# Sharp weighted CR trace Sobolev inequalities

Gunhee Cho  Department of Mathematics
University of California, Santa Barbara
552 University Rd, Isla Vista, CA 93117<EMAIL_ADDRESS>and  Zetian
Yan  Department of Mathematics
Penn State University
University Park
PA 16802
USA<EMAIL_ADDRESS>
###### Abstract.

We establish a sharp Sobolev trace inequality on the Siegel domain
$\Omega_{n+1}$ involving the weighted
norm-$W^{2,2}(\Omega_{n+1},\rho^{1-2[\gamma]})$. The inequality is closely
related the realization of fractional powers of the sub-Laplacian on
$\mathbb{H}^{n}=\partial\Omega_{n+1}$ as generalized Dirichlet-to-Neumann
operators associated to the weighted poly-sublaplacian, generalizing
observations of Frank–González–Monticelli–Tan.

###### Key words and phrases:

CR extension problem; CR fractional GJMS operators; trace Sobolev inequalities

###### 2020 Mathematics Subject Classification:

Primary 39B62; Secondary 32V20, 32V40, 35B38.

##  1\. Introduction

In [Cas20], Case recovered the fractional Laplacian
$(-\overline{\Delta})^{\gamma}$ on $\mathbb{R}^{n}$ as a Dirichlet-to-Neumann
operator associated to the weighted poly-Laplacian $L_{2k}:=\Delta_{m}^{k}$
determined by $\gamma\in(0,\infty)\backslash\mathbb{N}$, where
$\Delta_{m}:=\Delta+my^{-1}\partial_{y}$ is the weighted Laplacian on
$\mathbb{R}_{+}^{n+1}:=\mathbb{R}^{n}\times(0,\infty)$ for $m=1-2[\gamma]$ and
$k:=\lfloor\gamma\rfloor+1$. Specifically, given functions $f^{(2j)}\in
C^{\infty}(\mathbb{R}^{n})\cap H^{\gamma-2j}(\mathbb{R}^{n})$,
$j\in[0,\lfloor\gamma/2\rfloor]$ and $\phi^{(2j)}\in
C^{\infty}(\mathbb{R}^{n})\cap
H^{\lfloor\gamma\rfloor-[\gamma]-2j}(\mathbb{R}^{n})$,
$j\in[0,\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1]$, if $V$ is a solution
of

(1.1)
$\left\\{\begin{array}[]{ll}L_{2k}V=0,&~{}~{}\mbox{in}~{}~{}\mathbb{R}^{n+1}_{+},\\\
B_{2j}^{2\gamma}(V)=f^{(2j)},&~{}~{}\mbox{for}~{}~{}j\in[0,\lfloor\gamma/2\rfloor],\\\
B_{2[\gamma]+2j}^{2\gamma}(V)=\phi^{(2j)},&~{}~{}\mbox{for}~{}~{}j\in[0,\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1],\end{array}\right.$

there exists constants $c_{\gamma,j}$ and $d_{\gamma,j}$ such that

(1.2)
$\begin{split}B_{2\gamma-2j}^{2\gamma}(V)=c_{\gamma,j}(-\overline{\Delta})^{\gamma-2j}f^{(2j)},&~{}~{}\mbox{for}~{}~{}j\in[0,\lfloor\gamma/2\rfloor],\\\
B_{2\lfloor\gamma\rfloor-2j}^{2\gamma}(V)=d_{\gamma,j}(-\overline{\Delta})^{\lfloor\gamma\rfloor-[\gamma]-2j}\phi^{(2j)},&~{}~{}\mbox{for}~{}~{}j\in[0,\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1],\end{split}$

where $B_{2j}^{2\gamma}$ and $B_{2[\gamma]+2j}^{2\gamma}$ are boundary
operators constructed in [Cas20, Definition 3.1].

Applying the Dirichlet principle, one deduces that

(1.3)
$\begin{split}\mathcal{E}_{2\gamma}(U)\geqslant&\sum_{j=0}^{\lfloor\gamma/2\rfloor}c_{\gamma,j}\oint_{\mathbb{R}^{n}}f^{(2j)}(-\overline{\Delta})^{\gamma-2j}f^{(2j)}dx\\\
+&\sum_{j=0}^{\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1}d_{\gamma,j}\oint_{\mathbb{R}^{n}}\phi^{(2j)}(-\overline{\Delta})^{\lfloor\gamma\rfloor-[\gamma]-2j}\phi^{(2j)}dx,\end{split}$

for any $U\in\mathcal{C}^{2\gamma}\cap
W^{k,2}(\mathbb{R}^{n+1}_{+},y^{1-2[\gamma]})$, where the Dirichlet energy
$\mathcal{E}_{2\gamma}(U)$ is given by

$\begin{split}\mathcal{E}_{2\gamma}(U):=\int_{\mathbb{R}^{n+1}_{+}}UL_{2k}Uy^{m}dxdy&+\sum_{j=0}^{\lfloor\gamma/2\rfloor}\oint_{\mathbb{R}^{n}}B_{2j}^{2\gamma}(U)B_{2\gamma-2j}^{2\gamma}(U)dx\\\
&-\sum_{j=0}^{\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1}\oint_{\mathbb{R}^{n}}B_{2[\gamma]+2j}^{2\gamma}(U)B_{2\lfloor\gamma\rfloor-2j}^{2\gamma}(U)dx;\end{split}$

see [Cas20] for more details. We may regard (1.3) as a functional inequality
for the Sobolev trace embedding

$W^{k,2}(\mathbb{R}^{n+1}_{+},y^{1-2[\gamma]})\hookrightarrow\bigoplus_{j=0}^{\lfloor\gamma/2\rfloor}H^{\gamma-2j}(\mathbb{R}^{n})\oplus\bigoplus_{j=0}^{\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1}H^{\lfloor\gamma\rfloor-[\gamma]-2j}(\mathbb{R}^{n}).$

Combining (1.3) with Lieb’s sharp fractional Sobolev inequality [Lie83] yields
the sharp Sobolev trace inequality for the embedding

$W^{k,2}(\mathbb{R}^{n+1}_{+},y^{1-2[\gamma]})\hookrightarrow\bigoplus_{j=0}^{\lfloor\gamma/2\rfloor}L^{\frac{2n}{n-2\gamma+4j}}(\mathbb{R}^{n})\oplus\bigoplus_{j=0}^{\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1}L^{\frac{2n}{n-2\lfloor\gamma\rfloor+2[\gamma]+4j}}(\mathbb{R}^{n})$

when $n>2\gamma$.

The sharp version of (1.3) for $\gamma\in(0,1)$ stems from Caffarelli and
Silvestre’s seminal paper [CS07]. Furthermore, the special case
$\gamma=\frac{3}{2}$ was proven by Ache–Chang [AC17]; the special case
$\gamma=\mathbb{N}_{0}+\frac{1}{2}$ was prove by Yang [Yan19]; the general
case $\gamma\in(0,\infty)\backslash\mathbb{N}$ was proven by Case [Cas20].

The remarkable breakthrough in [Cas20] is the construction of boundary
operators associated to the weighted poly-Laplacian. They are defined
recursively in terms of the weighted Laplacian $\Delta_{m}$ and the weighted
derivative $y^{m}\partial_{y}$ in $\mathbb{R}^{n+1}_{+}$, and the induced
Laplacian $\overline{\Delta}$ on $\mathbb{R}^{n}$ as follows:

$\begin{split}\iota^{*}\circ\Delta_{m}^{j}&=\sum_{l=0}^{j}(-1)^{l}\binom{j}{l}\frac{(\lfloor\gamma\rfloor-l)!\Gamma(\gamma-
j-l)}{(\lfloor\gamma\rfloor-j)!\Gamma(\gamma-2l)}\overline{\Delta}^{j-l}B_{2l}^{2\gamma},\\\
\iota^{*}\circ
y^{m}\partial_{y}\Delta_{m}^{j}&=(-1)^{j+1}\sum_{l=0}^{j}\binom{j}{l}\frac{(\lfloor\gamma\rfloor-l)!\Gamma(1+2l-\lfloor\gamma\rfloor+[\gamma])}{(\lfloor\gamma\rfloor-j)!\Gamma(1+j+l-\lfloor\gamma\rfloor+[\gamma])}\overline{\Delta}^{j-l}B_{2[\gamma]+2l}^{2\gamma},\end{split}$

where $\iota^{*}:C^{\infty}(\overline{\mathbb{R}^{n+1}_{+}})\to
C^{\infty}(\mathbb{R}^{n})$ is the restriction operator. These definitions are
justified by three properties. First, they are such that the associated
Dirichlet form
$\mathcal{Q}_{2\gamma}(U,V):=\frac{1}{2}\left(\mathcal{E}_{2\gamma}(U+V)-\mathcal{E}_{2\gamma}(U)-\mathcal{E}_{2\gamma}(V)\right)$
is symmetric. Second, the boundary operators are covariant with respect to the
group of conformal isometries of $(\mathbb{R}^{n+1}_{+};\mathbb{R}^{n})$; see
[Cas20, Section 4] for the precise statement. Third, as shown in (1.2), they
are such that the generalized Dirichlet-to-Neumann operators recover the
fractional Laplacians.

In the CR setting, fractional covariant operators of order $2\gamma$ for
$\gamma\in\mathbb{R}$, was defined from scattering theory on a Kähler–Einstein
manifold $\mathcal{X}$ [GSB08, HPT08, EMM91]. They are pseudodifferential
operators whose principal symbol agress with the pure fractional powers of the
CR sub-Laplacian $(-\Delta_{b})^{\gamma}$ on the boundary
$M=\partial\mathcal{X}$. In the special case of the Heisenberg group
$\mathbb{H}^{n}$, they are simply the intertwining operators on the CR sphere
calculated in [BFM13] using classical representation theory.

In the same spirit of Caffarelli and Silvestre, Frank, González, Monticelli
and Tan formulated an extension problem on $\Omega_{n+1}$ and constructed the
pure fractional powers of the sub-Laplacian on $\mathbb{H}^{n}$ as the
Dirichlet-to-Neumann operator of a degenerate elliptic equation using the
Fourier transform and the spectral resolution of the operator. Moreover, we
have the following sharp Sobolev trace inequality.

