|_c \to 0.$$
Hence, for some $C\geq 0$ we have

\begin{align*}
& \int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k  \int_{\B_c(0,1)} |\phi(xy)-\phi(x)-D_h\phi(x)(y)| h(t,y)dy dt \\
\leq & \ C \int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k  \int_{\B_c(0,1)} \|y \|_c^2 h(t,y)dy dt \\
=& \ C \int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k t^{-\frac{Q}{2}} \int_{\B_c(0,1)} \|y \|_c^2 h(1,\delta_{\frac{1}{\sqrt{t}}}(y))dy dt,
\end{align*}

due to the homogeneity of $h$. Now, by a change of variables $z= \delta_{\frac{1}{\sqrt{t}}}(y)$ and by the homogeneity of the Lebesgue measure on $\G$ we get

\begin{align}
\label{dos}
& C\int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k t^{-\frac{Q}{2}} \int_{\B_c(0,1)} \|y \|_c^2 h(1,\delta_{\frac{1}{\sqrt{t}}}(y))dy dt  \notag \\
=& \ C\int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k  \int_{\B_c(0,\frac{1}{\sqrt{t}})} \|\delta_{\sqrt{t}}(z) \|_c^2 h(1,z)dz dt  \notag \\
=& \ C\int_0^{\infty} t^{\frac{\alpha}{2}-1} |\ln(t)|^k \left( \frac{1}{2} \right)^k  \int_{\B_c(0,\frac{1}{\sqrt{t}})} t\| z\|_c^2 h(1,z)dz dt \\
\leq &  \ C\int_0^{\infty} t^{\frac{\alpha}{2}} |\ln(t)|^k \left( \frac{1}{2} \right)^k  dt\int_{\G} \|z\|_c^2h(1,z)dz.   \notag
\end{align}

Notice that $\int_{\G} \|z\|_c^2 h(1,z) dz < \infty$ since $h(1,z) \in \mathcal{S}(\G)$. Now, observe that, according to (<ref>),
$$ b_k= \int_0^{\infty} t^{\frac{\alpha}{2}} |\ln(t)|^k \left( \frac{1}{2} \right)^k   dt,$$
when $N=1$, recalling that $N=1 >- \frac{\alpha}{2}$ if $\alpha>-2$. Hence, we can repeat the argument involving the estimate in (<ref>) obtaining that for every $k \in \N$

\begin{equation}
\label{tres}
\begin{split}
\int_0^{\infty} t^{\frac{\alpha}{2}} |\ln(t)|^k \left( \frac{1}{2} \right)^k   dt
\leq C \frac{2}{(\alpha+2)^{k+1}} k!.
\end{split}
\end{equation}

By combining (<ref>), (<ref>) and (<ref>) the proof is concluded.

Now, we are in position to prove Theorem <ref>.

Let $x \in \G$ and $\phi \in \mathcal{S}(\G)$. We rearrange the map $(0,Q) \ni \alpha \to \psi(x, \alpha)$ as

\begin{align}
\label{cavametti}
\psi(x,\alpha)&= \int_{\G} \phi(xy)P_{\alpha}(y) dy  \notag \\
&= \int_{\G \setminus \B_c(0,1)} \phi(xy)P_{\alpha}(y) dy +  \int_{\B_c(0,1)} \phi(xy)P_{\alpha}(y) dy  \notag \\
&= \int_{\G \setminus \B_c(0,1)} \phi(xy)P_{\alpha}(y) dy \\
&+ \int_{\B_c(0,1)} (\phi(xy)-\phi(x))P_{\alpha}(y) dy + \int_{\B_c(0,1)} \phi(x)P_{\alpha}(y) dy.  \notag
\end{align}

Since $\alpha \in (0,Q)$, we can compute the third integral of the sum in (<ref>) exploiting the coarea formula of Corollary <ref> and homogeneity properties, getting

\begin{align}\label{eq:dentropalla}
\int_{\B_c(0,1)} \phi(x)P_{\alpha}(y) dy &= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\B_c(0,1)} \| y \|_{\alpha}^{\alpha-Q} dy \notag\\ &= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{0}^{1} \int_{\partial \B_c(0,r)} \| y \|_{\alpha}^{\alpha-Q} d\mathcal{S}^{Q-1}_{\infty}(y)  dr \notag\\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{0}^{1} \int_{\partial \B_c(0,r)} \| \delta_r(z) \|_{\alpha}^{\alpha-Q} d\mathcal{S}_{\infty}^{Q-1}(\delta_r(z))  dr \notag\\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{0}^{1} \int_{\partial \B_c(0,1)} r^{\alpha-Q} \| z \|_{\alpha}^{\alpha-Q} r^{Q-1} d\mathcal{S}_{\infty}^{Q-1}(z)  dr \notag\\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})}  \int_{0}^{1} r^{\alpha-1} \int_{\partial \B_c(0,1)} \| z \|_{\alpha}^{\alpha-Q}  d\mathcal{S}_{\infty}^{Q-1}(z)  dr\\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \sigma(\alpha) \int_{0}^{1} r^{\alpha-1} dr \sigma(\alpha)\notag \\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \sigma(\alpha) \lim_{a \to 0}\bigg[ \frac{r^{\alpha}}{\alpha} \bigg]_{r=a}^1 \sigma(\alpha) \notag \\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha). \notag
\end{align}

Hence, for every $\alpha \in (0,Q)$ the equality

\begin{align}\label{patrono}
\psi(x,\alpha)= \int_{\G \setminus \B_c(0,1)} \phi(xy)P_{\alpha}(y) dy + \int_{\B_c(0,1)} (\phi(xy)-\phi(x))P_{\alpha}(y) dy + \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha)
\end{align}

By Remark <ref>, the first term of the sum in (<ref>) is analytic with respect to $\alpha$ on $(-\infty,Q)$.
On the other hand, the third term of the sum in (<ref>) is analytic with respect to $\alpha$ on $(-\infty,Q)$, since by Item <ref> in Section <ref>, $\alpha \to \sigma(\alpha)$ is analytic on $(-\infty,Q)$ and $\lim_{\alpha \to 0} \Gamma(\frac{\alpha}{2}) \alpha=2$.

We point out that by Lemma <ref>, the expression in (<ref>) is well-defined for every $\alpha \in (-2,Q)$. Eventually, by Lemma <ref>, the second term of the sum in (<ref>) is analytic with respect to $\alpha$ on $(-2,Q)$ and this concludes the proof.

Let us show that $\psi(x,0)=\phi(x)$.

* $\lim_{\alpha \to 0} \frac{1}{\alpha \Gamma(\frac{\alpha}{2})}=\frac{1}{2},$

* by Item <ref> of Section <ref>, $\lim_{\alpha \to 0} \sigma(\alpha)=2,$

$$\lim_{\alpha \to 0} \frac{1}{\alpha \Gamma(\frac{\alpha}{2})}\sigma(\alpha)=1$$

holds. Since $\lim_{\alpha \to 0} \frac{1}{\Gamma(\frac{\alpha}{2})}=0,$ then $\psi(x,0)=\lim_{\alpha \to 0} \psi(x, \alpha)$ is

\begin{align*}
\lim_{\alpha \to 0} \left( \frac{1}{\Gamma(\frac{\alpha}{2})}\int_{\G \setminus \B_c(0,1)} \phi(xy)\|y \|_{\alpha}^{\alpha-Q} dy + \frac{1}{\Gamma(\frac{\alpha}{2})}  \int_{\B_c(0,1)} (\phi(xy)-\phi(x))\|y \|_{\alpha}^{\alpha-Q}dy +\phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha) \right)=\phi(x).
\end{align*}

In the following proposition, we deduce an useful representation of $\psi(x,\alpha)$ valid for $\alpha \in (-2,0)$.

