# The optimal constant in the $L^{2}$ Folland-Stein inequality on the
quaternionic Heisenberg group

Stefan Ivanov  University of Sofia, Faculty of Mathematics and Informatics,
blvd. James Bourchier 5, 1164, Sofia, Bulgaria<EMAIL_ADDRESS>,
Ivan Minchev  University of Sofia
Sofia, Bulgaria
and Mathematik und Informatik
Philipps-Universität Marburg
Hans-Meerwein-Str. / Campus Lahnberge 35032 Marburg, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Dimiter Vassilev  Department
of Mathematics and Statistics
University of New Mexico
Albuquerque, New Mexico, 87131-0001<EMAIL_ADDRESS>
###### Abstract.

We determine the best (optimal) constant in the $L^{2}$ Folland-Stein
inequality on the quaternionic Heisenberg group and the non-negative functions
for which equality holds.

###### Key words and phrases:

Yamabe equation, quaternionic contact structures

###### 1991 Mathematics Subject Classification:

58G30, 53C17

###### Contents

  1. 1 Introduction
  2. 2 The model quaternionic contact structures
  3. 3 The best constant in the Folland-Stein inequality

##  1\. Introduction

The goal of this note is to determine the best (optimal) constant in the
$L^{2}$ Folland-Stein inequality on the quaternionic Heisenberg group and the
non-negative extremal functions, i.e., the functions for which equality holds.
Alternatively, this is equivalent to finding the Yamabe constant of the
standard quaternionic contact structure of the sphere. The proof relies on the
realization of Branson, Fontana and Morpurgo [2], recently remarkably used
also by Frank and Lieb [6], that the old idea of Szegö [18], see also Hersch
[14], can be used to find the sharp form of (logarithmic) Hardy-Littlewood-
Sobolev type inequalities on the Heisenberg group.

The conformal nature of the problem we consider is key to its solution. The
analysis is purely analytical. In this respect, even though the quaternionic
contact (qc) Yamabe functional is involved, the qc scalar curvature is used in
the proof without much geometric meaning. Rather, it is the conformal sub-
laplacian that plays a central role and the qc scalar curvature appears as a
constant determined by the Cayley transform and the left-invariant sub-
laplacian on the quaternionic Heisenberg group. In this respect, this method
does not give all solutions of the qc Yamabe equation on the quaternionic
contact sphere. The complete solution of the latter problem requires some
additional very non-trivial argument and it is at this place where the
geometric nature of the problem becomes even more important. In the CR
setting, the solution of the CR Yamabe problem was achieved with the help of
an ingenious divergence formula [13]. The other known sub-Riemannian case is
that of the qc Yamabe equation on the seven dimensional standard quaternionic
contact sphere [11]. Another relevant result appeared earlier [7], where the
sub-Riemannian Yamabe equation was solved in the unifying setting of groups of
Iwasawa type under an additional assumption of partial symmetry of the
solution. This result can be used at the final stage of all known proofs after
such symmetry has been shown to exist. We recall that the groups of Iwasawa
type comprise of the complex (=”usual”), quaternion and octonian Heisenberg
groups, which are defined by (1.4) replacing, correspondingly, the quaternions
$\mathbb{H}$ with the complex numbers $\mathbb{C}$, the quaternions
$\mathbb{H}$, and the octonians $\mathbb{O}$.

Given a compact quaternionic contact manifold $M$ of real dimension $4n+3$
with an $\mathbb{R}^{3}$-valued contact form
$\eta=\\{\eta_{1},\eta_{2},\eta_{3}\\}$, i.e. a codimension three horizontal
distribution $H$ determined as the kernel of $\eta$ such that $d\eta_{|_{H}}$
are the fundamental two forms of a quaternionic hermitian structure $({\bf
g},I_{1},I_{2},I_{3})$ on $H$, $(d\eta_{s})_{|_{H}}=2{\bf
g}(I_{s}.,.)=2\omega_{s},s=1,2,3$, a natural question is to determine the qc
Yamabe constant of the conformal class $[\eta]$ of $\eta$ defined as the
infimum

(1.1) $\lambda(M,[\eta])\ =\ \inf\\{\Upsilon(u):\
\int_{M}u^{2^{*}}\,Vol_{\eta}\ =\ 1,\ u>0\\},$

where $Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$
denotes the volume form determined by $\eta$. The qc Yamabe functional of the
conformal class of $\eta$ is defined by

$\Upsilon(u)\ =\ \int_{M}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla u\rvert^{2}\
+\ S\,u^{2}\Bigr{)}Vol_{\eta},\qquad\int_{M}u^{2^{*}}\,Vol_{\eta}\ =\ 1,\
u>0,$

denoting by $\nabla$ the Biquard connection [1] of $\eta$, and $S$ standing
for the qc scalar curvature of $(M,\,\eta)$. This is the so called _qc Yamabe
constant problem_. In this paper we shall find
$\lambda(S^{4n+3},[\tilde{\eta}])$, where $\tilde{\eta}$ is the standard qc
form on the unit sphere $S^{4n+3}$, see (2.1). The question is of course
related to the solvability of the qc Yamabe equation

(1.2) $\mathcal{L}u\ \equiv\ 4\frac{Q+2}{Q-2}\ \triangle u-\ S\,u\ =\ -\
\overline{S}\,u^{2^{*}-1},$

where $\triangle$ is the horizontal sub-Laplacian, $\triangle u\ =\
tr^{g}(\nabla du)$, $S$ and $\overline{S}$ are the qc scalar curvatures
correspondingly of $(M,\,\eta)$ and $(M,\,\bar{\eta})$,
$\bar{\eta}=u^{4/(Q-2)}\eta$, and $2^{*}=\frac{2Q}{Q-2}.$ Here, and throughout
the paper $Q=4n+6$ is the homogeneous dimension. The natural question is to
find all solutions of the qc Yamabe equation. This is the so called _qc Yamabe
problem_ , which is equivalent to finding all qc structures conformal to a
given structure $\eta$ (of constant qc scalar curvature) which also have
constant qc scalar curvature. As usual the two problems are related by noting
that on a compact quaternionic contact manifold $M$ with a fixed conformal
class $[\eta]$ the qc Yamabe equation characterizes the non-negative extremals
of the qc Yamabe functional.

The 4n+3 dimensional sphere is an important example of a locally quternionic
contact conformally flat qc structure characterized locally in [12] with the
vanishing of a curvature-type tensor invariant. From the point of view of the
qc Yamabe problem the sphere plays a role similar to its Riemannian and CR
counterparts. A solution of the qc Yamabe problem on the seven dimensional
sphere equipped with its natural quaternionic contact structure was given in
[11] where more details on the qc Yamabe problem can be found. The main result
of [11] is the following

######  Theorem ([11]).

Let $\tilde{\eta}=\frac{1}{2h}\eta$ be a conformal deformation of the standard
qc-structure $\tilde{\eta}$ on the quaternionic unit sphere $S^{7}$. If $\eta$
has constant qc scalar curvature, then up to a multiplicative constant $\eta$
is obtained from $\tilde{\eta}$ by a conformal quaternionic contact
automorphism. In particular, $\lambda(S^{7})=48\,(4\pi)^{1/5}$ and this
minimum value is achieved only by $\tilde{\eta}$ and its images under
conformal quaternionic contact automorphisms.