######  Theorem 1.1 ([FdMGMT15]).

Given $\gamma\in(0,1)$, for any $U\in\mathcal{C}_{D}^{2\gamma}$, $\phi\in
S^{\gamma,2}(\mathbb{H}^{n})$, we have

(1.4) $\mathcal{E}_{2\gamma}(U)\geqslant
2^{1-2\gamma}\gamma\frac{\Gamma(1-\gamma)}{\Gamma(1+\gamma)}\oint_{\mathbb{H}^{n}}\phi
P_{\gamma}^{\theta}\phi dzdt.,\quad\phi=\iota^{*}U,$

where $\iota^{*}:C^{\infty}(\overline{\Omega}_{n+1})\to
C^{\infty}(\mathbb{H}^{n})$ is the restriction operator, the corresponding
function space $\mathcal{C}_{D}^{2\gamma}$ is defined in Theorem 6.1 and the
Dirichlet energy $\mathcal{E}_{2\gamma}(U)$ is given by

$\mathcal{E}_{2\gamma}(U)=\int_{\Omega_{n+1}}\left(|\partial_{\rho}U|^{2}+\frac{1}{2}\sum_{j=1}^{n}|X_{j}(U)|^{2}+|Y_{j}(U)|^{2}+\rho^{2}|\partial_{t}U|^{2}\right)\rho^{1-2\gamma}dzdtd\rho.$

Equality is attained if and only if $U$ is the unique solution for the
extension problem

(1.5)
$\left\\{\begin{array}[]{ll}\left(\partial^{2}_{\rho}+(1-2\gamma)\rho^{-1}\partial_{\rho}+\rho^{2}\partial_{t}^{2}+\Delta_{b}\right)U=0,&~{}~{}\mbox{in}~{}~{}\Omega_{n+1},\\\
U=\phi,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n}.\end{array}\right.$

Our main result gives the sharp Sobolev trace inequality for $\gamma\in(1,2)$.

######  Theorem 1.2.

Given $\gamma\in(1,2)$, for any $U\in\mathcal{C}_{D}^{2\gamma}$, $\phi\in
S^{\gamma,2}(\mathbb{H}^{n})$, we have

(1.6) $\mathcal{E}_{2\gamma}(U)\geq
2^{3-2\gamma}\frac{\Gamma(2-\gamma)}{\Gamma(\gamma)}\oint_{\mathbb{H}^{n}}\phi
P^{\theta}_{\gamma}\phi dzdt,\quad\phi=\iota^{*}U,$

where $\iota^{*}:C^{\infty}(\overline{\Omega}_{n+1})\to
C^{\infty}(\mathbb{H}^{n})$ is the restriction operator, the corresponding
function space $\mathcal{C}_{D}^{2\gamma}$ is defined in Theorem 6.1. The
Dirichlet energy $\mathcal{E}_{2\gamma}(U)$ in (1.6) is given by

$\begin{split}\mathcal{E}_{2\gamma}(U):=&\int_{\Omega_{n+1}}UL_{4}U\rho^{1-2[\gamma]}dzdtd\rho\\\
&+\oint_{\mathbb{H}^{n}}B_{0}^{2\gamma}(U)B_{2\gamma}^{2\gamma}(U)dzdt-\oint_{\mathbb{H}^{n}}B_{2[\gamma]}^{2\gamma}(U)B_{2}^{2\gamma}(U)dzdt,\end{split}$

where the associated boundary operators are

(1.7)
$\begin{split}B_{0}^{2\gamma}=\iota^{*},\quad&B_{2}^{2\gamma}=-\iota^{*}\circ\mathcal{T}+\frac{1-[\gamma]}{[\gamma]}\Delta_{b}\circ
B_{0}^{2\gamma},\\\
B_{2[\gamma]}^{2\gamma}=-\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho},\quad&B_{2\gamma}^{2\gamma}=\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}\mathcal{T}-\frac{1+[\gamma]}{[\gamma]}\Delta_{b}\circ
B_{2[\gamma]}^{2\gamma},\end{split}$

if we set

$\mathcal{T}:=\partial^{2}_{\rho}+(1-2[\gamma])\partial_{\rho}+\rho^{2}\partial^{2}_{t}.$

Equality in (1.6) is attained if and only if $U$ is the unique solution for
the extension problem

(1.8)
$\left\\{\begin{array}[]{ll}L_{4}V=0,&~{}~{}\mbox{on}~{}~{}\Omega_{n+1},\\\
B_{0}^{2\gamma}(V)=\phi,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\\\
B_{2[\gamma]}^{2\gamma}(V)=0,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\end{array}\right.$

where

$\begin{split}L_{4}&:=L_{[\gamma]}^{2}+T^{2},\\\
L_{[\gamma]}=\partial^{2}_{\rho}+(1-2[\gamma])&\rho^{-1}\partial_{\rho}+\rho^{2}\partial_{t}^{2}+\Delta_{b},\quad
T=2\frac{\partial}{\partial t}.\end{split}$

In particular, $P^{\theta}_{\gamma}$ can be recovered from the solution of
(1.8) by

$P^{\theta}_{\gamma}\phi=2^{-3+2\gamma}\frac{\Gamma(\gamma)}{\Gamma(2-\gamma)}\lim_{\rho\to
0^{+}}\rho^{1-2[\gamma]}\partial_{\rho}L_{[\gamma]}U.$

The article is organized as follows:

In Section 2 we review some basic concepts on the Heisenberg group
$\mathbb{H}^{n}$ as a CR manifold and as the boundary of Siegel domain.

In Section 3 we recall the identification of fractional powers of the sub-
Laplacian on $\mathbb{H}^{n}$ via scattering theory for the hyperbolic
Laplacian on the Siegel domain $\Omega_{n+1}$ [HPT08]. In the same spirit in
[Cas20], we define the weighted poly-sublaplacian on $\Omega_{n+1}$ and prove
a nice factorization in Lemma 3.5. We also give a direct relationship between
the weighted poly-sublaplacian on $\Omega_{n+1}$ and the complex hyperbolic
poly-Poisson operator in hyperbolic space.

In Section 4 we introduce the boundary operators associated to the weighted
poly-sublaplacian for $\gamma\in(0,2)$ and study their role in recovering
fractional powers of the sub-Laplacian as generalized Dirichlet-to-Neumann
operators. The key results here link our boundary operators to the asymptotics
of solutions to a Poisson equation relevant to scattering theory [HPT08] and
show that the Dirichlet form determined by our boundary operators is
symmetric.

Due to the extra term $\rho^{2}\partial^{2}_{t}$ in extension problems, it is
hard to construct higher order boundary operators which satisfy the ‘mutual
exclusion’ property; see Remark 4.4 for a precise statement.

In Section 5 we prove the existence and uniqueness of solutions to extension
problems (5.1) and (5.2), which are degenerate elliptic boundary value
problems. This enables us to prove the main result of this section, Theorem
5.3.

In Section 6 we prove various sharp trace inequalities. First, Theorem 6.1
asserts a Dirichlet principle for solutions of the Dirichlet problem (5.1) and
(5.2). As a result, we recover Theorem 1.1 in Corollary 6.2 and prove the
sharp Sobolev trace inequality (1.6) as a direct consequence of Corollary 6.3.
Combining Corollary 6.2 and Corollary 6.3 with the sharp Sobolev inequalities
on $\mathbb{H}^{n}$ [FL12] we obtain a sharp Sobolev trace inequality stated
in Corollary 6.4.

Acknowledgements. We would like to thank Prof. Jeffrey S. Case for useful
discussion.

##  2\. Prelimiaries

###  2.1. The Heisenberg group as a CR manifold

The $n$-dimensional Heisenberg group $\mathbb{H}^{n}$ is defined as the set
$\mathbb{C}^{n}\times\mathbb{R}$ endowed with the group law

$\left(\hat{z},\hat{t}\right)\cdot\left(z,t\right)=\left(\hat{z}+z,\hat{t}+t+2\mathrm{Im}\langle\hat{z},z\rangle_{\mathbb{C}^{n}}\right)$

for $\left(\hat{z},\hat{t}\right),\left({z},{t}\right)\in\mathbb{H}^{n}$,
where $\langle\cdot,\cdot\rangle_{\mathbb{C}^{n}}$ is the standard inner
product in $\mathbb{C}^{n}$. The CR structure on the Heisenberg group
$\mathbb{H}^{n}$ is inherited from the embedding
$\mathbb{H}^{n}\hookrightarrow\mathbb{C}^{n+1}$. The holomorphic tangent
bundle $T^{1,0}\mathbb{H}^{n}$ is given by

$T^{1,0}\mathbb{H}^{n}=T^{1,0}\mathbb{C}^{n+1}\cap\mathbb{C}T\mathbb{H}^{n}=\mathrm{span}_{\mathbb{C}}\langle
Z_{1},\cdots,Z_{n}\rangle,$

where

(2.1) $Z_{j}=\frac{\partial}{\partial
z_{j}}+i\bar{z}_{j}\frac{\partial}{\partial
t},\quad\quad\bar{Z}_{j}=\frac{\partial}{\partial\bar{z}_{j}}-iz_{j}\frac{\partial}{\partial
t},\quad j=1,\cdots,n.$

We also set

(2.2) $X_{j}=\frac{\partial}{\partial x_{j}}+2y_{j}\frac{\partial}{\partial
t},\quad Y_{j}=\frac{\partial}{\partial y_{j}}-2x_{j}\frac{\partial}{\partial
t},\quad j=1,\cdots,n,\quad T=2\frac{\partial}{\partial t},$

which form a base of the Lie algebra of vector fields on $\mathbb{H}^{n}$ and
are left invariant with respect to the group action. We immediately observe
that the associated maximal complex distribution
$H(\mathbb{H}^{n})=\mathrm{Re}\left\\{T^{1,0}\mathbb{H}^{n}\oplus
T^{0,1}\mathbb{H}^{n}\right\\}$ is simply

$H(\mathbb{H}^{n})=\mathrm{span}_{\mathbb{C}}\langle
X_{1},\cdots,X_{n},Y_{1},\cdots,Y_{n}\rangle,$

which carries the complex structure $J_{b}:H(\mathbb{H}^{n})\to
H(\mathbb{H}^{n})$ defined by

$J_{b}X_{j}=Y_{j},\quad J_{b}Y_{j}=-X_{j},\quad j=1,\cdots,n.$

The real tangent bundle $T\mathbb{H}^{n}$ satisfies

(2.3) $T\mathbb{H}^{n}=H(\mathbb{H}^{n})\oplus\mathbb{R}T.$

The canonical contact form $\theta$ on $\mathbb{H}^{n}$ is precisely

(2.4)
$\theta=\frac{1}{2}\left[dt+i\sum_{j=1}^{n}\left(z_{j}d\bar{z}_{j}-\bar{z}_{j}dz_{j}\right)\right],$

and the vector field $T$ is the characteristic direction satisfying

$\theta(T)=1,\quad\quad T\rfloor d\theta=0.$

The Levi form $L_{\theta}$ associated to $\theta$ is given by

$L_{\theta}=-id\theta=\sum_{j=1}^{n}dz_{j}\wedge d\bar{z}_{j},$

which is a positive definite matrix. This tells us that
$(\mathbb{H}^{n},\theta)$ is strictly pseudoconvex.

