text:
\section{Introduction}\label{sec:Intro}

Let $\pi$ be an irreducible representation of a group $G$
 on a vector space $V$,
 and $G'$ a subgroup.
The $G$-module $(\pi,V)$ may be seen as a $G'$-module by restriction,
 for which we write $\pi|_{G'}$.
For an irreducible representation $(\rho,W)$
 of the subgroup $G'$,
 a {\it{symmetry breaking operator}} is
 a (continuous) linear map $V \to W$
 which intertwines $\pi|_{G'}$ and $\rho$.
In recent years {\it{individual}} symmetry breaking operators
 have been studied intensively
 in different settings
 ranging from automorphic form theory
 to conformal geometry,
 see \cite{Clerc, Cohen, FG, Juhl, KKPAdS, KP16b, KS, Z}
 and references therein.

In this article,
 we investigate a {\bf{collection}} of symmetry breaking operators,
 $$R_\ell\, \colon\, V\longrightarrow W_\ell,\,\ell\in\Lambda,
 $$
 referred to as a \emph{symmetry breaking transform},
 for a family of irreducible representations $\rho_\ell$ of the subgroup $G'$
 on vector spaces $W_\ell$ with parameter $\ell\in\Lambda$.

Various expansions in classical analysis
 can be interpreted through this paradigm:

\begin{example}
[$GL_n \downarrow GL_{n-1}$]
Arranging homogeneous polynomials of $x=(x_1,\cdots,x_n)$
 in descending order with respect to the power of $x_n$ is an example of symmetry breaking transform for $(G,G')=(GL_n, GL_{n-1})$. In fact, taking the $\ell$-th component in the expansion
 $$
 f(x)=\sum_{\ell=0}^k f_\ell(x') x^{k-\ell}_n,\quad\mathrm{for}\,\, x'=(x_1,\cdots,x_{n-1})
 $$
 defines a $G'$-homomorphism from
 $V:=\mathrm{Pol}^k[x]$ to $W_\ell:=\mathrm{Pol}^\ell[x']$ on which $G$ and $G'$, respectively,
act irreducibly.
\end{example}

Traditional representation-theoretic viewpoint tells
 that the Fourier series expansion or Fourier transform
 is the irreducible decomposition
 of the regular representation
 of the {\it{abelian}} group $G'=S^1$ or ${\mathbb{R}}$,
 whereas we make use of a \emph{hidden symmetry} of the {\it{noncommutative}} group $G=SL(2,{\mathbb{R}})$ in the sense that $G$ contains $G'$ as a subgroup and that $G$
 acts on the space of functions on $S^1$ or $\R$. The latter viewpoint brings us a new
 interpretation of the (classical) Fourier series or Fourier transform in the framework of
 ``symmetry breaking" as follows.

\begin{example}
\label{ex:FourierSL}
A spherical principal series representation $\pi_{\lambda}$
 of $G=SL(2,{\mathbb{R}})$ is realized
 on the vector space of homogeneous functions
$$
V_\lambda:=\left\{ f \in C^\infty (\R^2\setminus\{(0,0)\}) :
                   f(ax, ay)=|a|^{\lambda}f(x,y) \text { for all } a \in {\mathbb{R}}^{\times} \right\}.
$$
This representation is irreducible for all $\lambda\in\C\setminus\Z$.
\par\noindent
$\bullet$\enspace
{\rm{(Fourier series)}}\enspace
The representation $\pi_{\lambda}$ of $S L(2,{\mathbb{R}})$ can be realized
 in $C^{\infty}(S^1)$
 via the identification $V_{\lambda} \overset \sim \to C^{\infty}(S^1)$,
 $f(x,y) \mapsto h(\theta):=f(\cos \theta, \sin \theta)$,
 because any homogeneous function is determined by its restriction to the unit circle $S^1$.
Since $S^1$ is preserved
 by the subgroup
$G':=SO(2)$,
 the collection of the Fourier coefficients
\begin{eqnarray*}
        V_{\lambda}&\stackrel
{\sim}{\longrightarrow}& C^\infty(S^1)\longrightarrow\operatorname{Map}(\Z,\C),
\\
       f \mapsto h&\mapsto& \widehat h(\ell):=\frac1{2\pi}\int_0^{2\pi} h(\theta) e^{-i\ell\theta} d\theta, \quad\ell\in\Z
\end{eqnarray*}
gives a symmetry breaking transform from the infinite-dimensional representation
 $(\pi_{\lambda}, V_{\lambda})$
 of $G=S L(2,{\mathbb{R}})$
 to the collection of one-dimensional representations $\chi_\ell$
 of the abelian subgroup $G'=SO(2) \simeq S^1$
 indexed by $\ell \in {\mathbb{Z}}$.

\par\noindent
$\bullet$\enspace
{\rm{(Fourier transform)}}\enspace
Similarly, any function $f(x,y)\in V_\lambda$ is determined by its restriction
to the real line $y=1$,
which is preserved by the unipotent subgroup
$
   G'':=\left\{\begin{pmatrix}1&\xi\\0&1\end{pmatrix} : \xi\in\R\right\}\,(\simeq\R)
$.
Thus the Fourier transform
$$
  L^1(\R)\longrightarrow C(\R),
\quad
  F \mapsto (\mathcal F F)(\xi):=\int_\R F(x) e^{-i x \xi}d x,
$$
induces another symmetry breaking transform
 for the pair $(G, G'')=(SL(2,\R), \R)$.
\end{example}

\vskip10pt
\begin{example}
[spherical harmonics]
Expansion of functions on $S^n$
 by eigenfunctions of the Laplacian $\Delta_{S^n}$
 corresponds to a symmetry breaking transform
{}from a spherical principal series representation $\pi$
 of $G=SO(n+1,1)$
 to a collection
 of irreducible finite-dimensional representations
 of the compact subgroup $G'=O(n+1)$.
\end{example}

Reversing the arrows in the definition of a symmetry breaking operator
$R_\ell \colon V \to W_{\ell}$,
 we consider a $G'$-homomorphism $\Psi_\ell \colon W_{\ell} \to V$,
 going from smaller to larger representation space, and thus
 referred to as a \emph{holographic operator}.
As in the case of symmetry breaking,
 the collection of holographic operators $\{\Psi_\ell\}$
 is said to be a \emph{holographic transform.}
$$
 G \curvearrowright
       \xymatrix
       {
          V\ar@<0.5ex>[r]^{R_\ell}&
            W_\ell
              \ar@<0.5ex>[l]^{\Psi_\ell}
       }
  \curvearrowleft G'.
$$
To illustrate a holographic transform
 by an example
 with both $V$ and $W_{\ell}$ being infinite-dimensional,
 we recall that the classical Poisson integral (see \emph{e.g.} \cite[Sec. 0]{KKMOOT})
\begin{eqnarray*}
\mathcal P_\nu\colon C_c(\R)&\longrightarrow& C^\infty(\Pi)
\\
h(t) &\mapsto&
(\mathcal P_\nu h)(x, y) = \int_{-\infty}^{\infty} \frac{y^\nu}{((x-t)^2+y^2)^{\nu}} h(t) d t
\end{eqnarray*}
constructs eigenfunctions of the Laplace--Beltrami differential operator $\Delta= y^2\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)$ for the eigenvalue $\nu(\nu-2)$ on the upper-half plane $\Pi$
 endowed with Poincar\'e metric.
The group $S L(2,{\mathbb{R}})$ acts isometrically on $\Pi$
 and conformally on its boundary.
Traditionally,
 the Poisson integral was treated
 in the context of representations
 of $S L(2,{\mathbb{R}})$,
 however,
 we highlight the fact
 that the totality of functions on $\Pi$ admits a larger symmetry
 because the group $S L(2,{\mathbb{C}})$
 acts on the conformal compactification of $\Pi$.
Thus the Poisson integral can be interpreted as a particular occurrence
 of a holographic operator
 for the pair $(G,G')=(SL(2,\C), SL(2,\R))$
 as below.