Another motivation for studying the qc Yamabe equation and the qc Yamabe
constant of the qc sphere comes from its connection with the determination of
the norm and extremals in a relevant Sobolev-type embedding on the
quaternionic Heisenberg group [7] and [19] and [20]. As well known, the sub-
Riemannian Yamabe equation is also the Euler-Lagrange equation of the
extremals for the $L^{2}$ case of such embedding results. Recall the following
Theorem due to Folland and Stein [5].

######  Theorem (Folland and Stein).

Let $\Omega\subset{G}$ be an open set in a Carnot group ${G}$ of homogeneous
dimension $Q$ and Haar measure $dH$. For any $1<p<Q$ there exists
$S_{p}=S_{p}({G})>0$ such that for $u\in C^{\infty}_{o}(\Omega)$

(1.3) $\left(\int_{\Omega}\ |u|^{p^{*}}\ dH(g)\right)^{1/p^{*}}\leq\ S_{p}\
\left(\int_{\Omega}|Xu|^{p}\ dH(g)\right)^{1/p},$

where $|Xu|=\sum_{j=1}^{m}|X_{j}u|^{2}$ with $X_{1},\dots,X_{m}$ denoting a
basis of the first layer of ${G}$ and $p^{*}=\frac{pQ}{Q-p}$.

Let $S_{p}$ be the best constant in the Folland-Stein inequality, i.e., the
smallest constant for which (1.3) holds.

In [11] we determined all extremals, i.e., solutions of the qc Yamabe
equation, and the best constant in Folland and Stein’s theorem when $p=2$ in
the case of the seven dimensional quaternionic Heisenberg group. In the case
of the complex (i.e. ”usual”) Heisenberg group this was done earlier by
Jerison and Lee [13] who determined _all_ solutions to the CR Yamabe equation
on the CR sphere. In that setting, Frank and Lieb [6] determined the best
constant and found all functions for which the minimum is achieved, thus
simplifying parts of [13] while answering a less general question. However, in
[6] the authors also gave sharp forms of many Hardy-Littlewood-Sobolev type
inequalities on the Heisenberg group.

Following the idea of [6], the main result of this paper determines the best
constant in the Folland and Stein’s theorem when $p=2$ and the functions for
which it is achieved in the case of the quaternionic Heisenberg group
$\boldsymbol{G}$ of any dimension.

As a manifold $\boldsymbol{G}=\mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ with
the group law given by

(1.4) $(q^{\prime},\omega^{\prime})\ =\ (q_{o},\omega_{o})\circ(q,\omega)\ =\
(q_{o}\ +\ q,\omega\ +\ \omega_{o}\ +\ 2\ \text{Im}\ q_{o}\,\bar{q}),$

where $q,\ q_{o}\in\mathbb{H}^{n}$ and
$\omega,\omega_{o}\in\text{Im}\,\mathbb{H}$. The standard quaternionic
contact(qc) structure is defined by the left-invariant quaternionic contact
form

$\tilde{\Theta}\ =\ (\tilde{\Theta}_{1},\ \tilde{\Theta}_{2},\
\tilde{\Theta}_{3})\ =\ \frac{1}{2}\ (d\omega\ -\ q^{\prime}\cdot
d\bar{q}^{\prime}\ +\ dq^{\prime}\,\cdot\bar{q}^{\prime}),$

where $\cdot$ denotes the quaternion multiplication. The purpose of the
present note is to prove the next

######  Theorem 1.1.

a) Let $\boldsymbol{G}\ =\ \mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ be the
quaternionic Heisenberg group. The best constant in the $L^{2}$ Folland-Stein
embedding inequality (1.3) is

$S_{2}\ =\ \frac{\left[2^{-2n}\
\omega_{4n+3}\right]^{-1/(4n+6)}}{2\sqrt{n(n+1)}},$

where $\omega_{4n+3}=2\pi^{2n+2}/(2n+1)!$ is the volume of the unit sphere
$S^{4n+3}\subset\mathbb{R}^{4n+4}$. The non-negative functions for which (1.3)
becomes an equality are given by the functions of the form

(1.5) $\displaystyle F$ $\displaystyle=\ \gamma\left[(1+\lvert
q\rvert^{2})^{2}\ +\
\lvert\omega\rvert^{2}\right]^{-(n+1)},\qquad\gamma=const,$

and all functions obtained from $F$ by translations (3.2) and dilations (3.3).

b) The qc Yamabe constant of the standard qc structure of the sphere is

(1.6) $\lambda(S^{4n+3},[\tilde{\eta}])\ =\
16\,n(n+2)\,\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}.$

These constants are in complete agreement with the ones obtained in [11] and
[7] taking into account the next Remark and the well known formulas involving
the gamma function

$\displaystyle\Gamma(n+1)=n!,\qquad\Gamma(z+n)=z(z+1)\dots(z+n-1)\Gamma(z),\qquad
n\in\mathbb{N},$ $\displaystyle\omega_{n}=2\pi^{n/2}/\Gamma(n/2)\qquad\text{--
volume of unit $(n-1)$--dimensional sphere in }\mathbb{R}^{n},$
$\displaystyle\Gamma(2z)=2^{2z-1}\ \pi^{-1/2}\
\Gamma(z)\,\Gamma\left(z+\frac{1}{2}\right)\quad\text{ -- the Legendre
formula. }$

Our result partially confirms the Conjecture made after [7, Theorem 1.1]. In
addition, the fact that any function of the described form is a solution of
Yamabe problem was first noted in [8] in the setting of groups of Heisenberg
type. Of course, this class of groups is much wider than the class of groups
of Iwasawa type.

######  Remark 1.2.

With the left invariant basis of Theorem 1.1 the quaternionic Heisenberg group
is not a group of Heisenberg type. If we consider it as a group of Heisenberg
type then the best constant in the $L^{2}$ Folland-Stein embedding theorem is,
cf. [7, Theorem 1.6],

$S_{2}\ =\ \frac{1}{\sqrt{4n(4n+4)}}\ 4^{3/(4n+6)}\ \pi^{-(4n+3)/2(4n+6)}\
\left(\frac{\Gamma(4n+3)}{\Gamma((4n+3)/2)}\right)^{1/(4n+6)}.$

and extremals are given by dilations and translations of the function

$F(q,\omega)\ =\ \gamma\ \left[(1+|q|^{2})^{2}\ +\
16|\omega|^{2})\right]^{-(n+1)},\ (q,\omega)\in\boldsymbol{G}.$

It is worth pointing that studying the Yamabe extremals in the sub-Riemannian
setting has applications to sharp inequalities in the Euclidean setting. For
example, in the paper [21] are determined the extremals of some Euclidean
Hardy-Sobolev inequalities involving the distance to a $n-k$ dimensional
coordinate subspace of $\mathbb{R}^{n}$. This is achieved by relating
extremals on the Heisenberg groups to extremals in the Euclidean setting. In
the particular case when $k=n$ one obtains the Caffarelli-Kohn-Nirenberg
inequality, see [3], for which the optimal constant was found in [9].