For every $x \in \G$, for every $\phi \in \mathcal{S}(\G)$ the following representation holds

\begin{equation}\label{representation-2}
\psi(x, \alpha)= \int_{\G}(\phi(xy)-\phi(x))P_{\alpha}(y) dy,
\end{equation}

for every $\alpha \in (-2,0)$.

Let us compute for $-2<\alpha<0$ the following integral

\begin{align}\label{eq:fuoripalla}
\int_{\G \setminus \B_c(0,1)} \phi(x)P_{\alpha}(y) dy &= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(0,1)} \| y \|_{\alpha}^{\alpha-Q} dy \notag \\ &= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{1}^{\infty} \int_{\partial \B_c(0,r)} \| y \|_{\alpha}^{\alpha-Q} d\mathcal{S}^{Q-1}_{\infty}(y)  dr \notag \\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{1}^{\infty} \int_{\partial \B_c(0,r)} \| \delta_r(z) \|_{\alpha}^{\alpha-Q} d\mathcal{S}_{\infty}^{Q-1}(\delta_r(z))  dr\\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \lim_{a \to \infty} \bigg[ \frac{r^{\alpha}}{\alpha} \bigg]_{r=1}^a \int_{\partial \B_c(0,1)} \| z \|_{\alpha}^{\alpha-Q}  d\mathcal{S}_{\infty}^{Q-1}(z) \notag \\
&= - \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha). \notag
\end{align}

As usual, we have exploited the coarea formula of Corollary <ref> and the homogeneity properties. The proof follows by combining Theorem <ref> and (<ref>).

The results of the current subsection hold as well being true if $\phi \in C^{\infty}(\G)$ and $|\phi(z)| \leq K \|z\|_c^{-L}$ as $\|z\|_c>S$ for some $L, S, K>0$. In particular, under these hypotheses one can check that $\psi(x,\alpha)$ can be analytically continued on the interval $(-2,\min\{Q,L\})$.

§.§ Extension of the definition of $\mathcal{L}^s$ to $-\frac{Q}{2}<s<1$

In this section, we introduce a further notation. For every fixed $x \in \G$ and $\phi \in \mathcal{S}(\G)$ it is useful to denote the map $\psi(x,\alpha)$ also by $\psi_{\phi}(x,\alpha)$. This notation appears in the paper only when we wish to emphasize the dependence of $\psi(x,\alpha)$ on the choice of the map $\phi$.
By combining <cit.>, which states the pointwise integral representation of $\mathcal{L}^s\phi(x)$ in (<ref>), and the representation (<ref>) of $\psi_{\phi}(x,\alpha)$ showed in Proposition <ref>, we notice that for every $ s \in (0,1)$, $x \in \G $, and $\phi \in \mathcal{S}(\G)$ the equality

\begin{equation}
\label{fiuguale}
\psi_{\phi}(x, -2s)= \mathcal{L}^s\phi(x)
\end{equation}

holds. Since for every $x \in \G$ and $\phi \in \mathcal{S}(\G)$ we have analytically continued the map $\alpha \to \psi_{\phi}(x,\alpha)$ on $ (-2,Q)$, we can introduce for every $x \in \G$  a family of tempered distributions $\{ \widetilde{P}_{\alpha}^x \}_{\alpha \in (-2,Q)}$ by re-reading the map $\psi_{\phi}(x, \alpha)$ as a the action of $\widetilde{P}^x_{\alpha}$ on $\phi \in \mathcal{S}(\G)$, so that for every $\alpha \in (-2,Q)$ we have

$$\widetilde{P}_{\alpha}^x (\phi ):= \psi_{\phi}(x, \alpha).$$

Thus, for every $s \in (0,1)$ and $x \in \G$ we have

$$ \widetilde{P}_{-2s}^x (\phi  )= \psi_{\phi}(x, -2s)=\mathcal{L}^s\phi(x),$$

for every $\phi \in \mathcal{S}(\G)$.
We notice that, when $0<\alpha<Q$, $P_{\alpha}$ is a distribution as well and $\widetilde{P}^x_{\alpha}$ and $P_{\alpha}$ satisfy the following relation

\begin{equation}\label{peralfamag0}
\widetilde{P}_{\alpha}^x(\phi)=P_{\alpha}(\phi \circ l_x)= \int_{\G}\phi(xy)P_{\alpha}(y)dy,
\end{equation}

for every $x \in \G$ and $\phi \in \mathcal{S}(\G)$. It is natural to introduce the following notion.

For every $s \in (-\frac{Q}{2},1)$,

$$\mathcal{L}^s \phi(x):=  \psi_{\phi}(x,-2s)=\widetilde{P}_{-2s}^x(\phi),$$

for every $\phi \in \mathcal{S}(\G)$ and for every $x \in \G$.

In the following two subsections we study the properties of the operator $\mathcal{L}^s$ introduced in Definition <ref>.

§.§ An estimate of $x \to \psi(x, \alpha)$

First, we observe that $ \G \ni x \to \psi(x,\alpha)$ is a smooth map. With the following result, later we wish to prove the consistence of the extension of the operator $\mathcal{L}^s$ provided by Definition <ref>.

For every $\alpha>0$ and $\phi \in \mathcal{S}(\G)$
$$ \psi(x, \alpha) =O(\|x \|_c^{\alpha-Q})$$
as $\| x \|_c \to \infty$.

Recalling the estimate in (<ref>) and the fact that $\phi \in \mathcal{S}(\G)$, we obtain

\begin{align*}
&\int_{\G}\phi(xy) P_{\alpha}(y) dy = \ \frac{1}{\Gamma(\frac{\alpha}{2}) } \int_{\G} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dy
\leq  \  \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} \int_{\G} |\phi(xy) | \| y \|_{c}^{\alpha-Q} dy \\
= & \ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} \int_{\G} |\phi(z)| \| x^{-1} z \|_{c}^{\alpha-Q} dz
=  \ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} \int_{\G} |\phi(z)| \frac{(1+\| z \|_c^k)}{(1+\|z\|_c^k)} \| x^{-1} z \|_{c}^{\alpha-Q} dz \\
\leq & \ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\G} \frac{1}{(1+\|z\|_c^k)} \| x^{-1} z \|_{c}^{\alpha-Q} dz
=   \ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\G} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz \\
= & \ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\G \setminus \B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz
+ \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz,
\end{align*}

where $C_{\alpha}$ and $c_k$ are suitable positive constants.
Let us discuss separately the two terms of the last line. By exploiting homogeneity properties and the coarea formula of Corollary <ref> we get

\begin{align}
\label{same-comp}
\frac{1}{\Gamma(\frac{\alpha}{2}) } &C_{\alpha} c_k \int_{\G \setminus \B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dy  \leq  \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{{2}^{Q-\alpha}}{\| x \|_c^{Q-\alpha}} \int_{\G \setminus \B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)} dy  \notag \\
& \leq  \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{{2}^{Q-\alpha}}{\| x \|_c^{Q-\alpha}} \int_{\G } \frac{1}{(1+\|z\|_c^k)} dy\\
& \leq  \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{{2}^{Q-\alpha}}{\| x \|_c^{Q-\alpha}} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \int_{0}^{\infty} \frac{1}{(1+r^k)} r^{Q-1}dy . \notag
\end{align}

The integral converges for every $k>Q$. Hence, we proved that for every $k>Q$ the estimate

\begin{equation}
\label{uno}
\frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\G \setminus \B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dy  \leq c(k,\alpha) \| x \|_c^{\alpha-Q}
\end{equation}

holds for some suitable positive constant $c(k,\alpha)$.