The sub-Laplacian associated to $\theta$ is defined as

(2.5)
$\Delta_{b}u:=\sum_{j=1}^{n}\left[\bar{Z}_{j}Z_{j}+Z_{j}\bar{Z}_{j}\right]u=\frac{1}{2}\sum_{j=1}^{n}\left[X^{2}_{j}+Y^{2}_{j}\right]u;$

the differential operator is linear, second order, degenerate elliptic, and it
is hypoelliptic being the sum of squares of smooth vector fields satisfying
the Hörmander condition.

###  2.2. The Heisenberg group as the boundary of the Siegel domain

The Heisenberg group $\mathbb{H}^{n}$ may be identified as the boundary of a
domain in $\mathbb{C}^{n+1}$. Indeed, let $\Omega_{n+1}$ be the Siegel domain,
defined by

(2.6)
$\Omega_{n+1}:=\left\\{\zeta=(z,z_{n+1})\in\mathbb{C}^{n}\times\mathbb{C}|q(\zeta)>0\right\\},$

where

$q(\zeta)=\mathrm{Im}z_{n+1}-|z|^{2}.$

The boundary
$\partial\Omega_{n+1}=\left\\{\zeta\in\mathbb{C}^{n+1}|q(\zeta)=0\right\\}$ is
an oriented CR manifold of hypersurface type with the CR structure induced by
$\mathbb{C}^{n+1}$, which is CR equivalent to $\mathbb{H}^{n}$ by the CR
isomorphism $\mathcal{G}:\mathbb{H}^{n}\to\partial\Omega_{n+1}$,

$\begin{split}\mathcal{G}(z,t)&=\left(z,t+i|z|^{2}\right),\quad(z,t)\in\mathbb{H}^{n},\\\
\mathcal{G}^{-1}(\zeta)&=(z,\mathrm{Re}z_{n+1}),\quad\zeta\in\partial\Omega_{n+1}.\end{split}$

The boundary manifold $\partial\Omega_{n+1}$ inherits a natural CR structure
from the complex structure of the ambient manifold $\mathbb{C}^{n+1}$, $-q$ as
a defining function. The pseudohermitian structure on $\mathbb{H}^{n}$ is then
given (via pullback) by the contact form

$\theta=\frac{i}{2}\left(\bar{\partial}-\partial\right)(-q)=\frac{1}{4}(dz_{n+1}+d\bar{z}_{n+1})+\frac{i}{2}\sum_{j=1}^{n}\left(z_{j}d\bar{z}_{j}-\bar{z}_{j}dz_{j}\right),$

which agrees with the definition given in (2.4).

In order to complete a basis for $T\Omega_{n+1}$, we define

$\xi_{n+1}=\frac{1}{2}(N-iT),\quad\quad\bar{\xi}_{n+1}=\frac{1}{2}(N+iT),$

where

$T=2\frac{\partial}{\partial t},\quad\quad N=-2\frac{\partial}{\partial q}.$

Then a frame in $\Omega_{n+1}$ is given by
$\left\\{Z_{j},\bar{Z}_{j},\xi_{n+1},\bar{\xi}_{n+1}\right\\}_{j=1}^{n}$, and
the dual coframe of this basis is
$\left\\{\theta^{j},\theta^{\bar{j}},\theta,dq\right\\}$ with $dq=\partial
q+\bar{\partial}q$. In particular, the dual coframe in
$\mathbb{C}T\mathbb{H}^{n}$ of $\left\\{Z_{j},\bar{Z}_{j},T\right\\}$ is
simply $\left\\{\theta^{j},\theta^{\bar{j}},\theta\right\\}$.

On the other hand, the functions $t=\mathrm{Re}Z_{n+1}\in\mathbb{R}$,
$z_{1},\cdots,z_{n}\in\mathbb{C}^{n}$ and $\rho=(2q)^{1/2}\in\mathbb{R}_{+}$
give coordinates in $\Omega_{n+1}\simeq\mathbb{H}^{n}\times\mathbb{R}_{+}$.
For the defining function $-q$, we can construct a Kähler form in
$\Omega_{n+1}$ as

$\omega_{+}=-\frac{i}{2}\partial\bar{\partial}\log(q)=\frac{i}{2}\left(-\frac{\partial\bar{\partial}q}{q}+\frac{\partial
q\wedge\bar{\partial}q}{q^{2}}\right).$

By direct calculation, we observe that

(2.7) $-\partial\bar{\partial}q=\sum_{j=1}^{n}dz_{j}\wedge
d\bar{z}_{j},\quad\quad\partial
q\otimes\bar{\partial}q+\bar{\partial}q\otimes\partial
q=\frac{1}{2}\left[(dq)^{2}+4\theta^{2}\right].$

Then after the change $q=\frac{\rho^{2}}{2}$, the Riemannian metric $g^{+}$
associated to $\omega_{+}$ is given by

$g^{+}=\frac{1}{2}\left(\frac{d\rho^{2}}{\rho^{2}}+\frac{2\delta_{\alpha\bar{\beta}}\theta^{\alpha}\otimes\theta^{\bar{\beta}}}{\rho^{2}}+\frac{4\theta}{\rho^{2}}\right).$

Then $(\Omega_{n+1},\omega_{+})$ is a Kähler manifold with constant
holomorphic curvature. We recall from [FdMGMT15] that the Laplace operator on
the Kähler manifold $(\Omega_{n+1},\omega_{+})$ is calculated as

$\Delta_{+}u=\mathrm{Tr}\left(i\partial\bar{\partial}u\right),$

and the following formula decomposes the Laplacian into tangential and normal
components relative to the level sets of $q$:

(2.8)
$\Delta_{+}=q\left[q(\partial_{q}^{2}+\partial_{t}^{2})+\frac{1}{2}\Delta_{b}-n\partial_{q}\right],$

with $\Delta_{b}$ the sub-Laplacian (2.5). The spectrum of the Laplacian
$-\Delta_{+}$ consists of an absolutely continuous part

$\sigma_{ac}(-\Delta_{+})=\left[\frac{(n+1)^{2}}{4},\infty\right),$

and the pure point spectrum satisfying

$\sigma_{pp}(-\Delta_{+})\subset\left(0,\frac{(n+1)^{2}}{4}\right),$

and moreover it consists of a finite set of $L^{2}$-eigenvalues.

##  3\. Scattering theory on the Siegel domain

Given $\gamma\in(0,m)\backslash\mathbb{N}$, $m=n+1$, for each $f\in
C^{\infty}(\mathbb{H}^{n})\cap S^{\gamma,2}(\mathbb{H}^{n})$, where
$S^{\gamma,2}(\mathbb{H}^{n})$ is the Folland–Stein space characterized in
[BFM13], there exists a unique solution $u_{s}$ of the Poisson equation

(3.1) $\Delta_{+}u_{s}+s(m-s)u_{s}=0,\quad s=\frac{m+\gamma}{2}.$

Then the Poisson map is defined as
$\mathcal{P}(s):C^{\infty}(\mathbb{H}^{n})\to
C^{\infty}(\overline{\Omega}_{n+1})$ by $f\to u_{s}$; see [HPT08] for a
Poisson kernel for (3.1). For our purposes, it suffices to know that there are
functions $F,G\in C^{\infty}(\overline{\Omega}_{n+1})$ such that

(3.2) $\left\\{\begin{array}[]{lr}u_{s}=\mathcal{P}(s)(f)=q^{m-s}F+q^{s}G,&\\\
F|_{\mathbb{H}^{n}}=f.&\end{array}\right.$

Define the scattering operator $\mathcal{S}(s)(f):=G|_{\mathbb{H}^{n}}$. The
CR fractional GJMS operator on $(\mathbb{H}^{n},\theta)$ of order $2\gamma$ is
given by

(3.3)
$P_{\gamma}^{\theta}:=2^{\gamma}\frac{\Gamma(\gamma)}{\Gamma(-\gamma)}\mathcal{S}(\frac{m+\gamma}{2}),$

which satisfies

(3.4)
$P_{\gamma}^{e^{2w}\theta}f=e^{-(n+1+\gamma)w}P_{\gamma}^{\theta}\left(e^{(n+1-\gamma)w}f\right),\quad
w\in C^{\infty}(\mathbb{H}^{n}).$

It is known from [FdMGMT15] that

(3.5)
$P_{\gamma}^{\theta}=\left(2|T|\right)^{\gamma}\frac{\Gamma\left(\frac{1+\gamma}{2}+\frac{-\Delta_{b}}{2|T|}\right)}{\Gamma\left(\frac{1-\gamma}{2}+\frac{-\Delta_{b}}{2|T|}\right)},\quad\gamma\in(0,1).$

The function $F$ in (3.2) is determined modulo $O(q^{\infty})$ by $f$ by
finding the Taylor series solution to

(3.6)
$\left\\{\begin{array}[]{lr}\left(\Delta_{+}+\frac{m^{2}-\gamma^{2}}{4}\right)\left(q^{\frac{m-\gamma}{2}}F\right)=O(q^{\infty}),&\\\
F|_{\mathbb{H}^{n}}=f.&\end{array}\right.$

Similarly, the function $G$ in (3.2) is determined modulo $O(q^{\infty})$ by
$P_{\gamma}^{\theta}f$ by finding the Taylor series solution to

(3.7)
$\left\\{\begin{array}[]{lr}\left(\Delta_{+}+\frac{m^{2}-\gamma^{2}}{4}\right)\left(q^{\frac{m+\gamma}{2}}G\right)=O(q^{\infty}),&\\\
G|_{\mathbb{H}^{n}}=\mathcal{S}(\frac{m+\gamma}{2})(f).&\end{array}\right.$

For this reason, we would like to understand the formal solution of

(3.8) $\left(\Delta_{+}+s(m-s)\right)(V)=0,\quad
s=\frac{m\pm\gamma}{2},\quad\gamma\in(0,m)\backslash\mathbb{N}$

Because of a computational error in [HPT08, Lemma 3.4], we recast the
calculation for the correct expansion.

######  Proposition 3.1.

Suppose that $u_{s}\in C^{\infty}(\overline{\Omega}_{n+1})$ satisfying
$u_{s}=q^{m-s}F+q^{s}G$ is the solution of (3.2). Then we have

$F=\sum_{l=0}^{\infty}q^{l}f_{l},\quad\quad
f_{l}=(-1)^{l}\frac{\Gamma(m-2s+1)}{l!\Gamma(m-2s+l+1)}P^{s}_{l}(f_{0}),$

and

$G=\sum_{l=0}^{\infty}q^{l}g_{l},\quad\quad
g_{l}=(-1)^{l}\frac{\Gamma(2s-m+1)}{l!\Gamma(2s-m+l+1)}G^{s}_{l}(g_{0}),$

where $P^{s}_{l}$ and $G^{s}_{l}$ are differential operators of order $2l$
defined by (3.18) and (3.20), respectively, with leading symbol

(3.9) $\sigma(P^{s}_{l})=\sigma(G^{s}_{l})=\sigma(\Delta_{b}^{l}).$

In particular, we have $P^{s}_{l}=G^{m-s}_{l}$.