\begin{example}[Poisson integral]
\label{ex:Poisson}
A generic symmetry breaking operator $A_{\lambda,\nu}$ from the spherical principal series representation $\pi_\lambda$
 of $G=S L(2,{\mathbb{C}})$ on $C^\infty(S^2)$
 to the one $\varpi_\nu$ of the subgroup $G'=S L(2,{\mathbb{R}})$ on $C^\infty(S^1)$ takes the following form $($see \cite[(7.2)]{KS}$)$:
\begin{eqnarray*}
A_{\lambda,\nu}:
C_c^\infty(\R^2)&\longrightarrow& C^\infty(\R),
\\
f(x,y) &\mapsto& (A_{\lambda,\nu} f)(x)=\int_{\R^2} f(t,y) K_{\lambda,\nu}(x-t,y)dtdy
\end{eqnarray*}
in the flat coordinates
 where $K_{\lambda,\nu}$ is a distributional kernel
 given by
\begin{equation}
\label{eqn:Klmdnu}
K_{\lambda,\nu}(x,y)=(x^2+y^2)^{-\nu}\vert y\vert^{\lambda+\nu-2}.
\end{equation}
Then the dual map of $A_{\lambda,\nu}$ yields a holographic operator
 $\Psi_{\lambda,\nu}$ with the formula
$$
g(t) \mapsto \left(\Psi_{\lambda,\nu}g\right)(x,y):=\int_{\R}g(t) K_{\lambda,\nu}(x-t,y)dt.
$$
Thus
 the (classical) Poisson integral $\mathcal P_{\nu}$ can be viewed as the restriction
 of the holographic operator $\Psi_{\lambda,\nu}$
 with $\lambda=2$,
 namely,  {defn}
$
\mathcal P_\nu=\mathrm{Rest}_\Pi\circ\Psi_{2,\nu}.
$
\end{example}

With these interpretations of classical examples in mind,
 we raise the following two general problems
 for a symmetry breaking transform
$\displaystyle R(v)=\{R_\ell(v)\}_{\ell\in \Lambda},$
where $R_\ell
 \colon V\longrightarrow W_{\ell}$ ($\ell\in\Lambda$),
are symmetry breaking operators:
\begin{problem}
Can we recover an element $v$ of $V$ from its symmetry breaking transform
$R(v)=\{R_\ell(v)\}_{\ell\in\Lambda}$?
\end{problem}

Problem A includes the following subproblems:
\begin{itemize}
   \item[{\bf A.0.}] Tell \emph{a priori} if $\Lambda$ is sufficiently large for $R$ to be injective.
      \item[{\bf A.1.}] Construct a \lq\lq{holographic transform}\rq\rq.
   \item[{\bf A.2.}] Find an explicit inversion of the symmetry breaking transform $R$.
\end{itemize}

When $V$ is a Hilbert space on which $G$ acts unitarily,
 we also ask for a Parseval--Plancherel type theorem
 for the symmetry breaking transform:

\begin{problem}\label{prob:B}
  Find a closed formula for the norm of an element $v$ in $V$ in terms of its symmetry breaking transform  $\{R_\ell(v)\}_{\ell\in\Lambda}$.
\end{problem}

In this article, we investigate Problems A and B in the following two cases:
\begin{itemize}
  \item Rankin--Cohen transform (Section \ref{sec:RCT});
  \item Holomorphic Juhl transform (Section \ref{sec:HJT}).
\end{itemize}
In both cases,
 the transform is a collection of holomorphic differential operators
 between complex manifolds:
 the first case is associated with the family of
 the Rankin--Cohen operators that appeared in the theory of holomorphic modular forms \cite{Cohen},
 whereas the second case originated from Juhl's conformally covariant operators \cite{Juhl}.

These transforms can be analyzed in the framework of infinite dimensional representations of Lie groups, namely, the decomposition of the tensor product of two holomorphic discrete series
representations of $SL(2,\R)$ in the first case, and the branching laws of holomorphic discrete series
representations of the conformal Lie group $G=SO_o(2,n)$
 when restricted to a subgroup $G'=SO_o(2,n-1)$, in the second case.

The main goal here is to
give a solution
 to Problems A and \ref{prob:B} for the above two transforms.
We provide two types of integral expressions
 as a solution to Problem A1,
 see Theorems \ref{thm:psi} and \ref{thm:CIRM18}.
The main results are summarized as below.

{\small\begin{figure}[!h]
   \begin{center}
    \begin{tabular}{c|c|c}
       $G\supset G'$ &$SL_2\times SL_2\supset SL_2$  &$SO_o(2,n)\supset SO_o(2,n-1)$ \\
      &&\\
               \hline
         Problem A1 & &\\
         construction of & Theorem \ref{thm:psi}& Theorem \ref{thm:CIRM18}\\
         holographic transform &&\\
         \hline
         Problem A2 &&\\
         inversion of symmetry & Theorem \ref{thm:RCinv} &Theorem \ref{thm:170887}\\
         breaking transform&&\\
         \hline
         Problem B &&\\
         $L^2$-theory for &Theorem \ref{thm:PPTT4RCB}&Theorem \ref{thm:170887}\\
            &&
        \end{tabular}
   \end{center}
  \end{figure}%
}

The key idea of our approach is to
introduce \lq\lq{special orthogonal polynomials}\rq\rq\ $\{P_\ell\}$ associated to
symmetry breaking operators. This can be done
  via the F-method, which we developed in \cite{KP16a,KP16b},
 that analyzes the representations through the Fourier transform
 of their geometric realizations.
 In this article, we show for
 the Rankin--Cohen bidifferential operators
 $\{R_\ell\}$
that the polynomials $\{P_\ell\}$ are the Jacobi polynomials and that
 the holographic operators are given by the Jacobi transforms
 along the transversal direction to a codimension-one foliation of the symmetric cone
 (Section \ref{sec:RCT});
for the holomorphic Juhl operators $\{R_\ell\}$, the holographic operators are associated to the Gegenbauer polynomials $\{P_\ell\}$
(Section \ref{sec:HJT}).
Thus Problems A and \ref{prob:B}
 for symmetry breaking transforms
 can be studied
 as questions on special orthogonal polynomials via the F-method.

The table below shows some new links
 which the F-method provides
 between representations
 and special functions
 in this setting.

{\small
{\begin{figure}[!h]
   \begin{center}
    \begin{tabular}{c|c}
    Symmetry breaking operators $\{R_\ell\}$
& Special orthogonal polynomials $\{P_\ell\}$
\\
&\\
\hline
&\\
   $G'$-intertwining property
&  hypergeometric differential equations
\\
&\\
\hline
&\\
  operator norm of  $R_\ell$
&
 $L^2$-norm of $P_\ell$
\\
&\\
\hline
&\\
branching law $\pi|_{G'}$
&
$L^2$-completeness of $\{P_\ell\}$
\\
&\\
\hline
&\\
holographic transform
($L^2$-model)
&
integral transform
associated to $\{P_\ell\}$
\\

\end{tabular}
   \end{center}
  \end{figure}%
}
}
\vskip7pt

Analogously to the classical Poisson transform
 (Example \ref{ex:Poisson}),
 the holographic transform
 provides an integral expression
 of eigenfunctions
 of certain holomorphic differential operator.
We illustrate this idea with the example of the Rankin--Cohen operators, see Theorem \ref{thm:180297} which is proved as a byproduct of the main results.

In Section \ref{sec:conclusion} we discuss the background of Problems A and \ref{prob:B} from a viewpoint of the representation theory of real reductive Lie groups.

\vskip 10pt
Notation: $\N=\{0,1,2,\cdots\}$, $i=\sqrt{-1}$ (imaginary unit), $(x)_k=x(x+1)(x+2)\cdots(x+k-1)$ for $k\in\N$ (Pochhammer symbol), and $[x]$ is the largest integer that does not exceed $x\in\R$.

\vskip1pc

\emph{Acknowledgments}. The first author was partially supported by the JSPS under the
Grant-in-Aid for Scientific Research (A) (JP18H03669).
Both authors were partially supported by the CNRS grant PICS-7270 and
they
are grateful to Institut des Hautes \'Etudes Scientifiques
(Bures-sur-Yvette, France), Institut Henri Poincar\'e (Paris,
France)
and Centre International de Rencontres Math\'ematiques (Luminy,
France) where an important part of this work was done.

\vskip10pt

\section{Rankin--Cohen transform and its holographic transform}\label{sec:RCT}

The Rankin--Cohen bidifferential operators map functions of two variables to those of one variable,
respecting twisted actions of $SL(2,\R)$. In this section, we solve Problems A and B stated in Section \ref{sec:Intro} for the \emph{Rankin--Cohen transform} (Definition
\ref{def:RCT}), a collection of such operators.

\subsection{Rankin--Cohen bidifferential operators}
We begin with a quick review of the Rankin--Cohen bidifferential operators.