Concerning the second term of the sum, we have

\begin{align*}
&\frac{1}{\Gamma(\frac{\alpha}{2}) }  C_{\alpha} c_k \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz \\
& =\frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|xx^{-1}z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz \\
& \leq \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+(\|x\|_c-\|x^{-1}z\|_c)^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz\\
& \leq \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{1}{(1+(\frac{\|x\|_c}{2})^k)} \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz \\
& \leq \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{1}{(1+(\frac{\|x\|_c}{2})^k)} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1))\int_{0}^{\frac{\|x\|_c}{2}} r^{\alpha-Q} r^{Q-1} dz \\
& = \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k \frac{1}{(1+(\frac{\|x\|_c}{2})^k)} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \frac{(\frac{\|x\|_c}{2})^{\alpha}}{\alpha}\\
& \leq \frac{1}{\Gamma(\frac{\alpha}{2}) } C_{\alpha} c_k 2^k \|x\|_c^{-k} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \frac{\|x\|_c^{\alpha}}{2^{\alpha}\alpha}\\
& \leq \tilde{c}(k, \alpha) \|x \|_c^{\alpha-k},
\end{align*}

where $\tilde{c}(k,\alpha)$ is a suitable positive constant. Hence, we have proved that for every $k$ we have

\begin{equation}
\label{due}
\frac{1}{\Gamma(\frac{\alpha}{2}) }  C_{\alpha} c_k \int_{\B_c(x, \frac{\|x\|_c}{2})} \frac{1}{(1+\|z\|_c^k)}\frac{1}{ \| x^{-1} z \|_{c}^{Q-\alpha}} dz \leq \tilde{c}(k, \alpha) \|x \|_c^{\alpha-k}.
\end{equation}

By combining (<ref>) and (<ref>), we get that for every $k>Q$ the estimate

\begin{equation*}
\frac{1}{\Gamma(\frac{\alpha}{2}) } \int_{\G} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dy \leq c(k,\alpha) \| x \|_c^{\alpha-Q}+ \tilde{c}(k, \alpha) \|x \|_c^{\alpha-k} \leq C(\alpha, k) \| x \|_c^{\alpha-Q}
\end{equation*}

holds, where $C(k,\alpha)$ is a suitable positive constant, concluding the proof.

For every $-2 < \alpha <0$ and $\phi \in \mathcal{S}(\G)$

\begin{equation}
\label{thesis}
\psi(x, \alpha) =O(\|x \|_c^{\alpha-Q})
\end{equation}

as $\| x \|_c \to \infty$.

We can use the representation in (<ref>), i.e. for $\alpha \in (-2,0)$

$$ \psi(x, \alpha) = \int_{\G} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy.$$

Let $b=b(\G)>0$ the constant which appears in Theorem <ref> and consider

\begin{align}
\label{threeterms}
&\int_{\G} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy  \notag \\
=& \int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy+\int_{\B_c(0, \frac{\| x\|_c}{2b^2})} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy\\
=&\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(xy) P_{\alpha}(y) dy -\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(x) P_{\alpha}(y) dy +\int_{\B_c(0, \frac{\| x\|_c}{2b^2})} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy.  \notag
\end{align}

We deal with the three terms of the sum in (<ref>), separately. Let us consider the first one, in particular

\begin{align*}
\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(xy) P_{\alpha}(y) dy &= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(xy) \| y \|_{\alpha}^{\alpha-Q} dy\\
&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(x, \frac{\| x\|_c}{2b^2})} \phi(z) \| x^{-1}z \|_{\alpha}^{\alpha-Q} dz \\
&= \frac{1}{\Gamma(\frac{\alpha}{2})} \int_{\G \setminus \B_c(x, \frac{\| x\|_c}{2b^2})} \phi(z) \frac{(1+\|z \|_c^k)}{(1+\|z \|_c^k)} \| x^{-1}z \|_{\alpha}^{\alpha-Q} dz \\
&\leq \frac{1}{|\Gamma(\frac{\alpha}{2})|} c_k C_{\alpha} \int_{\G \setminus \B_c(x, \frac{\| x\|_c}{2b^2})}  \frac{1}{(1+\|z \|_c^k)} \frac{1}{\| x^{-1}z \|_{c}^{Q-\alpha}} dz ,
\end{align*}

holds for some positive constant $C_{\alpha}$.
Now, as in (<ref>) (since $\alpha<Q$), we get the estimate in (<ref>), i.e. for every $k>Q$ the estimate

\begin{equation}
\label{an}
\begin{split}
\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(xy) P_{\alpha}(y) dy\leq \frac{1}{|\Gamma(\frac{\alpha}{2})|} c_k c_{\alpha} \int_{\G \setminus \B_c(x, \frac{\| x\|_c}{2b^2})}  \frac{1}{(1+\|z \|_c^k)} \frac{1}{\| x^{-1}z \|_{c}^{Q-\alpha}} dz \leq  c(k,\alpha) \| x \|_c^{\alpha-Q}
\end{split}
\end{equation}

Let us consider the second term of (<ref>). By the coarea formula of Corollary <ref> and homogeneity properties we obtain

\begin{align*}
-\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(x) P_{\alpha}(y) dy &=- \phi(x) \int_{\frac{\|x \|_c}{2b^2}}^{\infty} \int_{\partial \B_c(0,r)} P_{\alpha}(y) d\mathcal{S}^{Q-1}_{\infty}(y) dr \\
&=- \phi(x) \int_{\frac{\|x \|_c}{2b^2}}^{\infty} r^{\alpha-Q} r^{Q-1} \int_{\partial \B_c(0,1)} P_{\alpha}(z) d\mathcal{S}^{Q-1}_{\infty}(z) dr \\
&= \phi(x) \frac{1}{\Gamma(\frac{\alpha}{2})} \frac{1}{\alpha} \sigma(\alpha) \frac{\|x\|_c^{\alpha}}{(2b^2)^{\alpha}}.
\end{align*}

We point out that we used that $\alpha \in (-2,0)$. Now, since $\phi(x)=O( \|x\|_c^N )$ as $\|x \|_c \to \infty$ for every negative integer $N $, fixing $N=-Q$ we get that for $\|x\|_c>1$ the estimate

\begin{equation}
\label{do}
\begin{split}
-\int_{\G \setminus \B_c(0, \frac{\| x\|_c}{2b^2})} \phi(x) P_{\alpha}(y) dy \leq \widehat{c}(\alpha)\|x \|_c^{\alpha-Q} \phi(x)
\end{split}
\end{equation}

holds. By a change of variables in the third term of the sum in (<ref>), we obtain

\begin{equation}\label{continue2}
\begin{split}
\int_{\B_c(0, \frac{\| x\|_c}{2b^2})} (\phi(xy)-\phi(x)) P_{\alpha}(y) dy = \int_{\B_c(x, \frac{\| x\|_c}{2b^2})} (\phi(z)-\phi(x)) P_{\alpha}(x^{-1}z) dz.
\end{split}
\end{equation}

