# A compactness result for the CR Yamabe problem in three dimensions

Claudio Afeltra

###### Abstract

We prove the compactness of the set of solutions to the CR Yamabe problem on a
compact strictly pseudoconvex CR manifold of dimension three whose blow-up
manifolds at every point have positive p-mass. As a corollary we deduce that
compactness holds for CR-embeddable manifolds which are not CR-equivalent to
$S^{3}$. The theorem is proved by blow-up analysis.

##  1 Introduction

After the Yamabe problem, that is the problem of finding a conformal metric
with constant scalar curvature on a compact Riemannian manifold, was solved,
the question of describing the set of solutions thereof naturally arose.

In negative or zero Yamabe class it is quite simple to prove that the solution
is unique up to normalization, so the problem is non trivial only in the case
of positive Yamabe class. In a topics course at Stanford in 1988 (whose notes
by Pollack are quite famous and easily found on the internet), Schoen
conjectured that, fixing the value of the curvature, the set of solutions of
the Yamabe problem is compact except when the manifold is conformally
equivalent to $S^{n}$, giving indications on how to prove it.

The conjecture was proved in dimension $n\leq 24$ by Khuri, Marques and Schoen
(see [KMS] for the proof and a more detailed history of the problem), while it
was proved to be false in dimension $n\geq 25$ (see [B] and [BM]). The theorem
has also been proved in general dimension for conformally flat manifolds (see
[S]), or under the hypothesis that the Weyl tensor $W$ satisfies $|W|+|\nabla
W|>0$ (see [LZha]).

It is natural to study the same question on CR manifolds. A CR manifold is a
$2n+1$-dimensional manifold (with $2n+1\geq 3$) endowed with an
$n$-dimensional subbundle $\mathscr{H}\subset TM\otimes\mathbf{C}$ such that
$[\Gamma(\mathscr{H}),\Gamma(\mathscr{H})]\subset\Gamma(\mathscr{H})$ and
$\mathscr{H}\cap\overline{\mathscr{H}}=\\{0\\}$. Such kind of manifolds arise
in the study of hypersurfaces of $\mathbf{C}^{n+1}$. Under a generic
hypothesis of nondegeneracy, CR manifolds have a canonical contact structure;
a CR manifold with a contact form on this contact structure is called
pseudohermitian manifold.

The choice of a contact form induces a rich geometric structure, including a
pseudoriemannian metric (which is Riemannian for “strictly pseudoconvex”
manifolds) and a connection called Tanaka-Webster connection. As in Riemannian
geometry, the Tanaka-Webster connection allows to define a curvature tensor,
and contracting it twice using the metric one gets a scalar quantity known as
Webster curvature.

Since the choice of a contact form is determined up to the multiplication by a
smooth function, problems from conformal geometry naturally extend to CR
geometry, and trying to apply techniques therefrom to this setting showed that
strong analogies hold.

In particular the problem of finding a contact form for compact pseudoconvex
CR manifolds, known as CR Yamabe problem, has a variational formulation very
similar to the one for the Yamabe problem: a contact form for the CR manifold
$M$ has constant Webster curvature if and only if it is stationary for the CR
Yamabe functional

$\mathfrak{Q}(\theta)=\frac{\int_{M}R\theta\wedge{(d\theta)}^{n}}{\left(\int_{M}\theta\wedge{(d\theta)}^{n}\right)^{\frac{n}{n+1}}}$

where $R$ is the Webster curvature. The infimum of $\mathfrak{Q}$, called CR
Yamabe class, is indicated by $\mathscr{Y}(M)$ 111$\mathfrak{Q}(\theta)$ and
$\mathscr{Y}(M)$ depend not only by $\theta$ and $M$ but also by the CR
structure; by abuse of notation sometimes we omit the latter , and it can be
shown with the same proof of the Riemannian case that if
$\mathscr{Y}(M)<\mathscr{Y}(S^{2n+1})$ then $\mathfrak{Q}$ has minimizers.

Fixing a reference contact form $\theta$ and calling
$\widetilde{\theta}=u^{\frac{2}{n}}\theta$,

$\mathfrak{Q}(\widetilde{\theta})=\frac{\int_{M}\left(b_{n}|\nabla^{H}u|^{2}+Ru^{2}\theta\wedge{(d\theta)}^{n}\right)}{\left(\int_{M}u^{2+\frac{2}{n}}\theta\wedge{(d\theta)}^{n}\right)^{\frac{n}{n+1}}}=\frac{\int_{M}uL_{\theta}u}{\left(\int_{M}u^{2+\frac{2}{n}}\theta\wedge{(d\theta)}^{n}\right)^{\frac{n}{n+1}}}$

where $\nabla^{H}$ is the subriemannian gradient, $b_{n}=2+\frac{2}{n}$ and
$L_{\theta_{0}}=-b_{n}\Delta_{b}+R$, $\Delta_{b}$ being the sublaplacian. In
particular the problem is equivalent to solving the equation
$L_{\theta}u=u^{1+\frac{2}{n}}$.

Jerison and Lee proved that $\mathfrak{Q}$ has minimizers, thus solving the CR
Yamabe problem, for non spherical (that is, not locally equivalent to
$S^{2n+1}$) compact manifolds of dimension $2n+1\geq 5$ using the strategy
that had been used for non conformally flat Riemannian manifolds of dimension
$n\geq 6$ (see [JL1], [JL3]). The remaining cases of the problem were solved
in [G] and [GY] using the method of critical points at infinity; the solutions
found in this way are not necessarily minimizers of $\mathfrak{Q}$.

In the study of the problem of compactness for solutions to the Yamabe
problem, two fundamental elements which do not have an obvious analogue in CR
geometry are used: the positive mass theorem and the classification of metrics
of constant positive scalar curvature on $\mathbf{R}^{n}$ conformal to the
Euclidean metric.

The theory of mass for asymptotically flat pseudohermitian manifolds has yet
to reach the extent of the analogous Riemannian theory; the techniques used in
the study of the latter, in particular the theory of minimal hypersurfaces, do
not yet have been developed to the required generality.

A positive mass theorem for spherical CR manifolds of dimension $2n+1\geq 7$
(and for $2n+1=5$ under a technical condition) was proved in [CCY]. A
definition of pseudohermitian mass, or p-mass, valid for general three
dimensional CR manifolds (but straightforwardly generalizable to higher
dimension) has been given in [CMY1], where a CR positive mass theorem under
certain hypotheses was proved. In [CC] a CR positive mass theorem was proved
for spin five dimensional manifolds was proved. A remarkable counterexample by
Cheng, Malchiodi and Yang (see [CMY2]) shows that, unlike in Riemannian
geometry, the p-mass can be negative for three dimensional CR manifolds. In
analogy with other phenomena in CR geometry, many believe that such a
counterexample can exist only in three dimension, and conjecture that the CR
positive mass theorem be true in higher dimensions.

Like in Riemannian geometry, these theorems have applications to the study of
the CR Yamabe functional $\mathfrak{Q}$: the aforementioned CR positive mass
theorems imply as corollaries that under the respective hypotheses
$\mathfrak{Q}$ has minimum, while the authors in [CMY2] proved that in their
counterexample thereto $\mathscr{Y}(M)$ is not attained.

Regarding metrics of constant positive scalar curvature on $\mathbf{R}^{n}$
conformal to the Euclidean metric, it was proved that they are the one
obtained by pulling back the standard metric of $S^{n}$ through the
stereographic projection, and translations and dilations thereof (Corollary
8.2 in [CGS]). This result is used in a fundamental way when rescaling
solutions tending to infinity in order to describe how they blow up. In order
to carry out the same strategy in CR geometry, an analogous theorem is needed
for the Heisenberg group $\mathbf{H}^{n}$, which is the space obtained blowing
up a pseudohermitian manifold at a point. As in the Euclidean case, a family
of solutions can be obtained by pulling back the standard contact form of
$S^{2n+1}$ through the Cayley transform, which is a map $\mathbf{H}^{n}\to
S^{2n+1}$ analogous to the stereographic projection, and it is natural to
conjecture that they are all the solutions. The problem is equivalent to the
classification of the positive solutions of the equation
$-\Delta_{b}u=u^{1+\frac{2}{n}}$. The proof by Caffarelli, Gidas and Spruck
uses the method of moving planes, but in $\mathbf{H}^{n}$ the symmetry group
is not big enough to apply this method.

In [JL2] Jerison and Lee classified the solutions of the CR Yamabe problem on
$S^{2n+1}$, and through a removable singularity theorem, they proved the
classification theorem under the hypothesis that $u\in L^{2+\frac{2}{n}}$,
which is equivalent to having finite volume, but this result is not strong
enough to carry out blow-up analysis, which requires the theorem at least for
bounded solutions.

Very recently, Catino, Li, Monticelli and Roncoroni proved the classification
theorem for $\mathbf{H}^{1}$ (see [CLMR]). This allows us to perform the blow-
up analysis and to prove a compactness theorem for the solutions of the CR
Yamabe problem.

Given a compact strictly pseudoconvex three-dimensional pseudohermitian
manifold with $\mathscr{Y}(M,\mathscr{H})>0$ and $x\in M$ we will denote by
$m_{x}$ the p-mass, as defined in Definition 2.3 in [CMY1], of the manifold
$(M\setminus\\{x\\},\mathscr{H})$ with the blow-up contact form as defined in
Subsection 2.1 in the same work.

Then we will prove the following theorem. As customary, due to the importance
of subcritical approximation, we will prove a result slightly more general
than the compactness of the solutions of the CR Yamabe problem, and which
comprises subcritical exponents as well.

######  Theorem 1.1.

Let $(M,\mathscr{H},\theta)$ be a three-dimensional strictly pseudoconvex
pseudohermitian manifold of positive CR Yamabe class such that, for every
$x\in M$, $m_{x}>0$. Then for every $\varepsilon>0$ and $k\in\mathbf{N}$ there
exists a constant $C$ such that

$\begin{cases}\frac{1}{C}\leq u\leq C\\\ \left\lVert
u\right\rVert_{\Gamma^{k,\alpha}}\leq C\end{cases}$

for every $u\in\cup_{1+\varepsilon\leq p\leq 3}\mathscr{M}_{p}$ and
$0<\alpha<1$, where

$\mathscr{M}_{p}=\left\\{u>0\;\middle|\;L_{\theta}u=u^{p}\right\\},$

$L_{\theta}=-4\Delta_{b}+R$ being the conformal sublaplacian, and
$\Gamma^{2,\alpha}$ is the Hölder space defined in Subsection 2.2.

In particular, $\cup_{1+\varepsilon\leq p\leq 3}\mathscr{M}_{p}$ is compact in
the $\Gamma^{k,\alpha}$ topology.

We point out that by known results in functional analysis (Theorem 2.3 below),
in the theorem above the pseudohermitian Hölder spaces spaces
$\Gamma^{k,\alpha}$ could be replaced by the standard Hölder spaces
$C^{k,\alpha}$.

Furthermore, we remark that when $\mathscr{Y}(M)\leq 0$ it is easy to prove
uniqueness up to normalization (Theorem 7.1 in [JL1]), so the problem is
trivial.

The hypothesis in the theorem above is quite hard to check directly, since the
computation of the p-mass can be rather complex even for simple CR manifolds
(see [CMY2]). Fortunately the aforementioned positive mass theorem of Cheng,
Malchiodi and Yang (Theorem 1.1 in [CMY1]), along with a later result of
Takeuchi which allows to replace the hypothesis thereof with a simpler one
(Theorem 1.1 in [T]), allow us to state the following corollary to Theorem
1.1. We recall that a CR manifold is said to be embeddable in $\mathbf{C}^{N}$
whenever there exists a CR embedding in $\mathbf{C}^{N}$, that is an embedding
$\Phi:M\to\mathbf{C}^{N}$ such that
$d\Phi(\mathscr{H})\subseteq\operatorname{span}\left\\{\frac{\partial}{\partial
z_{1}},\ldots,\frac{\partial}{\partial z_{N}}\right\\}$.

######  Corollary 1.2.

Let $(M,\mathscr{H},\theta)$ be a three-dimensional strictly pseudoconvex
pseudohermitian manifold with $\mathscr{Y}(M,\mathscr{H})>0$ which is CR-
embeddable in $\mathbf{C}^{N}$ for some $N$, and not CR-equivalent to $S^{3}$
with the standard CR structure.

Then the thesis of Theorem 1.1 holds.

######  Remark 1.3.

Unlike the case of low-dimensional Riemannian manifolds, Theorem 1.1 is not
true if the only hypothesis on $M$ is that it is not CR-equivalent to $S^{3}$.
In fact by a result mentioned above (Theorem 1.3 in [CMY2]) there exist three
dimensional CR manifolds for which $\mathscr{Y}(M)$ is not attained. It is
standard to prove that the subcritical functional

$\mathfrak{Q}_{p}(u)=\frac{\int_{M}uL_{\theta}u}{\left(\int_{M}u^{p}\theta\wedge{(d\theta)}^{n}\right)^{\frac{1}{2}}}$

has a minimum point $u_{p}$ for $p\in[2,3)$ (Theorem 6.2 in [JL1]). Were the
thesis of Theorem 1.1 true, $u_{p_{n}}\to u$ in $C^{2}$ for some $p_{n}\to 3$,
and by Lemma 6.4 (b) in [JL1], $u$ would be a minimum point for
$\mathfrak{Q}$.

We do not know whether there exists a CR manifold not CR-equivalent to $S^{3}$
for which the set of solution of the CR Yamabe problem
$\left\\{u>0\;\middle|\;L_{\theta}u=u^{3}\right\\}$ is not compact.

#### Acknowledgements.

The author is supported by the fund “MIUR PRIN 2017” - CUP:E64I19001240001. He
would like to thank Andrea Malchiodi for suggesting the problem and for useful
discussion.

##  2 A review of CR geometry

We recall some facts about CR geometry that we will need in this work. For a
complete introduction we refer to [DT].

A CR manifold is a manifold $M$ of dimension $2n+1$ endowed with a
$n$-dimensional subbundle $\mathscr{H}$ of $TM\otimes\mathbf{C}$ such that
$\mathscr{H}\cap\overline{\mathscr{H}}=\\{0\\}$ and
$[\mathscr{H},\mathscr{H}]\subseteq\mathscr{H}$.

The Levi distribution is
$H(M)=\mathfrak{Re}(\mathscr{H}\oplus\overline{\mathscr{H}})$. A CR manifold
is said nondegenerate if $H(M)$ is a contact distribution. In this article all
CR manifolds will be assumed to be nondegenerate. A contact form for
$\mathbf{H}(M)$ is called a pseudohermitian form or pseudohermitian structure.
A nondegenerate CR manifold equipped with a contact form is called
pseudohermitian manifold.