###### Proof.

We think of (2.8) as a variable-coefficient differential operator with respect
to vector field $q\frac{\partial}{\partial q}$ and vector fields tangent to
$\mathbb{H}^{n}$, i.e.,

(3.10) $\Delta_{+}=\sum_{i=0}^{2}q^{i}L_{i},$

where

(3.11) $L_{0}=\left(q\frac{\partial}{\partial
q}\right)^{2}-(n+1)q\frac{\partial}{\partial q},\quad\quad
L_{1}=\frac{\Delta_{b}}{2},\quad\quad L_{2}=\frac{T^{2}}{4}.$

We substitute $V=q^{m-s}F$ into (3.8) and obtain that

(3.12)
$\begin{split}&\left(\Delta_{+}+s(m-s)\right)(q^{m-s}F)=\left(\sum_{i=0}^{2}q^{i}L_{i}+s(m-s)\right)(q^{m-s}F)\\\
&=\left(L_{0}+s(m-s)\right)(q^{m-s}F)+qL_{1}(q^{m-s}F)+q^{2}L_{2}(q^{m-s}F).\end{split}$

We assume that $F$ has the formal expansion $F=\sum_{l=0}^{\infty}q^{l}f_{l}$.
Direct calculation shows that

(3.13)
$\begin{split}\left(L_{0}+s(m-s)\right)&(q^{m-s}F)=\left(m-2s+1\right)q^{m-s+1}\frac{\partial
F}{\partial q}+q^{m-s+2}\frac{\partial^{2}F}{\partial q^{2}}\\\
&=\sum_{l=1}^{\infty}l\left(m-2s+1\right)q^{m-s+l}f_{l}+\sum_{l=1}^{\infty}l\left(l-1\right)q^{m-s+l}f_{l}\\\
&=\sum_{l=1}^{\infty}l\left(m-2s+l\right)q^{m-s+l}f_{l}.\end{split}$

Plugging (3.13) into (3.12) yields

(3.14) $f_{-1}=0,\quad f_{0}=f,\quad
f_{l}=-\frac{L_{1}f_{l-1}+L_{2}f_{l-2}}{l\left(m-2s+l\right)},\quad l\geqslant
1.$

We assume that

(3.15) $f_{l}=(-1)^{l}\frac{\Gamma(m-2s+1)}{l!\Gamma(m-2s+l+1)}P^{s}_{l}(f),$

where $P^{s}_{l}$ is an operator depending on $L_{1},L_{2}$ and $s$.

Combining this with (3.14), we observe that

(3.16) $\begin{split}&P^{s}_{-1}=0,\quad P^{s}_{0}=\mathrm{id},\\\
P^{s}_{l}=L_{1}\circ P^{s}_{l-1}-&(l-1)(m-2s+l-1)L_{2}\circ P^{s}_{l-2},\quad
l\geqslant 1.\end{split}$

By induction one sees that $P^{s}_{l}$ is a homogeneous polynomial of degree
$l$ in the commuting variables $L_{1}$ and $\sqrt{L_{2}}$. Hence, the
recursion relation for $P^{s}_{l}$ reduces to the recursion for the
polynomials $p_{l}^{s}$ of one variable $x$ defined by

(3.17) $\begin{split}&p^{s}_{-1}=0,\quad p^{s}_{0}(x)=1,\\\
p^{s}_{l}=xp^{s}_{l-1}-&(l-1)(m-2s+l-1)p^{s}_{l-2},\quad l\geqslant
1.\end{split}$

Obviously, $p^{s}_{l}$ is a polynomial of degree $l$ with the leading
coefficient $1$. Moreover, $P^{s}_{l}$ is recovered from $p^{s}_{l}$ by

(3.18) $P^{s}_{l}=\left(\sqrt{L_{2}}\right)^{l}\circ
p^{s}_{l}\left(\frac{L_{1}}{\sqrt{L_{2}}}\right).$

Similarly, we have

(3.19)
$g_{l}=(-1)^{l}\frac{\Gamma(2s-m+1)}{l!\Gamma(2s-m+l+1)}G^{s}_{l}(\mathcal{S}(s)(f)),$

where $G^{s}_{l}$ is given by

(3.20) $G^{s}_{l}=\left(\sqrt{L_{2}}\right)^{l}\circ
g^{s}_{l}\left(\frac{L_{1}}{\sqrt{L_{2}}}\right),$

and the polynomial $g^{s}_{l}$ is determined by

(3.21) $\begin{split}&g^{s}_{-1}=0,\quad g^{s}_{0}(x)=1,\\\
g^{s}_{l}=xg^{s}_{l-1}-&(l-1)(2s-m+l-1)g^{s}_{l-2},\quad l\geqslant
1,\end{split}$

which implies that $P^{s}_{l}=G^{m-s}_{l}$. Finally, (3.9) follows from
(3.18), (3.20) immediately. ∎

######  Remark 3.2.

The analogue of the operator $P^{s}_{l}$ on $\mathbb{R}^{n}$ is precisely
$(-\overline{\Delta})^{l}$, independent of the value of $s$. However, from the
recursion relation (3.17), we observe that $P^{s}_{l}$ depends on the value of
$s$ when $l\geqslant 2$ and it is hard to solve the recursion relation (3.17)
explicitly.

There are two ways to study the CR fractional GJMS operators via an extension.
The first approach is to identify solutions of (3.2) as elements of the kernel
of a second-order “weighted” Laplacian on the Siegel domain.

######  Lemma 3.3.

Let $\gamma\in(0,m)\backslash\mathbb{N}$. We define

(3.22)
$L_{\gamma}=\partial^{2}_{\rho}+(1-2\gamma)\rho^{-1}\partial_{\rho}+\rho^{2}\partial_{t}^{2}+\Delta_{b}.$

Then

$L_{\gamma}\circ\rho^{-m+\gamma}\circ\mathcal{P}(\frac{m+\gamma}{2})=0.$

###### Proof.

After the change $q=\frac{\rho^{2}}{2}$, we have

(3.23)
$\Delta_{+}=\frac{\rho^{2}}{4}\left(\partial^{2}_{\rho}-(2n+1)\rho^{-1}\partial_{\rho}+\rho^{2}\partial_{t}^{2}+\Delta_{b}\right).$

A direct computation using (3.23) shows that

(3.24)
$\frac{1}{4}L_{\gamma}\left(\rho^{-m+\gamma}U\right)=\rho^{-m+\gamma-2}\left(\Delta_{+}+\frac{m^{2}-\gamma^{2}}{4}\right)U,$

for all $U\in C^{\infty}(\overline{\Omega}_{n+1})$. The conclusion follows
from the definition of the Poisson operator $\mathcal{P}(\frac{m+\gamma}{2})$.
∎

The second approach to studying the CR fractional GJMS operators via an
extension is to identify solutions of (3.2) as elements of the kernel of a
product of $L_{\gamma}$. This can be done as follows.

First, let $\gamma\in(0,m)\backslash\mathbb{N}$. Set
$k:=\lfloor\gamma\rfloor+1$. The weighted poly-sublaplacian determined by
$\gamma$ is

(3.25) $L_{2k}:=\prod_{j=0}^{k-1}L_{\gamma-2j}=L_{\gamma-2k+2}\circ\cdots\circ
L_{\gamma},$

where $L_{\mu}$ is defined in (3.22).

Second, let $\gamma\in(0,m)\backslash\mathbb{N}$. Set $s:=\frac{m+\gamma}{2}$
and

$D_{s}:=\Delta_{+}+s(m-s).$

The complex hyperbolic poly-Poisson operator determined by $\gamma$ is

(3.26) $L^{+}_{2k}:=\prod_{j=0}^{k-1}D_{s-j}.$

The following two lemmas capture the essential features of this relationship
as needed to study CR sharp Sobolev trace inequalities on the Siegel domain.

First, to understand the boundary operators, it is useful to rewrite the
factorization of $L_{2k}^{+}$.

######  Lemma 3.4.

Let $\gamma\in(0,m)\backslash\mathbb{N}$ and $L_{2k}^{+}$ be the weighted
poly-sublaplacian defined in (3.26). Then

(3.27)
$L_{2k}^{+}=\prod_{j=0}^{\lfloor\gamma/2\rfloor}D_{s_{j}}\circ\prod_{j=0}^{\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1}D_{\tilde{s}_{j}},$

where $s_{j}=\frac{m+\gamma}{2}-j$ and
$\tilde{s}_{j}=\frac{m+\lfloor\gamma\rfloor-[\gamma]}{2}-j$.

###### Proof.

Separating (3.26) into terms with $s-j>\frac{m}{2}$ and $s-j<\frac{m}{2}$, we
compute that

$L_{2k}^{+}=\prod_{j=0}^{\lfloor\gamma/2\rfloor}D_{s_{j}}\circ\prod_{j=\lfloor\gamma/2\rfloor+1}^{k-1}D_{s_{j}}$

Observe that if $j+i=\lfloor\gamma\rfloor$ for
$\lfloor\gamma/2\rfloor+1\leqslant j\leqslant\lfloor\gamma\rfloor$, we have

(3.28)
$D_{\frac{m+\gamma}{2}-j}=D_{\frac{m+\lfloor\gamma\rfloor-[\gamma]}{2}-i}.$

Using (3.28), the desired result follows by reindexing. ∎

Second, the operator $L_{2k}$ defined as a product of $L_{\gamma-2j}$ has a
nice factorization (3.29) which implies that $L_{2k}$ is formally self-adjoint
with respect to the weighted measure $\rho^{1-2[\gamma]}dzdtd\rho$ on
$\Omega_{n+1}$. The following result can be deduced from Theorem 1.3 [LY22]
easily. In order to make this paper self-contained, we prove it in an
alternative way.

######  Lemma 3.5.

Let $\gamma\in(0,m)\backslash\mathbb{N}$ and $L_{2k}$ be the weighted poly-
sublaplacian defined in (3.25). Then

(3.29)
$L_{2k}=\prod_{j=0}^{k-1}\left(L_{[\gamma]}+i\left(k-1-2j\right)T\right),$

where $L_{\mu}$ is given in (3.22) and $T=2\partial_{t}$.

###### Proof.

We prove the claim by induction. Direct calculation shows that (3.29) holds
for $\gamma\in(0,2)$. Suppose that (3.29) holds for $\gamma\in(0,k-2)$ with
$k\in\mathbb{N}$ and $k>2$.