Recalling the stratified mean value inequality of Theorem <ref> and the fact that $\phi$ is smooth, we have the following inequality

\begin{equation}
| \phi(xy)-\phi(x)-D_h\phi(x)(y) |=| \phi(xy)-\mathcal{T}_1(\phi,x)(xy) | \leq c \|y\|_c^2 \sup_{p \in \B_c(x,b^2\|y\|_c)} \mathfrak{d}^2_h\phi(p),
\end{equation}

where $ \mathfrak{d}^2_h\phi$ is $\mathfrak{d}_h^2\phi(z):\G \to \R, z \to \max_{|\alpha|=2} |(Z_1 \ldots Z_{m_1})^{\alpha}\phi(z)|$ for every $z \in \G$. We get

\begin{equation}\label{oss1}
| \phi(z)-\phi(x)-D_h\phi(x)(x^{-1}z) | \leq c \|x^{-1}z\|_c^2 \sup_{p \in \B_c(x,b^2\|x^{-1}z\|_c)} \mathfrak{d}^2_h\phi(p)=c \|x^{-1}z\|_c^2  \mathfrak{d}^2_h\phi(\xi),
\end{equation}

for some $\xi=\xi(z) \in \B_c(x,b^2\|x^{-1}z\|_c ) \subset \B_c(x, \frac{\|x\|_c}{2})$.
Now, considering the Pansu differential $D_h\phi(x)$ of $\phi$ at $x$, we notice that the following equality

\begin{equation}\label{oss2}
\begin{split}
&\int_{\B_c(x, \frac{\| x\|_c}{2b^2})}
D_h\phi(x)(x^{-1}z)  P_{\alpha}(x^{-1}z) dz=\int_{\B_c(0, \frac{\| x\|_c}{2b^2})}  D_h\phi(x)(y) P_{\alpha}(y) dy =0
\end{split}
\end{equation}

holds, by the same argument in (<ref>).
Hence, from (<ref>), exploiting (<ref>), (<ref>) and (<ref>) and the fact that $\phi \in \mathcal{S}(\G)$, we have

\begin{align}
\label{continue3}
&\int_{\B_c(x, \frac{\| x\|_c}{2b^2})} (\phi(z)-\phi(x)) P_{\alpha}(x^{-1}z) dz  \notag\\
= &\int_{\B_c(x, \frac{\| x\|_c}{2b^2})} (\phi(z)-\phi(x)-D_h \phi(x)(x^{-1}z)) P_{\alpha}(x^{-1}z) dz  \notag \\
\leq &\int_{\B_c(x, \frac{\| x\|_c}{2b^2})} c \|x^{-1}z\|_c^2 \mathfrak{d}^2_h\phi(\xi) |P_{\alpha}(x^{-1}z)| dz  \notag \\
\leq & \frac{1}{|\Gamma(\frac{\alpha}{2})|} c C_{\alpha} \int_{\B_c(x, \frac{\| x\|_c}{2b^2})}   \mathfrak{d}^2_h\phi(\xi) \|x^{-1}z\|_c^{\alpha-Q+2} dz\\
= & \frac{1}{|\Gamma(\frac{\alpha}{2})|}c C_{\alpha} \int_{\B_c(x, \frac{\| x\|_c}{2b^2})} \mathfrak{d}^2_h\phi(\xi) \frac{(1+ \| \xi \|_c^k)}{(1+ \| \xi \|_c^{k})} \|x^{-1}z\|_c^{\alpha-Q+2} dz  \notag \\
\leq & \frac{1}{|\Gamma(\frac{\alpha}{2})|}c C_{\alpha} c_k \int_{\B_c(x, \frac{\| x\|_c}{2b^2})} \frac{1}{(1+ \| \xi \|_c^{k})} \|x^{-1}z\|_c^{\alpha-Q+2} dz,  \notag
\end{align}

where $c$ and $c_{\alpha}$ are suitable positive constants.
Now, remarking that
$$\|  \xi \|_c \geq \|x\|_c-\|x^{-1}\xi\|_c \geq \|x\|_c - b^{2} \|x^{-1}z\|_c \geq \|x\|_c-b^2 \frac{\|x\|_c}{2b^2}= \frac{\|x\|_c}{2}, $$
we can continue from (<ref>) getting

\begin{align*}
&\int_{\B_c(x, \frac{\| x\|_c}{2b^2})} (\phi(z)-\phi(x)) P_{\alpha}(x^{-1}z) dz \\
\leq & \ \frac{1}{|\Gamma(\frac{\alpha}{2})|}c C_{\alpha} c_k \int_{\B_c(x, \frac{\| x\|_c}{2b^2})} \frac{1}{(1+ \| \xi \|_c^{k})} \|x^{-1}z\|_c^{\alpha-Q+2} dz\\
\leq & \ \frac{1}{|\Gamma(\frac{\alpha}{2})|}c C_{\alpha} c_k \int_{\B_c(x, \frac{\| x\|_c}{2b^2})} \frac{1}{(1+ \big( \frac{\|x\|_c}{2} \big)^{k})} \frac{1}{\|x^{-1}z\|_c^{Q-\alpha-2}} dz\\
\leq & \ \frac{1}{|\Gamma(\frac{\alpha}{2})|} c C_{\alpha} c_k \frac{1}{(1+ \big( \frac{\|x\|_c}{2} \big)^{k})} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \int_{0}^{\frac{\|x\|_c}{2}} r^{\alpha+2-Q+Q-1} dr\\
= & \ \frac{1}{|\Gamma(\frac{\alpha}{2})|}c C_{\alpha} c_k \frac{1}{(1+ \big( \frac{\|x\|_c}{2} \big)^{k})} \mathcal{S}^{Q-1}_{\infty}(\partial \B_c(0,1)) \frac{\|x\|_c^{\alpha+2}}{2^{\alpha+2}(\alpha+2)}\\
\leq & \ c(\alpha, k) \| x\|_c^{\alpha+2-k},
\end{align*}

where $c(\alpha,k)$ is a suitable positive constant.

Thus, by choosing $k>Q+2$ we get the following estimate

\begin{equation}
\label{trua}
\begin{split}
\int_{\B_c(0, \frac{\| x\|_c}{2})} (\phi(xy)-\phi(x)) P_{\alpha}(y) \leq \widetilde{c}(\alpha) \| x\|_c^{\alpha-Q},
\end{split}
\end{equation}

for a suitable constant $\widetilde{c}(\alpha)$.
Combining (<ref>), (<ref>) and (<ref>) we obtain (<ref>).

§.§ Some properties of $\mathcal{L}^s$, when $-\frac{Q}{2}<s<1$

Coherently with the continuation constructed in the previous subsections, we show that the definition of the fractional sub-Laplacian $\mathcal{L}^s$ with argument $s \in (-\frac{Q}{2},1)$, that we introduced in Definition <ref> is consistent.

First, we observe that for $s=0$, it holds that

$$\psi_{\phi}(x,0)= \phi(x)= \mathcal{L}^0\phi(x),$$

for every $\phi \in \mathcal{S}(\G)$ and $x \in \G$.

Moreover, $\mathcal{L}^s$ satisfies the semigroup property stated in the following proposition.