The sesquilinear form
$L_{\theta}\in\Gamma(\mathscr{H}^{*}\otimes\overline{\mathscr{H}}^{*})$
defined by

$L_{\theta}(Z,\overline{W})=i\theta([Z,\overline{W}])=-id\theta(Z,\overline{W})$

is called Levi form, and the degeneracy thereof is equivalent to the
nondegeneracy of the CR structure. A CR structure is said strictly
pseudoconvex if it admits a contact form with positive definite Levi form. In
such a case, all contact forms have definite Levi form, and $L_{\theta}$ is
positive definite if and only if $L_{-\theta}$ is negative definite. In a
certain sense strictly pseudoconvex CR manifolds occupy, among CR manifolds,
the role that Riemannian manifolds occupy among pseudoriemannian manifolds. In
the following we will assume all manifolds to be strictly pseudoconvex and all
contact forms to have a positive Levi form. Notice that when $n=1$, the main
focus of this article, all manifolds are strictly pseudoconvex, and so up to a
change of sign every contact form can be assumed to have positive Levi form.

The Levi form induces a symmetric form $G_{\theta}$ on $H(M)$ which is
positive definite if and only if the pseudohermitian manifold is pseudoconvex.
In such a case $(M,H(M),G_{\theta})$ is a subriemannian manifold. We will call
$d$ the Carnot-Carathéodory distance.

On a pseudohermitian manifold there exists a unique vector field $T$, called
Reeb field, such that $\theta(T)=1$ and $i_{T}d\theta=0$. It can be used to
extend $G_{\theta}$ to a pseudoriemannian metric (Riemannian in the strictly
pseudoconvex case) $g_{\theta}$, called the Webster metric, such that $T$ is
orthogonal to $H(M)$ and $g_{\theta}(T,T)=1$.

Given a real function $u$ of class $C^{1}$, we define $\nabla^{H}u$ as the
subriemannian gradient, that is the section of $H(M)$ such that
$G_{\theta}(\nabla^{H}u,X)=Xu$ for every section $X$ of $H(M)$.

On a pseudohermitian manifolds there are two natural volume forms, the
Riemannian one $dV$ associated to $g_{\theta}$ and the contact one
$\theta\wedge(d\theta)^{n}$. They just differ by a dimensional constant, and
so choosing one over the other does not create substantial difference. For
commodity to apply the divergence theorem, in this article we will always use
the first one.

Since we have a volume form, we can define the divergence of a vector field as

$(\operatorname{div}X)dV=\mathscr{L}_{X}(dV).$

We define the sublaplacian as the differential operator
$\Delta_{b}u=\operatorname{div}(\nabla^{H}u)$. $\Delta_{b}$ is not elliptic,
but it is degenerate elliptic and shares many properties with the Laplacian.

On any nondegenerate pseudohermitian manifold there exist a canonical
connection called Tanaka-Webster connection.

To perform local computations, consider a local frame of $\mathscr{H}$,
$(Z_{1},\ldots,Z_{n})$. Define $Z_{\overline{\alpha}}=\overline{Z}_{\alpha}$
and $Z_{0}=T$. Let
$(\theta^{1},\ldots,\theta^{n},\theta^{\overline{1}},\ldots,\theta^{\overline{n}},\theta^{0})$
be the dual frame of
$(Z_{1},\ldots,Z_{n},Z_{\overline{1}},\ldots,Z_{\overline{n}},Z_{0})$ (in
particular we take $\theta^{0}=\theta$). Given a tensor $\mathfrak{T}$ of type
$(k,\ell)$ and indexes
$A_{i},B_{i}\in\\{1,\ldots,n,\overline{1},\ldots,\overline{n},0\\}$ let us
define

${\mathfrak{T}^{A_{1}\ldots A_{k}}}_{B_{1}\ldots
B_{\ell}}=\mathfrak{T}(\theta^{A_{1}},\ldots,\theta^{A_{k}},Z_{B_{1}},\ldots,Z_{B_{\ell}}).$

In the sequel Greek letters will denote indexes in $\\{1,\ldots,n\\}$.
Let $h_{\alpha\overline{\beta}}=L_{\theta}(Z_{\alpha},Z_{\overline{\beta}})$
be the coefficients of the Levi form. $h_{\alpha\overline{\beta}}$ and its
inverse $h^{\alpha\overline{\beta}}$ can be used to raise and lower indexes.
Therefore when using the Einstein convention, two repeated indexes are summed
if one of them is barred and the other is non-barred when they are both up or
both up, while in the other case they are summed if they are both barred or
both non barred.

Since $\mathscr{H}$ is parallel, there exist 1-forms
${\omega_{\alpha}}^{\beta}$ such that

$\nabla Z_{\alpha}={\omega_{\alpha}}^{\beta}\otimes Z_{\beta}.$

We separate indexes relative to covariant derivatives with respect to the
Tanaka-Webster connection through a comma. Thus, for example, if
$X=X_{\alpha}\theta^{\alpha}$ we have

$X_{\alpha,\beta}=Z_{\beta}(X_{\alpha})-{\omega_{\alpha}}^{\gamma}(Z_{\beta})X_{\gamma}.$

The following formula for the sublaplacian holds:

$\Delta_{b}u={u^{,\alpha}}_{\alpha}+{u^{,\overline{\alpha}}}_{\overline{\alpha}}=u_{,\alpha\overline{\alpha}}+u_{,\overline{\alpha}\alpha}.$
(1)

The pseudo-Hermitian curvature tensor is the section of
$\mathscr{H}^{*}\otimes\overline{\mathscr{H}}^{*}\otimes\mathscr{H}^{*}\otimes\overline{\mathscr{H}}$
defined as

$R(Z,\overline{W})X=\nabla_{Z}\nabla_{\overline{W}}X-\nabla_{\overline{W}}\nabla_{Z}X-\nabla_{[Z,\overline{W}]}X$

for $Z,W,X\in\Gamma(T^{(1,0)}M)$. We indicate the coordinates thereof by

${R_{\alpha\overline{\beta}\mu}}^{\nu}=L_{\theta}(R(Z_{\alpha},Z_{\overline{\beta}})Z_{\mu},Z_{\nu}).$

By contracting the pseudo-Hermitian curvature tensor, we get the pseudo-
Hermitian Ricci tensor
$R_{\alpha\overline{\beta}}={R_{\mu\overline{\beta}\alpha}}^{\mu}$, and
contracting another time we get the Webster scalar curvature
$R=h^{\alpha\overline{\beta}}R_{\alpha\overline{\beta}}$.

The conformal sublaplacian $L_{\theta}=-b_{n}\Delta_{b}+R$, where
$b_{n}=2+\frac{2}{n}$, verifies the covariant transformation law

$L_{\widetilde{\theta}}\left(\frac{\varphi}{u}\right)=u^{-\frac{n+2}{n}}L_{\theta}\varphi$
(2)

where $\widetilde{\theta}=u^{\frac{2}{n}}\theta$. In particular we obtain the
formula for the transformation of the Webster curvature

$\widetilde{R}=u^{-\frac{n+2}{n}}L_{\theta}u=u^{-\frac{n+2}{n}}(-b_{n}\Delta_{b}+R)u.$
(3)

###  2.1 The Heisenberg group

The most important CR manifold is the Heisenberg group $\mathbf{H}^{n}$, that
is $\mathbf{C}^{n}\times\mathbf{R}$ with the group law
$(z,t)\cdot(w,s)=(z+w,t+s+2\mathfrak{Im}(z\cdot\overline{w}))$, endowed with
the left-invariant CR structure generated by the left-invariant vector fields

$Z_{\alpha}=\frac{\partial}{\partial
z_{\alpha}}+i\overline{z}_{\alpha}\frac{\partial}{\partial t}.$

Up to a constant there exist a unique left-invariant 1-form annihilating
$H(\mathbf{H}^{n})$, that is
$\theta=dt+i\sum_{\alpha}\left(z^{\alpha}d\overline{z}^{\alpha}-\overline{z}^{\alpha}dz^{\alpha}\right)$,
and with this choice it can be easily computed that $\mathbf{H}^{n}$ is
strictly pseudoconvex, and in particular that with respect to this frame
$h_{\alpha\overline{\beta}}=2\delta_{\alpha\overline{\beta}}$. Furthermore it
can be proved that the Reeb vector field is $T=\frac{\partial}{\partial t}$
and that the Webster curvature is zero.

The Korányi norm is defined as
$|(z,t)|=\left(|z|^{4}+t^{2}\right)^{\frac{1}{4}}$.

The fundamental solution for the CR sublaplacian of $\mathbf{H}^{n}$, which by
formula (1) is

$\Delta_{b}=\frac{1}{2}\sum_{\alpha=1}^{n}(Z_{\alpha}Z_{\overline{\alpha}}+Z_{\overline{\alpha}}Z_{\alpha}),$

has been computed by Folland and Stein (see Section 6 in [FS] or Theorem 3.9
in [DT]): it holds that

$-\Delta_{b}\left(\frac{\gamma_{n}}{|(z,t)|^{2n}}\right)=\delta_{0}$ (4)

with
$\gamma_{n}=2^{2n-2}\pi^{-n-1}\left|\Gamma\left(\frac{n}{2}\right)\right|^{2}$.

$\mathbf{H}^{n}$ admits a family of CR and group automorphisms analogous to
the dilations of $\mathbf{R}^{n}$, defined by $\delta_{\lambda}(z,t)=(\lambda
z,\lambda^{2}t)$, and called Heisenberg dilations.

Since $(\delta_{\lambda})^{*}dV=\lambda^{2n+2}dV$, the number $Q=2n+2$ is
called homogeneous dimension of $\mathbf{H}^{n}$.

$\mathbf{H}^{n}$ plays a role among pseudoconvex pseudohermitian manifolds
analogous to the role of $\mathbf{R}^{n}$ among Riemannian manifolds; indeed
every pseudoconvex pseudohermitian manifold can locally be appoximated with
$\mathbf{H}^{n}$, through coordinates analogous to the normal coordinates of
Riemannian geometry known as pseudohermitian normal coordinates.

Given $x\in M$, pseudohermitian normal coordinates are given by a function
$\Phi:\Omega\to M$ with $0\in\Omega$ and $\Phi(0)=x$ approximating the
pseudohermitian structure of $M$ as much as possible (see [JL3] and the
appendix for precise statements). When using pseudohermitian normal
coordinates, we will generally omit $\Phi$ and, with abuse of notation, we
will identify $\Omega$ and $\Phi(\Omega)$. We will indicate by a circle the
pseudohermitian objects coming from $\mathbf{H}^{n}$ through normal
coordinates
($\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\alpha},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b},d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt},\ldots$).
Balls will be meant with respect to the Korányi distance, not to the Carnot-
Carathéodory distance $d$. Anyways it is known that on $\mathbf{H}^{n}$ (or on
balls thereof) the two distances are equivalent.

######  Remark 2.1.

When working in pseudohermitian coordinate around some point $\overline{x}$,
we will often use dilations and translations to rescale the equation. If $u$
solves

$L_{\theta}u=\widetilde{R}u^{p}$

in $\Omega$, $x\in\Omega$ and $\lambda>0$ then
$\widetilde{u}=\lambda^{\frac{2}{p_{i}-1}}u\circ\delta_{\lambda}\circ L_{x}$
(where $L_{x_{i}}(x)=x_{i}^{-1}x$ is the left translation) solves

$L_{\widetilde{\theta}}\widetilde{u}=(\widetilde{R}\circ\delta_{\lambda}\circ
L_{x})\widetilde{u}^{p}$

on $\delta_{\frac{1}{\lambda}}(L_{x^{-1}}(\Omega))$, where
$\widetilde{\theta}=\frac{1}{\lambda^{2}}(\delta_{\lambda}\circ
L_{x})^{*}\theta$ and $L_{\widetilde{\theta}}$ is the conformal sublaplacian
associated to the pseudohermitian manifold
$(\delta_{\frac{1}{\lambda}}(L_{x^{-1}}(\Omega)),\widetilde{\theta},(\delta_{\lambda}\circ
L_{x})^{*}\mathscr{H})$. Notice that as $x\to\overline{x}$ and
$\lambda\to\infty$,

$\left(\delta_{\frac{1}{\lambda}}(L_{x^{-1}}(\Omega)),\widetilde{\theta},(\delta_{\lambda}\circ
L_{x})^{*}\mathscr{H}\right)\to(\mathbf{H}^{n},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\mathscr{H}}\rule{0.0pt}{9.90276pt})$
(5)

smoothly on compact sets.

In conformal geometry there exist local coordinates adapted to the conformal
structure obtained using normal coordinates with respect to a metric in the
conformal class that is “as flat as possible” around the base point, known as
“conformal normal coordinates” and introduced by Lee and Parker in [LP].
Analogously in CR geometry local coordinates adapted to the CR structure,
obtained using pseudohermitian normal coordinates with respect to an
appropriate contact form, were introduced by Jerison and Lee in [JL3]. They
are known as CR normal coordinates.

We recall the recent classification theorem for contact forms with constant
positive Webster curvature in $\mathbf{H}^{1}$ proved by Catino, Li,
Monticelli and Roncoroni in [CLMR].

######  Theorem 2.2.

If $u$ is a positive solution to

$-\Delta_{b}u=u^{3}$

on $\mathbf{H}^{1}$, then $u=U\circ\delta_{\lambda}\circ L_{x_{0}}$ for some
$\lambda\in(0,\infty)$ and $x_{0}\in\mathbf{H}^{1}$, where

$U(z,t)=\frac{c_{1}}{\left(t^{2}+(1+|z|^{2})^{2}\right)^{\frac{1}{2}}}$

is the Jerison-Lee solution and $L_{x_{0}}(x)=x_{0}^{-1}\cdot x$ is the left
translation.

###  2.2 Function spaces and estimates for the sublaplacian

In order to study analytical problems on CR manifolds, function spaces adapted
to their structure are needed. We refer to Section 5 in [JL1] or Subsection
3.4 in [DT] for their definition.

The Folland-Stein spaces $S^{k,p}(M)$ are analogous to the Sobolev spaces, but
using only horizontal derivatives (the ones in the directions of $H(M)$).

Analogously, using the Carnot-Carathéodory distance, Hölder spaces which
consider only horizontal derivatives, denoted by $\Gamma^{k,\alpha}(M)$, can
be defined.

Let $C^{k,\alpha}(M)$ be the standard Hölder space relative to the Riemannian
metric $g_{\theta}$.

Then the following immersions and regularity results hold. These results were
firstly stated in Section 5 in [JL1] noticing that the proofs for analogous
results by Folland and Stein in [FS] for the Kohn Laplacian $\square_{b}$ are
valid also for $\Delta_{b}$.

######  Theorem 2.3.

Let $U$ be a relatively compact open subset of a pseudohermitian manifold
$(M,\mathscr{H},\theta)$. Then, if $\alpha\in(0,1)$, $k,\ell\in\mathbf{N}$,
$p\in(1,\infty)$:

  1. 1.