######  Convention 1.3.

We use the following conventions:

  * •

the abbreviation qc will stand for quaternionic contact;

  * •

$\boldsymbol{G}$ will denote the qc Heisenberg group;

  * •

$\tilde{\eta}$ will denote the standard qc form on the unit sphere $S^{4n+3}$,
see (2.1). Note that this form is actually twice the 3-Sasakain qc form on
$S^{4n+3}$;

  * •

$Vol_{\eta}$ will denote the volume form determined by the qc form $\eta$,
thus $Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$,
see [10, Chapter 8].

Acknowledgments. Research was partially supported by Contract “Idei”, DO
02-257/18.12.2008 and DID 02-39/21.12.2009. S.I. and I. M. are partially
supported by the Contract 082/2009 with the University of Sofia
‘St.Kl.Ohridski’.

##  2\. The model quaternionic contact structures

In this section we review the standard quaternionic contact structure on the
quaternionic Heisenberg group and the $4n+3$-dimensional unit sphere. We will
rely heavily on [10], but prefer to repeat some key points in order to make
the current paper somewhat self-contained. Besides serving as a background,
this section will supply some key numerical constants - the qc scalar
curvature and the first eigenvalue of the sub-laplacian of the standard qc
form of the sphere. This will be achieved using the conformal sub-laplacian
and the properties of the Cayley transform.

First let us recall the quaternionic Heisenberg group [10, Section 5.2]. We
remind the following model of the quaternionic Heisenberg group
$\boldsymbol{G}$. Define $\boldsymbol{G}\ =\
\mathbb{H}^{n}\times\text{Im}\,\mathbb{H}$ with the group law given by
$(q^{\prime},\omega^{\prime})\ =\ (q_{o},\omega_{o})\circ(q,\omega)\ =\
(q_{o}\ +\ q,\omega\ +\ \omega_{o}\ +\ 2\ \text{Im}\ q_{o}\,\bar{q}),$ where
$q,\ q_{o}\in\mathbb{H}^{n}$ and $\omega,\omega_{o}\in\text{Im}\,\mathbb{H}$.
In coordinates, with $\omega=ix+jy+kz$ and
$q_{\alpha}=t_{\alpha}+ix_{\alpha}+jy_{\alpha}+kz_{\alpha}$, $\alpha=1,\dots
n$, a basis of left invariant horizontal vector fields
$T_{\alpha},X_{\alpha}=I_{1}T_{\alpha},Y_{\alpha}=I_{2}T_{\alpha},Z_{\alpha}=I_{3}T_{\alpha},\alpha=1\dots,n$
is given by

$\displaystyle T_{\alpha}\ =\ {\partial_{t_{\alpha}}}{}\
+2x_{\alpha}{\partial_{x}}{}+2y_{\alpha}{\partial_{y}}{}+2z_{\alpha}{\partial_{z}}{}\,\qquad
X_{\alpha}\ =\ {\partial_{x_{\alpha}}}{}\
-2t_{\alpha}{\partial_{x}}{}-2z_{\alpha}{\partial_{y}}{}+2y_{\alpha}{\partial_{z}}{}$
$\displaystyle Y_{\alpha}\ =\ {\partial_{y_{\alpha}}}{}\
+2z_{\alpha}{\partial_{x}}{}-2t_{\alpha}{\partial_{y}}{}-2x_{\alpha}{\partial_{z}}{}\,\qquad
Z_{\alpha}\ =\ {\partial_{z_{\alpha}}}{}\
-2y_{\alpha}{\partial_{x}}{}+2x_{\alpha}{\partial_{y}}{}-2t_{\alpha}{\partial_{z}}{}\,.$

The above vectors generate the _horizontal space_ , denoted as usual by $H$.
In addition, by declaring them to be an orthonormal basis we obtain a metric
on the horizontal space, which is the so called _horizontal metric_. The
central (vertical) vector fields $\xi_{1},\xi_{2},\xi_{3}$ are described as
follows

$\xi_{1}=2{\partial_{x}}{}\,\quad\xi_{2}=2{\partial_{y}}{}\,\quad\xi_{3}=2{\partial_{z}}{}\,.$

The standard quaterninic contact form, written as a purely imaginary
quaternion valued form
$\tilde{\Theta}=i\tilde{\Theta}_{1}+j\tilde{\Theta}_{2}+k\tilde{\Theta}_{3})$,
is

$2\tilde{\Theta}\ =\ \ d\omega\ -\ q^{\prime}\cdot d\bar{q}^{\prime}\ +\
dq^{\prime}\,\cdot\bar{q}^{\prime},$

where $\cdot$ denotes the quaternion multiplication. The Biquard connection
coincides with the flat left invariant connection on $\boldsymbol{G}$, in
particular the qc scalar curvature vanishes.

Following [10], we give another model of the Heiseneberggroup, which is the
one we will use in this paper. Let us identify $\boldsymbol{G}$ with the
boundary $\Sigma$ of a Siegel domain in $\mathbb{H}^{n}\times\mathbb{H}$,

$\Sigma\ =\ \\{(q^{\prime},p^{\prime})\in\mathbb{H}^{n}\times\mathbb{H}\ :\
\Re{\ p^{\prime}}\ =\ \lvert q^{\prime}\rvert^{2}\\},$

by using the map $(q^{\prime},\omega^{\prime})\mapsto(q^{\prime},\lvert
q^{\prime}\rvert^{2}-\omega^{\prime})$. Since $dp^{\prime}\ =\ q^{\prime}\cdot
d\bar{q}^{\prime}\ +\ dq^{\prime}\,\cdot\bar{q}^{\prime}\ -\
d\omega^{\prime},$ under the identification of $\boldsymbol{G}$ with $\Sigma$
we have also $2\tilde{\Theta}\ =\ -dp^{\prime}\ +\
2dq^{\prime}\cdot\bar{q}^{\prime}.$ Taking into account that $\tilde{\Theta}$
is purely imaginary, the last equation can be written also in the following
form

$4\,\tilde{\Theta}\ =\ (d\bar{p}^{\prime}\ -\ dp^{\prime})\ +\
2dq^{\prime}\cdot\bar{q^{\prime}}\ -\ 2q^{\prime}\cdot d\bar{q}^{\prime}.$