For every $s,m \in (-\frac{Q}{2},1)$ such that $-\frac{Q}{2}<m+s<1 $ the equality

$$\mathcal{L}^s \circ \mathcal{L}^m=\mathcal{L}^{s+m}$$

Let us consider $\alpha \in (0,Q)$ and $ \beta \in (0,Q)$ such that $\alpha+\beta<Q$. We recall that by (<ref>), the convolution rule
$$P_{\alpha+\beta}=P_{\alpha}\star P_{\beta}$$

holds. For every $x \in \G$ and $\phi \in \mathcal{S}(\G)$, exploiting the relation (<ref>), we obtain that

\begin{align}\label{assumption}
\widetilde{P}_{\alpha+\beta}^x(\phi)&=P_{\alpha+\beta}(\phi  \circ l_x )= \int_{\G} P_{\alpha+\beta}(y)\phi(xy)dy \notag \\
&= \int_{\G} P_{\alpha}\star P_{\beta}(y)\phi(xy)dy = \int_{\G} \int_{\G} P_{\alpha}(z) P_{\beta}(z^{-1}y) dz \phi(xy)dy \notag \\
&= \int_{\G } P_{\alpha}(z)  \int_{\G} P_{\beta}(z^{-1}y) \phi(xy)dy dz = \int_{\G } P_{\alpha}(z)  \int_{\G} P_{\beta}(z') \phi(xzz')dz' dz\\
&= \int_{\G } P_{\alpha}(z) \psi_{\phi}(xz,\beta) dz =\widetilde{P}_{\alpha}^x(\psi_{\phi}(\cdot,\beta)) =\widetilde{P}_{\alpha}^x(\widetilde{P}_{\beta}^{xz}(\phi)) = \psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha).  \notag
\end{align}

We wish to show that the equality in (<ref>) holds for every $\alpha, \beta \in (-2,Q)$ such that $-2<\alpha+\beta<Q$. This conclude the proof by the uniqueness of analytic continuation.

First, we observe that, as a consequence of our analytic continuation, the map $\widetilde{P}_{\alpha+\beta}^x(\phi)=\psi_{\phi}(x,\alpha+\beta)$ is analytic with respect to $\alpha$ and $\beta$ on the set of $\alpha$ and $\beta$ such that $-2<\alpha+\beta<Q$.

Let us show that the map $\psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)$ is analytic as well with respect to $\alpha$ and $\beta$ for $\alpha, \beta \in (-2,Q)$, $\alpha+\beta<Q$.

In order to do this, we fix $\beta \in(-2,Q)$ and consider $\alpha \to \psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)$.
We need to apply the results of Proposition <ref> and Section <ref>. In fact we proved that, if one considers a map $\phi \in C^{\infty}(\G)$ such that $|\phi(z)|\leq K \|z\|_c^{-L}$ when $\|z\|_c \geq S$ for some $L,S,K>0$, then one can carry out the analytic continuation of the map $\psi_{\phi}(x, \alpha)$ for any $\alpha<\min\{L,Q\}$.
We apply this result to the map $\alpha \to \psi_{\psi(\cdot,\beta)}(x,\alpha)$. In fact, by the estimates of Section <ref> we know that for every $\phi \in \mathcal{S}(\G)$ we have $\psi_{\phi}(z,\beta)=O(\|z\|_c^{\beta-Q})$ as $\|z\|_c \to \infty$.

Hence, by Proposition <ref>, we conclude that $\alpha \to \psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)$ is analytic up to $-2<\alpha<\min\{Q-\beta,Q\}$, i.e. for every $\alpha \in (-2,Q)$ such that $\alpha+\beta<Q$.

Clearly, the role of $\alpha$ and $\beta$ can be exchanged since $\widetilde{P}_{\alpha+\beta}^x(\phi)=\widetilde{P}_{\beta+\alpha}^x(\phi)$, and then $\psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)=\psi_{\psi_{\phi}(\cdot,\alpha)}(x,\beta)$. Thus, for every $\alpha \in (-2,Q)$, the map $\beta \to \psi_{\psi(\cdot,\beta)}(x,\alpha)$ is analytic for $\beta \in (-2,Q)$ as $\alpha+\beta <Q$. To sum up, we proved that for every $x \in \G$ and $\phi \in \mathcal{S}(\G)$,

\begin{equation}
\psi_{\phi}(x,\alpha+\beta)=\widetilde{P}_{\alpha+\beta}^x(\phi)=\psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)
\end{equation}

hold for every $\alpha, \beta \in (-2,Q)$ such that $-2<\alpha+\beta<Q$.

Let us now consider $x \in \G$ and $\phi \in \mathcal{S}(\G)$ and set $\alpha, \beta \in (-2,Q)$ such that $-2<\alpha+\beta<Q$. Set $\alpha=-2s$ and $\beta=-2m$ for suitable $s,m \in (-\frac{Q}{2},1)$. We have

\begin{align*}
\mathcal{L}^{s+m} \phi(x)&=\widetilde{P}^{x}_{\alpha+\beta}(\phi)=\psi_{\psi_{\phi}(\cdot,\beta)}(x,\alpha)=\psi_{\mathcal{L}^m\phi(\cdot)}(x,\alpha)= \mathcal{L}^s (\mathcal{L}^m \phi)(x).
\end{align*}

This concludes the proof.

§ THE HEISENBERG GROUP $\H^N$

In this section we recall the setting of the Heisenberg group $\H^n$ and we analyse some suitable families of integrals. This analysis will be an auxiliary tool to construct an analytic continuation of the map $\alpha \to \psi(x,\alpha)$ on $(-\infty, Q)$ in the Heisenberg group.

§.§ The Heisenberg group: definition and some properties

We discuss the Heisenberg group according to Section <ref>, referring to the representation of a Carnot group $\G$ in (<ref>). For every $n \in \N$, independently of the choice of $n$, the Heisenberg group can be represented as a vector space

\[
\mathbb{H}^n = H_1 \oplus H_2,
\]

with dim($H_1$)$=2n$ and dim($H_2$)$=1$, endowed with a symplectic form $\omega$ on $H_1$ and a fixed nonvanishing element
\begin{equation}
\label{edueennepiuuno}
e_{2n+1} \in H_2.
\end{equation}

We denote by $\pi_{H_1}$ and $\pi_{H_2}$ the canonical projections on $H_1$ and $H_2$, respectively, associated with the direct sum.
The space $\mathbb{H}^n$ has a structure of Lie algebra given by

\begin{equation}
[ x,y]= \omega(\pi_{H_1}(x),\pi_{H_1}(y)) \ e_{2n+1}.
\end{equation}

Then, by the Baker-Campbell-Hausdorff formula, the Lie group product is

\begin{equation}\label{lieprod}
x y = x+ y + \frac{[x,y]}{2}
\end{equation}

for every $x,y \in \H^n$.
Notice that $q=2n+1$ and $Q=2n+2$.
We fix a symplectic basis $ (e_1, \dots, e_{2n} )$ of $(H_1,\omega)$, namely

\[
\omega(e_i,e_{n+j})=\delta_{ij}, \qquad \omega(e_i,e_j)=\omega(e_{n+i},e_{n+j})=0,
\]

for every $i,j=1,\ldots,n$, where $\delta_{ij}$ is the Kronecker delta.
Thus, considering the vector $e_{2n+1}$ given in (<ref>), we obtain a basis

\begin{equation}
\label{heisbasis}
\mathcal{B}= ( e_1, \dots, e_{2n+1} ).
\end{equation}