$\left\lVert u\right\rVert_{\Gamma^{k,\alpha}(U)}\leq C\left\lVert
u\right\rVert_{S^{\ell,p}(M)}$ if $\frac{1}{p}=\frac{\ell-k-\alpha}{2n+2}$;

  2. 2.

$C_{1}\left\lVert u\right\rVert_{C^{\lfloor
k/2\rfloor,\lfloor\alpha/2\rfloor+\\{k/2\\}}(U)}\leq\left\lVert
u\right\rVert_{\Gamma^{k,\alpha}(M)}\leq C_{2}\left\lVert
u\right\rVert_{C^{k,\alpha}(M)}$ where $\\{k\\}$ is the fractional part of
$k$;

  3. 3.

$\left\lVert u\right\rVert_{S^{k+2,p}(U)}\leq
C\left(\left\lVert\Delta_{b}u\right\rVert_{S^{k,p}(M)}+\left\lVert
u\right\rVert_{S^{k,p}(M)}\right)$;

  4. 4.

$\left\lVert u\right\rVert_{\Gamma^{k+2,\alpha}(U)}\leq
C\left(\left\lVert\Delta_{b}u\right\rVert_{\Gamma^{k,\alpha}(M)}+\left\lVert
u\right\rVert_{\Gamma^{k,\alpha}(M)}\right)$.

Furthermore for any of the above estimates there exists a $m$ such that in a
$C^{m}$ neighborhood of the pseudohermitian structure $(\mathscr{H},\theta)$,
the immersion holds with uniform constant.

The uniformity of the constant is not stated in [FS], but follows from the
proof. As in the case of the Laplacian, one can take $M=U$ in the above
estimates if some regularity condition is assumed on $U$, but we will not need
this.

As a last remark, we point out that the weak maximum principle holds for
operators of the form $-\Delta_{b}+f$ with $f\geq 0$ because $\Delta_{b}$ is
degenerate elliptic; furthermore for operators of the form $-\Delta_{b}+f$ the
Harnack inequality holds with a constant depending on the pseudohermitian data
and on $\left\lVert f\right\rVert_{L^{\infty}}$, and it can be proved by
following the proof for elliptic operators from Theorem 8.20 in [GT] (see for
example [GuL] ).

##  3 A Pohozaev identity

As in the Riemannian case, where the blow-up analysis requires using the
Pohozaev identity for $\mathbf{R}^{n}$, in the CR case we will need a Pohozaev
formula for the Heisenberg group. Pohozaev-type formulas have already been
studied on the Heisenberg group (see [GaL]) and on more general Carnot groups
(see [GV]). We recall the formula that we will need, and for utility of the
reader we include the proof.

We warn the reader that in the literature generally the subriemannian metric
used is $\frac{1}{4}G_{\theta}$ instead of $G_{\theta}$, and this could cause
unsubstantial differences in the computations.

In this section we will use the standard real frame for $H(\mathbf{H}^{n})$

$X_{k}=\frac{\partial}{\partial x_{k}}+2y_{k}\frac{\partial}{\partial
t},\;\;\;Y_{k}=X_{n+k}=\frac{\partial}{\partial
y_{k}}-2x_{k}\frac{\partial}{\partial t}.$

With respect to this frame the subriemannian gradient is

$\nabla^{H}f=\frac{1}{4}\sum_{k=1}^{n}(X_{k}fX_{k}+Y_{k}fY_{k})$

and the sublaplacian

$\Delta_{b}=\frac{1}{4}\sum_{k=1}^{n}(X_{k}^{2}+Y_{k}^{2}).$

In order to state the formula, we define $\Xi$ as the vector field whose flow
is the semigroup $\varphi_{t}=\delta_{\log t}$, given explicitely by

$\Xi=\sum_{k=1}^{n}\left(x_{k}\frac{\partial}{\partial
x_{k}}+y_{k}\frac{\partial}{\partial y_{k}}\right)+2t\frac{\partial}{\partial
t}=\sum_{k=1}^{n}\left(x_{k}X_{k}+y_{k}Y_{k}\right)+2tT=$
$=\sum_{\alpha=1}^{n}\left(z_{\alpha}Z_{\alpha}+\overline{z}_{\alpha}Z_{\overline{\alpha}}\right)+2tT.$
(6)

In the following proofs we will use some easily verified properties of $\Xi$.

######  Lemma 3.1.

  1. 1.

$\operatorname{div}\Xi=Q$;

  2. 2.

$[X_{k},\Xi]=X_{k}$, $[Y_{k},\Xi]=Y_{k}$;

  3. 3.

$f$ is homogeneous of degree $\alpha$ with respect to the Heisenberg dilations
if and only if $\Xi f=\alpha f$.

Note that the third of the above properties corresponds to Euler’s theorem on
$\mathbf{R}^{n}$.

We will denote by $d\sigma$ the Riemannian volume form of a submanifold and by
$\nu$ the Riemannian normal.

######  Proposition 3.2.

If $\Omega\subset\mathbf{H}^{n}$ is a bounded domain with $C^{1}$ boundary and
$u\in C^{2}(\overline{\Omega})$ verifies the equation

$-\Delta_{b}u=f(x,u)$ (7)

where $f\in C^{1}(\mathbf{H}^{n}\times\mathbf{R})$, then, defining
$F(x,t)=\int_{0}^{t}f(x,s)ds$, the following formula holds:

$\int_{\Omega}\left(QF(x,u)-\frac{Q-2}{2}uf(x,u)+(\Xi_{x}F)(x,u)\right)dV=$
$=\int_{\partial\Omega}\left(\left(F(x,u)-\frac{1}{2}|\nabla^{H}u|^{2}\right)\Xi\cdot\nu+\Xi
u\nabla^{H}u\cdot\nu+\frac{Q-2}{2}u\nabla^{H}u\cdot\nu\right)d\sigma.$ (8)

###### Proof.

Multiplying equation (7) by $\Xi u$ and integrating we get

$-\int_{\Omega}\Xi u\Delta_{b}u=\int_{\Omega}f(x,u)\Xi
u=\int_{\Omega}\left[\Xi(F(x,u))-(\Xi_{x}F)(x,u)\right]=$
$=\int_{\partial\Omega}F(x,u)\Xi\cdot\nu-Q\int_{\Omega}F(x,u)-\int_{\Omega}(\Xi_{x}F)(x,u)$
(9)

where we used that $\operatorname{div}\Xi=Q$. Integrating by parts the right-
hand side we get

$-\int_{\Omega}\Xi u\Delta_{b}u=-\int_{\partial\Omega}\Xi
u\nabla^{H}u\cdot\nu+\int_{\Omega}\nabla^{H}u(\Xi u).$ (10)

Integrating by parts the last term we get, using that $[X_{k},\Xi]=X_{k}$,

$\int_{\Omega}\nabla^{H}u(\Xi
u)=\frac{1}{4}\sum_{k=1}^{2n}\int_{\Omega}X_{k}u\cdot X_{k}(\Xi
u)=\frac{1}{4}\sum_{k=1}^{2n}\int_{\Omega}X_{k}u\cdot(\Xi X_{k}+X_{k})u=$
$=\frac{1}{4}\sum_{k=1}^{2n}\int_{\Omega}\frac{1}{2}\Xi|X_{k}u|^{2}+\frac{1}{4}\sum_{k=1}^{2n}\int_{\Omega}|X_{k}u|^{2}=\frac{1}{2}\int_{\Omega}\Xi|\nabla^{H}u|^{2}+\int_{\Omega}|\nabla^{H}u|^{2}=$
$=\frac{1}{2}\int_{\partial\Omega}|\nabla^{H}u|^{2}\Xi\cdot\nu-\frac{Q}{2}\int_{\Omega}|\nabla^{H}u|^{2}+\int_{\Omega}|\nabla^{H}u|^{2}=$
$=\frac{1}{2}\int_{\partial\Omega}|\nabla^{H}u|^{2}\Xi\cdot\nu+\frac{2-Q}{2}\int_{\Omega}|\nabla^{H}u|^{2}.$
(11)

Multiplying equation (7) by $u$ and integrating we get

$\int_{\Omega}uf(x,u)=-\int_{\Omega}u\Delta_{b}u=-\int_{\partial\Omega}u\nabla^{H}u\cdot\nu+\int_{\Omega}|\nabla^{H}u|^{2}.$
(12)

Putting together formulas (9), (10), (11) and (12) we get the thesis. ∎

Now, using pseudohermitian normal coordinates, we can deduce a formula valid
in small neighborhoods of any point of a pseudohermitian manifold.

######  Proposition 3.3.

Let $M$ be a pseudohermitian manifold, $\overline{x}\in M$, and let $u$ be a
solution of the equation

$-b_{n}\Delta_{b}u+Ru=\widetilde{R}u^{p}.$

Then in pseudohermitian normal coordinates around $\overline{x}$ the following
formula holds

$\int_{B_{r}}\left(\frac{1}{b_{n}}\left(\frac{Q}{p+1}-\frac{Q-2}{2}\right)\widetilde{R}u^{p+1}-\frac{1}{b_{n}}Ru^{2}+\frac{1}{b_{n}}\frac{1}{p+1}\Xi(\widetilde{R})u^{p+1}+\right.$
$\left.-\frac{1}{b_{n}}\frac{1}{2}\Xi(R)u^{2}-\left(\Xi
u+\frac{Q-2}{2}u\right)(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt}=$
$=\int_{\partial
B_{r}}\left(\left(\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}-\frac{1}{b_{n}}\frac{1}{2}Ru^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\mathscr{B}(x,u,\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u)\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}$

where

$\mathscr{B}(x,u,\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u)=\frac{Q-2}{2}u(x)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u(x)\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}_{B_{d(x,0)}}(x)-\frac{1}{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u(x)|^{2}\Xi(x)\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}_{B_{d(x,0)}}(x)+$
$+\Xi
u(x)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u(x)\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}_{B_{d(x,0)}}(x)$

and $B_{r}$ denotes the ball with respect to the Korányi norm.

###### Proof.

$u$ verifies

$-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u=-\frac{1}{b_{n}}Ru+\frac{1}{b_{n}}\widetilde{R}u^{p}+(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u).$

Defining

$f(x,u)=-\frac{1}{b_{n}}R(x)u(x)+\frac{1}{b_{n}}\widetilde{R}(x)u^{p}+(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u(x))$

we have

$F(x,u)=-\frac{1}{b_{n}}R(x)u(x)u+\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}(x)u^{p+1}+(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u(x))u,$

and thus, applying formula (8), we get

$\int_{B_{r}}\left(-\frac{Q}{b_{n}}Ru^{2}+\frac{Q}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}+Q(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)u+\right.$
$+\frac{Q-2}{2}\frac{1}{b_{n}}Ru^{2}-\frac{Q-2}{2}\frac{1}{b_{n}}\widetilde{R}u^{p+1}-\frac{Q-2}{2}(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)u+$
$\left.-\frac{1}{b_{n}}\Xi(Ru)u+\frac{1}{b_{n}}\frac{1}{p+1}\Xi(\widetilde{R})u^{p+1}+u\Xi(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)\right)=$
$=\int_{\partial
B_{r}}\left(\left(-\frac{1}{b_{n}}Ru^{2}+\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}+(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)u-\frac{1}{2}|\nabla^{H}u|^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\right.$
$\left.+\Xi
u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{Q-2}{2}u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}\right)=$
$=\int_{B_{r}}\left(u\Xi(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)+(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)\Xi
u+Q(\Delta_{b}u-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)u\right)+$
$+\int_{\partial
B_{r}}\left(\left(-\frac{1}{b_{n}}Ru^{2}+\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}-\frac{1}{2}|\nabla^{H}u|^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\right.$
$\left.+\Xi
u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{Q-2}{2}u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}\right).$
(13)

With some computations we can get

$-\frac{1}{b_{n}}\int_{B_{r}}\Xi(Ru)u=-\frac{1}{b_{n}}\int_{B_{r}}\left(\Xi(R)u^{2}+\frac{1}{2}R\Xi(u^{2})\right)=$
$=-\frac{1}{b_{n}}\int_{B_{r}}\Xi(R)u^{2}-\frac{1}{b_{n}}\frac{1}{2}\int_{\partial
B_{r}}Ru^{2}\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{1}{b_{n}}\frac{1}{2}\int_{B_{r}}\Xi(R)u^{2}+\frac{1}{b_{n}}\frac{Q}{2}\int_{B_{r}}Ru^{2}=$
$=-\frac{1}{b_{n}}\frac{1}{2}\int_{B_{r}}\Xi(R)u^{2}-\frac{1}{b_{n}}\frac{1}{2}\int_{\partial
B_{r}}Ru^{2}\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{1}{b_{n}}\frac{Q}{2}\int_{B_{r}}Ru^{2}.$

Thus, substituting in formula (13), we get

$\int_{B_{r}}\left(-\frac{1}{b_{n}}\frac{Q+2}{2}Ru^{2}+\frac{1}{b_{n}}\left(\frac{Q}{p+1}-\frac{Q-2}{2}\right)\widetilde{R}u^{p+1}+\right.$
$\left.-\left(\Xi
u+\frac{Q-2}{2}u\right)(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)+\frac{1}{b_{n}}\frac{1}{p+1}\Xi(\widetilde{R})u^{p+1}\right)+$
$-\frac{1}{b_{n}}\frac{1}{2}\int_{B_{r}}\Xi(R)u^{2}-\frac{1}{b_{n}}\frac{1}{2}\int_{\partial
B_{r}}Ru^{2}\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{1}{b_{n}}\frac{Q}{2}\int_{B_{r}}Ru^{2}=$
$=\int_{\partial
B_{r}}\left(\left(-\frac{1}{b_{n}}Ru^{2}+\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}-\frac{1}{2}|\nabla^{H}u|^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\right.$
$\left.+\Xi
u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{Q-2}{2}u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}\right)$

that is

$\int_{B_{r}}\left(\frac{1}{b_{n}}\left(\frac{Q}{p+1}-\frac{Q-2}{2}\right)\widetilde{R}u^{p+1}-\frac{1}{b_{n}}Ru^{2}+\right.$
$\left.-\left(\Xi
u+\frac{Q-2}{2}u\right)(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)+\frac{1}{b_{n}}\frac{1}{p+1}\Xi(\widetilde{R})u^{p+1}-\frac{1}{b_{n}}\frac{1}{2}\Xi(R)u^{2}\right)=$
$=\int_{\partial
B_{r}}\left(\left(-\frac{1}{b_{n}}\frac{1}{2}Ru^{2}+\frac{1}{b_{n}}\frac{1}{p+1}\widetilde{R}u^{p+1}-\frac{1}{2}|\nabla^{H}u|^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\right.$
$\left.+\Xi
u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\frac{Q-2}{2}u\nabla^{H}u\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}\right)$

which is the thesis. ∎

In the following we will need an estimate of the term involving $\mathscr{B}$.