For convenience, we denote the operator $\rho^{-1}\partial_{\rho}$ by $Y$.
First observe the commutator identity

(3.30)
$\left[Y,L_{\mu}\right]=2\left(Y^{2}+\partial_{t}^{2}\right),\quad\forall\mu\in\mathbb{R}.$

A direct computation using (3.25) yields

(3.31)
$\begin{split}L_{2k}=&L_{\gamma-2k+2}\circ\prod_{j=1}^{k-2}\left(L_{\gamma-k+1}+i\left(k-1-2j\right)T\right)\circ
L_{\gamma}\\\ =&\left(L_{[\gamma]}+2(k-1)Y\right)\circ
L_{2(k-2)}\circ\left(L_{[\gamma]}-2(k-1)Y\right)\\\ =&L_{2(k-2)}\circ
L_{[\gamma]}^{2}+2(k-1)\left[Y,L_{2(k-2)}\circ L_{[\gamma]}\right]\\\
&-4(k-1)^{2}Y\circ L_{2(k-2)}\circ Y.\end{split}$

Hence, it suffices to show that for $k\in\mathbb{N}$ and $k>2$

(3.32) $\left[Y,L_{2(k-2)}\circ L_{[\gamma]}\right]=2(k-1)Y\circ
L_{2(k-2)}\circ Y+2(k-1)L_{2(k-2)}\circ\partial_{t}^{2}.$

By induction, suppose that (3.32) holds for $k-2$, then

(3.33) $\begin{split}\left[Y,L_{2k}\circ
L_{[\gamma]}\right]&=\left[Y,\left(L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right)\circ
L_{2(k-2)}\circ L_{[\gamma]}\right]\\\
&=\left[Y,L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right]\circ L_{2(k-2)}\circ
L_{[\gamma]}\\\
&+\left(L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right)\circ\left[Y,L_{2(k-2)}\circ
L_{[\gamma]}\right].\end{split}$

Notice that $T$ can commute with $Y$ and $L_{\mu}$ for all $\mu\in\mathbb{R}$.
Hence

(3.34)
$\begin{split}\left[Y,L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right]&=\left[Y,L_{[\gamma]}\right]\circ
L_{[\gamma]}+L_{[\gamma]}\circ\left[Y,L_{[\gamma]}\right]\\\
&=2\left(Y^{2}\circ L_{[\gamma]}+L_{[\gamma]}\circ
Y^{2}\right)+4L_{[\gamma]}\circ\partial_{t}^{2}.\end{split}$

Plugging into (3.34) and (3.32) into (3.33) yields

(3.35) $\begin{split}\left[Y,L_{2k}\circ L_{[\gamma]}\right]&=2Y^{2}\circ
L_{[\gamma]}^{2}\circ L_{2(k-2)}+2L_{[\gamma]}\circ Y^{2}\circ L_{2(k-2)}\circ
L_{[\gamma]}\\\ &+2(k-1)\left(L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right)\circ
Y\circ L_{2(k-2)}\circ Y\\\ &+4L_{[\gamma]}^{2}\circ
L_{2(k-2)}\circ\partial_{t}^{2}+2(k-1)L_{2k}\circ\partial_{t}^{2}.\end{split}$

Besides, we have

(3.36) $Y^{2}\circ L_{[\gamma]}^{2}\circ L_{2(k-2)}=Y\circ
L_{[\gamma]}^{2}\circ L_{2(k-2)}\circ Y+Y\circ\left[Y,L_{[\gamma]}^{2}\circ
L_{2(k-2)}\right],$ (3.37) $\begin{split}L_{[\gamma]}\circ
Y^{2}\circ&L_{2(k-2)}\circ L_{[\gamma]}=Y\circ L_{[\gamma]}^{2}\circ
L_{2(k-2)}\circ Y\\\ &+Y\circ L_{[\gamma]}\circ\left[Y,L_{2(k-2)}\circ
L_{[\gamma]}\right]\\\ &+\left[L_{[\gamma]},Y\right]\circ Y\circ
L_{[\gamma]}\circ L_{2(k-2)},\end{split}$

and

(3.38) $\left(L_{[\gamma]}^{2}+(k-1)^{2}T^{2}\right)\circ Y\circ
L_{2(k-2)}\circ Y=Y\circ L_{2k}\circ Y+\left[L_{[\gamma]}^{2},Y\right]\circ
L_{2(k-2)}\circ Y.$

Combining them together yields

(3.39) $\begin{split}\left[Y,L_{2k}\circ L_{[\gamma]}\right]=&2(k+1)Y\circ
L_{2k}\circ Y+\mathrm{I}+\mathrm{II}+\mathrm{III}\\\ &+4L_{[\gamma]}^{2}\circ
L_{2(k-2)}\circ\partial_{t}^{2}+2(k-1)L_{2k}\circ\partial_{t}^{2}\\\
&-4(k-1)^{2}Y\circ L_{2(k-2)}\circ Y\circ T^{2},\end{split}$

where

(3.40) $\begin{split}\mathrm{I}=&2Y\circ\left[Y,L_{[\gamma]}^{2}\circ
L_{2(k-2)}\right]\\\ \mathrm{II}=&2Y\circ
L_{[\gamma]}\circ\left[Y,L_{2(k-2)}\circ L_{[\gamma]}\right]\\\
&+2\left[L_{[\gamma]},Y\right]\circ Y\circ L_{[\gamma]}\circ L_{2(k-2)}\\\
\mathrm{III}=&2(k-1)\left[L_{[\gamma]}^{2},Y\right]\circ L_{2(k-2)}\circ
Y.\end{split}$

Using (3.32) and (3.30), we find that

(3.41) $\begin{split}\mathrm{I}+\mathrm{II}=&8(k-1)Y\circ L_{[\gamma]}\circ
Y\circ L_{2(k-2)}\circ Y\\\ &+8(k-1)Y\circ L_{[\gamma]}\circ
L_{2(k-2)}\circ\partial_{t}^{2}\\\ \mathrm{III}=&-4(k-1)Y^{2}\circ
L_{[\gamma]}\circ L_{2(k-2)}\circ Y\\\ &-4(k-1)L_{[\gamma]}\circ Y^{2}\circ
L_{2(k-2)}\circ Y\\\ &-8(k-1)L_{[\gamma]}\circ L_{2(k-2)}\circ
Y\circ\partial_{t}^{2}\end{split}$

Notice that

(3.42) $\begin{split}Y\circ L_{[\gamma]}\circ Y\circ L_{2(k-2)}\circ
Y=&Y^{2}\circ L_{[\gamma]}\circ L_{2(k-2)}\circ Y\\\
&-2Y\circ\left(Y^{2}+\partial_{t}^{2}\right)\circ L_{2(k-2)}\circ Y\\\
-L_{[\gamma]}\circ Y^{2}\circ L_{2(k-2)}\circ Y=&-Y^{2}\circ L_{[\gamma]}\circ
L_{2(k-2)}\circ Y\\\ &+4Y\circ\left(Y^{2}+\partial_{t}^{2}\right)\circ
L_{2(k-2)}\circ Y.\end{split}$

We conclude that

(3.43)
$\begin{split}\mathrm{I}+\mathrm{II}+\mathrm{III}&=8(k-1)\left[Y,L_{2(k-2)}\circ
L_{[\gamma]}\right]\circ\partial_{t}^{2}\\\ &=16(k-1)^{2}Y\circ
L_{2(k-2)}\circ
Y\circ\partial_{t}^{2}+16(k-1)^{2}L_{2(k-2)}\circ\partial_{t}^{2}.\end{split}$

Observe that

(3.44) $\begin{split}&4L_{[\gamma]}^{2}\circ
L_{2(k-2)}\circ\partial_{t}^{2}+2(k-1)L_{2k}\circ\partial_{t}^{2}\\\
&=2(k+1)L_{2k}\circ\partial_{t}^{2}-16(k-1)^{2}L_{2(k-2)}\circ\partial_{t}^{2}.\end{split}$

Finally, (3.32) holds for $k$. ∎

We now prove that elements of the kernel $D_{s-j}$ are also in the kernel of
$L_{2k}$ when weighted against a suitable power of $\rho$.

######  Lemma 3.6.

Let $\gamma\in(0,m)\backslash\mathbb{N}$ and set $k:=\lfloor\gamma\rfloor+1$.
Denote

$\mathcal{I}^{\gamma}:=\left\\{\gamma-2j|j\in[0,\lfloor\gamma/2\rfloor]\right\\}\cup\left\\{\lfloor\gamma\rfloor-[\gamma]-2j|j\in[0,\lfloor\gamma\rfloor-\lfloor\gamma/2\rfloor-1]\right\\}.$

For each $\tilde{\gamma}\in\mathcal{I}^{\gamma}$ it holds that

$L_{2k}\circ\rho^{-m+\gamma}\circ\mathcal{P}(\frac{m+\tilde{\gamma}}{2})=0.$

###### Proof.

It is clear from Lemma 3.6 that

$L_{2k}^{+}\circ\mathcal{P}(\frac{m+\tilde{\gamma}}{2})=0,$

for each $\tilde{\gamma}\in\mathcal{I}^{\gamma}$. The desired result follows
from a repeated application of the transformation law (3.24):

(3.45) $4^{-k}L_{2k}\circ\rho^{-m+\gamma}=\rho^{-m+\gamma-2k}\circ
L^{+}_{2k}.$

∎

##  4\. Boundary operators on the Siegel domain

In this section we introduce the boundary operators associated to the weighted
ploy-sublaplacian $L_{2k}$ determined by $\gamma\in(0,2)\backslash\mathbb{N}$.
By Lemma 3.6, the kernel of $L_{2k}$ contains solutions of the Poisson
equation (3.2) for any $\tilde{\gamma}\in\mathcal{I}^{\gamma}$. Our boundary
operators are designed to pick out the functions $F(\cdot,0)$ and $G(\cdot,0)$
of solutions to (3.2). They also give rise to formally self-adjoint boundary
value problems. To that end, it is convenient to introduce the space

$\mathcal{C}^{2\gamma}=C^{\infty}_{\mathrm{even}}(\overline{\Omega}_{n+1})+\rho^{2[\gamma]}C^{\infty}_{\mathrm{even}}(\overline{\Omega}_{n+1})$

associated to a given $\gamma\in(0,m)\backslash\mathbb{N}$, where
$C^{\infty}_{\mathrm{even}}$ denotes the space of smooth functions on
$\overline{\Omega}_{n+1}$ whose Taylor series expansions in $\rho$ at $\rho=0$
contain only even terms. Note that
(1) if $\gamma\in\frac{1}{2}+\mathbb{N}_{0}$, then
$\mathcal{C}^{2\gamma}=C^{\infty}(\overline{\Omega}_{n+1})$; and
(2) for any $\tilde{\gamma}\in\mathcal{I}^{\gamma}$, it holds that
$\mathcal{P}\left(\frac{m+\tilde{\gamma}}{2}\right):C^{\infty}(\mathbb{H}^{n})\cap
S^{\tilde{\gamma},2}(\mathbb{H}^{n})\to\mathcal{C}^{2\gamma}$.