Now, consider the Cayley transform, see [15] and [4], as the map $\
\mathcal{C}:S\mapsto\Sigma\ $ from the sphere $S\ =\ \\{\lvert
q\rvert^{2}+\lvert p\rvert^{2}=1\\}\subset\mathbb{H}^{n}\times\mathbb{H}$
minus a point to the Heisenberg group $\Sigma$, with $\mathcal{C}$ defined by

$(q^{\prime},p^{\prime})\ =\ \mathcal{C}\ \Big{(}(q,p)\Big{)},\qquad
q^{\prime}\ =\ (1+p)^{-1}\ q,\qquad p^{\prime}\ =\ (1+p)^{-1}\ (1-p)$

and with an inverse map $(q,p)\ =\
\mathcal{C}^{-1}\Big{(}(q^{\prime},p^{\prime})\Big{)}$ given by

$q\ =\ \ 2(1+p^{\prime})^{-1}\ q^{\prime},\qquad p\ =\ (1+p^{\prime})^{-1}\
(1-p^{\prime}).$

The Cayley transform maps $S^{4n+3}\setminus\\{(-1,0)\\}$,
$(-1,0)\in\mathbb{H}^{n}\times\mathbb{H}$, to $\Sigma$ since

$\Re{\ p^{\prime}}\ =\ \Re{\frac{(1+\bar{p})(1-p)}{\lvert 1+p\,\rvert^{2}}}\
=\ \Re{\frac{1-\lvert p\rvert}{\lvert 1+p\,\rvert^{2}}}\ =\ \frac{\lvert
q\rvert^{2}}{\lvert 1+p\,\rvert^{2}}\ =\ \lvert q^{\prime}\rvert^{2}.$

Writing the Cayley transform in the form $(1+p)q^{\prime}\ =\ \
q,\quad(1+p)p^{\prime}\ =\ 1-p,$ gives

$dp\cdot q^{\prime}\ +\ (1+p)\cdot dq^{\prime}\ =\ dq,\hskip 36.135ptdp\cdot
p^{\prime}\ +\ (1+p)\cdot dp^{\prime}\ =\ -dp,$

from where we find

$\displaystyle dp^{\prime}$ $\displaystyle=\ -2(1+p)^{-1}\cdot
dp\cdot(1+p)^{-1}$ $\displaystyle dq^{\prime}$ $\displaystyle=\
(1+p)^{-1}\cdot[dq\ -\ dp\cdot(1+p)^{-1}\cdot q].$

The Cayley transform is a conformal quaternionic contact diffeomorphism
between the quaternionic Heisenberg group with its standard quaternionic
contact structure $\tilde{\Theta}$ and the sphere minus a point with its
standard structure $\tilde{\eta}$. In fact, by [10, Section 8.3] we have

$\Theta\ \overset{def}{=}\ \lambda\ \cdot(\mathcal{C}^{-1})^{*}\,\tilde{\eta}\
\cdot\bar{\lambda}\ =\ \frac{8}{\lvert
1+p^{\prime}\,\rvert^{2}}\,\tilde{\Theta}.$

where $\lambda\ ={\lvert 1+p\,\rvert}\,{(1+p)^{-1}}$ is a unit quaternion and
$\tilde{\eta}$ is the standard contact form on the sphere,

(2.1) $\tilde{\eta}\ =\ dq\cdot\bar{q}\ +\ dp\cdot\bar{p}\ -\ q\cdot
d\bar{q}-\ p\cdot d\bar{p}.$

######  Lemma 2.1.

The qc scalar curvature $\tilde{S}$ of the standard qc structure (2.1) on
$S^{4n+3}$ is

(2.2) $\tilde{S}\ =\ \frac{1}{2}(Q+2)(Q-6)=8n(n+2).$

######  Remark 2.2.

Notice that the standard qc contact form we consider here is twice the
3-Sasakian form on $S^{4n+3}$, which has qc scalar curvature equal to 16n(n+2)
[10].

###### Proof.

Let us introduce the functions

(2.3) $\displaystyle h$
$\displaystyle=\frac{1}{16}|1+p^{\prime}|^{2}=\frac{1}{16}\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right],\qquad(q^{\prime},p^{\prime})\in\Sigma\subset\mathbb{H}^{n}\times\mathbb{H},\quad
p^{\prime}=|q^{\prime}|^{2}+\omega^{\prime},$ and $\displaystyle\Phi$
$\displaystyle=\left({2h}\right)^{-(Q-2)/4}=8^{(Q-2)/4}\
{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4},$

so that now we have

$\Theta=\frac{1}{2h}\tilde{\Theta}=\Phi^{4/(Q-2)}\tilde{\Theta}.$

With the help of [10, Section 5.2] a small calculation shows that the sub-
laplacian of $h$ w.r.t. $\tilde{\Theta}$ is given by $\triangle
h=\frac{Q-6}{4}+\frac{Q+2}{4}|q^{\prime}|^{2}$ and thus $\Phi$ is a solution
of the qc Yamabe equation on the Heisenberg group $\Sigma$

(2.4) $\triangle\Phi=-K\,\Phi^{2^{*}-1},\qquad K=(Q-2)(Q-6)/8,$

where $\triangle$ is the sub-laplacian on the quaternionic Heisenberg group.
Denoting with $\mathcal{L}$ and $\tilde{\mathcal{L}}$ the conformal sub-
laplacians of $\Theta$ and $\tilde{\Theta}$, respectively, we have

$\Phi^{-1}\mathcal{L}(\Phi^{-1}u)=\Phi^{-2^{*}}\tilde{\mathcal{L}}u.$

We remind, cf. [1] and [11], that for a qc contact form $\Theta$ the conformal
sublaplacian is,

$\mathcal{L}=a\triangle_{\Theta}-S_{\Theta},\qquad a=4\frac{Q+2}{Q-2},$

where $\triangle_{\Theta}$ is the sub-laplacian associated to $\Theta$, i.e.,
$\triangle_{\Theta}u=tr(\nabla^{\Theta}du)$–the horizontal trace of the
Hessian of $u$, using the Biquard connection $\nabla^{\Theta}$ of $\Theta$,
and $S_{\Theta}$ is the qc scalar curvature of $\Theta$. Thus, letting
$u=\Phi$ we come to $\mathcal{L}(1)=\Phi^{1-2^{*}}\tilde{\mathcal{L}}\Phi$,
which shows $-S_{\Theta}=-4\frac{Q+2}{Q-2}K$. The latter is the same as that
of $\tilde{\eta}$ since the two structures are isomorphic via the
diffemorphism $\mathcal{C}$, or rather its extension, since we can consider
$\mathcal{C}$ as a quaternionic contact conformal transformation between the
whole sphere $S^{4n+3}$ and the compactification $\hat{\Sigma}\cup{\infty}$ of
the quaternionic Heisenberg group by adding the point at infinity, cf. [11,
Section 5.2]. ∎

We turn to the task of determining the first eigenvalue of the sub-laplacian
on $S^{4n+3}$. In fact, we shall need only the fact that the restriction of
every coordinate function is an eigenvalue. The proof of this fact can be seen
directly without any reference to the Biquard connection, but this will
require setting a lot of notation, so we prefer to use a result from [10].