######  Proposition 3.4.

There exists a constant $c_{n}>0$ such that if $u$ is a function defined on a
neighborhood of $0$ in $\mathbf{H}^{n}$ such that
$u(x)=\frac{\gamma_{n}}{|x|^{Q-2}}+A+\alpha(x)$ with $\alpha(x)=o(1)$ for
$x\to 0$, where $\gamma_{n}$ is the constant from formula (4), then

$\lim_{\sigma\to 0}\sigma^{Q-1}\int_{\partial
B_{1}(0)}\mathscr{B}(x,u\circ\delta_{\sigma},\nabla^{H}(u\circ\delta_{\sigma}))=-c_{n}A.$

###### Proof.

$\sigma^{Q-1}\int_{\partial
B_{1}}\mathscr{B}(x,u\circ\delta_{\sigma},\nabla^{H}(u\circ\delta_{\sigma}))=\sigma^{Q-1}\int_{\partial
B_{1}}\left(\frac{Q-2}{2}u\circ\delta_{\sigma}\nabla^{H}(u\circ\delta_{\sigma})\cdot\nu+\right.$
$\left.-\frac{1}{2}|\nabla^{H}(u\circ\delta_{\sigma})|^{2}\Xi\cdot\nu+\Xi(u\circ\delta_{\sigma})\nabla^{H}(u\circ\delta_{\sigma})\cdot\nu\right)=$
$=\sigma^{Q-1}\int_{\partial
B_{1}}\left[\frac{Q-2}{2}\left(\frac{\gamma_{n}}{\sigma^{Q-2}|x|^{Q-2}}+A\right)\cdot\left(\frac{1}{\sigma^{Q-1}}\nabla^{H}\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+\right.$
$-\frac{1}{2}\left|\nabla^{H}\left(\frac{\gamma_{n}}{\sigma^{Q-2}|x|^{Q-2}}\right)\right|^{2}\Xi\cdot\nu+$
$\left.+\left(-(Q-2)\frac{\gamma_{n}}{\sigma^{Q-2}|x|^{Q-2}}\right)\left(\frac{1}{\sigma^{Q-1}}\nabla^{H}\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu\right]+o(1)=$
$=\frac{Q-2}{2}A\int_{\partial
B_{1}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+$
$-\frac{1}{\sigma^{Q-2}}\left[\int_{\partial
B_{1}}\frac{Q-2}{2}\frac{\gamma_{n}}{|x|^{Q-2}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu-\frac{1}{2}\int_{\partial
B_{1}}\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\Xi\cdot\nu\right]+o(1),$

where we used that $\Xi(|x|^{-Q+2})=(-Q+2)|x|^{-Q+2}$, which is a consequence
of point 3 of Lemma 3.1. Applying the divergence theorem we get, for $r>1$,

$\left(\int_{\partial B_{r}}-\int_{\partial
B_{1}}\right)\left((Q-2)\frac{\gamma_{n}}{|x|^{Q-2}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\Xi\cdot\nu\right)=$
$=\int_{B_{r}\setminus
B_{1}}\left((Q-2)\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}+(Q-2)\frac{\gamma_{n}}{|x|^{Q-2}}\Delta_{b}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)+\right.$
$\left.+Q\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}+\Xi\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\right)=$
$=\int_{B_{r}\setminus
B_{1}}\left((2Q-2)\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}+(-2(Q-1))\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\right)=0.$

where we used Lemma 3.1 and the fact that
$\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}$ is
homogeneous of degree $-2(Q-1)$.

Since, if $r\to\infty$,

$\int_{\partial
B_{r}}\left((Q-2)\frac{\gamma_{n}}{|x|^{Q-2}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\Xi\cdot\nu\right)\to
0$

then

$\int_{\partial
B_{1}}\left((Q-2)\frac{\gamma_{n}}{|x|^{Q-2}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+\left|\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\right|^{2}\Xi\cdot\nu\right)=0.$

Therefore

$\sigma^{Q-1}\int_{\partial
B_{1}}\mathscr{B}(x,u\circ\delta_{\sigma},\nabla^{H}(u\circ\delta_{\sigma}))=\frac{Q-2}{2}A\int_{\partial
B_{1}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu+o(1).$

Using that $\nu=\frac{\nabla_{g_{\theta}}|x|}{|\nabla_{g_{\theta}}|x|}$ and
that for every $f$, $g$ one has
$\nabla_{g_{\theta}}f\cdot\nabla^{H}g=\nabla^{H}f\cdot\nabla^{H}g$ as an easy
consequence of the definitions, it is easy to prove that

$c_{n}=-\frac{Q-2}{2}\int_{\partial
B_{1}}\nabla^{H}\left(\frac{\gamma_{n}}{|x|^{Q-2}}\right)\cdot\nu$

is strictly positive, and so the thesis is proved. ∎

##  4 Blow-up analysis

In this section we carry out the blow-up analysis necessary in order to prove
Theorem 1.1. This analysis follows closely the analogous analysis for
Riemannian manifolds, which has be performed in various works: we refer, for
example, to [L], [LZhu] and [M].

We start by introducing in the context of CR manifolds some definitions due to
Schoen about blow-up points in conformal geometry which proved themselves
fundamental in the study of nonlinear equations.

Let $M$ be a three-dimensional CR manifold endowed with a pseudohermitian
structure $\theta$, $p_{i}\to 3$ a sequence with $1<p_{i}\leq 3$ for every
$i$, and $u_{i}\in C^{2}(M)$ a sequence of positive solutions of

$L_{\theta}u_{i}=\widetilde{R}u_{i}^{p_{i}},$ (14)

where $\widetilde{R}$ is a positive function of class $C^{1}$.

######  Definition 4.1.

A point $\overline{x}\in M$ is called a blow-up point if there exist a
sequence $x_{i}\to\overline{x}$ such that $M_{i}=u_{i}(x_{i})\to\infty$.

######  Definition 4.2.

A point $\overline{x}\in M$ is called an isolated blow-up point if there exist
$\overline{r}>0$, a constant $C$, and a sequence $x_{i}\to\overline{x}$ such
that $x_{i}$ is a local maximum of $u_{i}$, $u_{i}(x_{i})\to\infty$, and

$u_{i}(x)\leq Cd(x,x_{i})^{-\frac{2}{p_{i}-1}}$

for every $x\in B_{\overline{r}}(x_{i})$.

Given an isolated blow-up point $\overline{x}$, define

$\overline{u}_{i}(r)=\int_{\partial
B_{1}(x_{i})}u_{i}\circ\delta_{r}d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}$

in pseudohermitian normal coordinates, and
$\overline{w}_{i}(r)=r^{\frac{2}{p_{i}-1}}\overline{u}_{i}(r)$.

######  Definition 4.3.

An isolated blow-up point $\overline{x}$ is called an isolated simple blow-up
point $\overline{x}$ if there exists $\rho\in(0,\overline{r})$ (independent of
$i$) such that $\overline{w}_{i}$ has exactly one critical point in
$(0,\rho)$.

Notice that taking the definition from conformal geometry in a straightforward
manner would not be appropriate because in $\mathbf{H}^{1}$, unlike in
$\mathbf{R}^{n}$, the family of surface measures on $\partial B_{r}$ does not
have homogeneity properties, so we have to take this slight variation suited
to the blow-up procedure, which relies on dilations in a fundamental manner.

######  Lemma 4.4.

If $\overline{x}$ is an isolated blow-up point then there exists $C$ such that
for $0<r<\frac{\overline{r}}{3}$ it holds that

$\max_{B_{2r}\setminus B_{r/2}}u_{i}\leq C\min_{B_{2r}\setminus
B_{r/2}}u_{i}.$

###### Proof.

Let $w_{i}=r^{\frac{2}{p_{i}-1}}u_{i}\circ\delta_{r}$. If
$\theta_{r}=\frac{1}{r^{2}}(\delta_{r})^{*}\theta$, using the notation of
Remark 2.1, $w_{i}$ verifies

$L_{\theta_{r}}w_{i}=(\widetilde{R}\circ\delta_{r})w_{i}^{p_{i}}$

on $\delta_{\frac{1}{r}}B_{\overline{r}}$, and the definition of isolated
blow-up and the properties of pseudohermitian normal coordinates imply that
$w_{i}(x)\leq\frac{C}{|x|^{\frac{2}{p_{i}-1}}}$. So $w_{i}$ is uniformly
bounded outside some neighborhood of $0$ with bound independent by $r$, and
the thesis follows the Harnack inequality, which we can apply with constant
independent by $L_{\theta_{r}}$ thanks to formula (5). ∎

In the following, given an isolated blow-up point $x_{i}\to\overline{x}$, in
order to study the blowing up sequence of functions we do a rescaling defining
$M_{i}=u_{i}(x_{i})$ and

$v_{i}=\frac{1}{M_{i}}u_{i}\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}\circ
L_{x_{i}}$

defined on $B_{\overline{r}M_{i}^{\frac{p_{i}-1}{2}}}(\overline{x}$). 222By
our notation this denotes the ball in the Korányi distance, while the
definition of simple blow-up involves the Carnot-Carathéodory distance, but
since the two are equivalent, up to changing $\overline{r}$ there is no
substantial difference in using one or the other. Following the notation of
Remark 2.1, $v_{i}$ verifies

$L_{\theta_{i}}v_{i}=(\widetilde{R}\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}})v_{i}^{p_{i}}$

where $L_{\theta_{i}}$ is the sublaplacian with respect to the rescaled
contact form
$\theta_{i}=M_{i}^{-(p_{i}-1)}\left(\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}\circ
L_{x_{i}}\right)^{*}\theta$ on the rescaled CR structure.

In the following all covariant derivatives applied to $v_{i}$ are meant with
respect to this rescaled pseudohermitian structure.

######  Proposition 4.5.

If $\overline{x}$ is an isolated blow-up point then for any $R_{i}\to\infty$,
$\varepsilon_{i}\to 0$ and $k\in\mathbf{N}$, up to subsequences, in
pseudohermitian normal coordinates around $\overline{x}$ it holds that

$\left\lVert\frac{1}{M_{i}}u_{i}\left(\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}(x_{i}^{-1}\cdot
x)\right)-(U\circ\delta_{\widetilde{R}(0)^{1/2}})(x)\right\rVert_{\Gamma^{k,\alpha}(B_{R_{i}}(0))}\leq\varepsilon_{i},$

where $U$ is defined in Theorem 2.2, $M_{i}=u_{i}(x_{i})$, and

$\frac{R_{i}}{\log M_{i}}\to 0.$

###### Proof.

Using the notation before the statement of this theorem, $v_{i}$ satisfies

$\begin{cases}L_{\theta_{i}}v_{i}=(\widetilde{R}\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}})v_{i}^{p_{i}}\\\
v_{i}(0)=1\\\ \nabla^{H}_{\theta_{i}}v_{i}(0)=0\\\
0<v_{i}(x)<Cd(x,0)^{-\frac{2}{p_{i}-1}}.\end{cases}$ (15)

Thanks to Lemma 4.4, there exists $C$ such that

$\max_{\partial B_{r}}v_{i}\leq C\min_{\partial B_{r}}v_{i}$

for $r<1$. Let $\psi>0$ be a function such that $\psi^{2}\theta$ has positive
Webster curvature in some ball $B_{\widetilde{r}}(0)$ (for example
$\varphi=U\circ\delta_{\lambda}$ for $\lambda$ big enough) and define
$\psi_{i}=\psi\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}$. The operator
$L_{\psi_{i}^{2}\theta_{i}}$ satisfies the maximum principle, and since, by
formula (2),

$L_{\psi_{i}^{2}\theta_{i}}\left(\psi_{i}^{-1}v_{i}\right)=\psi_{i}^{-3}L_{\theta_{i}}v_{i}>0,$

then for any $r\leq\widetilde{r}$ we have
$\min_{B_{r}}\psi_{i}^{-1}v_{i}\geq\min_{\partial B_{r}}\psi_{i}^{-1}v_{i}$,
that is

$\min_{B_{r}}v_{i}\geq C\min_{\partial B_{r}}v_{i}$

for some $C>0$. Therefore if $r\leq\widetilde{r}$,

$\max_{\partial B_{r}}v_{i}\leq C\min_{\partial B_{r}}v_{i}\leq
C_{1}\min_{B_{r}}v_{i}\leq C_{1}v_{i}(0)=C_{1}.$

Thanks to this and to the last inequality in equation (15) we get that
$v_{i}\leq C$ in $B_{\overline{r}M_{i}^{\frac{p_{i}-1}{2}}}$ for any $i$ and
some $C$.

Applying Theorem points 3 and 1 of Theorem 2.3, we can deduce that for any
bounded open subset $V\subset\mathbf{H}^{1}$, eventually $\left\lVert
v_{i}\right\rVert_{\Gamma^{1,\alpha}_{\theta_{i}}(V)}\leq C_{V}$ for every
$\alpha\in(0,1)$. By a standard bootstrap argument using point 4 of Theorem
2.3, for any $k$ we have $\left\lVert
v_{i}\right\rVert_{\Gamma^{k,\alpha}_{\theta_{i}}(V)}\leq C_{V,k}$, and this
implies, by point 2 of the same theorem, the uniformity statement thereof and
formula (5), that $\left\lVert v_{i}\right\rVert_{C^{k,\alpha}(V)}\leq
C^{\prime}_{V,k}$ for every $k$.

This implies that, for any $k$ and $R>0$, up to subsequences $v_{i}$ tends to
some limit $v$ in $C^{k,\alpha}(B_{R})$. By a diagonal argument, up to a
further passage to subsequences we get that $v_{i}\to v$ in
$C^{k,\alpha}(B_{R})$ for any $k$ and $R$, with $v$ defined on
$\mathbf{H}^{1}$ and satisfying:

$\begin{cases}L_{\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{6.93195pt}}v=\widetilde{R}(0)v^{3}\\\
v(0)=1\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}v(0)=0\\\
v>0.\end{cases}$ (16)

The thesis of the theorem follows from Catino-Li-Monticelli-Roncoroni’s
classification theorem (Theorem 2.2). ∎

We point out that in the proof of the theorem above we had to be slightly more
careful than in the proof of the analogous Riemannian result in handling
regularity theorems because, while Sobolev and Hölder spaces with respect to
different Riemannian metrics on a compact manifold coincide, this is not true
for the spaces $S^{k,\alpha}$ and $\Gamma^{k,\alpha}$ with respect to two
different pseudohermitian structures $(\mathscr{H}_{1},\theta_{1})$ and
$(\mathscr{H}_{2},\theta_{2})$ on the same manifold, because they are defined
through derivatives along the respective Levi distributions $H_{1}(M)$ and
$H_{2}(M)$, which do not necessarily coincide.