The second point means that the space $\mathcal{C}^{2\gamma}$ is well-suited
to studying all of the scattering problems formed from the factors of the
complex hyperbolic poly-Poisson operator $L_{2k}^{+}$. Let
$\iota^{*}:\mathcal{C}^{2\gamma}\to C^{\infty}(\mathbb{H}^{n})$ denote the
restriction operator, $\iota^{*}U=U|_{\mathbb{H}^{n}}$. Our boundary operators
are elements of the set

$\begin{split}\mathcal{B}^{2\gamma}:=&\left\\{B_{2j}^{2\gamma}:\mathcal{C}^{2\gamma}\to
C^{\infty}(\mathbb{H}^{n})|j\in[0,\lfloor\gamma\rfloor]\right\\}\\\
&\cup\left\\{B_{2[\gamma]+2j}^{2\gamma}:\mathcal{C}^{2\gamma}\to
C^{\infty}(\mathbb{H}^{n})|j\in[0,\lfloor\gamma\rfloor]\right\\}\end{split}$

defined as follows:

######  Definition 4.1.

Set

$\mathcal{R}:=\partial^{2}_{\rho}+(1-2[\gamma])\partial_{\rho},\quad\quad\mathcal{T}:=\mathcal{R}+\rho^{2}\partial^{2}_{t}.$

For $\gamma\in(0,1)$, we define associated boundary operators by

(4.1) $B_{0}^{2\gamma}=\iota^{*},\quad
B_{2\gamma}^{2\gamma}=-\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}.$

For $\gamma\in(1,2)$, we define associated boundary operators by

(4.2)
$\begin{split}B_{0}^{2\gamma}=\iota^{*},\quad&B_{2}^{2\gamma}=-\iota^{*}\circ\mathcal{T}+\frac{1-[\gamma]}{[\gamma]}\Delta_{b}\circ
B_{0}^{2\gamma},\\\
B_{2[\gamma]}^{2\gamma}=-\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho},\quad&B_{2\gamma}^{2\gamma}=\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}\mathcal{T}-\frac{1+[\gamma]}{[\gamma]}\Delta_{b}\circ
B_{2[\gamma]}^{2\gamma}.\end{split}$

The first goal of this section is to show that the boundary operators
$\mathcal{B}^{2\gamma}$ are relevant for picking out the Dirichlet data
$F(\cdot,0)$ and the Neumann data $G(\cdot,0)$ of solutions of the Poisson
equation $\mathcal{P}\left(\frac{m+\tilde{\gamma}}{2}\right)$ for any
$\tilde{\gamma}\in\mathcal{I}^{\gamma}$. This is accomplished by the following
two propositions.

######  Proposition 4.2.

Let $\gamma\in(0,2)\backslash\mathbb{N}$. Let
$V=\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\gamma}}{2}\right)(f)$ for some
$f\in C^{\infty}(\mathbb{H}^{n})\cap S^{\gamma,2}(\mathbb{H}^{n})$. It holds
that

(4.3) $B_{0}^{2\gamma}(V)=2^{\frac{\gamma-m}{2}}f,$ (4.4)
$B_{2\gamma}^{2\gamma}(V)=(-1)^{\lfloor\gamma\rfloor+1}2^{-\frac{m+\gamma}{2}}2^{2\lfloor\gamma\rfloor+1}\lfloor\gamma\rfloor!\frac{\Gamma(\gamma+1)}{\Gamma([\gamma])}\hat{f}$

where $\hat{f}=\mathcal{S}\left(s\right)f$ with $s=\frac{m+{\gamma}}{2}$.
Moreover, $B_{2\alpha}^{2\gamma}(V)=0$ for all
$B_{2\alpha}^{2\gamma}\in\mathcal{B}^{2\gamma}\backslash\\{B_{0}^{2\gamma},B_{2\gamma}^{2\gamma}\\}$.

###### Proof.

To begin, note that

(4.5)
$\mathcal{R}(\rho^{2j})=4j(j-[\gamma])\rho^{2j-2},\quad\quad\mathcal{R}(\rho^{2[\gamma]+2j})=4j(j+[\gamma])\rho^{2[\gamma]+2j-2}.$

By (3.2), it holds that

(4.6)
$V=\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\gamma}}{2}\right)f=\left(\frac{1}{2}\right)^{\frac{m-\gamma}{2}}F+\left(\frac{1}{2}\right)^{\frac{m+\gamma}{2}}\rho^{2\gamma}G,$

where by Proposition 3.1,

$F=\sum_{l=0}^{\infty}(-1)^{l}\frac{\Gamma(1-\gamma)}{2^{l}l!\Gamma(l+1-\gamma)}\rho^{2l}P^{s}_{l}(f),$

and

$G=\sum_{l=0}^{\infty}(-1)^{l}\frac{\Gamma(1+\gamma)}{2^{l}l!\Gamma(l+1+\gamma)}\rho^{2l}G^{s}_{l}(\hat{f}).$

We separate the proof into two cases:

First consider $B_{2l}^{2\gamma}$ for $l\in[0,\lfloor\gamma\rfloor]$. By the
definition of $B_{0}^{2\gamma}$, we know that (4.3) holds. When
$\gamma\in(1,2)$ and $l=1$, using (4.3) and (3.18),we compute that

(4.7)
$B_{2}^{2\gamma}(V)=-2^{\frac{\gamma-m}{2}}2\frac{\Gamma(2-[\gamma])}{\Gamma(1-[\gamma])}f_{1}-\frac{\Gamma(2-[\gamma])\Gamma(1-\gamma)}{\Gamma(1-[\gamma])\Gamma(2-\gamma)}\Delta_{b}\circ
B_{0}^{2\gamma}(V)=0.$

Next consider $B_{2[\gamma]+2l}^{2\gamma}$ for $l\in[0,\lfloor\gamma\rfloor]$.
Notice that from (4.5) and (4.6) we immediately deduce that

(4.8)
$\begin{split}B_{2\gamma}^{2\gamma}(V)=-2^{-\frac{m+\gamma}{2}}2\gamma\hat{f},\quad\gamma\in(0,1);\end{split}$

i.e., (4.4) holds. When $\gamma\in(1,2)$, by the definition of
$B_{2[\gamma]+2l}^{2\gamma}$, we immediately have
$B_{2[\gamma]}^{2\gamma}(V)=0$. Besides, we compute that

(4.9)
$B_{2\gamma}^{2\gamma}(V)=2^{-\frac{m+\gamma}{2}}2^{3}\frac{\Gamma([\gamma]+2)}{\Gamma([\gamma])}g_{0};$

i.e., (4.4) holds. ∎

######  Proposition 4.3.

Let $\gamma\in(1,2)$. Let
$\tilde{V}=\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\tilde{\gamma}}}{2}\right)\tilde{f}$
for some $\tilde{f}\in C^{\infty}(\mathbb{H}^{n})\cap
S^{\tilde{\gamma},2}(\mathbb{H}^{n})$. It holds that

(4.10)
$B_{2[\gamma]}^{2\gamma}(\tilde{V})=-2^{\frac{\tilde{\gamma}-m}{2}}2[\gamma]\tilde{f}$
(4.11)
$B_{2}^{2\gamma}(\tilde{V})=-2^{-\frac{m+\tilde{\gamma}}{2}}4(1-[\gamma])\hat{\tilde{f}},$

where $\hat{\tilde{f}}=\mathcal{S}(\tilde{s})\tilde{f}$ with
$\tilde{s}=\frac{m+\tilde{\gamma}}{2}$, $\tilde{\gamma}=1-[\gamma]$. Moreover,
$B_{2\alpha}^{2\gamma}(V)=0$ for all
$B_{2\alpha}^{2\gamma}\in\mathcal{B}^{2\gamma}\backslash\\{B_{2[\gamma]}^{2\gamma},B_{2}^{2\gamma}\\}$.

###### Proof.

It follows from (3.2) that

(4.12)
$\tilde{V}=\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\tilde{\gamma}}}{2}\right)f=\left(\frac{1}{2}\right)^{\frac{m-\tilde{\gamma}}{2}}\rho^{2[\gamma]}\tilde{F}+\left(\frac{1}{2}\right)^{\frac{m+\tilde{\gamma}}{2}}\rho^{2}\tilde{G}$

where by Proposition 3.1,

(4.13)
$\tilde{F}=\sum_{l=0}^{\infty}(-1)^{l}\frac{\Gamma(1-\tilde{\gamma})}{2^{l}l!\Gamma(l+1-\tilde{\gamma})}\rho^{2l}P^{\tilde{s}}_{l}(\tilde{f}),$

and

$\tilde{G}=\sum_{l=0}^{\infty}(-1)^{l}\frac{\Gamma(1+\tilde{\gamma})}{2^{l}l!\Gamma(l+1+\tilde{\gamma})}\rho^{2l}G^{\tilde{s}}_{l}(\hat{\tilde{f}}).$

Using (4.5), (4.12) and Definition 4.1, direct computation yields the desired
result. ∎

######  Remark 4.4.

Our boundary operators are designed to pick out the functions $F(\cdot,0)$ and
$G(\cdot,0)$ of solutions of (3.2). To guarantee that $B_{2l}^{2\gamma}(V)=0$
for $l>j$ (see [Cas20, Propositon 3.2] for more details), we need to make the
cancellation by composing $B_{2j}^{2\gamma}$ with
$P^{s}_{l-j}=(-\overline{\Delta})^{l-j}$ because the output of $\iota^{*}\circ
T^{l}$ is exactly $f_{l-j}$. However, in the CR setting, the presence of the
term $\rho^{2}\partial^{2}_{t}$ affects the output of leading terms
$\iota^{*}\circ\mathcal{T}^{l}$ and
$\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}\mathcal{T}^{l}$, $l\geqslant
2$, which implies that the operator to be composed is no longer $P^{s}_{l-j}$.