We assume that on $\H^n$ is fixed the scalar product $\langle \cdot, \cdot \rangle$ that makes $\mathcal{B}$ orthonormal.
We can read the product (<ref>) on $\H^n$ in coordinates with respect to $\cB$ (i.e. on $\mathbb{R}^{2n+1}$) as follows:

\begin{equation}
\label{productheis}
(x_1, \dots, x_{2n+1})  (y_1, \dots, y_{2n+1})= \pa{x_1 + y_1, \ldots, x_{2n+1} + y_{2n+1} + \sum_{i=1}^n \frac{x_iy_{i+n}-x_{i+n} y_i}2},
\end{equation}

for every $(x_1, \ldots, x_{2n+1}), (y_1, \ldots, y_{2n+1}) \in \H^n$.
Notice that $H_2= \mathrm{span}(e_{2n+1})$ is the center of $\H^n$.
Moreover, taking in consideration the product (<ref>), in our coordinates the Jacobian basis of left invariant vector fields is the following

\begin{equation}
\label{eqbase}
\begin{aligned}
X_j(x) &= \partial_{x_j}-\frac{1}{2} x_{j+n} \partial_{x_{2n+1}} \ \ \ \ \ \ j=1, \dots, n\\
Y_j(x) &= \partial_{x_{n+j}} + \frac{1}{2}x_j \partial_{x_{2n+1}} \ \ \ \ \ \ j=1, \dots, n\\
T(x)&= \partial_{x_{2n+1}},
\end{aligned}
\end{equation}

for every $x=(x_1, \dots, x_{2n+1}) \in \H^n$.
We will also use the following notation: for $i=1, \ldots, 2n+1$ we denote

$$\mathrm{Lie}(\H^n) \ni Z_i:=
\begin{cases}
X_i \qquad \mathrm{if \ }i=1 \ldots n\\
Y_{i-n} \qquad \mathrm{if \ }i=n+1, \ldots, 2n\\
T \qquad \mathrm{if \ }i=2n+1.\\
\end{cases}

With a slight abuse of notation, we denote by $\pi_{H_1}$ and $\pi_{H_2}$ also the corresponding projections read in coordinates with respect to the basis $\mathcal{B}$ i.e.

\begin{align*}
&\pi_{H_1}:\H^n \to \R^{2n},
\pi_{H_1}(x_1, \ldots, x_{2n+1})=(x_1, \ldots, x_{2n}),\\
& \pi_{H_2}:\H^n \to \R, \pi_{H_2}(x_1, \ldots, x_{2n+1})=x_{2n+1},
\end{align*}

for every $(x_1, \ldots, x_{2n+1}) \in \H^n$.

The structure of the Lie algebra of $\H^n$ reflects on the explicit form of the $Z$-Taylor polynomials. For example, we consider below the explicit form of the $Z$-Taylor Polynomial of $\G$-degree $n=3$ related to a map $\phi \in C^{\infty}(\H^n)$ in the Heisenberg group $\H^n$. Let $x \in \H^n$, then for every $y \in \H^n$, according to Theorem <ref>, we have

\begin{equation}\label{Taylor3}
\begin{split}
\mathcal{T}_3(\phi,x)(xy)=&\phi(x)+\sum_{i=1}^{2n}Z_i\phi(x)y_i+ T\phi(x)y_{2n+1}+ \frac{1}{2}\sum_{i,j=1}^{2n} Z_iZ_j\phi(x)y_iy_j\\
&+\frac{1}{2}\sum_{i=1}^{2n}Z_i T\phi(x)y_iy_{2n+1}+\frac{1}{6}\sum_{i,j,k=1}^{2n} Z_iZ_jZ_k\phi(x)y_iy_jy_k.
\end{split}
\end{equation}

Moreover, by applying Theorem <ref>, we get the existence of constants $c=c(\H^n)>0$ and $b=b(\H^n)>0$ such that

\begin{equation}
\label{utile}
|\phi(xy)-\mathcal{T}_3(\phi,x)(xy)| \leq c \|y\|_c^{4} \sup_{\{\|z\|_c \leq b^{4} \|y\|_c\}} \sup_{\{|\beta|=4\}} |(Z_1 \cdots Z_{2n})^{\beta} \phi(xz)|,
\end{equation}

for every $x,y \in \G$. Here $\beta$ denotes a $2n$-multi-index.

We recall also some metric properties on $\H^n$. Referring to Theorem <ref>, if $\G=\H^n$, then the Carnot-Carathéodory distance $d_c(x,0) \in C^1(\H^n \setminus H_2)$ (see for example <cit.>) and $ | \nabla_H d_c(x, 0)| =1$ for every $x \in \H^n \setminus H_2$.

In the following two subsections, we introduce some properties of $h$ and $ \B_c(0,1)$ in $\H^n$.

§.§ Invariance of the heat kernel under horizontal rotations

In many papers, like [22, 26, 23, 32],
the authors studied the heat kernel $h$ in the Heisenberg group $\H^n$ and, in general, in stratified groups of step 2.
In particular, in the Heisenberg group $\H^n$, for every $t>0$ and $x=(x_1, \dots, x_{2n}, x_{2n+1}) \in \H^n$, the heat kernel $h$ depends on $t$, on the last component $x_{2n+1}$ of $x$ and on the norm $|(x_1, \dots, x_{2n})|=|\pi_{H_1}(x)|$.
This allows to deduce that the map $h$ is invariant under the action of horizontal rotations, i.e. the rotations which do not involve the $x_{2n+1}$-axis $H_2$.

As a consequence, for every $t>0$, for every $x=(x_1, \dots, x_{2n+1}) \in \H^n$ and for every $j \in \{1, \dots, 2n \}$ the equality

\begin{equation}
\label{parita-heat}
h(t,(x_1, \dots, x_j, \dots, x_{2n}, x_{2n+1}))=h(t,(x_1, \dots, -x_j, \dots, x_{2n}, x_{2n+1}))
\end{equation}

Clearly, (<ref>) yields that for every $\alpha<Q$, for every $t>0$ and for every $x=(x_1, \dots, x_{2n+1}) \in \H^n$ the following equalities

\begin{align}
\label{parita-kernel}
\|(x_1, \dots, x_j, \dots, x_{2n}, x_{2n+1})\|_{\alpha}^{\alpha-Q}&=\|(x_1, \dots, -x_j, \dots, x_{2n}, x_{2n+1})\|_{\alpha}^{\alpha-Q}, \notag \\
P_{\alpha}(x_1, \dots, x_j, \dots, x_{2n}, x_{2n+1})&=P_{\alpha}(x_1, \dots, -x_j, \dots, x_{2n}, x_{2n+1})
\end{align}

hold for every $j \in \{1, \dots, 2n \}$.
We notice that, a priori, this properties are not proved in arbitrary stratified groups, included the stratified groups of step 2. For the explicit form of the heat kernel we address the reader to [8]. For further details about this topic, we point out <cit.>.

§.§ Invariance of $\B_c(0,1)$ under horizontal rotations

In the Heisenberg group the Carnot-Carathéodory ball $\B_c(0,R)$, of radius $R$, for $R>0$, centered at $0 \in \H^n$ is invariant under horizontal rotations, i.e. the rotations preserving the center of the group $H_2$ (see for instance <cit.>).