######  Lemma 4.6.

Let $\overline{x}$ an isolated simple blow-up point, $R_{i}\to\infty$, and
suppose that the thesis of Lemma 4.5 holds for some $\varepsilon_{i}\to 0$.
Then, given a fixed, sufficiently small $\delta>0$, there exists
$\rho_{1}\in(0,\rho)$ ($\rho$ being the one from Definition 4.3) such that

$u_{i}(x)\leq CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-2+\delta},$
$|(u_{i})_{,1}(x)|\leq CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-3+\delta},$
$|(u_{i})_{,11}(x)|\leq CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-4+\delta}$
$|(u_{i})_{,1\overline{1}}(x)|\leq
CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-4+\delta},$ $|(u_{i})_{,0}(x)|\leq
CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-4+\delta}$ $|(u_{i})_{,01}(x)|\leq
CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-5+\delta},$ $|(u_{i})_{,00}(x)|\leq
CM_{i}^{-\lambda_{i}}d(x,x_{i})^{-6+\delta},$

for $R_{i}M_{i}^{-\frac{p_{i}-1}{2}}\leq d(x,x_{i})\leq\rho_{1}$, where
$\lambda_{i}=(2-\delta)\frac{p_{i}-1}{2}-1$.

###### Proof.

The proof is analogous to the proof of Lemma 3.3 in [LZhu]. ∎

######  Lemma 4.7.

In the hypotheses of Lemma 4.6 for $|x|\leq\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}$
the following estimates hold:

$v_{i}(x)\leq CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-2},$
$|(v_{i})_{,1}(x)|\leq CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-3},$
$|(v_{i})_{,11}(x)|\leq CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-4},$
$|(v_{i})_{,1\overline{1}}(x)|\leq
CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-4},$ $|(v_{i})_{,0}(x)|\leq
CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-4},$ $|(v_{i})_{,01}(x)|\leq
CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-5},$ $|(v_{i})_{,00}(x)|\leq
CM_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-6}.$

###### Proof.

It follows from Lemmas 4.5 and 4.6. ∎

######  Lemma 4.8.

If $\overline{x}$ is an isolated blow-up point then, with the notation of
Lemma 4.6, in pseudohermitian normal coordinates around $x_{i}$

$\left|\int_{B_{\rho_{1}}(x_{i})}\left(u_{i}+\Xi
u_{i}\right)(\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u_{i}-\Delta_{b}u_{i})\right|\leq
CM_{i}^{-2+2\delta+o(1)}$

###### Proof.

Thanks to formulas (6) and (29) and Lemma 4.7, and the fact that $v_{i}$ is
real, we have 333Estimate (29) has been proved for a fixed pseudohermitian
structure, while the functions $v_{i}$ are differentiated with respect to a
sequence of pseudohermitian structures, but since that sequence converges in
$C^{\infty}$, the estimate hold with an uniform constant. This fact is used
also later.

$\left|v_{i}+\Xi v_{i}\right|\lesssim
v_{i}+|x||\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}v_{i}|+|x|^{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}v_{i}|\lesssim
v_{i}+|x||(v_{i})_{,1}|+|x|^{2}|(v_{i})_{,0}|\lesssim$ $\lesssim
M_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-2}+|x|M_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-3}+|x|^{2}M_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-4}\lesssim$
$\lesssim M_{i}^{\frac{p_{i}-1}{2}\delta}(1+|x|)^{-2}.$

Furthermore define
$v_{i}=\frac{1}{M_{i}}u_{i}\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}$. Since
$u_{i}$ is real, using Lemmas A.6 and 4.7

$|\Delta_{b}u_{i}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u_{i}|\lesssim|x||(u_{i})_{,1}|+|x|^{2}|(u_{i})_{,0}|+|x|^{2}|(u_{i})_{,1\overline{1}}|+|x|^{2}|(u_{i})_{,11}|+$
$+|x|^{3}|(u_{i})_{01}|+|x|^{6}|(u_{i})_{,00}|\lesssim$ $\lesssim
M_{i}\left(M_{i}^{-\frac{p_{i}-1}{2}}|x|M_{i}^{\frac{p_{i}-1}{2}}|(v_{i})_{,1}|+M_{i}^{-(p_{i}-1)}|x|^{2}M_{i}^{p_{i}-1}|(v_{i})_{,0}|+\right.$
$+M_{i}^{-(p_{i}-1)}|x|^{2}M_{i}^{p_{i}-1}|(v_{i})_{,1\overline{1}}|+M_{i}^{-(p_{i}-1)}|x|^{2}M_{i}^{p_{i}-1}|(v_{i})_{,11}|+$
$\left.+M_{i}^{-3\frac{p_{i}-1}{2}}|x|^{3}M_{i}^{3\frac{p_{i}-1}{2}}|(v_{i})_{01}|+M_{i}^{-3(p_{i}-1)}|x|^{6}M_{i}^{2(p_{i}-1)}|(v_{i})_{,00}|\right)\circ\delta_{M_{i}^{\frac{p_{i}-1}{2}}}\lesssim$
$\lesssim
M_{i}M_{i}^{\frac{p_{i}-1}{2}\delta}\left(|x|(1+|x|)^{-3}+|x|^{2}(1+|x|)^{-4}+|x|^{2}(1+|x|)^{-4}+\right.$
$\left.+|x|^{2}(1+|x|)^{-4}+|x|^{3}(1+|x|)^{-5}+M_{i}^{-(p_{i}-1)}|x|^{6}(1+|x|)^{-6}\right)\circ\delta_{M_{i}^{\frac{p_{i}-1}{2}}}\lesssim$
$\lesssim
M_{i}M_{i}^{\frac{p_{i}-1}{2}\delta}\left(|x|(1+|x|)^{-3}+M_{i}^{-(p_{i}-1)}|x|^{6}(1+|x|)^{-6}\right)\circ\delta_{M_{i}^{\frac{p_{i}-1}{2}}}.$

So

$\left|\int_{B_{\rho_{1}}(x_{i})}\left(u_{i}+\Xi
u_{i}\right)(\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u_{i}-\Delta_{b}u_{i})\right|=$
$=M_{i}M_{i}^{-2(p_{i}-1)}\left|\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}(0)}\left(v_{i}+\Xi
v_{i}\right)\cdot(\Delta_{b}u_{i}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u_{i})\circ\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}\right|\lesssim$
$\lesssim
M_{i}^{2}M_{i}^{-2(p_{i}-1)}M_{i}^{(p_{i}-1)\delta}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}(0)}(1+|x|)^{-2}\cdot\left[|x|(1+|x|)^{-3}+\right.$
$\left.+M_{i}^{-(p_{i}-1)}|x|^{6}(1+|x|)^{-6}\right]\lesssim$ $\lesssim
M_{i}^{4-2p_{i}}M_{i}^{(p_{i}-1)\delta}\left[\log
M_{i}+M_{i}^{-(p_{i}-1)}M_{i}^{2}\right]=M_{i}^{-2+2\delta+o(1)}.$

∎

######  Lemma 4.9.

In the hypotheses of Lemma 4.6, if $\tau_{i}=\frac{Q+2}{Q-2}-p_{1}=3-p_{i}$
then

$\tau_{i}=O(u_{i}(x_{i})^{-2+2\delta+o(1)})$

and in particular $u_{i}(x_{i})^{\tau_{i}}\to 1$.

###### Proof.

Let us apply the Pohozaev identity of Proposition 3.3 with respect to the base
point $x_{i}$ with $r=\rho_{1}$:

$\int_{B_{\rho_{1}}}\left(\left(\frac{1}{p_{i}+1}-\frac{1}{4}\right)\widetilde{R}u_{i}^{p_{i}+1}-\frac{1}{4}Ru_{i}^{2}+\frac{1}{4}\frac{1}{p_{i}+1}\Xi(\widetilde{R})u_{i}^{p_{i}+1}+\right.$
$\left.-\frac{1}{8}\Xi(R)u_{i}^{2}-\left(\Xi
u_{i}+u_{i}\right)(\Delta_{b}u(x)-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u)\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt}=$
$=\int_{\partial
B_{\rho_{1}}}\left(\left(\frac{1}{4}\frac{1}{p_{i}+1}\widetilde{R}u_{i}^{p_{i}+1}-\frac{1}{8}Ru_{i}^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\mathscr{B}(x,u_{i},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u)\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}.$

Let us estimate the various terms.

$\int_{B_{\rho_{1}}}\Xi(\widetilde{R})u_{i}^{p_{i}+1}\lesssim$ $\lesssim
M_{i}^{p_{i}+1}M_{i}^{-2(p_{i}-1)}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}(M_{i}^{-\frac{p_{i}-1}{2}}|x|+M_{i}^{-(p_{i}-1)}|x|^{2})v_{i}^{p_{i}+1}\lesssim$
$\lesssim
M_{i}^{3-p_{i}}\left(\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}M_{i}^{-\frac{p_{i}-1}{2}}|x|v_{i}^{p_{i}+1}+\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}M_{i}^{-(p_{i}-1)}|x|^{2}v_{i}^{p_{i}+1}\right)\lesssim$
$\lesssim
M_{i}^{\frac{7-3p_{i}}{2}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}|x|(1+|x|)^{-2(p_{i}+1)}+$
$+M_{i}^{4-2p_{i}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}|x|^{2}(1+|x|)^{-2(p_{i}+1)}\lesssim$
$\lesssim
M_{i}^{\frac{7-3p_{i}}{2}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{\frac{p_{i}-1}{2}\left[-2(p_{i}+1)+5\right]}+M_{i}^{4-2p_{i}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{\frac{p_{i}-1}{2}\left[-2(p_{i}+1)+6\right]}=$
$=M_{i}^{\frac{7-3p_{i}}{2}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{\frac{p_{i}-1}{2}\left[-2(p_{i}+1)+5\right]}+M_{i}^{4-2p_{i}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{\frac{p_{i}-1}{2}\left[-2(p_{i}+1)+6\right]}=$
$=M_{i}^{\frac{7-3p_{i}}{2}+5\frac{p_{i}-1}{2}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{-(p_{i}^{2}-1)}+M_{i}^{4-2p_{i}+6\frac{p_{i}-1}{2}}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{-(p_{i}^{2}-1)}=$
$=M_{i}^{p_{i}+1}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta}M_{i}^{-(p_{i}^{2}-1)}=M_{i}^{-(p_{i}+1)(p_{i}-2)}M_{i}^{\frac{p_{i}^{2}-1}{2}\delta},$

$\int_{B_{\rho_{1}}}\left(\frac{1}{4}R+\frac{1}{8}\Xi(R)\right)u_{i}^{2}\lesssim
M_{i}^{2}u_{i}(x_{i})^{-2(p_{i}-1)}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}v_{i}^{2}\lesssim$
$\lesssim
M_{i}^{4-2p_{i}}M_{i}^{(p_{i}-1)\delta}\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}(1+|x|)^{-4}\lesssim
M_{i}^{4-2p_{i}}M_{i}^{(p_{i}-1)\delta}\log M_{i},$

and thanks to Lemma 4.6 it is obvious that

$\int_{\partial
B_{\rho_{1}}}\left(\left(\frac{1}{4}\frac{1}{p_{i}+1}\widetilde{R}u_{i}^{p_{i}+1}-\frac{1}{8}Ru_{i}^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\mathscr{B}(x,u_{i},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}u)\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}=$
$=O(M_{i}^{(-2+\delta)(p_{i}-1)+2}).$

Finally by Proposition 4.5 it holds that

$\int_{B_{\rho_{1}M_{i}^{\frac{p_{i}-1}{2}}}}u_{i}^{p_{i}+1}\gtrsim
u_{i}(x_{i})^{p_{i}+1}u_{i}(x_{i})^{-2(p_{i}-1)}\cdot$
$\cdot\int_{B_{R_{i}}}\frac{1}{u_{i}(x_{i})^{p_{i}+1}}u_{i}\left(\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}(x_{i}^{-1}\cdot
x)\right)^{p_{i}+1}\gtrsim u_{i}(x_{i})^{3-p_{i}}\gtrsim 1.$

So, using also Lemma 4.8, and the fact that
$\frac{1}{p_{i}+1}-\frac{1}{4}=\frac{\tau_{i}}{4(p_{i}+1)}$, we get the
thesis. ∎

######  Lemma 4.10.

In the hypotheses of Lemma 4.6, if $\overline{x}$ is an isolated simple blow-
up point then for every $\sigma\in(0,\frac{\overline{r}}{2})$

$\limsup_{i\to\infty}\max_{\partial B_{\sigma}}M_{i}u_{i}(x)\leq C(\sigma).$

###### Proof.

Thanks to Lemma 4.4, it is sufficient to prove the thesis for $\sigma$ small
enough. In particular, reasoning as in the proof of Proposition 4.5, we can
suppose that $R>0$.

Let $x_{\sigma}$ such that $d(x_{\sigma},x_{i})=\sigma$, and define Let
$w_{i}(x)=u_{i}(x_{\sigma})^{-1}u_{i}(x)$. Then $w_{i}$ satisfies

$L_{\theta}w_{i}=u_{i}(x_{\sigma})^{p_{i}-1}w_{i}^{p_{i}}.$ (17)

Thanks to Lemma 4.6 and to Harnack inequality, for every compact $K\subset
B_{\rho_{1}}(\overline{x})\setminus\\{\overline{x}\\}$ there exists $C_{K}$
such that $C_{K}^{-1}\leq w_{i}\leq C_{K}$. Therefore, applying the regularity
theory from Theorem 2.3, we can deduce that up to subsequences $w_{i}\to w$ in
$C^{2}_{\mathrm{loc}}(B_{\rho_{1}}(\overline{x})\setminus\\{\overline{x}\\})$,
and since, by Lemma 4.6, $u_{i}(x_{\sigma})\to 0$, passing to the limit
equation (17) one gets that $L_{\theta}w=0$.