Let

(4.14)
$\begin{split}\mathcal{Q}_{2\gamma}(U,V):=&-\int_{\Omega_{n+1}}UL_{2}V\rho^{1-2\gamma}dzdtd\rho+\oint_{\mathbb{H}^{n}}B_{0}^{2\gamma}(U)B_{2\gamma}^{2\gamma}(V)dzdt,\\\
\mathcal{Q}_{2\gamma}(U,V):=&\int_{\Omega_{n+1}}UL_{4}V\rho^{1-2[\gamma]}dzdtd\rho\\\
&+\oint_{\mathbb{H}^{n}}B_{0}^{2\gamma}(U)B_{2\gamma}^{2\gamma}(V)dzdt-\oint_{\mathbb{H}^{n}}B_{2[\gamma]}^{2\gamma}(U)B_{2}^{2\gamma}(V)dzdt\end{split}$

be the associated Dirichlet energy determined by $\gamma\in(0,1)$ and
$\gamma\in(1,2)$, respectively. The second goal of this section is to prove
that $\mathcal{Q}_{2\gamma}$ is symmetric. This implies that the boundary
value problem involving $L_{2k}$ and $\mathcal{B}^{2\gamma}$ are variational.
The proof that $\mathcal{Q}_{2\gamma}$ is symmetric is essentially integration
by parts on the Siegel domain $\Omega_{n+1}$ with flat metric. To that end, it
is useful to express $\iota^{*}\circ L_{[\gamma]}$ and
$\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}L_{[\gamma]}$ in terms of
boundary operators.

######  Proposition 4.5.

For $\gamma\in(1,2)$, it holds that

(4.15) $\begin{split}\iota^{*}=B_{0}^{2\gamma},\quad&\iota^{*}\circ
L_{[\gamma]}=-B_{2}^{2\gamma}+\frac{1}{[\gamma]}\Delta_{b}\circ
B_{0}^{2\gamma},\\\
\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}=-B_{2[\gamma]}^{2\gamma},\quad&\iota^{*}\circ\rho^{1-2[\gamma]}\partial_{\rho}L_{[\gamma]}=B_{2\gamma}^{2\gamma}+\frac{1}{[\gamma]}\Delta_{b}\circ
B_{2[\gamma]}^{2\gamma}.\end{split}$

###### Proof.

Note that

(4.16) $\iota^{*}\circ
L_{[\gamma]}=\iota^{*}\circ\left(\mathcal{T}+\Delta_{b}\right)=\iota^{*}\circ\mathcal{T}+\iota^{*}\circ\Delta_{b}=\iota^{*}\circ\mathcal{T}+\Delta_{b}\circ\iota^{*}.$

Then, (4.15) follows from Definition 4.1 directly. ∎

We now prove that $\mathcal{Q}_{2\gamma}$ is symmetric by giving an explicit
formula for $\mathcal{Q}_{2\gamma}$. Especially notable in this formula is
that the boundary integration involves only the Dirichlet data

$\mathcal{B}^{2\gamma}_{D}:=\left\\{B^{2\gamma}_{2\alpha}\in\mathcal{B}^{2\gamma}|0\leqslant\alpha<\gamma/2\right\\}$

of the inputs.

######  Theorem 4.6.

Given $\gamma\in(0,1)$, we have

(4.17) $\begin{split}&\mathcal{Q}_{2\gamma}(U,V)\\\
&=\int_{\Omega_{n+1}}\left(\partial_{\rho}U\partial_{\rho}V+\frac{1}{2}\sum_{j=1}^{n}X_{j}(U)X_{j}(V)+Y_{j}(U)Y_{j}(V)+\rho^{2}\partial_{t}U\partial_{t}V\right)\rho^{1-2\gamma}dzdtd\rho.\end{split}$

Likewise, given $\gamma\in(1,2)$, we have

(4.18)
$\begin{split}\mathcal{Q}_{2\gamma}(U,V)&=\int_{\Omega_{n+1}}\left(L_{[\gamma]}(U)L_{[\gamma]}(V)-T(U)T(V)\right)\rho^{1-2[\gamma]}dzdtd\rho\\\
&-\frac{1}{[\gamma]}\oint_{\mathbb{H}^{n}}\Delta_{b}\left(B_{0}^{2\gamma}(U)B_{2[\gamma]}^{2\gamma}(V)+B_{2[\gamma]}^{2\gamma}(U)B_{0}^{2\gamma}(V)\right)dzdt\end{split}$

###### Proof.

Given $\gamma\in(0,1)$, using the integration by parts on the Siegel domain
$\Omega_{n+1}$ with flat metric, we have

(4.19)
$\begin{split}&-\int_{\Omega_{n+1}}U\left(\partial^{2}_{\rho}+(1-2\gamma)\rho^{-1}\partial_{\rho}\right)V\rho^{1-2\gamma}dzdtd\rho\\\
&=\int_{\Omega_{n+1}}\partial_{\rho}U\partial_{\rho}V\rho^{1-2\gamma}dzdtd\rho-\oint_{\mathbb{H}^{n}}B_{0}^{2\gamma}(U)B_{2\gamma}^{2\gamma}(V)dzdt.\end{split}$
(4.20)
$-\int_{\Omega_{n+1}}U\Delta_{b}V\rho^{1-2\gamma}dzdtd\rho=\frac{1}{2}\int_{\Omega_{n+1}}\sum_{j=1}^{n}X_{j}(U)X_{j}(V)+Y_{j}(U)Y_{j}(V)\rho^{1-2\gamma}dzdtd\rho.$
(4.21)
$-\int_{\Omega_{n+1}}U\rho^{2}\partial^{2}_{t}V\rho^{1-2\gamma}dzdtd\rho=\int_{\Omega_{n+1}}\rho^{2}\partial_{t}U\partial_{t}V\rho^{1-2\gamma}dzdtd\rho.$

Combining (4.19), (4.20) and (4.21) yields (4.17).

Similarly, given $\gamma\in(1,2)$, by Lemma 3.5, we know that
$L_{4}=L^{2}_{[\gamma]}+T^{2}$. Notice that $L_{[\gamma]}$ is formally self-
adjoint with respect to the weighted measure $\rho^{1-2[\gamma]}dzdtd\rho$. By
the integration by parts, we have that

(4.22)
$\begin{split}&\int_{\Omega_{n+1}}UL^{2}_{[\gamma]}(V)\rho^{1-2[\gamma]}dzdtd\rho=\int_{\Omega_{n+1}}L_{[\gamma]}(U)L_{[\gamma]}(V)\rho^{1-2[\gamma]}dzdtd\rho\\\
&+\oint_{\mathbb{H}^{n}}B_{0}^{2\gamma}(U)B_{2[\gamma]}^{2\gamma}(L_{[\gamma]}V)dzdt-\oint_{\mathbb{H}^{n}}B_{2[\gamma]}^{2\gamma}(U)B_{0}^{2\gamma}(L_{[\gamma]}V)dzdt,\\\
&\int_{\Omega_{n+1}}UT^{2}(V)\rho^{1-2[\gamma]}dzdtd\rho=-\int_{\Omega_{n+1}}T(U)T(V)\rho^{1-2[\gamma]}dzdtd\rho.\end{split}$

By Propositon 4.5, we know that

(4.23) $\begin{split}B_{0}^{2\gamma}\circ
L_{[\gamma]}&=-B_{2}^{2\gamma}+\frac{1}{[\gamma]}\Delta_{b}\circ
B_{0}^{2\gamma},\\\ B_{2[\gamma]}^{2\gamma}\circ
L_{[\gamma]}&=-B_{2\gamma}^{2\gamma}-\frac{1}{[\gamma]}\Delta_{b}\circ
B_{2[\gamma]}^{2\gamma}.\end{split}$

Plugging (4.23) into (4.22) yields (4.18). ∎

##  5\. The generalized extension problem

The main result of this section is that solutions of the Dirichlet boundary
value problems

(5.1)
$\left\\{\begin{array}[]{ll}L_{2}V=0,&~{}~{}\mbox{on}~{}~{}\Omega_{n+1},\quad\gamma\in(0,1),\\\
B_{0}^{2\gamma}(V)=\phi,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\end{array}\right.$

and

(5.2)
$\left\\{\begin{array}[]{ll}L_{4}V=0,&~{}~{}\mbox{on}~{}~{}\Omega_{n+1},\quad\gamma\in(1,2),\\\
B_{0}^{2\gamma}(V)=\phi,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\\\
B_{2[\gamma]}^{2\gamma}(V)=\psi,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\end{array}\right.$

can be used to recover the CR fractional GJMS operators. To that end, we first
characterize the solutions of (5.1) and (5.2).

######  Theorem 5.1.

Let $\gamma\in(0,1)$. Given functions $\phi\in C^{\infty}(\mathbb{H}^{n})\cap
S^{\gamma,2}(\mathbb{H}^{n})$, there is a unique solution $V$ of (5.1).
Indeed,

(5.3)
$V=2^{-\frac{\gamma-m}{2}}\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\gamma}}{2}\right)\phi.$

###### Proof.

It follows from Lemma 3.6 and Proposition 4.2 that $V$ satisfies (5.1). The
uniqueness follows from the fact that $\mathcal{Q}_{2\gamma}(W,W)\geqslant 0$
with equality holds if and only if $W=0$. ∎

######  Theorem 5.2.

Let $\gamma\in(1,2)$. Given $\phi\in C^{\infty}(\mathbb{H}^{n})\cap
S^{\gamma,2}(\mathbb{H}^{n})$ and $\psi\in C^{\infty}(\mathbb{H}^{n})\cap
S^{1-[\gamma],2}(\mathbb{H}^{n})$, there is a unique solution $V$ of (5.2).
Indeed,

(5.4)
$V=2^{-\frac{\gamma-m}{2}}\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+{\gamma}}{2}\right)\phi-2^{-\frac{\tilde{\gamma}-m}{2}}\frac{1}{2[\gamma]}\rho^{-m+\gamma}\mathcal{P}\left(\frac{m+1-[\gamma]}{2}\right)\psi,$

where $\tilde{\gamma}=1-[\gamma]$.

###### Proof.

It follows from Lemma 3.6 and Proposition 4.3 that $V$ satisfies (5.2).
Suppose now that $U$ is a solution of (5.2). Then $W:=U-V$ solves

(5.5)
$\left\\{\begin{array}[]{ll}L_{4}W=0,&~{}~{}\mbox{on}~{}~{}\Omega_{n+1},\quad\gamma\in(1,2),\\\
B_{0}^{2\gamma}(W)=0,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n},\\\
B_{2[\gamma]}^{2\gamma}(W)=0,&~{}~{}\mbox{on}~{}~{}\mathbb{H}^{n}.\end{array}\right.$

From Theorem 4.6 and [LY22, Theorem 1.5], we know that

$\mathcal{Q}_{2\gamma}(W,W)=\int_{\Omega_{n+1}}\left(L_{[\gamma]}(W)L_{[\gamma]}(W)-T(W)T(W)\right)\rho^{1-2[\gamma]}dzdtd\rho\geqslant
0,$

with equality holds if and only if $W=0$. Therefore, $U=V$. ∎

We now present our generalized extension problem. In fact, the following
result implies that the CR fractional GJMS operators can be determined without
fully specifying the Dirichlet data.