\subsubsection{Holomorphic discrete series representations of ${SL(2,\R)}{\;}\widetilde{}$}\label{subsec:pilmd}
Let $\Pi=\{z=x+iy\in\C:\; x\in\R,\;y>0\}$ be the upper half-plane, and $\mathcal O(\Pi)$ the space
 of holomorphic functions on $\Pi$.
For $\lambda\in\Z$
 we define a representation $\pi_\lambda$ of ${SL(2,\R)}$  on $\mathcal O(\Pi)$ by
$$
\pi_\lambda(g)f(z)=(cz+d)^{-\lambda}f\left(\frac{az+b}{cz+d}\right)
\qquad\mathrm{for}\quad
g^{-1}=\left(
\begin{array}{cc}
  a   & b  \\
  c   & d
\end{array}
\right).
$$
Viewed as a representation of the universal covering group
 ${SL(2,\R)}{\;}\widetilde{}$,
 the representation $\pi_{\lambda}$ is well-defined
 for all $\lambda \in {\mathbb{C}}$.
There is a canonical perfect pairing between
$(\pi_\lambda,\mathcal O(\Pi))$ and the Verma module
$$
M_{-\lambda}:= U(\mathfrak g_\C)\otimes_{U(\mathfrak b)}\C_{-\lambda},
$$
where $U(\mathfrak g_\C)$ denotes the universal enveloping algebra of
$\mathfrak g_\C=\mathfrak{sl}(2,\C)$ and $\mathfrak b$ is a Borel subalgebra
 containing
$\mathfrak k_\C=\mathfrak{so}(2,\C)$.
Therefore,
 $(\pi_\lambda,\mathcal O(\Pi))$ is irreducible
 if and only if $\lambda \in {\mathbb{C}} \setminus (-{\mathbb{N}})$ because
 the $\mathfrak g$-module $M_\nu$ is reducible if and only if $\nu\in\N$.

Let $p \colon {SL(2,{\mathbb{R}}}){\;}\widetilde{} \to SL(2,{\mathbb{R}})$
 be the covering homomorphism,
 and set ${SO(2)}{\;}\widetilde{}=p^{-1}(SO(2))$.
For every $\lambda\in\C$, we can form a homogeneous holomorphic line bundle $\mathcal L_\lambda$ over $\Pi\simeq
{SL(2,\R)}{\;}\widetilde{}/{SO(2)}{\;}\widetilde{}$ associated  to a character
$\C_\lambda$ of ${SO(2)}{\;}\widetilde{}$, and the multiplier representation $(\pi_\lambda, \mathcal O(\Pi))$ is equivalent to the natural action
 of ${SL(2,\R)}{\;}\widetilde{}\;$
 on the space $\mathcal O(\Pi, \mathcal L_\lambda)$ of holomorphic sections of  $\mathcal L_\lambda$.

\subsubsection{Holomorphic model ${\mathcal{H}}^2(\Pi)_{\lambda}$}
For $\lambda>1$ the weighted Bergman space $\mathcal H^2(\Pi)_\lambda:=(\mathcal O\cap L^2)(\Pi, y^{\lambda-2}d x d y)$ is nonzero, and the Hilbert space $\mathcal H^2(\Pi)_\lambda$ admits a reproducing kernel
$K_\lambda(z,w)=\frac{\lambda-1}{4\pi}\left(\frac{z-\bar w}{2i}\right)^{-\lambda}$, see \cite[Prop. XIII.1.2]{FK}.
The representation
$(\pi_\lambda,\mathcal O(\Pi))$ yields an irreducible unitary representation of ${SL(2,\R)}{\;}\widetilde{}{\;}$ on
$\mathcal H^2(\Pi)_\lambda$,
which descends to $SL(2,\R)$ when $\lambda\in\Z$. The set of equivalence classes of
irreducible unitary representations (\emph{unitary dual}) of $SL(2,\R)$ contains a family of those with continuous parameter (\emph{e.g.} principal series representations, complementary series representations), whereas $\pi_\lambda$ ($\lambda= 2,3,\cdots$)
form a countable family of irreducible unitary representations realized in the kernel of the Cauchy--Riemann operator. Thus
$\pi_\lambda$ ($\lambda= 2,3,\cdots$)
is referred to as a \emph{holomorphic discrete series representations} of $SL(2,\R)$,
and $\pi_\lambda$ ($\lambda>1$)
  as a relative holomorphic discrete series representation of the covering group
 ${SL(2,\R)}{\;}\widetilde{}$.
We call the realization on $\mathcal H^2(\Pi)_{\lambda}$ \emph{holomorphic model} of the representation $\pi_\lambda$.
Similarly, the direct product group ${SL(2,\R)}{\;}\widetilde{}\times
{SL(2,\R)}{\;}\widetilde{}\;$ acts on $\mathcal H^2(\Pi\times\Pi)_{(\lambda',\lambda'')}\simeq
\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}$ as an irreducible unitary representation if $\lambda',\lambda''>1$, where $\widehat\otimes$ stands for the completion of the algebraic tensor product.

We shall deal with another realization ($L^2$-model) of the same representation $\pi_\lambda$
in Section \ref{subsec:phi}.
$$
$$
\subsubsection{Rankin--Cohen bidifferential operators}
~~~\par
Consider $\lambda',\lambda'',\lambda'''\in\C$ such that  $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$ and
define a differential operator
$\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}:\mathcal O(\Pi\times\Pi)\longrightarrow \mathcal O(\Pi\times\Pi)$ by
\begin{equation}\label{rcb}
\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}(f)(\zeta_1,\zeta_2):=
\sum_{j=0}^\ell (-1)^{j}
\frac{
\left(\lambda'+\ell-j\right)_j \left(\lambda''+j \right)_{\ell-j}}
{j!(\ell-j)!}
\frac{\partial^\ell f}{\partial \zeta_1^{\ell-j}\partial \zeta_2^{j}}(\zeta_1,\zeta_2).
\end{equation}

 The {\it{Rankin--Cohen bidifferential operator}} is a linear map
$$
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}: \mathcal O(\Pi\times
\Pi)\longrightarrow \mathcal O(\Pi),
$$
defined by $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}:=
\mathrm{Rest}\circ \mathcal{R}_{\lambda',\lambda''}^{\lambda'''}$,
 where
 $\mathrm{Rest}$ stands for the restriction map $f(\zeta_1,\zeta_2)\mapsto f(\zeta,\zeta)$ to the diagonal.

The Rankin--Cohen bidifferential operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
 is a symmetry breaking operator from the tensor product representation $\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}$ to
$\pi_{\lambda'''}$ with respect to the diagonal embedding
${SL(2,\R)}{\;}\widetilde{}\hookrightarrow {SL(2,\R)}{\;}\widetilde{}\times
{SL(2,\R)}{\;}\widetilde{},$ and such a symmetry breaking operator
is unique up to scalar multiplication for generic parameters (see \cite[Cor. 9.3]{KP16b} for the precise condition).
Moreover,
 $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ induces a continuous map from the weighted Bergman space
  ${\mathcal{H}}^2(\Pi \times \Pi)_{(\lambda',\lambda'')}$
 to ${\mathcal{H}}^2(\Pi)_{\lambda'''}$
 if $\lambda', \lambda'' >1$
 (\cite[Thm.~5.13]{KP16a}, see also Proposition \ref{prop:RCopnorm} below
 for an explicit formula of its operator norm).

\subsection{Notations and two constants $c_\ell(\lambda',\lambda'')$ and $r_\ell(\lambda',\lambda'')$}\label{sec:constants}
The parameter set in Section \ref{sec:RCT} is $(\lambda',\lambda'',\lambda''')\in\C^3$ with
$\lambda'''-\lambda'-\lambda''\in2\N$. Throughout this section, we use the following notation:
\begin{equation}\label{eqn:abl}
\alpha=\lambda'-1,\quad \beta=\lambda''-1,\quad 2\ell=\lambda'''-\lambda'-\lambda''.
\end{equation}
The main results involve the following two constants
\begin{eqnarray}\label{eqn:v-ell}
c\equiv c_\ell(\lambda',\lambda'') &:=& \nonumber
\frac1{2^{\alpha+\beta+1}}\int_{-1}^1\vert P_\ell^{\alpha,\beta}(v)\vert^2 (1-v)^\alpha (1+v)^\beta dv\\
&=&
\frac{\Gamma(\lambda'+\ell) \Gamma(\lambda''+\ell)}{(\lambda'+\lambda''+2\ell-1) \Gamma(\lambda'+\lambda''+\ell-1)\ell!},
\\
\label{eqn:b-ell}
r\equiv r_\ell(\lambda',\lambda'') &:=& \nonumber
\frac{b(\lambda''')}{b(\lambda')b(\lambda'')}\\
&=&
\frac{\Gamma(\lambda'+\lambda''+2\ell-1)}{2^{2\ell+2}\pi \Gamma(\lambda'-1)\Gamma(\lambda''-1)},
\end{eqnarray}
where $P_\ell^{\alpha,\beta}(v)$ is the Jacobi polynomial (see \eqref{eqn:Pnorm} in Appendix), and
$b(\lambda)=2^{2-\lambda}\pi\Gamma(\lambda-1)$ is a Plancherel density (see Fact \ref{fact:Laplace} below).
We note that $c_{\ell}(\lambda',\lambda'') \ne 0$
 if $\operatorname{Re}\lambda',\operatorname{Re}\lambda'' >0$
 and $\ell \in {\mathbb{N}}$.

\subsection{Integral formula for holographic operators}
~~~

In this section,
 we introduce integral transforms $\Psi_{\lambda',\lambda''}^{\lambda'''}$
 ({\it{holographic operator}})
 that realize  irreducible summands in the tensor
product representations $\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}$.