Since the blow-up is isolated simple, and since Proposition 4.5 implies that
$r^{\frac{2}{p_{i}-1}}\overline{u}_{i}$ has a critical point in
$(0,R_{i}M_{i}^{-\frac{p_{i}-1}{2}})$ after which it is decreasing,
$r^{\frac{2}{p_{i}-1}}\overline{u}_{i}$ is decreasing in
$(R_{i}M_{i}^{-\frac{p_{i}-1}{2}},\rho)$, and because

$u_{i}(x_{\sigma})^{-1}r^{\frac{2}{p_{i}-1}}\overline{u}_{i}(r)=r^{\frac{2}{p_{i}-1}}\overline{w}_{i}(r)\to
r\overline{w}(r),$

$r\overline{w}(r)$ is decreasing on $(0,\rho)$. Since $w>0$, $w$ must be
singular at $\overline{x}$. Corollary 9.1 in [LZhu] can be extend to
pseudohermitian geometry repeating the proof with minor adaptations; applying
it we get that

$-\int_{B_{\sigma}(x_{i})}\Delta_{b}w_{i}=-\int_{\partial
B_{\sigma}(x_{i})}\nabla^{H}w_{i}\cdot\nu=-\int_{\partial
B_{\sigma}(\overline{x})}\nabla^{H}w\cdot\nu+o(1)=c+o(1)>0$ (18)

while integrating equation (17) we get

$-4\int_{B_{\sigma}(x_{i})}\Delta_{b}w_{i}=\int_{B_{\sigma}(x_{i})}\left(-Rw_{i}+u_{i}(x_{\sigma})^{p_{i}-1}w_{i}^{p_{i}}\right)\leq
u_{i}(x_{\sigma})^{-1}\int_{B_{\sigma}(x_{i})}u_{i}^{p_{i}}.$ (19)

But, calling $r_{i}=R_{i}M_{i}^{-\frac{p_{i}-1}{2}}$, by Lemmas 4.5, 4.6 and
4.9 eventually

$\int_{B_{\sigma}(x_{i})}u_{i}^{p_{i}}=\left(\int_{B_{r_{i}}(x_{i})}+\int_{B_{\sigma}(x_{i})\setminus
B_{r_{i}}(x_{i})}\right)u_{i}^{p_{i}}\lesssim$ $\lesssim
M^{-2(p_{i}-1)}M_{i}^{p_{i}}\int_{B_{R_{i}}(0)}(1+|x|)^{-2p_{i}}+M_{i}^{-\lambda_{i}p_{i}}\int_{B_{\sigma}(x_{i})\setminus
B_{r_{i}}(x_{i})}|x|^{(-2+\delta)p_{i}}\lesssim$ $\lesssim
M_{i}^{2-p_{i}}+M_{i}^{-\lambda_{i}p_{i}}(R_{i}M_{i}^{-\frac{p_{i}-1}{2}})^{(-2+\delta)p_{i}+4}\lesssim
M_{i}^{-1}.$ (20)

Putting together formulas (18), (19) and (20) we get that
$u_{i}(x_{i})u_{i}(x_{\sigma})$ is a bounded sequence. The thesis follows from
Lemma 4.4. ∎

######  Proposition 4.11.

If $\overline{x}$ is an isolated simple blow-up point then there exists $C$
such that

$M_{i}u_{i}(x)\leq Cd(x,x_{i})^{-2}$

if $d(x,x_{i})\leq\frac{\rho}{2}$. Furthermore, up to subsequences, there
exists $a>0$ such that

$M_{i}u(x)\to aG_{\overline{x}}(x)+b$

in
$C^{2}_{\mathrm{loc}}(B_{\frac{\rho}{2}}(\overline{x})\setminus\\{\overline{x}\\})$,
where $G_{\overline{x}}$ is the Green function of $L_{\theta}$ (which exists
because $M$ has positive CR Yamabe class) and $L_{\theta}b=0$ on
$B_{\frac{\rho}{2}}(\overline{x})$.

###### Proof.

If this were not the case, then, up to subsequences, there would exist a
sequence $\widetilde{x}_{i}$ with

$M_{i}u_{i}(\widetilde{x}_{i})d(x_{i},\widetilde{x}_{i})^{2}\to\infty.$ (21)

Define $\widetilde{r}_{i}=d(x_{i},\widetilde{x}_{i})$.

Since
$\sup_{\lambda>0}\lambda^{2}U(\delta_{\lambda}(x))\leq\frac{C}{|x|^{2}}$, it
is easy to verify using Proposition 4.5 and Lemma 4.9 that
$R_{i}M_{i}^{-\frac{p_{i}-1}{2}}\leq\widetilde{r}_{i}\leq\rho$.

Define
$\widetilde{u}_{i}=\widetilde{r}_{i}^{\frac{2}{p_{i}-1}}u_{i}\circ\delta_{\widetilde{r}_{i}}\circ
L_{x_{i}}$ in $B_{2}$. $\widetilde{u}_{i}$ satisfies

$L_{\theta_{i}}\widetilde{u}_{i}=\widetilde{R}\widetilde{u}_{i}^{p_{i}}$

and verifies the hypotheses of Lemma 4.10, therefore $\max_{\partial
B_{1}}\widetilde{u}_{i}(0)\widetilde{u}_{i}<\infty$. Using the definition of
$\widetilde{u}_{i}$ and Lemma 4.9, this goes in contradiction with formula
(21).

Hence $M_{i}u_{i}$ is locally bounded in
$B_{\frac{\rho}{2}}(\overline{x})\setminus\\{\overline{x}\\}$, and satisfies

$L_{\theta}(M_{i}u_{i})=M_{i}^{1-p_{i}}(M_{i}u_{i})^{p_{i}},$

therefore, applying regularity theory (see Theorem 2.3),

$M_{i}u_{i}\to
v\;\;\;\text{in}\;\;\;C^{2}_{\mathrm{loc}}(B_{\frac{\rho}{2}}(\overline{x})\setminus\\{\overline{x}\\})$

with $v$ satisfying $L_{\theta}v=0$. Known results about singular solutions
(see for example Proposition 9.1 in [LZhu], which can adapted without
difficulty to pseudohermitian geometry) imply the rest of the thesis, except
the fact that $a>0$. This can be proved by proving that $v$ must be singular
with the same proof of Lemma 4.10. ∎

Using Proposition 4.11 and Lemma 4.9 we can improve the estimates of Lemma 4.6
and Lemma 4.8 with the same proofs (which we do not repeat). We state only the
estimates that we will need. The last statement of the following lemma can be
proved by repeating the proof of Lemma 4.8 using the estimates for CR normal
coordinates (see Proposition A.5 in [CMY1]).

######  Lemma 4.12.

If $\overline{x}$ is an isolated blow-up point then, with the notation of
Lemma 4.6 there exists $\rho_{1}\in(0,\rho)$ such that

$u_{i}(x)\leq CM_{i}^{-1}d(x,x_{i})^{-2},$ $|(u_{i})_{,1}(x)|\leq
CM_{i}^{-1}d(x,x_{i})^{-3},$

for $R_{i}M_{i}^{-\frac{p_{i}-1}{2}}\leq d(x,x_{i})\leq\rho_{1}$, and in CR
normal coordinates around $x_{i}$, for $\sigma\leq\rho_{1}$,

$\left|\int_{B_{\sigma}(x_{i})}\left(u_{i}+\Xi
u_{i}\right)(\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}u_{i}-\Delta_{b}u_{i})\right|\leq
C\sigma M_{i}^{-2}.$

From now on we will focus on the case of constant $\widetilde{R}$, and by
homogeneity we can suppose that $\widetilde{R}\equiv 1$. So the equation
becomes

$L_{\theta}u_{i}=u_{i}^{p_{i}}.$ (22)

######  Remark 4.13.

In the following we will need to use CR normal coordinates, which involve a
change of contact form. If $\varphi$ is such that
$\widetilde{\theta}=\varphi^{2}\theta$ defines CR normal coordinates around
some point, and $\widetilde{u}_{i}=\frac{u_{i}}{\varphi}$, then, thanks to
formula (2), $\widetilde{u}_{i}$ verifies

$L_{\widetilde{\theta}}\widetilde{u}_{i}=\varphi^{-\tau_{i}}\widetilde{u}_{i}^{p_{i}}.$

It is easy to prove that the estimates proved until now for $u_{i}$, in
particular the ones in Proposition 4.5, Proposition 4.11 and Lemma 4.12, hold
for $\widetilde{u}_{i}$ too. It is also easy to prove that a point
$\overline{x}$ is an isolated blow-up point for the sequence $u_{i}$ if and
only if it is one for the sequence $\widetilde{u}_{i}$. We will say that
$\overline{x}$ is an isolated blow-up point in CR normal coordinates if it is
an isolated blow-up point for $\widetilde{u}_{i}$ with respect to the contact
form $\widetilde{\theta}$.

At some point we will need to consider CR coordinates around $x_{i}$ where
$x_{i}\to\overline{x}$, defined by forms
$\widetilde{\theta}_{i}=\varphi_{i}^{2}\theta$. In such case
$\widetilde{u}_{i}=\frac{u_{i}}{\varphi_{i}}$ verifies the same estimates for
analogous reasons because $\varphi_{i}\to\varphi$ smoothly on compact sets,
because their construction involves only algebraic operations with objects
depending smoothly on the base point (see [JL3]).

######  Lemma 4.14.

If $\overline{x}$ is an isolated simple blow-up point for equation (22) and
$M_{i}u_{i}\to h$ in
$C^{2}_{\mathrm{loc}}(B_{r}(\overline{x})\setminus\\{\overline{x}\\})$, and in
CR normal coordinates around $\overline{x}$ one has
$h(x)=\frac{a}{|x|^{2}}+A+o(1)$ with $a>0$, then $A\leq 0$.

###### Proof.

Using the notation introduced in Remark 4.13 for CR normal coordinates around
$\overline{x}$, given $\sigma\in(0,\rho_{1})$ the function
$\widetilde{u}_{i}\circ\delta_{\sigma}$ verifies the equation

$L_{\theta^{\prime}_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})=(\varphi\circ\delta_{\sigma})^{-\tau_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}}$

where
$\theta^{\prime}_{i}=\frac{1}{\sigma^{2}}\delta_{1/\sigma}^{*}\widetilde{\theta}_{i}$.

Apply the Pohozaev identity to $\widetilde{u}_{i}\circ\delta_{\sigma}$ on
$B_{1}(\overline{x})$:

$\int_{B_{1}(\overline{x})}\left(\left(\frac{1}{p_{i}+1}-\frac{1}{4}\right)(\varphi\circ\delta_{\sigma})^{-\tau_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}-\frac{1}{4}(R\circ\delta_{\sigma})(\widetilde{u}_{i}\circ\delta_{\sigma})^{2}+\right.$
$+\frac{1}{4}\frac{1}{p_{i}+1}\Xi((\varphi\circ\delta_{\sigma})^{-\tau_{i}})(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}-\frac{1}{8}\Xi(R\circ\delta_{\sigma})(\widetilde{u}_{i}\circ\delta_{\sigma})^{2}+$
$\left.-\left(\Xi(\widetilde{u}_{i}\circ\delta_{\sigma})+(\widetilde{u}_{i}\circ\delta_{\sigma})\right)(\Delta_{b}(\widetilde{u}_{i}\circ\delta_{\sigma})-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}(\widetilde{u}_{i}\circ\delta_{\sigma}))\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt}=$
$=\int_{\partial
B_{1}(\overline{x})}\left(\left(\frac{1}{4}\frac{1}{p_{i}+1}(\varphi\circ\delta_{\sigma})^{-\tau_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}-\frac{1}{8}(R\circ\delta_{\sigma})(\widetilde{u}_{i}\circ\delta_{\sigma})^{2}\right)\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}+\right.$
$\left.+\mathscr{B}(x,\widetilde{u}_{i}\circ\delta_{\sigma},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}(\widetilde{u}_{i}\circ\delta_{\sigma}))\right)d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}$
(23)

Since $u_{i}\circ\delta_{\sigma}=M_{i}v_{i}\circ L_{x_{1}}\circ\delta_{\sigma
M_{i}^{\frac{p_{i}-1}{2}}}$, $x_{i}\to\overline{x}$ and $R=O(|x|^{2})$ and
$R_{,1}=O(|x|)$ (see Appendix A.2 in [CMY1]), and thanks to Lemma 4.12, we
have

$\int_{B_{1}(\overline{x})}(2R\circ\delta_{\sigma}+\Xi(R\circ\delta_{\sigma}))(u_{i}\circ\delta_{\sigma})^{2}d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt}\lesssim$
$\lesssim\sigma^{-4}M_{i}^{-2(p_{i}-1)}M_{i}^{2}M_{i}^{-(p_{i}-1)}\int_{B_{\sigma
M_{i}^{\frac{p_{i}-1}{2}}}(0)}|x|^{2}(1+|x|)^{-4}d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{V}\rule{0.0pt}{9.90276pt}\lesssim\sigma^{-2}M_{i}^{-2},$
$\int_{\partial
B_{1}(\overline{x})}(\varphi\circ\delta_{\sigma})^{-\tau_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}\Xi\cdot\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nu}\rule{0.0pt}{9.90276pt}d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}\lesssim
M_{i}^{-(p_{i}-1)}\sigma^{-2(p_{i}-1)},$

and since $\frac{4}{p_{i}+1}-1=\frac{\tau_{i}}{p_{i}+1}$,

$\int_{B_{1}(\overline{x})}\left[\left(\frac{4}{p_{i}+1}-1\right)(\varphi\circ\delta_{\sigma})^{-\tau_{i}}(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}+\frac{1}{p_{i}+1}\Xi((\varphi\circ\delta_{\sigma})^{-\tau_{i}})(\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}\right]=$
$=\frac{\tau_{i}}{p_{i}+1}\int_{B_{1}(\overline{x})}(\varphi\circ\delta_{\sigma})^{-\tau_{i}-1}\left[\varphi\circ\delta_{\sigma}+\Xi(\varphi\circ\delta_{\sigma})\right](\widetilde{u}_{i}\circ\delta_{\sigma})^{p_{i}+1}\geq
0$

if $\sigma$ is small enough.

Multiplying formula (23) by $\sigma^{3}M_{i}^{2}$, taking $\lim_{\sigma\to
0}\limsup_{i\to\infty}$ and using the above estimates and the last statement
in Lemma 4.12, one gets that

$\lim_{\sigma\to 0}\limsup_{i\to\infty}\sigma^{3}M_{i}^{2}\int_{\partial
B_{1}(\overline{x})}\mathscr{B}(x,\widetilde{u}_{i}\circ\delta_{\sigma},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}(\widetilde{u}_{i}\circ\delta_{\sigma}))d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}\geq
0.$

But by Proposition 3.4

$\lim_{\sigma\to 0}\limsup_{i\to\infty}\sigma^{3}M_{i}^{2}\int_{\partial
B_{1}(\overline{x})}\mathscr{B}(x,\widetilde{u}_{i}\circ\delta_{\sigma},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}(\widetilde{u}_{i}\circ\delta_{\sigma}))d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}=$
$=\lim_{\sigma\to 0}\sigma^{3}\int_{\partial
B_{1}(\overline{x})}\mathscr{B}(x,h\circ\delta_{\sigma},\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\nabla}\rule{0.0pt}{9.90276pt}^{H}(h\circ\delta_{\sigma}))d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}=-\frac{c_{n}a}{\gamma_{n}}A$

and therefore $A\leq 0$. ∎

######  Proposition 4.15.

Isolated blow-up points for equation (22) are isolated simple blow-up points
in CR normal coordinates.