As a consequence, let $(x_1, \dots, x_{2n}, x_{2n+1}) \in \B_c(0,R)$. Then, for every point $(y_1, \dots, y_{2n}, x_{2n+1}) \in \H^n$ such that the following equality

$$| (x_1, \dots, x_{2n})|=|(y_1, \dots, y_{2n})|$$

holds, then $(y_1, \ldots, y_{2n}, x_{2n+1}) \in \B_c(0,R)$.
Let us now introduce for every $j \in \{1, \dots, 2n \}$ the following projection

\begin{equation}\label{pij}
\pi_j: H_1 \to \R^{2n-1}, \pi_j\left( \sum_{i=1}^{2n}x_ie_i \right)=(x_1, \dots, x_{j-1}, x_{j+1}, \dots, x_{2n}, x_{2n}).
\end{equation}

Notice that $\pi_{H_2}$ and $\pi_j$, for $j= \{1, \dots, 2n \}$, are Lipschitz map in the Euclidean sense and their (Euclidean) Jacobian is unitary. Hence, for every $r \in \pi_{H_2}(\B_c(0,1)) \subset \R$ and for every $s=(s_1, \dots, s_{2n-1}) \in \pi_j (\pi_{H_2}^{-1}(r) \cap \B_c(0,1)) \subset \R^{2n-1}$, the invariance of $\B_c(0,R)$ with respect to horizontal rotations implies that
the set
\begin{equation}
\label{simmetriai}
\pi_j^{-1}(s) \cap \pi_{H_2}^{-1}(r) \cap \B_c(0,R)
\end{equation}

is symmetric in the $x_j$-direction with respect to the point $(s_1, \dots, s_{j-1}, 0, s_j, \dots, s_{2n-1},r)$.

More precisely, for every $r \in \R$, every $s=(s_1, \dots, s_{2n-1}) \in \R^{2n-1}$ and for every $x_j \in \R$:

\begin{equation*}
(s_1, \dots, s_{j-1}, x_j, s_j, \dots, s_{2n-1},r) \in \B_c(0,R) \mathrm{ \ if \ and \ only \ if \ }(s_1, \dots, s_{j-1}, -x_j, s_j, \dots, s_{2n-1},r) \in \B_c(0,R).
\end{equation*}

In fact, the previous equivalence follows by
$$|(s_1, \dots, s_{j-1}, x_j, s_j, \dots, s_{2n-1})|=|(s_1, \dots, s_{j-1}, -x_j, s_j, \dots, s_{2n-1})|,$$
combined with the fact that $\B_c(0,R)$ is invariant under horizontal rotations.

In addition, also $\partial \B_c(0,1)$ is invariant under horizontal rotations.

§.§ Auxiliary tools

In this subsection we study some specific families of integrals. In particular, we collect some technical results which will be used later on in the paper.
First, we fix $m \in \N$, $\alpha<Q$ and we collect few useful representations of the following integral

\begin{equation}
\label{diff-taylor}
\int_{\B_c(0,1)}\big(\mathcal{T}_{2m+1}(\phi,x)(xy)-\mathcal{T}_{2m-1}(\phi,x)(xy) \big) \|y\|_{\alpha}^{\alpha-Q} dy.
\end{equation}

We recall that the object $\mathcal{T}_n(\phi,x)$ denotes the $Z$-Taylor Polynomial of $\G$-degree $n$ related to $\phi$ centered at $x$, for every $n \in \N$.

§.§.§ Representation, I

First of all, we exploit the explicit representations of $\mathcal{T}_{2m+1}(\phi,x)(xy)$ and $\mathcal{T}_{2m-1}(\phi,x)(xy)$ given by Theorem <ref> which allows to write the integral in (<ref>) as

\begin{equation}\label{prima}
\sum_{h=2m}^{2m+1} \sum_{\substack{k=1, \ldots, h\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=h}} \frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy.
\end{equation}

Now, for $h \in \{ 2m, 2m+1\}$, $k \in \{1, \ldots, h\}$ and  $i_1, \ldots,i_k \leq 2n+1$ such that $  \mathrm{deg}(Z_{i_1})+ \ldots + \mathrm{deg}(Z_{i_k})=h$, we focus on the integral

\begin{equation}\label{integral}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy
\end{equation}

appearing in the sum (<ref>).
We observe one more time that $\B_c(0,1)$ and the map $\|y\|_{\alpha}^{\alpha-Q}$ are symmetric and invariant with respect to the group inversion. In fact,

* by the homogeneity of $d_c$, if a point $y \in \B_c(0,1)$, then $y^{-1} \in \B_c(0,1)$.
* by the property of the heat kernel $h$ stated in Theorem <ref>(iii), it holds that

for every $y \in \H^n$, $y \neq 0$.

Moreover, for every $y \in \H^n$, $y^{-1}=-y$. Therefore, we know that, if the number $k \in \{1, \ldots, h\}$ of the involved coordinates $y_{i_1} \ldots y_{i_k}$ is odd, then the integral in (<ref>) vanishes

\begin{equation}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy=0.
\end{equation}

As a consequence, we can rewrite (<ref>) as follows

\begin{equation}
\sum_{h=2m}^{2m+1} \sum_{\substack{k=1, \ldots, h, k  \ \mathrm{even}\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=h}} \frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy.
\end{equation}

Now, we consider again the integral in (<ref>)
and we make the following hypothesis

\begin{equation}\label{hyp}
\mathrm{there \ exists \ } j \in \{1, \dots, k\} \mathrm{ \ such \ that \ }\mathrm{deg}(Z_{i_j})=1 \mathrm{ \ and \ }\sharp \{ \ell \in \{1, \dots, k \} : i_j=i_{\ell} \} \mathrm{ \ is \ odd }.
\end{equation}

Now, if we apply the Euclidean coarea formula to the integral in (<ref>) we get

\begin{align*}
&\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \| y \|_{\alpha}^{\alpha-Q} dy =\int_{\pi_{H_2}(\B_c(0,1))} \int_{\pi_{H_2}^{-1}(y_{2n+1}) \cap \B_c(0,1)} y_{i_1} \ldots y_{i_k} \| y \|_{\alpha}^{\alpha-Q} dy_1, \dots dy_{2n} \ dy_{2n+1} \\
=&\int_{\pi_{H_2}(\B_c(0,1))} \int_{\pi_j(\pi_{H_2}^{-1}(y_{2n+1}) \cap \B_c(0,1))} \int_{\pi_j^{-1}(r) \cap \pi_{H_2}^{-1}(y_{2n+1}) \cap \B_c(0,1)} y_{i_1} \ldots y_{i_k} \| y \|_{\alpha}^{\alpha-Q} dy_j dr \ dy_{2n+1}.
\end{align*}

By combining the symmetry of $\B_c(0,1)$ discussed in Section <ref>, the symmetry of $\|y\|_{\alpha}^{\alpha-Q}$ discussed in Section <ref> and our hypothesis (<ref>), we conclude that

\begin{align*}
\int_{\pi_j^{-1}(r) \cap \pi_{H_2}^{-1}(y_{2n+1}) \cap \B_c(0,1)} y_{i_1} \ldots y_{i_k} \| y \|_{\alpha}^{\alpha-Q} dy_j=0,
\end{align*}

for every $y_{2n+1} \in \R$ and $r \in \R^{2n-1}$.
Hence, under the hypothesis (<ref>), the integral in (<ref>) vanishes. Then, the integral in (<ref>) can be non-zero only if

\begin{equation}\label{coi-multi}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy= \int_{\B_c(0,1)} (y_1, \dots, y_{2n})^{\gamma} y_{2n+1}^{\gamma_{2n+1}} \|y\|_{\alpha}^{\alpha-Q} dy,
\end{equation}

for some even $2n$-multi-index $\gamma \in (\N \cup \{0\})^{2n}$ and for some $\gamma_{2n+1} \in \N \cup \{0\}$.