######  Lemma 2.3.

If $\zeta$ is any of the (real) coordinate functions in
$\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$, then

(2.5)
$\tilde{\triangle}\zeta=-\lambda_{1}\zeta,\quad\lambda_{1}=\frac{\tilde{S}}{Q+2}={2n}$

for the horizontal trace of the Hessian, where $\tilde{\triangle}$ is the sub-
laplacian of the standard qc form $\tilde{\eta}$ of $S^{4n+3}$.

###### Proof.

It is enough to furnish a proof for the sub-laplacian on the 3-Sasakain sphere
since the two qc forms defer by a constant. We can see that every $\zeta$ of
the considered type is an eigenfunction by using [10, Corollary 6.24]. It will
be enough to see it for one coordinate function provided the sub-laplacian on
the sphere is rotation invariant. Thus, let us take $\zeta=t_{1}$. Notice that
$\zeta$ is quaternionic pluri-harmonic [10, Definition 6.7] since it is the
real part of the anti-regular function $t_{1}+ix_{1}-jy_{1}-kz_{1}$. So, its
restriction to the 3-Sasakain sphere is the real part of an anti-CRF function.
Therefore we apply [10, Corollary 6.24] which gives $tr(\nabla
d\zeta)=4\lambda n$ for the sub-laplacian of the 3-sasakain qc structure on
the sphere. Next, we compute $\lambda$, which can be found in [10, Theorem
6.20]. Using that the sphere is 3-Sasakian it follows the Reeb vector fields
are obtained from the outward pointing unit normal vector $N$ as follows,
$\xi_{1}=iN$, $\xi_{2}=jN$ and $\xi_{3}=kN$, where for a point on the sphere
we have $N(q)=q\in\mathbb{H}^{n+1}$. Therefore $\lambda=-t_{1}=-\zeta$. To see
this easier notice that only the first four coordinates of $N$ matter. So, if
we assume $n=0$ we have $iN=-x+it+ky-jz$, $jN=-y+iz+jt-kx$ and
$kN=-z-iy+jx+kt$, so we need to sum the _real dot product_ of these vectors
with $i$, $j$ and $k$, respectively, which gives $-t$. Thus, for the sub-
laplacian on the 3-Sasakian sphere we have

$tr(\nabla d\zeta)=-4n\zeta,$

where $\zeta$ is the restriction any of the coordinate functions of
$\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$. Since the qc contact form
$\tilde{\Theta}$ is twice the 3-Sasakain qc contact form on the sphere it
follows $\tilde{\triangle}$ is $1/2$ of the 3-Sasakain sub-laplacian. Thus

$\tilde{\triangle}=-2n\zeta,$

which shows $\lambda_{1}=2n=\frac{1}{2}(Q-6)={\tilde{S}}/{(Q+2)}$. ∎

We finish this section with a simple Lemma which will be used to relate the
various explicit constants. Its claim also follows from the conformal
invariance of the Yamabe equation, but we prefer to give a proof, which is
independent of the notion of qc scalar curvature. We recall, see [10, Chapter
8], that $Vol_{\eta}$ will denote the volume form determined by the qc form
$\eta$, thus
$Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$.
Also, for a qc form $\eta$ we let
$|\nabla^{\eta}F|^{2}=\sum_{\alpha=1}^{4n}|dF(e_{\alpha})|^{2}$ be the square
of the length of the horizontal gradient of a function $F$ taken with respect
to an orthonormal basis of the horizontal space $H=Ker\,\eta$ and the metric
determined by $\eta$.

######  Lemma 2.4.

Let $F\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$, cf. (3.1), be a
positive function with $\int_{\boldsymbol{G}}\
F^{2^{*}}Vol_{\tilde{\Theta}}=1$. Then we have

(2.6) $\int_{\boldsymbol{G}}\ a|\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}\ =\ \int_{S^{4n+3}}\
\left(a|\nabla^{\tilde{\eta}}g|^{2}+\tilde{S}g^{2}\right)\
Vol_{\tilde{\eta}},\qquad a=4(2^{*}-1),$

and

$\int_{\boldsymbol{G}}\ g^{2^{*}}Vol_{\tilde{\eta}}=1,$

where

(2.7) $g=\mathcal{C}^{*}(F\Phi^{-1}),$

and, as before, $\mathcal{C}:S^{4n+3}\rightarrow\Sigma$ is the Cayley
transform, $\Theta=\Phi^{4/(Q-2)}\tilde{\Theta}$, cf. (2.3).

######  Remark 2.5.

Notice that $Vol_{\tilde{\Theta}}=2^{-3}\,(2n)!\,dH$, where $dH$ is the
Lebesgue measure in $\mathbb{R}^{4n+3}$, which is a Haar measure on the group.

###### Proof.

It will be convenient for the remaining of this proof to denote by small
letters the pull-back by the Cayley transform of a function denoted with the
corresponding capital letter. Thus, $f=\mathcal{C}^{*}F=F\circ\mathcal{C}$,
$\phi=\mathcal{C}^{*}(\Phi)$ and $g=f\phi^{-1}$. By the conformality of the qc
structures on the group and the sphere we have

(2.8) $Vol_{\Theta}=\Phi^{2^{*}}Vol_{\tilde{\Theta}}$

By (2.8) we have
$F^{2^{*}}Vol_{\tilde{\Theta}}=f^{2^{*}}\phi^{-2^{*}}Vol_{\tilde{\eta}}$,
which motivates the definition (2.7) of the function $g$ which is defined on
the sphere and should be regarded as corresponding to the function $F$. Thus,
we have for example $F=G\Phi$. By definition we have

$\int_{\boldsymbol{G}}\ g^{2^{*}}Vol_{\tilde{\eta}}=1,$

so our next task is to see that the Yamabe integral is preserved

(2.9) $\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}\ =\ \int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+Kg^{2}\right)\ Vol_{\tilde{\eta}}.$

Here is where we shall exploit that a power of the conformal factor of the
Cayley transform is a solution of the Yamabe equation. Let
$\left<\nabla^{\Theta}\Phi,\nabla^{\Theta}G\right>=\sum_{a=1}^{4n}(e_{a}\Phi)\,(\textbf{e}_{a}G)$
where $\\{e_{1},\dots,e_{4n}\\}$ is an orthonormal basis of the horizontal
space $H$. Using the divergence formula from [10, Section 8.1] we find