\subsubsection{Construction of holographic operators for the tensor product}
~~~\par
\begin{defn}[holographic operators]\label{def:HOtensor}
For $\lambda',\lambda'',\lambda'''\in\C$ we set  $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')$. Assume that
\begin{equation}\label{eqn:HOconverge}
\mathrm{Re}(\lambda'+\ell)>0,\quad\mathrm{Re}(\lambda''+\ell)>0,\quad \mathrm{and}\quad\ell\in\N.
\end{equation}
 For a holomorphic function $g$ on the upper half plane $\Pi$, we define a holomorphic function on $\Pi\times\Pi$ by
 the line integral:
\begin{align}\label{eqn:psi}
&\left(\Psi_{\lambda',\lambda''}^{\lambda'''} g\right)(\zeta_1,\zeta_2):=
\frac{(\zeta_1-\zeta_2)^\ell}{2^{\lambda'+\lambda''+2\ell-1}\ell!}\times\\
&\int_{-1}^1
g\left(\frac{(\zeta_2-\zeta_1)v+(\zeta_1+\zeta_2)}{2}\right)
(1-v)^{\lambda'+\ell-1}(1+v)^{\lambda''+\ell-1}
dv.\nonumber
\end{align}

\end{defn}
We note that the set $\left\{\frac{(\zeta_2-\zeta_1)v+(\zeta_1+\zeta_2)}{2} : -1\leq v\leq 1\right\}$ is
the line segment connecting the two points $\zeta_1$ and $\zeta_2$ in $\Pi$.

\subsubsection{Basic properties of $\Psi_{\lambda',\lambda''}^{\lambda'''}$}

The integral transform $\Psi_{\lambda',\lambda''}^{\lambda'''}$
in \eqref{eqn:psi} provides
 a holographic operator in the following sense:

\begin{thm}[holographic operator in the upper half plane]
\label{thm:psi}
Suppose ${\lambda',\lambda''},{\lambda'''}\in\C$ satisfy \eqref{eqn:HOconverge}.
\begin{itemize}
\item[(1)] The map
$\Psi_{\lambda',\lambda''}^{\lambda'''}:\mathcal O(\Pi)\longrightarrow
\mathcal O(\Pi\times\Pi)$
intertwines the action of ${SL(2,\R)}{\;}\widetilde{}$\,\, from $\pi_{\lambda'''}$ to the tensor product representation
$\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$.

\item[(2)] Moreover, if
both $\lambda'$ and $\lambda''$ are real
 and greater than 1,
 then the linear map $\Psi_{\lambda',\lambda''}^{\lambda'''}$ induces an isometric embedding
 (up to rescaling)
 of the weighted Bergman space:
$$
\mathcal H^2(\Pi)_{\lambda'''}\longrightarrow\mathcal H^2(\Pi\times\Pi)_{(\lambda',\lambda'')}.
$$
\end{itemize}
\end{thm}
The image of the holographic operator $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is characterized by a
differential equation of second order on $\Pi\times\Pi$
associated to the Casimir element under the diagonal action,
see Theorem \ref{thm:180297}.
For ${\lambda',\lambda''}>1$,
the operator
 $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is a scalar multiple
 of the adjoint $\left(\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\right)^*$
 of the Rankin--Cohen bidifferential operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
 (see Proposition \ref{prop:PsiRC}), and its operator norm
 is given in Theorem \ref{thm:PPTT4RCB} (2).

Theorem \ref{thm:psi} will be proved in Section \ref{subsec:pfthmpsi}.

\begin{rem}\label{rem:2integral}
In Section \ref{sec:HJT}, we introduce {\it{relative}} reproducing kernels to construct irreducible summands in the holomorphic model. The integral formula given there (see
Theorem \ref{thm:CIRM18}) is different from the one introduced in Definition \ref{def:HOtensor}.
The advantage of the definition \eqref{eqn:psi} is
 that the holographic operator $\Psi_{\lambda',\lambda''}^{\lambda'''}$
 is defined also for the nonunitary case, see Theorem \ref{thm:psi} (1).
\end{rem}

\subsection{The Rankin--Cohen transform and its inversion}

In this section we introduce the Rankin--Cohen transform $\mathcal{RC}_{\lambda',\lambda''}$ as the collection of individual operators
$\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ for fixed $\lambda'$ and $\lambda''$. Its inversion formula
is proved in Theorem \ref{thm:RCinv} by using the holographic operators,  giving a solution to Problem A
in Section 1.

\begin{defn}[Rankin--Cohen transform]\label{def:RCT}
For $\lambda',\lambda''\in\C$, the \emph{Rankin--Cohen transform} $\mathcal{RC}_{\lambda',\lambda''}$ is
a linear map
\begin{equation}\label{eqn:RCtrans}
\mathcal{RC}_{\lambda',\lambda''}\colon \mathcal O(\Pi\times\Pi)\longrightarrow\operatorname{Map}(\mathbb N, \mathcal O(\Pi)),
\quad
f \mapsto \left(\ell\mapsto\mathcal{RC}_{\lambda',\lambda''}(f)_{\ell}\right)
\end{equation}
defined by
$
\left(\mathcal{RC}_{\lambda',\lambda''}(f)\right)_\ell:=\mathcal{RC}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell} f\qquad\mathrm{for}\;\ell\in\N.
$
\end{defn}

The Rankin--Cohen transform $\mathcal{RC}_{\lambda',\lambda''}$ intertwines $(\pi_{\lambda'}\widehat\otimes\pi_{\lambda''},\mathcal O(\Pi\times\Pi))$ with the formal direct
sum
$\widehat\bigoplus_{\ell\in\N}(\pi_{\lambda'+\lambda''+2\ell},\mathcal O(\Pi))$, and
can be inverted by using the integral operators $\Psi_{\lambda',\lambda''}^{\lambda'''}$ as follows.

\begin{thm}[inversion of the Rankin--Cohen transform]\label{thm:RCinv}
Suppose $\lambda',\lambda''>1.$ Then for any $f\in \mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}$ one has
$$
f=\sum_{\ell=0}^\infty \frac1{c_\ell(\lambda',\lambda'')}\,
\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}\left(\mathcal{RC}_{\lambda',\lambda''}(f)\right)_\ell.
$$
\end{thm}
Theorem \ref{thm:RCinv} will be proved in Section \ref{subsec:pfthmRCinv}.

\subsection{Parseval--Plancherel type theorem for the Rankin--Cohen transform and its holographic transform}

In this section we develop an $L^2$-theory for the Rankin--Cohen transform (Definition \ref{def:RCT}) and for the
holographic transform (Theorem \ref{thm:PPTT4RCB} (2)), thus providing an answer to Problem B for these two transforms.

\subsubsection{Weighted Hilbert sums}

In order to formulate the Parseval--Plancherel type theorem, we fix some notations for the Hilbert direct sum.
\begin{defn}
[weighted Hilbert sum]
\label{def:compsum}
Let$\{V_\ell\}_{\ell\in\N}$ be a family of Hilbert spaces and $\{a_\ell\}_{\ell\in\N}$ a sequence  of positive numbers.
The Hilbert sum $\displaystyle{\sum\limits_{\ell\in\N}}^\oplus V_\ell$  associated to the {\it{weights}} $\{a_\ell\}_{\ell\in\N}$ is the Hilbert
completion of the
algebraic direct sum $\displaystyle\bigoplus\limits_{\ell\in\N}V_\ell$ equipped with the pre-Hilbert structure given by
$$
(v,v'):=\sum_{\ell=0}^\infty a_\ell(v_\ell,v'_\ell)_{V_\ell}
\quad
\text{for $v=(v_\ell)_{\ell\in\N}$ and $v'=(v'_\ell)_{\ell\in\N}$}.
$$
\end{defn}

\subsubsection{Parseval--Plancherel type theorem}\label{subsec:PP-RC}

For
$\lambda',\lambda''>1$ the bidifferential
operators $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
extend to a continuous map between Hilbert spaces.
Now, we formulate a Parseval--Plancherel type theorem for the Rankin--Cohen transform as well as the ``holographic transform'', hence answer Problem B for these transforms.

\begin{thm}[Parseval--Plancherel theorem]\label{thm:PPTT4RCB}

Suppose $\lambda',\lambda''>1$.