Let us now consider again the integral in (<ref>) and assume that $h=2m+1$. Hence we have that

\begin{equation}\label{hyp2}
\mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m+1.
\end{equation}

Since we are working in $\H^n$, for every $j=1, \dots, k$, necessarily $\mathrm{deg}(Z_{i_j})$ is 1 or 2. Hence

\begin{equation}\label{dispari} \sharp\{ j \in \{1, \dots, k\}: \ \mathrm{deg} (Z_{i_j})=1\} = 2m+1- 2 \cdot \sharp\{ j \in \{1, \dots, k\}: \ \mathrm{deg} (Z_{i_j})=2\}.
\end{equation}

Thus, $\sharp\{ j \in \{1, \dots, k\}: \ \mathrm{deg} (Z_{i_j})=1\} $ is odd, and therefore hypothesis (<ref>) is verified, hence, also under the hypothesis (<ref>), we have

$$\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy=0.$$

Therefore, we can conclude that the integral in (<ref>) can be non-zero only if
$$ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m.$$
As a consequence, in (<ref>) $\gamma_{2n+1}$ has to be even too. Hence, the integral in (<ref>) can be non-zero only if

\begin{equation}\label{dariportare}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy= \int_{\B_c(0,1)} y^{\gamma} \|y\|_{\alpha}^{\alpha-Q} dy,
\end{equation}

for some even $(2n+1)$-multi-index $\gamma \in (\N \cup \{0\})^{2n+1}$.

We observe that our argument allows to state the equality

\begin{equation}\label{comb1}
\begin{split}
&\int_{\B_c(0,1)}\big(\mathcal{T}_{2m+1}(\phi,x)(xy)-\mathcal{T}_{2m-1}(\phi,x)(xy) \big) \|y\|_{\alpha}^{\alpha-Q} dy\\
=&   \sum_{\substack{k=1, \ldots, 2m, k  \ \mathrm{even}\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m}} \frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy.
\end{split}
\end{equation}

§.§.§ Representation, II

According to the previous subsection, we fix $m \in \N$, $k\in \{1, \ldots, 2m \}$ and $ i_1, \ldots,i_k \leq 2n+1$ such that

$$\mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m.$$
Applying the coarea formula of Corollary <ref>, we obtain

$$\int_{\B_c(0,1)} y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q} dy= \int_0^1 \int_{\partial \B_c(0,r)}  y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q}d\mathcal{S}^{Q-1}_{\infty}(y) dr.$$
By the homogeneity of the norm $\| \cdot \|_{\alpha}^{\alpha-Q}$ and the homogeneity of $\mathcal{S}^{Q-1}_{\infty}$, we get for every $-2m<\alpha<Q$

\begin{align*}
&\int_0^1 \int_{\partial \B_c(0,r)}  y_{i_1} \ldots y_{i_k} \|y\|_{\alpha}^{\alpha-Q}d\mathcal{S}^{Q-1}_{\infty}(y) dr\\
= & \int_0^1 \int_{\partial \B_c(0,1)}  (\delta_rz)_{i_1} \ldots (\delta_rz)_{i_k} \|\delta_r z\|_{\alpha}^{\alpha-Q} r^{Q-1} d\mathcal{S}^{Q-1}_{\infty}(z) dr\\
= & \int_0^1 \int_{\partial \B_c(0,1)}  r^{\mathrm{deg}(Z_{i_1})} z_{i_1} \ldots r^{ \mathrm{deg}(Z_{i_k})} z_{i_k} r^{\alpha-Q} \| z\|_{\alpha}^{\alpha-Q} r^{Q-1} d \mathcal{S}^{Q-1}_{\infty}(z)dr\\
= & \int_0^1 r^{\alpha-Q+Q-1+\mathrm{deg}(Z_{i_1})+\ldots \mathrm{deg}(Z_{i_k})} \int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z) dr\\
= & \int_0^1 r^{\alpha-1+2m} dr \int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z)\\
= & \frac{1}{\alpha+2m} \int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z).
\end{align*}

Finally, for every $-2m < \alpha<Q$ we can rewrite (<ref>) as

\begin{align}\label{comb2}
&\int_{\B_c(0,1)}\big(\mathcal{T}_{2m+1}(\phi,x)(xy)-\mathcal{T}_{2m-1}(\phi,x)(xy) \big) \|y\|_{\alpha}^{\alpha-Q} dy  \notag  \\
=&   \sum_{\substack{k=1, \ldots, 2m, k  \ \mathrm{even}\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m}}  \frac{1}{\alpha+2m} \frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z).
\end{align}

As a consequence, for every $-2m < \alpha<Q$ we also have that

\begin{align*}
&\int_{\B_c(0,1)}\big(\mathcal{T}_{2m+1}(\phi,x)(xy)-\mathcal{T}_{2m-1}(\phi,x)(xy) \big) P_{\alpha}(y) dy\\
=&   \sum_{\substack{k=1, \ldots, 2m, k  \ \mathrm{even}\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m}} \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha}{2})}\frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z).
\end{align*}

§.§.§ Regularity

Here, we prove that the following map is analytic

\begin{equation}\label{diff-analitica}
\begin{split}
(-\infty,Q) \ni \alpha \to &   \sum_{\substack{k=1, \ldots, 2m, k  \ \mathrm{even}\\ i_1, \ldots,i_k \leq 2n+1 \\ \mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k})=2m}} \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha}{2})}\frac{ Z_{i_1} \ldots Z_{i_k} \phi(x)}{k!}
\int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z).
\end{split}
\end{equation}

Let us first consider the map $\alpha \to \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha}{2})}$. By the standard properties of the Euler gamma function we have

\begin{align*}
&\lim_{\alpha \to -2m} \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha}{2})}\\
=& \lim_{\alpha \to -2m} \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha+2m}{2})}\left(\frac{\alpha}{2}+(m-1)\right)\left(\frac{\alpha}{2}+(m-2)\right) \ldots \left(\frac{\alpha}{2}\right) = \frac{1}{2}(-1)^m m!,
\end{align*}

then $\alpha \to \frac{1}{\alpha+2m}\frac{1}{\Gamma(\frac{\alpha}{2})}$ is analytic on $\R$.
Indeed, it is enough to show that for every even $k \in \{1, \ldots, 2m\}$ and every $i_1, \ldots i_k \leq 2n+1$ such that $\mathrm{deg}(Z_{i_1})+ \ldots \mathrm{deg}(Z_{i_k}) =2m$, the map

\begin{equation}\label{mappaalfa}
\begin{split}
(-\infty,Q) \ni \alpha \to \int_{\partial \B_c(0,1)}  z_{i_1} \ldots  z_{i_k}  \| z\|_{\alpha}^{\alpha-Q} d \mathcal{S}^{Q-1}_{\infty}(z)
\end{split}
\end{equation}