$\int_{\boldsymbol{G}}\ |\tilde{\nabla}^{\Theta}F|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
|\nabla^{\tilde{\Theta}}(G\Phi)|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
\Big{(}G^{2}|\nabla^{\tilde{\Theta}}\Phi|^{2}+\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}+\left<\Phi\nabla^{\tilde{\Theta}}\Phi,\nabla^{\tilde{\Theta}}G^{2}\right>\Big{)}\
Vol_{\tilde{\Theta}}\\\ =\int_{\boldsymbol{G}}\
\Big{(}\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}-G^{2}\Phi\triangle_{\tilde{\Theta}}\Phi\
\Big{)}Vol_{\tilde{\Theta}}.$

Now, the Yamabe equation (2.4) gives

$\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}F|^{2}\
Vol_{\tilde{\Theta}}=\int_{\boldsymbol{G}}\
\left(\Phi^{2}|\nabla^{\tilde{\Theta}}G|^{2}+KG^{2}\Phi^{2^{*}}\
\right)Vol_{\tilde{\Theta}}\\\ =\int_{S^{4n+3}}\
\left(\phi^{2-2^{*}}(|\nabla^{\tilde{\Theta}}G|\circ\mathcal{C})^{2}+Kg^{2}\
\right)Vol_{\tilde{\eta}}=\int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+Kg^{2}\ \right)Vol_{\tilde{\eta}},$

taking into account that $\mathcal{C}$ is a qc conformal map. Finally, a
glance at (2.4) and (2.2) shows $\tilde{S}/K=4(2^{*}-1)=(4(Q+2)/(Q-2)$ which
allows to put (2.9) in the form (2.6). ∎

##  3\. The best constant in the Folland-Stein inequality

In this section, following [6], we prove the main Theorem. It is important to
observe that a suitable adaptation of the method of concentration of
compactness due to P. L. Lions [16], [17] allows to prove that in any Carnot
group the Yamabe constant and optimal constant in the Folland-Stein inequality
is achieved in the space $\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$,
see [19] and [20]. Here

$\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})=\overline{C^{\infty}_{o}(\boldsymbol{G})}^{||\cdot||_{\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})}}.$

The space $\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})$ is endowed with
the norm

(3.1) $||u||_{\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G})}\ =\
||\,|\nabla u|\,||_{L^{2^{*}}(\boldsymbol{G})}.$

where $\nabla u$ is the horizontal gradient of $u$ and $|\nabla
u|^{2}=\sum_{a=1}^{4n}(e_{a}u)^{2}$ for an orthonormal basis
$\\{e_{1},\dots,e_{4n}\\}$ of horizontal left invariant vector fields.

In this regard an elementary, yet crucial observation, is that if $u$ is an
entire solution to the Yamabe equation, then such are also the two functions

(3.2) $\tau_{h}u\ \overset{def}{=}\ u\circ\tau_{h},\quad\quad\quad
h\in\boldsymbol{G},$

where $\tau_{h}:\boldsymbol{G}\to\boldsymbol{G}$ is the operator of left-
translation $\tau_{h}(g)=hg$, and

(3.3) $u_{\lambda}\ \overset{def}{=}\ \lambda^{(Q-2)/2}\
u\circ\delta_{\lambda},\quad\quad\quad\lambda>0.$

The Heisenberg dilations are defined by

$\delta_{\lambda}\left((q^{\prime},\omega^{\prime})\right)=\left((\lambda
q^{\prime},\lambda^{2}\omega^{\prime})\right),\qquad(q^{\prime},\omega^{\prime})\in\boldsymbol{G}$

It is also well known, [19] and [20], that there are smooth positive minimizer
of the Folland-Stein inequality on the quaternionic Heisenberg group
$\boldsymbol{G}$. These facts will be used without further notice on
regularity and existence.

We start with the ”new” key, see [2] and [6], allowing the ultimate solution
of the considered problem.

######  Lemma 3.1.

For every $v\in L^{1}(S^{4n+3})$ with $\int_{S^{4n+3}}v\ Vol_{\tilde{\eta}}=1$
there is a quaternionic contact conformal transformation $\psi$ such that

$\int_{S^{4n+3}}\psi\,v\ Vol_{\tilde{\eta}}=0.$

###### Proof.

Let $P\in S^{4n+3}$ be any point of the quaternionic sphere and $N$ be its
antipodal point. Let us consider the local coordinate system near $P$ defined
by the Cayley transform $\mathcal{C}_{N}$ from $N$. It is known that
$\mathcal{C}_{N}$ is a quaternionic contact conformal transformation between
$S^{4n+3}\setminus{N}$ and the quaternionic Heisenberg group. Notice that in
this coordinate system $P$ is mapped to the identity of the group. For every
$r$, $0<r<1$, let $\psi_{r,P}$ be the qc conformal transformation of the
sphere, which in the fixed coordinate chart is given on the group by a
dilation with center the identity by a factor $\delta_{r}$. If we select a
coordinate system in $\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}$ so
that $P=(1,0)$ and $N=(-1,0)$ and then apply the formulas for the Cayley
transform from [10, Section 8.2] the formula for
$(q^{*},p^{*})=\psi_{r,P}(q,p)$ becomes

$\displaystyle q^{*}$
$\displaystyle=2r\left(1+r^{2}(1+p)^{-1}(1-p)\right)^{-1}\left(1+p\right)q$
$\displaystyle p^{*}$
$\displaystyle=\left(1+r^{2}(1+p)^{-1}(1-p)\right)^{-1}\left(1-r^{2}(1+p)^{-1}(1-p)\right),i.e,$

We can define then the map $\Psi:B\rightarrow\bar{B}$, where $B$ ( $\bar{B}$ )
is the open (closed) unit ball in $\mathbb{R}^{4n+4}$, by the formula

$\Psi(rP)=\int_{S^{4n+3}}\psi_{1-r,P}\,v\ Vol_{\tilde{\eta}}.$

Notice that $\Psi$ can be continuously extended to $\bar{B}$ since for any
point $P$ on the sphere, where $r=1$, we have $\psi_{1-r,P}(Q)\rightarrow P$
when $r\rightarrow 1$. In particular, $\Psi=id$ on $S^{4n+3}$. Since the
sphere is not a homotopy retract of the closed ball it follows that there are
$r$ and $P\in S^{4n+3}$ such that $\Psi(rP)=0$, i.e.,
$\int_{S^{4n+3}}\psi_{1-r,P}\,v\ Vol_{\tilde{\eta}}=0$. Thus,
$\psi=\psi_{1-r,P}$ has the required property. ∎

In the next step we prove that we can assume that the minimizer of the
Folland-Stein inequality satisfies the zero center of mass condition. A number
of well known invariance properties of the Yamabe functional will be
exploited.