\begin{itemize}
\item[(1)]
The Rankin--Cohen transform $\mathcal{RC}_{\lambda',\lambda''}$ (Definition \ref{def:RCT})
induces an ${SL(2,\R)}{\;}\widetilde{}$-equivariant unitary operator
$$
\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}  \stackrel
{\sim}{\longrightarrow}
{\sum_{\ell\in\N}}^\oplus  \mathcal H^2(\Pi)_{\lambda'+\lambda''+2\ell}
$$
to the Hilbert sum associated to weights $\left
\{
\frac1{r_\ell(\lambda',\lambda'')c_\ell(\lambda',\lambda'')}\right\}_{\ell\in\N}$.
 Thus, for every $f\in \mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}$,
\begin{eqnarray*}
&&\Vert f\Vert_{\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}}^2\\
&=&
\sum_{\ell=0}^\infty \frac1{r_\ell(\lambda',\lambda'')c_\ell(\lambda',\lambda'')}
\Vert \left(\mathcal{RC}_{\lambda',\lambda''}(f)\right)_\ell\Vert_{\mathcal H^2(\Pi)_{\lambda'+\lambda''+2\ell}}^2.
\end{eqnarray*}
\item[(2)]
Collecting the holographic operators
$\Psi_{\lambda',\lambda''}^{\lambda'''}$, we define
the holographic
 transform
 $$
 \Psi_{\lambda',\lambda''}:\bigoplus_{\ell\in\N}
 \mathcal H^2(\Pi)_{\lambda'+\lambda''+2\ell}{\longrightarrow}
\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}
$$
by
 $$
 \Psi_{\lambda',\lambda''}:=\bigoplus_{\ell=0}^\infty \Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}.
 $$
  Then $\Psi_{\lambda',\lambda''}$
induces an ${SL(2,\R)}{\;}\widetilde{}$-equivariant unitary operator
$$
\sum_{\ell=0}^\infty{}^{\oplus} \mathcal H^2(\Pi)_{\lambda'+\lambda''+2\ell}\stackrel
{\sim}{\longrightarrow}
\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}
$$
from the Hilbert sum associated to the weights $\left\{\frac{c_\ell(\lambda',\lambda'')}{r_\ell(\lambda',\lambda'')}\right\}_{\ell\in\N}$.
Thus,
$$
\Vert \Psi_{\lambda',\lambda''} g\Vert^2_{\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}}=
\sum_{\ell=0}^\infty
\frac{c_\ell(\lambda',\lambda'')}{r_\ell(\lambda',\lambda'')}\Vert g_\ell\Vert^2_{\mathcal H^2(\Pi)_{\lambda'+\lambda''+2\ell}}
$$
for $g=(g_\ell)_{\ell\in\N}$.
\end{itemize}
\end{thm}

Theorem \ref{thm:PPTT4RCB} will be proved in Section \ref{subsec:pfthmPPTT4RCB}. It
gives quantitative information on the classical
branching law (\emph{fusion rule})
of the tensor product of two holomorphic discrete series representations $\pi_{\lambda'}$ and $\pi_{\lambda''}$
that decomposes  into a multiplicity-free direct Hilbert sum of irreducible unitary representations
 when $\lambda',\lambda''>1$ \cite{Mol,Repka}:
\begin{equation}\label{eqn:absbra}
\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}
\simeq
{\sum_{\ell\in\N}}^{\oplus}\pi_{\lambda'+\lambda''+2\ell}.
\end{equation}

The projection to each irreducible summand in the decomposition \eqref{eqn:absbra} is given as the composition of
the corresponding Rankin--Cohen operator and the holographic operator
in the holomorphic model.
Thus Theorem \ref{thm:RCinv} (and Proposition
\ref{prop:PsiRC} below) shows the following corollary.
\begin{cor}
[projection operator]
\label{cor:20181103}
Suppose $\lambda', \lambda'', \lambda'''>1$
 and $\ell :=\frac 1 2(\lambda'''- \lambda'- \lambda'') \in {\mathbb{N}}$.
Then
\[\frac{1}{c_{\ell}(\lambda', \lambda'')}
\Psi_{\lambda',\lambda''}^{\lambda'''}
\circ
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}
=
\frac{1}{r_{\ell}(\lambda', \lambda'')c_{\ell}(\lambda', \lambda'')}
(\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''})^{\ast}
\circ
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}
\]
is the projection operator of the Hilbert space
${\mathcal{H}}^2(\Pi)_{\lambda'}
\widehat{\otimes}
{\mathcal{H}}^2(\Pi)_{\lambda''}$
onto the irreducible summand
 which is isomorphic to
${\mathcal{H}}^2(\Pi)_{\lambda'''}$, see \eqref{eqn:absbra}.
\end{cor}

\subsection{Holographic transform in the $L^2$-model}

By the Fourier--Laplace transform, the weighted Bergman space $\mathcal H^2(\Pi)_\lambda$ realized in the space
of holomorphic functions on the upper half plane $\Pi$ is mapped into the space of functions supported on the
positive axis $\R_+$, more precisely, onto the Hilbert space $L^2(\R_+, x^{1-\lambda}d x)$, giving thus rise to an
$L^2$-\emph{model} of the same representation of ${SL(2,\R)}{\;}\widetilde{}$ (Fact \ref{fact:Laplace}).
We shall find closed formul{\ae} for
the symmetry breaking transform and the holographic transform also in this model and give an answer to Problems A and B,
see Theorems \ref{thm:phi}, \ref{thm:FRCinv} and \ref{thm:CL2SL2}. The results in the $L^2$-model
give a new interpretation of the classical theory of the Jacobi transform,
 and also play a key role in proving the theorems for the holomorphic model, see Section \ref{subsec:proofs}.

\subsubsection{$L^2$-model of holomorphic discrete series}

 For $\lambda>1$ we consider the Hilbert space $L^2(\R_+)_\lambda:=L^2(\R_+,x^{1-\lambda}d x)$.

\begin{fact}\label{fact:Laplace}
Suppose $\lambda>1$.
The Fourier--Laplace transform
$$
\mathcal F\colon F\mapsto \mathcal F F(\zeta):= \int_0^\infty F(z)e^{i\zeta z} dz,
$$
is an isometry from $L^2(\R_+)_\lambda$ onto the weighted Bergman space $\mathcal H^2(\Pi)_\lambda$ up to scalar multiplication.
To be precise, we have
 $$
\Vert\mathcal FF\Vert^2_{\mathcal H^2(\Pi)_\lambda}=b(\lambda)\Vert F\Vert^2_{L^2(\R_+)_\lambda}
$$
for all $F\in L^2(\R_+)_\lambda$ $($see \emph{e.g.} \cite[Thm.~XIII.1.1]{FK}$)$, where
$$
b(\lambda):=2^{2-\lambda}\pi\Gamma(\lambda-1).
$$
\end{fact}

For $\lambda>1$, via the unitary (up to scaling) map $\mathcal F \colon L^2(\R_+)_\lambda \stackrel
{\sim}{\longrightarrow}
\mathcal H^2(\Pi)_\lambda$, we define an irreducible unitary representation of
${SL(2,\R)}{\;}\widetilde{}$ on $L^2(\R_+)_\lambda$, which
 is referred to as the \emph {$L^2$-model} of the holomorphic discrete series representation $\pi_\lambda$.

 We shall write
 $$\mathcal F_1\equiv\mathcal F\quad \rm{and}\quad\mathcal F_2:=\mathcal F
\otimes \mathcal F
$$
in order to distinguish the framework  of functions of one or two variables, respectively,
and we write, by abuse of notations,
\begin{equation}\label{eqn:L2two}
L^2(\R_+^2)_{\lambda',\lambda''}:
=L^2(\R_+\times\R_+,x^{1-\lambda'}y^{1-\lambda''}dx\;dy)\simeq
L^2(\R_+)_{\lambda'}\widehat\otimes L^2(\R_+)_{\lambda''}.
\end{equation}

\subsubsection{Construction of discrete summands in the $L^2$-model}\label{subsec:phi}

Via the Fourier--Laplace transform, we can define the counterpart for the $L^2$-model of the Rankin--Cohen
bidifferential operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ and the holographic integral operator
$\Psi_{\lambda',\lambda''}^{\lambda'''}$ \eqref{eqn:psi} by
\begin{equation}\label{eqn:RCFourier}
   \widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}:=\mathcal F_1^{-1}\circ
   {\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}}\circ \mathcal F_2.
\end{equation}
\begin{equation}\label{eqn:FPsi}