###### Proof.

If this is not the case, up to subsequences there exists
$\overline{\mu}_{i}\to 0$ such that $r^{\frac{2}{p_{i}-1}}\overline{u}_{i}(r)$
has at least two critical points in $(0,\overline{\mu}_{i})$. Up to further
passage to subsequences, one can assume that the thesis of Proposition 4.5
holds. Then $r^{\frac{2}{p_{i}-1}}\overline{\widetilde{u}}_{i}(r)$ has exactly
one critical point in $(0,r_{i})$ where
$r_{i}=R_{i}M_{i}^{-\frac{p_{i}-1}{2}}$.

Since critical points form a closed set, let
$\mu_{i}\in(r_{i},\overline{\mu}_{i})$ be the second critical point.

We pass in CR normal coordinates around $x_{i}$ using the notation from Remark
4.13, and we define
$\xi_{i}=\mu_{i}^{\frac{2}{p_{i}-1}}\widetilde{u}_{i}\circ\delta_{\mu_{i}}$ on
$B_{1/\mu_{i}}(\overline{x})$. Then, following Remark 2.1, $\xi_{i}$ satisfies
the equation

$L_{\theta_{i}}\xi_{i}=(\varphi_{i}\circ\delta_{\mu_{i}})^{-\tau_{i}}\xi_{i}^{p_{i}}$

where
$\theta_{i}=\mu_{i}^{2}\left(\delta_{\mu_{i}}\right)^{*}\widetilde{\theta}_{i}$.

Furthermore $\xi_{i}(x)\leq Cd(x,x_{i})^{-\frac{2}{p_{i}-1}}$ by definition of
isolated blow-up point,

$\xi_{i}(0)=\mu_{i}^{\frac{p_{i}-1}{2}}\widetilde{u}_{i}(x_{i})\geq
R_{i}^{\frac{p_{i}-1}{2}}M_{i}^{-1}\widetilde{u}_{i}(x_{i})\to\infty,$

$r^{\frac{2}{p_{i}-1}}\overline{\xi}_{i}(r)$ has exactly one critical point on
$(0,1)$,

$\left.\frac{d}{dr}\right|_{r=1}\left(r^{\frac{2}{p_{i}-1}}\xi_{i}(0)\overline{\xi}_{i}(r)\right)=0,$
(24)

and the domains of $\xi_{i}$ exaust $\mathbf{H}^{1}$.

Proposition 4.11 and an application of Lemma 4.4 to increasing annuli permit
to prove that on every compact $K\subset\mathbf{H}^{1}\setminus\\{0\\}$ there
exists $C_{K}$ such that

$C_{K}^{-1}\leq\xi(0)\xi\leq C_{K}$

on K. This and Proposition 4.11 allow to conclude that, up to subsequences,

$\xi_{i}(0)\xi(x)\to h(x)=\frac{a}{|x|^{2}}+b(x)$

on compact subsets of $\mathbf{H}^{1}\setminus\\{0\\}$, with $a>0$ and $b$
satisfying $\Delta_{b}b=0$ on $\mathbf{H}^{1}$. Since $h$ is positive, then
$\liminf_{x\to\infty}b(x)\geq 0$, and thus by the maximum principle $b\geq 0$,
and therefore by the Liouville theorem on $\mathbf{H}^{n}$ (which, for
example, can be derived by the Harnack inequality) $b(x)\equiv b$ is a
constant.

Furthermore, passing to the limit in equation (24) allows to deduce that

$0=\left.\frac{d}{dr}\right|_{r=1}(r\overline{h}(r))=C\left.\frac{d}{dr}\right|_{r=1}\left(\frac{a}{r}+br\right)=(-a+b)C$

(where $C=\int_{\partial
B_{1}(0)}d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\sigma}\rule{0.0pt}{9.90276pt}$)
therefore $b=a>0$. But this goes against Lemma 4.14 (here we are in a
situation slightly different from the hypotheses of that lemma because the
contact form $\widetilde{\theta}_{i}$ is not fixed, but the proof can be
adapted straightforwardly). ∎

######  Proposition 4.16.

For every $\varepsilon>0$, $R>1$, there exist $C_{0}$ e $C_{1}$ such that if
$u$ solves equation (22) with $\max u>C_{0}$, then there exist
$x_{1},\ldots,x_{k}$ verifying:

  1. 1.

$0\leq\tau_{i}<\varepsilon$;

  2. 2.

the $x_{i}$ are local maxima of $u$, and calling
$\overline{r}_{j}=Ru(x_{j})^{-(p-1)/2}$, the balls
$B_{\overline{r}_{j}}(x_{j})$ are disjoint and in pseudohermitian coordinates
around $x_{i}$

$\left\lVert\frac{1}{u(x_{j})}u\left(\delta_{M_{i}^{-\frac{p_{i}-1}{2}}}(x_{i}^{-1}\cdot
x)\right)-U(x)\right\rVert_{C^{2}(B_{R}(0))}<\varepsilon;$

  3. 3.

$d(x,\\{x_{1},\ldots,x_{k}\\})^{\frac{2}{p-1}}\leq C_{1}$ per ogni $x\in M$.

###### Proof.

The proof is equal to the analogous statement for Riemannian manifolds, but
using Theorem 2.2 instead of the corresponding result in $\mathbf{R}^{n}$ by
Caffarelli, Gidas and Spruck (Corollary 8.2 in [CGS]) and Ma-Ou’s theorem
(Theorem 1.1 in [MO]) instead of the corresponding result in $\mathbf{R}^{n}$
by Gidas and Spruck (Theorem 1.1 in [GS]). See for example Proposition 5.1 in
[LZhu]. ∎

######  Proposition 4.17.

In the same hypotheses of the Proposition 4.16, there exists $\delta^{*}$
depending only by $(M,\mathscr{H},\theta)$, $R$ and $\varepsilon$ such that,
in the notation of the same proposition, if $\max u>C_{0}$ then
$d(x_{j},x_{\ell})\geq\delta^{*}$ for every $j\neq\ell$.

###### Proof.

Were the thesis false, there would exist $\varepsilon$, $R$ and sequences
$p_{i}$ and $u_{i}$ of solutions of equation (22) with local maxima
$x_{1}^{i},\ldots,x_{N_{i}}^{i}$ such that (up to relabeling)

$\sigma_{i}=d(x_{1}^{i},x_{2}^{i})=\min_{j,\ell}d(x_{j}^{i},x_{\ell}^{i})\to
0.$

By point 1 of Proposition 4.16, $p_{i}\to 3$. Since by point 2 in the same
proposition $B_{Ru_{i}(x_{1}^{i})}\cap B_{Ru_{i}(x_{2}^{i})}=\emptyset$, then
$u_{i}(x_{1}^{i}),u_{i}(x_{2}^{i})\to\infty$.

Using CR normal coordinates around $x_{1}^{i}$ with the notation from Remark
4.13, we have

$L_{\widetilde{\theta}_{i}}\widetilde{u}_{i}=\varphi_{i}^{-\tau_{i}}\widetilde{R}\widetilde{u}_{i}^{p_{i}}.$

Let
$w_{i}=\sigma_{i}^{\frac{2}{p_{i}-1}}\widetilde{u}_{i}\circ\delta_{\sigma_{i}}$
defined on $B_{1/\sigma}\subset\mathbf{H}^{1}$. Then $w_{i}$ satisfies

$L_{\theta^{\prime}_{i}}w_{i}=(\varphi_{i}\circ\delta_{\sigma_{i}})^{-\tau_{i}}\widetilde{R}w_{i}^{p_{i}}$

(where
$\widetilde{\theta}_{i}=\sigma_{i}^{2}\left(\delta_{\sigma_{i}}\right)^{*}\widetilde{\theta}$).

Calling $\widetilde{x}_{i}\in\mathbf{H}^{1}$ the point corresponding to
$x_{2}^{i}$ in CR coordinates, since the Carnot-Carathéodory distance of $M$
converges to the one of $\mathbf{H}^{1}$, $d(\widetilde{x}_{i},0)\to 1$, and
therefore up to subsequences $\widetilde{x}_{i}\to\widetilde{x}$ with
$d(\widetilde{x},0)=1$.

Let $S\subset\mathbf{H}^{1}$ be the set of blow-up points for $w_{i}$. By
construction

$\min_{x,y\in S,x\neq y}d(x,y)\geq 1,$

and thus, using point 3 in Proposition 4.16, they are isolated blow up points.
It is easy to see that if $w_{i}(0)\to\infty$ then $0\in S$, and similarly for
$\widetilde{x}$.

Since, by point 2 in Proposition 4.16,

$\sigma_{i}\geq\max\left\\{Ru_{i}(x_{i}^{1})^{-\frac{p_{i}-1}{2}},Ru_{i}(x_{i}^{2})^{-\frac{p_{i}-1}{2}}\right\\},$

then

$w_{i}(0),w_{i}(\widetilde{x}_{i})\geq R^{\frac{2}{p_{i}-1}}.$ (25)

We want to prove that up to subsequences
$w_{i}(0),w_{i}(\widetilde{x}_{i})\to\infty$.

Suppose that $w_{i}(0)\to\infty$ but $w_{i}(\widetilde{x}_{i})$ stays bounded.
Then, by Proposition 4.15, $0$ would be an isolated simple blow-up point, and
using Proposition 4.11, and the maximum principle and Harnack inequality in a
way similar to the proof of Proposition 4.5, it could be proved that
$w_{i}(\widetilde{x}_{i})\to 0$, against formula (25). If
$w_{i}(\widetilde{x}_{i})\to\infty$ but $w_{i}(0)$ stays bounded, the
reasoning is analogous.

Suppose that $w_{i}(0)$ and $w_{i}(\widetilde{x}_{i})$ both stay bounded.
Reasoning as in the other cases, if $S\neq\emptyset$ then it can be proved
that $w_{i}(0),w_{i}(\widetilde{x}_{i})\to 0$, against formula (25). Hence
$S=\emptyset$, and therefore $w_{i}$ is locally bounded. Therefore, using
regularity theory as in the proof of Proposition 4.5, it can be proved that up
to subsequences $w_{i}\to v$ in $C^{2}_{\mathrm{loc}}$, with $v$ satisfying
$-4\Delta_{b}v=v^{3}$, $v\geq 0$ and
$\nabla^{H}v(0)=\nabla^{H}v(\widetilde{x})=0$. By Theorem 2.2 $v\equiv 0$.
This goes yet again against formula (25).

So $0,\widetilde{x}\in S$. Now, arguing as before with the maximum principle
and Harnack inequality, it can be proved that $w_{i}(0)w_{i}$ is locally
bounded in $\mathbf{H}^{1}\setminus S$, and thus, up to subsequences, it
converges in $C^{2}_{\mathrm{loc}}(\mathbf{H}^{1}\setminus S)$ to a function
$h$, and because of Proposition 4.11

$h(x)=\frac{a}{|x|^{2}}+\frac{b}{|\widetilde{x}^{-1}x|^{2}}+c(x)$

where $a,b>0$, $c$ is smooth in
$\mathbf{H}^{1}\setminus(S\setminus\\{0,\widetilde{x}\\})$ and verifies
$\Delta_{b}c=0$. Since $h>0$, by the maximum principle $c\geq 0$. Therefore

$h(x)=\frac{a}{|x|^{2}}+A+o(1)$

with $A>0$, but, arguing as in the proof of Proposition 4.15, this is in
contradiction with Proposition 4.14. ∎

Now we can finally prove the main result of this work.

###### Proof of Theorem 1.1.

Using subelliptic estimates and the Harnack inequality as in the proof of
Proposition 4.5, it is sufficient to prove that there exists no blow-up point.

If this is not the case, by point 1 of Proposition 4.16 we know that $p_{i}\to
3$, and by Proposition 4.17 and point 3 of Proposition 4.16 there exist a
finite number of isolated blow-up points
$x^{1}_{i}\to\overline{x}^{1},\ldots,x_{i}^{N}\to\overline{x}^{N}$, which, by
Proposition 4.15, are isolated simple in CR normal coordinates.

Up to subsequences and renaming of the blow-up points, we can suppose that
$u_{i}(x_{i}^{1})\leq u_{i}(x_{i}^{k})$ for every $i$ and $k$. Define
$w_{i}=u_{i}(x_{i}^{1})u_{i}$. By Proposition 4.11,

$w_{i}(x)\leq Cd(x,x_{i}^{k})\;\;\text{in}\;\;B_{\frac{\rho}{2}}(x_{i}^{k})$
(26)

By point 3 in Proposition 4.16, $u_{i}$ is uniformly bounded in
$M\setminus\cup_{k}B_{\frac{\rho}{2}}(x^{k})$. Since the CR manifold has
positive CR Yamabe class, there exists a metric of positive Webster curvature
(this follows either by the solution of the CR Yamabe problem, or more
elementarily, thanks to formula (3), by taking $\varphi_{1}^{2}\theta$ where
$\varphi_{1}$ is the first eigenfunction of $L_{\theta}$). Then, arguing as in
the proof of Proposition 4.5, it can be proved that

$\max_{M\setminus\cup_{k}B_{\frac{\rho}{2}}(x^{k})}u_{i}\leq
C\min_{\partial\left(M\setminus\cup_{k}B_{\frac{\rho}{2}}(x^{k})\right)}u_{i}$

which along with (26) shows that $w_{i}$ is uniformly bounded in
$M\setminus\cup_{k}B_{\frac{\rho}{2}}(x^{k})$. This, inequality (26), Theorem
2.3 and Proposition 4.11 imply that, up to subsequences,

$w_{i}(x)=u_{i}(x_{i}^{1})u_{i}(x)\to
h(x)=\sum_{k=1}^{N}a_{k}G_{x^{k}}(x)+b(x)$

where $a_{k}\geq 0$ are constants, $a_{1}>0$ and $b$ is a function satisfying
$L_{\theta}b=0$. Since $M$ has positive CR Yamabe class, $L_{\theta}$ has
trivial kernel, therefore $b=0$.

By Proposition 5.3 in [CMY1], thanks to hypothesis that $m_{x^{1}}>0$, in CR
normal coordinates around $x^{1}$ we have

$G_{x^{1}}(x)=\frac{1}{4\pi|x|^{2}}+A+w(x)$

with $A>0$, $w(0)=0$, $w(x)=o(1)$. Thus

$h(x)=\frac{a_{1}}{4\pi|x|^{2}}+A^{\prime}+o(1)$

for $x\to 0$, with $A^{\prime}>0$. But this goes against Lemma 4.14. ∎

##  Appendix A Some computations in pseudohermitian normal coordinates

Let $M$ be a three-dimensional pseudoconvex pseudohermitian manifold and $x\in
M$.