######  Lemma 3.2.

Let $v$ be a smooth positive function on the sphere with
$\int_{S^{4n+3}}v^{2^{*}}\,Vol_{\tilde{\eta}}=1$. There is a smooth positive
function $u$ such that $\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla
u\rvert^{2}\ +\ \tilde{S}\,u^{2}\Bigr{)}\,Vol_{\tilde{\eta}}\
=\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla v\rvert^{2}\ +\
\tilde{S}\,v^{2}\Bigr{)}\,Vol_{\tilde{\eta}}$ and
$\int_{S^{4n+3}}u^{2^{*}}\,Vol_{\tilde{\eta}}=1$. In addition,

(3.4) $\int_{S^{4n+3}}P\,u^{2^{*}}(P)\,Vol_{\tilde{\eta}}=0,\qquad
P\in\mathbb{R}^{4n+4}=\mathbb{H}^{n}\times\mathbb{H}.$

In particular, the Yamabe constant

(3.5) $\lambda(S^{4n+3},[\tilde{\eta}])\ =\
\inf\\{\int_{S^{4n+3}}\Bigl{(}4\frac{Q+2}{Q-2}\ \lvert\nabla v\rvert^{2}\ +\
\tilde{S}\,v^{2}\Bigr{)}Vol_{\tilde{\eta}}:\ \int_{S^{4n+3}}v^{2^{*}}\
Vol_{\tilde{\eta}}\ =\ 1,\ v>0\\}$

is achieved for a positive function $u$ with a zero center of mass, i.e., for
a function $u$ satisfying (3.4).

###### Proof.

By [10, Section 8.1],
$Vol_{\eta}=\eta_{1}\wedge\eta_{2}\wedge\eta_{3}\wedge(\omega_{1})^{2n}$ is a
volume form on a qc manifold with contact form $\eta$. Thus if $\eta$ is a qc
structure on the sphere which is qc conformal to the standard qc structure
$\tilde{\eta}$, $\eta=\phi^{4/(Q-2)}\tilde{\eta}$, then
$Vol_{\eta}=\phi^{2^{*}}Vol_{\tilde{\eta}}$. This allows to cast equation
(1.2) in the form

$\phi^{-1}v\,\mathcal{L}(\phi^{-1}v)\ Vol_{\eta}=v\mathcal{\tilde{L}}(v)\
Vol_{\tilde{\eta}}.$

Therefore, if we take a positive function $v$ on the sphere
$\int_{S^{4n+3}}v^{2^{*}}\,Vol_{\tilde{\eta}}=1$ and then consider the
function

(3.6) $u=\phi^{-1}(v\circ\psi^{-1}),$

where $\psi$ is the qc conformal map of Lemma 3.1,
$\eta\equiv(\psi^{-1})^{*}\tilde{\eta}$, and $\phi$ is the corresponding
conformal factor of $\psi$, we can see that $u$ achieves the claim of the
Lemma. ∎

We shall call a function $u$ on the sphere a well centered function when (3.4)
holds true. In the next step we show that a well centered minimizer has to be
constant.

######  Lemma 3.3.

If $u$ is a well centered local minimum of the problem (3.5), then $u\equiv
const$.

###### Proof.

Let $\zeta$ be a smooth function on the sphere $S^{4n+3}$. After applying the
divergence formula [10, Section 8] we obtain the formula

(3.7) $\Upsilon(\zeta u)=\int_{S^{4n+3}}\zeta^{2}\Bigl{(}4\frac{Q+2}{Q-2}\
\lvert\tilde{\nabla}u\rvert^{2}\ +\ \tilde{S}\,u^{2}\Bigr{)}\
Vol_{\tilde{\eta}}\ -\
4\frac{Q+2}{Q-2}\int_{S^{4n+3}}u^{2}\zeta{\tilde{\triangle}}\zeta\
Vol_{\tilde{\eta}}.$

This suggests to take as a test function $\zeta$ an eigenfunction of the sub-
laplacian ${\tilde{\triangle}}$ of the standard qc structure. In particular,
we can let $\zeta$ be any of the coordinate functions in
$\mathbb{H}^{n}\times\mathbb{H}$ in which case
${\tilde{\triangle}}\zeta=-\lambda_{1}\zeta$.

It will be useful to introduce the functional
$N(v)=\left(\int_{S^{4n+3}}v^{2^{*}}\ Vol_{\tilde{\eta}}\right)^{2/2^{*}}$ so
that

(3.8) $\lambda(S^{4n+3},[\tilde{\eta}])=\inf\\{\mathcal{E}(v):\,v\in
D\,(S^{4n+3})\\},\qquad\mathcal{E}(v)\overset{def}{=}\Upsilon(v)/N(v).$

Computing the second variation
$\delta^{2}\mathcal{E}(u)v=\frac{d^{2}}{dt^{2}}\mathcal{E}(u+tv)_{|_{t=0}}$ of
$\mathcal{E}(u)$ we see that the local minimum condition
$\delta^{2}\mathcal{E}(u)v\geq 0$ implies

$\Upsilon(v)-(2^{*}-1)\Upsilon(u)\int_{S^{4n+3}}u^{2^{*}-2}v^{2}\
Vol_{\tilde{\eta}}\ \geq 0$

for any function $v$ such that $\int_{S^{4n+3}}u^{2^{*}-1}v\
Vol_{\tilde{\eta}}=0$. Therefore, for $\zeta$ being any of the coordinate
functions in $\mathbb{H}^{n}\times\mathbb{H}$ we have

$\Upsilon(\zeta u)-(2^{*}-1)\Upsilon(u)\int_{S^{4n+3}}u^{2^{*}}\zeta^{2}\
Vol_{\tilde{\eta}}\ \geq 0,$

which after summation over all coordinate functions taking also into account
(3.7) gives

$\Upsilon(u)-(2^{*}-1)\Upsilon(u)-4\lambda_{1}(2^{*}-1)\int_{S^{4n+3}}u^{2}\
Vol_{\tilde{\eta}}\geq 0,$

which implies, recall $2^{*}-1=({Q+2})/({Q-2})$,

$0\leq 4(2^{*}-1)\left(2^{*}-2\right)\int_{S^{4n+3}}|\tilde{\nabla}u|^{2}\
Vol_{\tilde{\eta}}\\\ \leq\
\left(4\lambda_{1}(2^{*}-1)-\left(2^{*}-2\right)\tilde{S}\right)\int_{S^{4n+3}}u^{2^{*}}\
Vol_{\tilde{\eta}}.$

Thus, our task of showing that $u$ is constant will be achieved once we see
that

(3.9) $4\lambda_{1}(2^{*}-1)-\left(2^{*}-2\right)\tilde{S}\leq 0,\ \text{ i.e,
}\ \lambda_{1}\leq\tilde{S}/(Q+2).$

By Lemma 2.5 we have actually equality $\lambda_{1}={\tilde{S}}/{(Q+2)}$,
which completes the proof. It is worth observing that inequality (3.9) can be
written in the form

$\lambda_{1}\,a\leq(2^{*}-2)\,\tilde{S},$

where $a$ is the constant in front of the (sub-)laplacian in the conformal
(sub-)laplacian, i.e., $a=4\frac{Q+2}{Q-2}$ in our case. ∎

At this point the proof of our main Theorem 1.1 follows easily as follows.