   \widehat{\Psi}_{\lambda',\lambda''}^{\lambda'''}:= \mathcal F_2^{-1}\circ
    {\Psi_{\lambda',\lambda''}^{\lambda'''}}\circ \mathcal F_1.
\end{equation}
We know from \cite{KP16a} that $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
is continuous between the weighted Bergman spaces, and so is
$\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$.
In turn, $\widehat{\Psi}_{\lambda',\lambda''}^{\lambda'''}$ is continuous between the Hilbert spaces by \eqref{eqn:3holograph} below, hence so is
${\Psi}_{\lambda',\lambda''}^{\lambda'''}$. Alternatively, the
continuity of ${\Psi}_{\lambda',\lambda''}^{\lambda'''}$ is also given by that of another
holographic operator
${\Phi}_{\lambda',\lambda''}^{\lambda'''}$ introduced in
Definition \ref{def:Phi} (Proposition \ref{prop:phinorm}).
The
 following commutative diagrams summarize these definitions:
$$
\small{
\xymatrix{
 \ar@/_5pc/[dd]_{\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}}
  L^2(\R_+^2,x^{1-\lambda'} y^{1-\lambda''}dxdy)
 \ar[r]^-{\mathcal F_{2}}
 \ar @{} [dr] |{\circlearrowright}
  \ar[d]
 & \mathcal H^2_{\lambda'}(\Pi) \widehat\otimes \mathcal H^2_{\lambda''}(\Pi)
 \ar[d]^{\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}}
  \ar@/^5pc/[dd]^{\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}}
 \\
 L^2(\R_+^2,x^{1-\lambda'}
y^{1-\lambda''}dx d y)
\ar[r]^-{\mathcal F_{2}}
 \ar @{} [dr] |{\circlearrowright}
 \ar[d]
&\mathcal H^2_{\lambda'}(\Pi) \widehat\otimes \mathcal H^2_{\lambda''}(\Pi)
\ar[d]^{\mathrm{Rest}}
\\
L^2(\R_+,z^{1-\lambda'''}dz)\ar[r]_-{\mathcal F_{1}}
 &
 \mathcal H^2_{\lambda'''}(\Pi)
 }
 }
$$
{\centerline{ Diagram 2.1. Symmetry breaking operators
$\pi_{\lambda'}\widehat\otimes\,\pi_{\lambda''}\twoheadrightarrow\pi_{\lambda'''}$}
{\centerline{ for holomorphic
 and $L^2$-models.}}

$$
\small{
\xymatrix{
L^2(\R_+^2,x^{1-\lambda'} y^{1-\lambda''}dx d y)
 \ar[r]^-{\mathcal F_{2}}
\ar @{}[rd] |{\circlearrowright}
& \mathcal H^2_{\lambda'}(\Pi) \widehat\otimes \mathcal H^2_{\lambda''}(\Pi)
 \\
 L^2(\R_+,z^{1-\lambda'''}dz)
 \ar[r]^-{\mathcal F_{1}}
  \ar[u]^{\qquad \widehat{\Psi}_{\lambda',\lambda''}^{\lambda'''}=(c'\Phi_{\lambda',\lambda''}^{\lambda'''})}
 & \mathcal H^2_{\lambda'''}(\Pi)
\ar[u]_{\Psi_{\lambda',\lambda''}^{\lambda'''}(=c''(\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''})^*)}
}
}
$$

{\centerline{Diagram 2.2. Holographic operators $\pi_{\lambda'''}\hookrightarrow
\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$}}
{\centerline{
 for holomorphic and $L^2$-models.}}
\vskip10pt

We shall give an explicit integral formula of the symmetry breaking operator $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$
 in the $L^2$-model in Proposition \ref{lem:1}.
On the other hand, we observe the holographic operator in the $L^2$-model
 has the following three important characteristics:
\begin{itemize}
    \item[(1)] the Fourier transform $\widehat \Psi_{\lambda',\lambda''}^{\lambda'''}$
    of the holographic operator $\Psi_{\lambda',\lambda''}^{\lambda'''}$, see \eqref{eqn:FPsi};
    \item[(2)] the adjoint of $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ (Proposition \ref{lem:2});
    \item[(3)] the multiplication operator $\Phi_{\lambda',\lambda''}^{\lambda'''}$, see \eqref{eqn:defPhi} below for definition.
\end{itemize}

These three approaches may be summarized as the following identities:
\begin{equation}\label{eqn:3holograph}
\widehat\Psi_{\lambda',\lambda''}^{\lambda'''}
=\left(\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}\right)^*
=i^\ell\Phi_{\lambda',\lambda''}^{\lambda'''},
\end{equation}
see Propositions \ref{lem:2} and \ref{prop:PhiPsi}.
The third characteristic is remarkable as it does not involve
any integration or differentiation.
For this reason,
 we adopt it as our definition
of holographic transform in the $L^2$-model, see Definition \ref{def:Phi} below.

For $\alpha,\beta\in\C$ and $\ell\in\N$,  let $P_\ell^{\alpha,\beta}(x)$ be the Jacobi polynomial of degree $\ell$, see \eqref{eqn:Jacobi} in Appendix.
\begin{defn}\label{def:Phi}
Retain the setting
 that $\lambda',\lambda'',\lambda'''\in\C$ with $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$.
  For a function $h(z)$ of one variable $z$ ($z>0$), we define a function of two variables $x,y$ ($x,y>0$) by
   \begin{equation}\label{eqn:defPhi}
      \left(\Phi_{\lambda',\lambda''}^{\lambda'''}h\right)(x,y):=
      \frac{x^{\lambda'-1}y^{\lambda''-1}}{(x+y)^{\lambda'+\lambda''+\ell-1}} P_\ell^{\lambda'-1,\lambda''-1}
      \left(\frac{y-x}{x+y}\right)\cdot h(x+y).
   \end{equation}
\end{defn}

\begin{thm}[holographic operator in the $L^2$-model]\label{thm:phi}
Suppose ${\lambda',\lambda''},{\lambda'''}>1$ such that $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$.
Then, $\Phi_{\lambda',\lambda''}^{\lambda'''}$ induces
an ${SL(2,\R)}{\;}\widetilde{}$-equivariant continuous homomorphism between the Hilbert spaces:
\begin{eqnarray*}
\Phi_{\lambda',\lambda''}^{\lambda'''}\colon L^2(\R_+,z^{1-\lambda'''}dz)&\longrightarrow&L^2(\R_+,x^{1-\lambda'}d x)\widehat\otimes L^2(\R_+,
y^{1-\lambda''}d y).
\end{eqnarray*}

\end{thm}
Theorem \ref{thm:phi} will be proved in Section \ref{subsec:pfthmphi}.
\begin{rem}
Using  the notation \eqref{eqn:Pxy} below, we may also write:
$$
(\Phi_{\lambda',\lambda''}^{\lambda'''}h)(x,y)=(-1)^\ell\frac{x^{\lambda'-1}y^{\lambda''-1}}{(x+y)^{\lambda'''-1}} \widetilde P_\ell^{\lambda'-1,\lambda''-1}\left(x,y\right)\cdot
h(x+y).
$$
\end{rem}

\subsubsection{Symmetry breaking transform in the $L^2$-model and its inversion}

In this subsection, we give an inversion formula of the
symmetry breaking operator
$\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ in the $L^2$-model by using the holographic operators
$\Phi_{\lambda',\lambda''}^{\lambda'''}$ (Definition \ref{def:Phi}). The symmetry breaking operator
$\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ was defined originally as the Fourier transform of the
Rankin--Cohen bidifferential operator ${\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ (see \eqref{eqn:RCFourier}) but we give a simpler expression
as an integral operator (\emph{Jacobi transform}).
\begin{prop}\label{lem:1}
Suppose $\lambda',\lambda''>1$ and $\ell\in\N$. Then for any $F\in C_c(\R_+\times\R_+)$, the following identity holds:
$$
\left(\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}F\right)(z)
= \frac{z^{\ell+1}}{2i^\ell}\int_{-1}^1 P_\ell^{\lambda'-1,\lambda''-1}(v) F\left(\frac z2(1-v),\frac z2(1+v)\right)dv.
$$
\end{prop}
See Section \ref{subsec:FTRC} for a proof.

Collecting the operators $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$, we define a
 symmetry breaking transform
\begin{equation}\label{eqn:defRCTL2}
    \widehat{\mathcal{RC}}_{\lambda',\lambda''}\colon L^2(\R_+)_{\lambda'} \widehat\otimes