######  Proposition A.1.

In pseudohermitian normal coordinates

$\theta^{1}=(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{\overline{1}}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}$
$\theta=(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}+O(|x|^{3})\theta^{1}+O(|x|^{3})\theta^{\overline{1}}.$
${\omega_{1}}^{1}=O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{1}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{\overline{1}}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}$
(27)

###### Proof.

It follows from Proposition 2.5 in [JL3]. ∎

######  Lemma A.2.

In pseudohermitian normal coordinates

$\begin{cases}Z_{1}=(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\\\
Z_{\overline{1}}=O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\\\
T=O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\end{cases},$
(28)

and

$\begin{cases}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}=(1+O(|x|^{2}))Z_{1}+O(|x|^{2})Z_{\overline{1}}+O(|x|^{3})T\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}=O(|x|^{2})Z_{1}+(1+O(|x|^{2}))Z_{\overline{1}}+O(|x|^{3})T\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}=O(|x|)Z_{1}+O(|x|)Z_{\overline{1}}+(1+O(|x|^{2}))T\end{cases}$
(29)

###### Proof.

Letting
$Z_{1}=a\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+b\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+c\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}$
and applying $\theta^{1}$, $\theta^{\overline{1}}$ and $\theta$, we get

$\begin{cases}1=(1+O(|x|^{2}))a+O(|x|^{2})b+O(|x|)c,\\\
0=(1+O(|x|^{2}))b+O(|x|^{2})a+O(|x|)c,\\\
0=(1+O(|x|^{2}))c+O(|x|^{3})a+O(|x|^{3})b\end{cases}$

respectively. The third one implies that $c=O(|x|^{3})$, and using this in the
other two allows to deduce that $a=1+O(|x|^{2})$ and $b=O(|x|^{2})$. Therefore

$Z_{1}=(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}.$

Letting
$T=d\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+e\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+f\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}$
and applying $\theta^{1}$, $\theta^{\overline{1}}$ and $\theta$, we get

$\begin{cases}0=(1+O(|x|^{2}))d+O(|x|^{2})e+O(|x|)f,\\\
0=O(|x|^{2})d+(1+O(|x|^{2}))e+O(|x|)f\\\
1=(1+O(|x|^{2}))f+O(|x|^{3})d+O(|x|^{3})e.\end{cases}$

Arguing as before, these imply that $d=O(|x|)$, $e=O(|x|)$ and
$f=1+O(|x|^{2})$. Therefore

$T=O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}.$

So we proved the first part of the Lemma.

We can write the formulas we proved as

$\left(\begin{array}[]{c}Z_{1}\\\ Z_{\overline{1}}\\\
T\end{array}\right)=\left(I+\left(\begin{array}[]{ccc}O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|)&O(|x|)&O(|x|^{2})\end{array}\right)\right)\left(\begin{array}[]{c}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\end{array}\right).$

Applying the Taylor expansion for the matrix inverse
$(I+A)^{-1}=I-A+A^{2}+O(\left\lVert A\right\rVert^{3})$, we get

$\left(\begin{array}[]{c}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\end{array}\right)=\left(I+\left(\begin{array}[]{ccc}O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|)&O(|x|)&O(|x|^{2})\end{array}\right)+\right.$
$+\left.\left(\begin{array}[]{ccc}O(|x|^{4})&O(|x|^{4})&O(|x|^{5})\\\
O(|x|^{4})&O(|x|^{4})&O(|x|^{5})\\\
O(|x|^{3})&O(|x|^{3})&O(|x|^{4})\end{array}\right)+O(|x|^{3})\right)\left(\begin{array}[]{c}Z_{1}\\\
Z_{\overline{1}}\\\ T\end{array}\right)=$
$=\left(I+\left(\begin{array}[]{ccc}O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|^{2})&O(|x|^{2})&O(|x|^{3})\\\
O(|x|)&O(|x|)&O(|x|^{2})\end{array}\right)\right)\left(\begin{array}[]{c}Z_{1}\\\
Z_{\overline{1}}\\\ T\end{array}\right).$

This implies the second part of the thesis. ∎

######  Lemma A.3.

In pseudohermitian normal coordinates

$\begin{cases}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}^{2}=(1+O(|x|^{2}))Z_{1}^{2}+O(|x|)Z_{1}+O(|x|)Z_{\overline{1}}+O(|x|^{2})T+\\\
+O(|x|^{2})Z_{\overline{1}}Z_{1}+O(|x|^{3})Z_{1}T+O(|x|^{4})Z_{\overline{1}}^{2}+O(|x|^{5})Z_{\overline{1}}T+O(|x|^{6})T^{2}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}=(1+O(|x|^{2}))Z_{\overline{1}}Z_{1}+O(|x|)Z_{1}+O(|x|)Z_{\overline{1}}+O(|x|^{2})T+\\\
+O(|x|^{2})Z_{1}^{2}+O(|x|^{2})Z_{\overline{1}}^{2}+O(|x|^{3})Z_{1}T+O(|x|^{3})Z_{\overline{1}}T+O(|x|^{6})T^{2}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}=(1+O(|x|^{2}))Z_{1}T+O(1)Z_{1}+O(1)Z_{\overline{1}}+O(|x|)T+\\\
+O(|x|)Z_{1}^{2}+O(|x|)Z_{\overline{1}}Z_{1}+O(|x|^{2})Z_{\overline{1}}T+O(|x|^{3})T^{2}+O(|x|^{3})Z_{\overline{1}}^{2}\\\
\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}^{2}=(1+O(|x|^{2}))T^{2}+O(1)Z_{1}+O(1)Z_{\overline{1}}+O(|x|)T+O(|x|)Z_{1}T+\\\
+O(|x|)Z_{\overline{1}}T+O(|x|^{2})Z_{\overline{1}}Z_{1}+O(|x|^{2})Z_{1}^{2}+O(|x|^{2})Z_{\overline{1}}^{2}\end{cases}$
(30)

###### Proof.

It follows from Lemma A.2 and some computations. ∎

######  Lemma A.4.

For any function $f$ in CR normal coordinates around $\overline{x}$

$|f_{,1}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|\lesssim(|f_{,1}|+f_{,\overline{1}}|)|x|^{2}+|f_{,0}||x|^{3},$
$|f_{,0}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}f|\lesssim(|f_{,1}|+f_{,\overline{1}}|)|x|+|f_{,0}||x|^{2},$
$|f_{,11}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}^{2}f|\lesssim(|f_{,1}|+|f_{,\overline{1}}|)|x|+(|f_{,0}|+|f_{,11}|+|f_{,1\overline{1}}|)|x|^{2}+|f_{,01}||x|^{3}+$
$+|f_{,\overline{1}\overline{1}}||x|^{4}+|f_{,0\overline{1}}||x|^{5}+|f_{,00}||x|^{6},$
$|f_{,1\overline{1}}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|\lesssim(|f_{,1}|+f_{,\overline{1}}|)|x|+(|f_{,0}|+|f_{,11}|+|f_{,1\overline{1}}|+|f_{,\overline{1}\overline{1}}|)|x|^{2}+$
$+(|f_{,01}|+|f_{,0\overline{1}}|)|x|^{3}+|f_{,00}||x|^{6},$
$|f_{,01}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}f|\lesssim|f_{,1}|+|f_{,\overline{1}}|+(|f_{,0}|+|f_{,11}|+|f_{,1\overline{1}}|)|x|+(|f_{,01}|+|f_{,0\overline{1}}|)|x|^{2}+$
$+(|f_{,00}|+|f_{,\overline{1}\overline{1}}|)|x|^{3},$
$|f_{,00}-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}^{2}f|\lesssim|f_{,1}|+|f_{,\overline{1}}|+(|f_{,0}|+|f_{,01}|+|f_{,0\overline{1}}|)|x|+(|f_{,11}|+|f_{,1\overline{1}}|+|f_{,\overline{1}\overline{1}}|+|f_{,00}|)|x|^{2}.$

###### Proof.

The first two estimates follow from formulas (29) and (27). The other ones
from formulas (30) and (27). ∎

######  Lemma A.5.

In pseudohermitian normal coordinates around a point $x$

$\Delta_{b}=\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+$
$+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{6})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}^{2}$

###### Proof.

Using formulas (28) we get

$\Delta_{b}=Z_{1}Z_{\overline{1}}+Z_{\overline{1}}Z_{1}-{\omega_{1}}^{1}(Z_{\overline{1}})Z_{1}-{\omega_{\overline{1}}}^{\overline{1}}(Z_{1})Z_{\overline{1}}=$
$=\left((1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\right)\cdot$
$\cdot\left(O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\right)+$
$-\left(O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{1}+O(|x|)_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}^{\overline{1}}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\theta}\rule{0.0pt}{9.90276pt}\right)\cdot$
$\cdot\left(O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+(1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\right)\cdot$
$\cdot\left((1+O(|x|^{2}))\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\right)+c.c.=$
$=\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|)\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+$
$+O(|x|^{2})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}+O(|x|^{3})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}+O(|x|^{6})\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}^{2}$

where “$+c.c.$” means that the complex conjugate of the previous terms is to
be added. ∎

######  Lemma A.6.

For any function $f$ in CR normal coordinates around $\overline{x}$

$|\Delta_{b}f-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}f|\lesssim|x||f_{,1}|+|x||f_{,\overline{1}}|+|x|^{2}|f_{,0}|+|x|^{2}|f_{,1\overline{1}}|+|x|^{2}|f_{,11}|+$
$+|x|^{2}|f_{,\overline{1}\overline{1}}|+|x|^{3}|f_{01}|+|x|^{3}|f_{,0\overline{1}}|+|x|^{6}|f_{,00}|$

###### Proof.

By Lemma A.5

$|\Delta_{b}f-\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{\Delta}\rule{0.0pt}{9.90276pt}_{b}f|\lesssim|x||\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|+|x||\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}f|+|x|^{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}f|+|x|^{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|+|x|^{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|+$
$+|x|^{2}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}f|+|x|^{3}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{1}f|+|x|^{3}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{Z}\rule{0.0pt}{9.90276pt}_{\overline{1}}f|+|x|^{6}|\accentset{\smash{\raisebox{-0.36165pt}{$\scriptstyle\circ$}}}{T}\rule{0.0pt}{9.90276pt}^{2}f|$

and the thesis follows from Lemma A.4 and some computations. ∎

## References

  * [B] Brendle, Simon; Blow-up phenomena for the Yamabe equation, J. Amer. Math. Soc. 21 (2008), no. 4, 951–979.
  * [BM] Brendle, Simon; Marques, Fernando C.; Blow-up phenomena for the Yamabe equation. II, J. Differential Geom. 81 (2009), no. 2, 225–250.
  * [CGS] Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel; Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), no. 3, 271–297.
  * [CLMR] Giovanni Catino; Yanyan Li; Dario D. Monticelli; Alberto Roncoroni; A Liouville theorem in the Heisenberg group, arXiv:2310.10469 (2023).
  * [CC] Cheng, Jih-Hsin; Chiu, Hung-Lin; Positive mass theorem and the CR Yamabe equation on 5-dimensional contact spin manifolds, Adv. Math. 404 (2022), part B, Paper No. 108446, 50 pp.
  * [CCY] Cheng, Jih-Hsin; Chiu, Hung-Lin; Yang, Paul; Uniformization of spherical CR manifolds, Adv. Math. 255 (2014), 182–216.
  * [CMY1] Cheng, Jih-Hsin; Malchiodi, Andrea; Yang, Paul; A positive mass theorem in three dimensional Cauchy-Riemann geometry, Adv. Math. 308 (2017), 276–347.
  * [CMY2] Cheng, Jih-Hsin; Malchiodi, Andrea; Yang, Paul; On the Sobolev quotient of three-dimensional CR manifolds, Rev. Mat. Iberoam. 39 (2023), no. 6, 2017–2066.
  * [DT] Dragomir, Sorin; Tomassini, Giuseppe; Differential geometry and analysis on CR manifolds, Progress in Mathematics, 246. Birkhäuser Boston, Inc., Boston, MA, 2006. xvi+487 pp.
  * [FS] Folland, G. B.; Stein, E. M.; Estimates for the $\bar{\partial}_{b}$ complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429–522.
  * [G] Gamara, Najoua; The CR Yamabe conjecture—the case n=1, J. Eur. Math. Soc. (JEMS) 3 (2001), no. 2, 105–137.
  * [GY] Gamara, Najoua; Yacoub, Ridha; CR Yamabe conjecture—the conformally flat case, Pacific J. Math. 201 (2001), no. 1, 121–175.
  * [GaL] Garofalo, Nicola; Lanconelli, Ermanno; Existence and nonexistence results for semilinear equations on the Heisenberg group, Indiana Univ. Math. J. 41 (1992), no. 1, 71–98.
  * [GV] Garofalo, Nicola; Vassilev, Dimiter; Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups Math. Ann. 318 (2000), no. 3, 453–516.
  * [GS] Gidas, B.; Spruck, J.; Global and local behavior of positive solutions of nonlinear elliptic equations Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
  * [GT] Gilbarg, David; Trudinger, Neil S.; Elliptic partial differential equations of second order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp.
  * [GuL] Gutiérrez, Cristian E.; Lanconelli, Ermanno; Maximum principle, nonhomogeneous Harnack inequality, and Liouville theorems for X-elliptic operators, Comm. Partial Differential Equations 28 (2003), no. 11-12, 1833–1862.
  * [JL1] Jerison, David; Lee, John M.; The Yamabe problem on CR manifolds, J. Differential Geom. 25 (1987), no. 2, 167–197.
  * [JL2] Jerison, David; Lee, John M.; Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), no. 1, 1–13.
  * [JL3] Jerison, David; Lee, John M.; Intrinsic CR normal coordinates and the CR Yamabe problem, J. Differential Geom. 29 (1989), no. 2, 303–343.
  * [KMS] Khuri, M. A.; Marques, F. C.; Schoen, R. M.; A compactness theorem for the Yamabe problem, J. Differential Geom. 81 (2009), no. 1, 143–196.
  * [LP] Lee, John M.; Parker, Thomas H.; The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91.
  * [L] Li, Yan Yan; Prescribing scalar curvature on Sn and related problems. I, J. Differential Equations 120 (1995), no. 2, 319–410.
  * [LZhu] Li, Yanyan; Zhu, Meijun; Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1 (1999), no. 1, 1–50
  * [LZha] Li, Yan Yan; Zhang, Lei; Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations 24 (2005), no. 2, 185–237.
  * [MO] Ma, Xi-Nan; Ou, Qianzhong; A Liouville theorem for a class semilinear elliptic equations on the Heisenberg group, Adv. Math. 413 (2023), Paper No. 108851, 20 pp.
  * [M] Marques, Fernando Coda; A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom. 71 (2005), no. 2, 315–346.
  * [S] Schoen, Richard M.; On the number of constant scalar curvature metrics in a conformal class, Differential geometry, 311–320, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991.
  * [T] Takeuchi, Yuya; Nonnegativity of the CR Paneitz operator for embeddable CR manifolds, Duke Math. J. 169 (2020), no. 18, 3417–3438.

Claudio Afeltra, Department of Mathematics, University of Trento, Via
Sommarive 14, 38123 Povo (Trento), Italy

Email address<EMAIL_ADDRESS>