######  Theorem 5.3.

  1. (1)

Given $\gamma\in(0,1)$, suppose that $V$ is a solution of (5.1). It holds that

$B_{2\gamma}^{2\gamma}(V)=2^{1-2\gamma}\frac{\Gamma(1-\gamma)}{\Gamma(\gamma)}P^{\theta}_{\gamma}\phi.$

  2. (2)

Given $\gamma\in(1,2)$, $\tilde{\gamma}=1-[\gamma]$, suppose that $V$ is a
solution of (5.2). It holds that

$\begin{split}B_{2}^{2\gamma}(V)&=2^{1-2\tilde{\gamma}}\frac{\tilde{\gamma}}{1-\tilde{\gamma}}\frac{\Gamma(-\tilde{\gamma})}{\Gamma(\tilde{\gamma})}P^{\theta}_{\tilde{\gamma}}\psi,\\\
B_{2\gamma}^{2\gamma}(V)&=2^{3-2\gamma}\frac{\Gamma(2-\gamma)}{\Gamma(\gamma)}P^{\theta}_{\gamma}\phi.\end{split}$

###### Proof.

It follows from (3.3), Proposition 4.2, Proposition 4.3, Theorem 5.1 and
Theorem 5.2 directly. ∎

##  6\. The sharp CR trace Sobolev inequalities

The purpose of this section is to use the boundary operators to prove CR sharp
Sobolev trace inequalities. A key tool in this endeavor is the Dirichlet
energy

$\mathcal{E}_{2\gamma}(U):=\mathcal{Q}_{2\gamma}(U,U),$

where $\mathcal{Q}_{2\gamma}$ is givey by (4.14). Our first result is a
Dirichlet principle for solutions of (5.1) and (5.2).

######  Theorem 6.1.

If $\gamma\in(0,1)$, fix a function $\phi\in
C^{\infty}\left(\mathbb{H}^{n}\right)\cap S^{\gamma,2}(\mathbb{H}^{n})$ and
denote

$\begin{gathered}\mathcal{C}_{D}^{2\gamma}:=\left\\{U\in\mathcal{C}^{2\gamma}\mid
B_{0}^{2\gamma}(U)=\phi\right\\}\end{gathered}$

Moreover, if $\gamma\in(1,2)$, then fix functions $\phi\in
C^{\infty}\left(\mathbb{H}^{n}\right)\cap S^{\gamma,2}(\mathbb{H}^{n})$, and
$\psi\in C^{\infty}\left(\mathbb{H}^{n}\right)\cap
S^{1-[\gamma],2}(\mathbb{H}^{n})$, and denote

$\begin{gathered}\mathcal{C}_{D}^{2\gamma}:=\left\\{U\in\mathcal{C}^{2\gamma}\mid
B_{0}^{2\gamma}(U)=\phi,\text{ and
}B_{2[\gamma]}^{2\gamma}(U)=\psi\right\\}.\end{gathered}$

Then it holds that

$\mathcal{E}_{2\gamma}(U)\geq\mathcal{E}_{2\gamma}\left(U_{D}\right)$

for all $U\in\mathcal{C}_{D}^{2\gamma}$, where
$U_{D}\in\mathcal{C}_{D}^{2\gamma}$ is the unique solution of (5.1) and (5.2).

###### Proof.

For $\gamma\in(0,1)$, fix an element $U_{0}\in\mathcal{C}_{D}^{2\gamma}$ and
set

$\displaystyle\mathcal{C}_{0}^{2\gamma}=\left\\{U\in\mathcal{C}^{2\gamma}\mid
B_{0}^{2\gamma}(U)=0,\text{ and }B_{2[\gamma]}^{2\gamma}(U)=0\right\\}.$

Observe that

$\mathcal{C}_{D}^{2\gamma}=U_{0}+\mathcal{C}_{0}^{2\gamma}$

Let $V\in\mathcal{C}_{0}^{2\gamma}$. It follows from (4.14) and Theorem 4.6
that

$\begin{split}\mathcal{E}_{2\gamma}(V):=&-\int_{\Omega_{n+1}}UL_{2}V\rho^{1-2\gamma}dzdtd\rho,\quad\gamma\in(0,1),\\\
\mathcal{E}_{2\gamma}(V):=&\int_{\Omega_{n+1}}UL_{4}V\rho^{1-2[\gamma]}dzdtd\rho,\quad\gamma\in(1,2),\end{split}$

and that

$\frac{d}{dt}\mathcal{E}_{2\gamma}(U+tV)=2\mathcal{E}_{2\gamma}(V)\geq 0.$

Moreover, by [LY22, Theorem 1.5], equality holds if and only if $V\equiv 0$,
and hence $\mathcal{E}_{2\gamma}$ is strictly convex in
$\mathcal{C}_{D}^{2\gamma}$. Since the solution
$U_{D}\in\mathcal{C}_{D}^{2\gamma}$ of (5.1) and (5.2) is a critical point of
$\mathcal{E}_{2\gamma}:\mathcal{C}_{D}^{2\gamma}\rightarrow\mathbb{R}$, the
result follows. ∎

The following corollary, obtained by evaluating $\mathcal{E}_{2\gamma}(U)$
using Theorem 5.3 gives a sharp Sobolev trace inequality for the corresponding
embedding.

######  Corollary 6.2.

Let $\gamma\in(0,1)$. Then

(6.1) $\mathcal{E}_{2\gamma}(U)\geqslant
2^{1-2\gamma}\frac{\Gamma(1-\gamma)}{\Gamma(\gamma)}\oint_{\mathbb{H}^{n}}\phi
P^{\theta}_{\gamma}\phi dzdt,$

for all $U\in\mathcal{C}_{D}^{2\gamma}$, where $\phi=B_{0}^{2\gamma}(U)$.
Moreover, equality holds if and only if $U$ is the solution of (5.1).

###### Proof.

Set $\mathcal{C}_{D}^{2\gamma}=U+\mathcal{C}_{0}^{2\gamma}$. By Theorem 6.1
there is a unique minimizer $U_{D}$ of
$\mathcal{E}_{2\gamma}:\mathcal{C}_{D}^{2\gamma}\rightarrow\mathbb{R}$. Since
$U_{D}$ satisfies (5.1), we deduce from Theorem 5.3 that

$\displaystyle\mathcal{E}_{2\gamma}\left(U_{D}\right)=2^{1-2\gamma}\frac{\Gamma(1-\gamma)}{\Gamma(\gamma)}\oint_{\mathbb{H}^{n}}\phi
P^{\theta}_{\gamma}\phi dzdt.$

The conclusion readily follows. ∎

Note that Corollary 6.2 recovers the main result in [FdMGMT15]. Similarly, we
have the following corollary.

######  Corollary 6.3.

Let $\gamma\in(1,2)$, $\tilde{\gamma}=1-[\gamma]$. Then

$\displaystyle\mathcal{E}_{2\gamma}(U)\geq
2^{1-2\tilde{\gamma}}\frac{\tilde{\gamma}}{1-\tilde{\gamma}}\frac{\Gamma(-\tilde{\gamma})}{\Gamma(\tilde{\gamma})}\oint_{\mathbb{H}^{n}}\psi
P^{\theta}_{\tilde{\gamma}}\psi
dzdt+2^{3-2\gamma}\frac{\Gamma(2-\gamma)}{\Gamma(\gamma)}\oint_{\mathbb{H}^{n}}\phi
P^{\theta}_{\gamma}\phi dzdt,$

for all $U\in\mathcal{C}_{D}^{2\gamma}$, where

$\psi=B_{2[\gamma]}^{2\gamma}(U),\quad\phi=B_{0}^{2\gamma}(U).$

Moreover, equality holds if and only if $U$ is the solution of (5.2).

Combining Corollary 6.2 and Corollary 6.3 with the sharp Sobolev inequalities
on $\mathbb{H}^{n}$ [FL12] we obtain the following inequality

######  Corollary 6.4.

We denote $C_{n,2\gamma}$ the sharp constant in CR Sobolev inequalities

(6.2) $\left(\int_{\mathbb{H}^{n}}|f|^{p}dzdt\right)^{\frac{2}{p}}\leqslant
C_{n,2\gamma}\int_{\mathbb{H}^{n}}\bar{f}P^{\theta}_{\gamma}(f)dzdt,\quad
p=\frac{2m}{m-\gamma}.$

  1. (1)

Let $\gamma\in(0,1)$. Then

(6.3) $\mathcal{E}_{2\gamma}(U)\geq
2^{1-2\gamma}\frac{\Gamma(1-\gamma)}{\Gamma(\gamma)}C^{-1}_{n,2\gamma}\left(\oint_{\mathbb{H}^{n}}|\phi|^{\frac{2m}{m-\gamma}}dzdt\right)^{\frac{m-\gamma}{m}},$

for all $U\in\mathcal{C}_{D}^{2\gamma}$, where $\phi=B_{0}^{2\gamma}(U)$.
Moreover, equality holds if and only if $U$ is the solution of (5.1) and

$\phi(z,t)=C\left(\frac{1}{(1+|z|^{2})^{2}+t^{2}}\right)^{\frac{m-\gamma}{2}}$

for some $C\in\mathbb{C}$ and up to CR automorphisms on $\mathbb{H}^{n}$.

  2. (2)

Let $\gamma\in(1,2)$, $\tilde{\gamma}=1-[\gamma]$. Then

$\begin{split}\mathcal{E}_{2\gamma}(U)\geq&2^{1-2\tilde{\gamma}}\frac{\tilde{\gamma}}{1-\tilde{\gamma}}\frac{\Gamma(-\tilde{\gamma})}{\Gamma(\tilde{\gamma})}C^{-1}_{n,2\tilde{\gamma}}\left(\oint_{\mathbb{H}^{n}}|\phi|^{\frac{2m}{m-\tilde{\gamma}}}dzdt\right)^{\frac{m-\tilde{\gamma}}{m}}\\\
&+2^{3-2\gamma}\frac{\Gamma(2-\gamma)}{\Gamma(\gamma)}C^{-1}_{n,2\gamma}\left(\oint_{\mathbb{H}^{n}}|\phi|^{\frac{2m}{m-\gamma}}dzdt\right)^{\frac{m-\gamma}{m}},\end{split}$

for all $U\in\mathcal{C}_{D}^{2\gamma}$, where

$\psi=B_{2[\gamma]}^{2\gamma}(U),\quad\phi=B_{0}^{2\gamma}(U).$

Moreover, equality holds if and only if $U$ is the solution of (5.2) and

$\begin{split}\psi(z,t)&=C_{1}\left(\frac{1}{(1+|z|^{2})^{2}+t^{2}}\right)^{\frac{m-\tilde{\gamma}}{2}},\\\
\phi(z,t)&=C_{2}\left(\frac{1}{(1+|z|^{2})^{2}+t^{2}}\right)^{\frac{m-\gamma}{2}}\end{split}$

for some $C_{1},C_{2}\in\mathbb{C}$ and up to CR automorphisms on
$\mathbb{H}^{n}$.

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