    L^2(\R_+)_{\lambda''}
      \longrightarrow
      \sum_{\ell\in\N}{}^{\oplus} L^2(\R_+)_{\lambda'+\lambda''+2\ell}
      \end{equation}
 by
$
\left(  \widehat{\mathcal{RC}}_{\lambda',\lambda''} (F)\right)_\ell
         :=
         \widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}(F)\quad\mathrm{for}\,\,\ell\in\N.
$
Then it can be inverted by using the holographic operators $\Phi_{\lambda',\lambda''}^{\lambda'''}$ (Definition \ref{def:Phi}) as follows.

\begin{thm}\label{thm:FRCinv}
    Suppose $\lambda', \lambda''>1$. Then for any element $F\in L^2(\R_+)_{\lambda'} \widehat\otimes
    L^2(\R_+)_{\lambda''}$, one has
    $$
       F=\sum_{\ell=0}^\infty \frac{i^\ell }{c_\ell(\lambda',\lambda'')}
       \Phi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}
         \left(  \widehat{\mathcal{RC}}_{\lambda',\lambda''} (F)\right)_\ell,
     $$
     and
     $$
     \Vert F\Vert^2_{L^2(\R_+)_{\lambda'}\widehat\otimes L^2(\R_+)_{\lambda''}}
     =
    \sum_{\ell=0}^\infty
    \frac1{c_\ell(\lambda',\lambda'')}\left\Vert\left(\widehat{\mathcal{RC}}_{\lambda',\lambda''}(F)\right)_\ell\right\Vert^2_{L^2(\R_+)_{\lambda'+\lambda''+2\ell}}.
     $$
\end{thm}

Theorem \ref{thm:FRCinv} will be proved in Section \ref{subsec:pfthmFRCinv}.
It gives an answer to Problem A.2 and Problem B in the $L^2$-model.

\begin{rem}\label{rem:180716}
 The Jacobi transform (see \emph{e.g.} \cite[Chap. 15]{DB}) defined by
$$
H(v)\mapsto \left(J^{\alpha,\beta}(H)\right)_\ell:= \int_{-1}^1 H(v) P^{\alpha,\beta}_\ell(v)(1-v)^\alpha(1+v)^\beta dv
$$
is inverted by the following formula:
\begin{equation}\label{eqn:invJacobi}
H(v)=\sum_{\ell=0}^\infty d_\ell(\alpha,\beta)  \left(J^{\alpha,\beta}(H)\right)_\ell  P^{\alpha,\beta}_\ell(v),
\end{equation}
where we set
\begin{eqnarray*}
d_\ell(\alpha,\beta)&:=&\frac{\ell! (\alpha+\beta+2\ell+1)\Gamma(\alpha+\beta+\ell+1)}{2^{\alpha+\beta+1}\Gamma(\alpha+\ell+1)\Gamma(\beta+\ell+1)}
\\
&=&\frac1{2^{\alpha+\beta+1}c_\ell(\alpha+1,\beta+1)}.
\end{eqnarray*}
By change of variables, we can see that Theorem \ref{thm:FRCinv} is equivalent to \eqref{eqn:invJacobi}
applied to
$$
H(v)=(1-v)^{-\alpha}(1+v)^{-\beta} F\left(\frac z2(1-v),\frac z2(1+v)\right)
$$
with $\alpha=\lambda'-1$ and $\beta=\lambda''-1$.
\end{rem}

\subsubsection{Parseval--Plancherel type theorem for the holographic transform in the $L^2$-model}

Collecting the holographic operators $ \Phi_{\lambda',\lambda''}^{\lambda'''}$, we define the holographic
transform
\begin{equation}\label{eqn:defPhiT}
\Phi_{\lambda',\lambda''}\colon \bigoplus_{\ell\in\N}  L^2(\R_+)_{\lambda'+\lambda''+2\ell} \longrightarrow
 L^2(\R_+)_{\lambda'} \widehat \otimes L^2(\R_+)_{\lambda''}
\end{equation}
by
$$
\Phi_{\lambda',\lambda''}:=\bigoplus_{\ell=0}^\infty \Phi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}.
$$

This transform is the counterpart in the $L^2$-model of the holographic transform
$ \Psi_{\lambda',\lambda''}$ (Theorem \ref{thm:PPTT4RCB} (2)) defined in the holomorphic model.

\begin{thm}\label{thm:CL2SL2}
Suppose $\lambda',\lambda''>1$ and $\ell\in\N$. Then, the holographic transform
$
\Phi_{\lambda',\lambda''}
$
induces an ${SL(2,\R)}{\;}\widetilde{}$ -equivariant unitary operator
$$
\sum_{\ell=0}^\infty{}^{\oplus}L^2(\R_+)_{\lambda'+\lambda''+2\ell}\stackrel
{\sim}{\longrightarrow}
L^2(\R_+)_{\lambda'}\widehat\otimes L^2(\R_+)_{\lambda''}
$$
subject to the following Parseval--Plancherel type formula:

$$
\Vert \Phi_{\lambda',\lambda''} h\Vert^2_{L^2(\R_+^2)_{\lambda',\lambda''}}=
\sum_{\ell=0}^\infty
c_\ell(\lambda',\lambda'')
\Vert h_\ell\Vert^2_{L^2(\R_+)_{\lambda'+\lambda''+2\ell}},
$$
for $h=(h_\ell)_{\ell\in\N}$ with $h_\ell\in L^2(\R_+)_{\lambda'+\lambda''+2\ell}$.
\end{thm}

Theorem \ref{thm:CL2SL2} will be proved in Section \ref{subsec:PPTPHI}.

\subsubsection{Representation theoretic interpretation of the Plancherel density}

The weights $c_\ell(\lambda',\lambda'')$ in the Plancherel formula (Theorem \ref{thm:CL2SL2}) are obviously positive when $\lambda',\lambda''>1$. We discuss the zeros of the meromorphic continuation of $c_\ell(\lambda',\lambda'')$ when we allow $\lambda'$ and
$\lambda''$ to wander outside the region $\lambda',\lambda''>1$, so that
$\pi_{\lambda'}$ and $\pi_{\lambda''}$ may not be (relative) holomorphic discrete series representations.

Assume furthermore that $\lambda',\lambda'',\lambda'''\in\Z$ such that
 $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$. Then the following four conditions on
$(\lambda',\lambda'',\lambda''')$ are equivalent (see \cite[Thm.~9.1]{KP16b}):
\begin{itemize}
\item[(i)] $c_\ell(\lambda',\lambda'')=0$;
\item[(ii)]
$2\geq\lambda'+\lambda''+\lambda'''$ and $\lambda'''\geq\vert\lambda'-\lambda''\vert+2$;
\item[(iii)] the Rankin--Cohen bilinear operator
$\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ vanishes;
\item[(iv)] $
\dim\mathrm{Hom}_{{SL(2,\R)}{\;}\widetilde{}}\left(
\mathcal O(\Pi\times\Pi,\mathcal L_{\lambda'}\,\boxtimes \,\mathcal L_{\lambda''}),
\mathcal O(\Pi,\mathcal L_{\lambda'''})\right)=2.
$
\end{itemize}

\subsection{Proof of Theorems \ref{thm:psi} and \ref{thm:phi}} \label{subsec:proofs}

In this section, we derive from the Rankin--Cohen bidifferential operators
$\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ the integral intertwining
operators that embed irreducible representations of ${SL(2,\R)}{\;}\widetilde{}$
into the tensor product representations, and give a proof of Theorems \ref{thm:psi} and \ref{thm:phi}.

The key idea is to use symmetry breaking operators $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$
in the $L^2$-model, which fits well into the F-method connecting the Rankin--Cohen operators with the
Jacobi polynomials. The scheme of the proof is summarized in the following diagram:
\begin{equation}\label{diagram:squig}
\xymatrix{
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}
\ar@{~>}[d]_{\mathrm{Proposition}\, \ref{lem:1}}
&& \Psi_{\lambda',\lambda''}^{\lambda'''} \ar@{~>}[ll]_{\mathrm{Proposition}\, \ref{prop:PsiRC}}
& (\mathrm{Theorem}\,\ref{thm:psi})\\
\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}
\ar@{~>}[rr]_{\mathrm{Proposition}\, \ref{lem:2}}&&  \Phi_{\lambda',\lambda''}^{\lambda'''}
\ar@{~>}[u]_{\mathrm{Proposition}\, \ref{prop:PhiPsi}}
& (\mathrm{Theorem}\,
\ref{thm:phi})
}
\end{equation}

\subsubsection{Jacobi polynomials and Rankin--Cohen bidifferential operators}

We retain the notation and assumption
that
$
\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N.
$
{lem}

The nature of the bidifferential symmetry breaking operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ is explained
 in \cite[Thm.\ 8.1]{KP16b} by the F-method, which we recall now.
We inflate  the Jacobi polynomial $P_\ell^{\alpha,\beta}(t)$
 (see \eqref{eqn:Jacobi}) into a homogeneous polynomial of degree $\ell$