###### Proof of Theorem 1.1.

Let $F$ be a minimizer (local minimum) of the Yamabe functional $\mathcal{E}$
on $\boldsymbol{G}$ and $g$ the corresponding function on the sphere defined
in Lemma 2.4. By Lemma 3.2 and (3.6) the function
$g_{0}=\phi^{-1}(g\circ\psi^{-1})$ will be well centered and a minimizer
(local minimum) of the Yamabe functional $\mathcal{E}$ on $S^{4n+3}$. The
latter claim uses also the fact that the map $v\mapsto u$ of equation (3.6) is
one-to-one and onto on the space of smooth positive functions on the sphere.
Now, from Lemma 3.3 we conclude that $g_{o}=const$. Looking back at the
corresponding functions on the group we see that

$F_{0}=\gamma\,{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4}$

for some $\gamma=const.>0.$ Furthermore, the proof of Lemma 3.1 shows that
$F_{0}$ is obtained from $F$ by a translation (3.2) and dilation (3.3).
Correspondingly, any positive minimizer (local maximum) of problem (3.11) is
given up to dilation or translation by the function

(3.10)
$F=\gamma\,{\left[(1+|q^{\prime}|^{2})^{2}+|\omega^{\prime}|^{2}\right]}^{-(Q-2)/4},\qquad\gamma=const.>0.$

Of course, translations (3.2) and dilations (3.3) do not change the value of
$\mathcal{E}$. Incidentally, this shows that any local minimum of the Yamabe
functional $\mathcal{E}$ on the sphere or the group has to be a global one.

We turn to the determination of the best constant. Let us define the constants

(3.11) $\displaystyle\Lambda_{\tilde{\Theta}}\ \overset{def}{=}\ inf\
\left\\{\underset{\boldsymbol{G}}{\int}|\nabla v|^{2}\ Vol_{\tilde{\Theta}}\
:\ v\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G}),\ v\geq 0,\
\underset{\boldsymbol{G}}{\int}|v|^{2^{*}}\ Vol_{\tilde{\Theta}}\ =\
1\right\\}$ and $\displaystyle\Lambda\overset{def}{=}\ inf\
\left\\{\underset{\boldsymbol{G}}{\int}|\nabla v|^{2}\ dH\ :\
v\in\overset{o}{\mathcal{D}}\,^{1,2}(\boldsymbol{G}),\ v\geq 0,\
\underset{\boldsymbol{G}}{\int}|v|^{2^{*}}\ dH\ =\ 1\right\\}.$

Clearly, $\Lambda_{\tilde{\Theta}}\ {=}\ S_{\tilde{\Theta}}^{-2}$, where
$S_{\tilde{\Theta}}$ is the best constant in the $L^{2}$ Folland-Stein
inequality

(3.12) $\left(\int_{\boldsymbol{G}}\ |u|^{2^{*}}\
Vol_{\tilde{\Theta}}\right)^{1/2^{*}}\leq\ S_{\tilde{\Theta}}\
\left(\int_{\boldsymbol{G}}\ |\nabla^{\tilde{\Theta}}u|^{2}\
Vol_{\tilde{\Theta}}\right)^{1/2},$

while $\Lambda\ {=}\ S_{2}^{-2}$ is the best constant in the $L^{2}$ Folland-
Stein inequality (1.3) (taken with respect to the Lebesgue measure !). By
Remark 2.5 we have

$\Lambda_{\tilde{\Theta}}\ {=}\ \left[2^{-3}(2n)!\right]^{1/(2n+3)}\ \Lambda.$

Furthermore, by Lemma 3.3 and equations (2.6) and (2.7) with $g=const$, we
have

$\Lambda_{\tilde{\Theta}}\ =\ \frac{1}{S_{2}^{2}}=\frac{\int_{\boldsymbol{G}}\
|\nabla^{\tilde{\Theta}}F|^{2}\,Vol_{\tilde{\Theta}}}{\left[\int_{\boldsymbol{G}}|F|^{2^{*}}\,Vol_{\tilde{\Theta}}\right]^{2/2^{*}}}\\\
=\ \frac{\int_{S^{4n+3}}\
\left(|\nabla^{\tilde{\eta}}g|^{2}+\frac{\tilde{S}}{a}\,g^{2}\right)\,Vol_{\tilde{\eta}}}{\left[\int_{S^{4n+3}}|g|^{2^{*}}\,Vol_{\tilde{\eta}}\right]^{2/2^{*}}}\
=\ 4n(n+1)\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}.$

Here,

$\omega_{4n+3}=2\pi^{2n+2}/\Gamma(2n+2)=2\pi^{2n+2}/(2n+1)!$

is the volume of the unit sphere $S^{4n+3}\subset\mathbb{R}^{4n+4}$ and we
also took into account Remark 2.2 which shows that $Vol_{\tilde{\eta}}$ gives
$2^{2n+3}\left((2n)!\right)\omega_{4n+3}$ for the volume of $S^{4n+3}$. Thus,

$S_{\tilde{\Theta}}\ =\
\left(4n(n+1)\left[\left((2n)!\right)\omega_{4n+3}\right]^{1/(2n+3)}\right)^{-1/2}\
=\
\frac{\left[\left((2n)!\right)\omega_{4n+3}\right]^{-1/(4n+6)}}{2\sqrt{n(n+1)}},$

which completes the proof of part a).

b) The Yamabe constant of the sphere is calculated immediately by taking a
constant function in (3.8)

(3.13) $\lambda(S^{4n+3},[\tilde{\eta}])=a\,\Lambda_{\tilde{\Theta}},\qquad
a=4\frac{Q+2}{Q-2}=4\frac{n+2}{n+1}.$

This completes the proof of Theorem 1.1. ∎

######  Remark 3.4.

In view of the above Lemmas it follows that in the conformal class of the
standard qc structure on the sphere (or the quaternionic Heisenberg group)
there is an extremal qc contact form for problem (1.1) which is also qc-
Einstein, see [10, Definition 4.1], and has partial symmetry, see [7,
Definition 1.2], if viewed as a qc structure on the group. Thus, the above
precise constants and extremals can also be taken directly from [10, Theorem
1.1 and 1.2] or the result of [7]. However, the functions (3.10) depend on one
more arbitrary multiplicative parameter $\gamma$ since in the current paper we
are dealing with the functions realizing the infimum of (3.8) rather than with
the qc Yamabe equation with a fixed qc scalar curvature.

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