 by
\begin{eqnarray}\label{eqn:Pxy}
\nonumber\widetilde{P}_\ell^{\alpha,\beta}(x,y)&:=&(-1)^\ell(x+y)^\ell P_\ell^{\alpha,\beta}\left(\frac{y-x}{x+y}\right)\\
&=&
\sum_{j=0}^\ell
\frac{(-1)^{\ell-j}(\alpha+\beta+\ell+1)_j(\alpha+j+1)_{\ell-j}}{(\ell-j)!j!}(x+y)^{\ell-j}x^j.
\end{eqnarray}
Then we have the following
\begin{prop}\label{prop:RCPnew}
Suppose $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$. Then the Rankin--Cohen
bidifferential operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ (see \eqref{rcb}) is given by
$
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}=\mathrm{Rest}\circ
\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}
$
with
\begin{equation}\label{eqn:RCPnew}
\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}=\widetilde{P}_\ell^{\lambda'-1,\lambda''-1}\left(\frac\partial{\partial \zeta_1},\frac\partial{\partial \zeta_2}\right).
\end{equation}
\end{prop}

\begin{rem}
In  \cite[(9.9)]{KP16b}, we gave a similar formula
\begin{equation}\label{eqn:RCPsel}
\mathcal{R}_{\lambda',\lambda''}^{\lambda'''}={P}_\ell^{\lambda'-1,1-\lambda'''}\left(\frac\partial{\partial \zeta_1},\frac\partial{\partial \zeta_2}\right)
\end{equation}
by using another two-variable function
$P_\ell^{\alpha,\beta}(x,y):=y^\ell P_\ell^{\alpha,\beta}\left(1+\frac{2x}y\right)$. Our expression \eqref{eqn:RCPnew} is symmetric with respect to the first and second variables.
\end{rem}
\begin{proof}
[Proof of Proposition \ref{prop:RCPnew}]
According to the first Kummer's relation for the hypergeometric function we get (see for instance \cite[8.962]{GR}):
$$
P_\ell^{\alpha,\beta}(x)=\left(\frac{1+x}2\right)^\ell P_\ell^{\alpha,-\alpha-\beta-2\ell-1}\left(\frac{3-x}{1+x}\right),
$$
and therefore
$$
P_\ell^{\lambda'-1,1-\lambda'''}(1-2s)=(1-s)^\ell P_\ell^{\lambda'-1,\lambda''-1}\left(\frac{1+s}{1-s}\right).
$$
Hence the right-hand sides of \eqref{eqn:RCPnew} and \eqref{eqn:RCPsel} are equal to each other.
\end{proof}

\subsubsection{Coordinate change in the $L^2$-model}\label{sec:coord-changeSL2}

For the study of symmetry breaking in the $L^2$-model, we introduce the following coordinates:
\begin{equation}\label{eqn:SLiota}
\iota \colon \R_+\times (-1,1) \stackrel{\sim}{\longrightarrow} \R_+^2,
\quad
(z,v) \mapsto (x,y):=\left(\frac z2(1-v), \frac z2(1+v)\right).
\end{equation}

 Then, $\iota$ is a diffeomorphism with $d x d y=\frac z2 d z d v$. With the convention \eqref{eqn:abl}
in Section \ref{sec:constants}, we set
\begin{equation}\label{eqn:Azv}
M(z,v)\equiv M_{\lambda',\lambda'',\lambda'''}(z,v)
      := 2^{\alpha+\beta}z^{\ell+1}
(1-v)^{-\alpha}(1+v)^{-\beta}.
\end{equation}
If $(x,y)=\iota (z,v)$, then we have
\begin{equation}\label{eqn:mmAQ-SL}
\frac{z^{1-\lambda'''}}{x^{1-\lambda'}y^{1-\lambda''}}M(z,v)=z^{-\ell},
\end{equation}
\begin{equation}\label{eqn:imeasure}
x^{1-\lambda'}y^{1-\lambda''}d x d y=M(z,v)^2z^{-\alpha-\beta-\lambda'''}(1-v)^\alpha(1+v)^\beta dzd v,
\end{equation}
whereas the holographic operator $\Phi_{\lambda',\lambda''}^{\lambda'''}$ (see \eqref{eqn:defPhi})
takes the form
\begin{equation}\label{eqn:iholograph}
\left(\Phi_{\lambda',\lambda''}^{\lambda'''}h\right)\circ\iota(z,v)= M(z,v)^{-1} P^{\alpha,\beta}_\ell(v)h(z).
\end{equation}

\subsubsection{Fourier transform of the Rankin--Cohen bidifferential operators}\label{subsec:FTRC}

We are ready to prove Proposition \ref{lem:1} for an integral expression of the symmetry breaking operator
$\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$
 (see \eqref{eqn:RCFourier}).

\begin{proof}[Proof of Proposition \ref{lem:1}]
For a function $F\in L^2(\R_+ ^2)_ {\lambda',\lambda''}$ we set
$$
G(x,y):=\widetilde{P}_\ell^{\lambda'-1,\lambda''-1}(x,y) F(x,y).
$$
By Proposition \ref{prop:RCPnew} the F-method shows that the
 Rankin--Cohen bidifferential operator is induced from the multiplication
 by the polynomial $\widetilde P_\ell^{\lambda'-1,\lambda''-1}(x,y)$, namely,
\begin{equation}\label{eqn:RCFG}
 (\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\mathcal F_2F)(\zeta)=i^\ell(\mathrm{Rest}\circ
\mathcal F_2G)(\zeta).
\end{equation}
The left-hand side of \eqref{eqn:RCFG} equals $\left(\mathcal F_1 \widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''} F\right)(\zeta)$
by the definition \eqref{eqn:RCFourier}.
 We compute the right-hand side of \eqref{eqn:RCFG}.
Via the diffeomorphism \eqref{eqn:SLiota},
we have  $\widetilde{P}_\ell^{\alpha,\beta}\circ\iota(z,v)=(-1)^\ell z^\ell
{P}_\ell^{\alpha,\beta}(v)$.
Thus we get
\begin{eqnarray*}
\left(\mathrm{Rest}\circ \mathcal F_{2}\right)G(\zeta)&=&
\int_0^\infty\int_0^\infty G(x,y) e^{i(x+y)\zeta} d x d y\\
&=&\frac12\int_0^\infty\int_{-1}^1G\circ\iota( z,v)e^{i z\zeta}zdzdv\\
&=&\frac12\mathcal F_{1}(\mathcal JF)(\zeta),
\end{eqnarray*}
where
\begin{eqnarray*}
\mathcal JF(z)&:=&z\int_{-1}^1G\circ\iota(z,v)dv\\
&=&(-1)^\ell z^{\ell+1}\int_{-1}^1
P_\ell^{\lambda'-1,\lambda''-1}(v) F\circ\iota(z,v)dv.
\end{eqnarray*}
Hence Proposition \ref{lem:1} is proved.
\end{proof}

\subsubsection{Three characteristics of holographic operators in the $L^2$-model}

In Section \ref{subsec:phi}, we discussed the three characteristics (1), (2), and (3) of holographic operators in the $L^2$-model. These three
characteristics play a key role in the proof of main theorems. In this subsection we explicate the relationship between
\begin{itemize}
\item[] (2) and (3) in Proposition \ref{lem:2}, and
\item[] (1) and (3) in Proposition \ref{prop:PhiPsi},
\end{itemize}
and prove the formula \eqref {eqn:3holograph}.

\begin{prop}\label{lem:2}
The adjoint of the holographic operator
 $\Phi_{\lambda',\lambda''}^{\lambda'''}$
 (Definition \ref{def:Phi})
is proportional to the Fourier transform of the Rankin--Cohen operator ${\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$:
$$
\left(\Phi_{\lambda',\lambda''}^{\lambda'''}\right)^*=i^\ell\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}.
$$
\end{prop}
\begin{proof}
We have already seen in Section \ref{subsec:phi} that $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ is a continuous map between the Hilbert spaces.
Hence we shall work with dense subspaces $C_c(\R_+)$ and $C_c(\R_+^2)$
in $L^2(\R_+)_{\lambda'''}$ and $L^2(\R_+^2)_{\lambda',\lambda''}$, respectively.
Take  $h\in C_c(\R_+)$ and $F\in C_c(\R_+^2)$.
By the integral expression of $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ given in Proposition \ref{lem:1}, we have
\begin{eqnarray*}
&&(h,\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}F)
_{L^2(\R_+,z^{1-\lambda'''}dz)}\\
&=&\frac{i^\ell}2\int_0^\infty
h(z)\overline{ z^{\ell+1}\int_{-1}^1 P_\ell^{\lambda'-1,\lambda''-1}(v) F\circ\iota(z,v)dv} z^{1-\lambda'''}dz
\\
&=&i^\ell\int_0^\infty\int_0^\infty (\Phi_{\lambda',\lambda''}^{\lambda'''}h)(x,y)\overline{F(x,y)} x^{1-\lambda'}y^{1-\lambda''}dxdy
\\
&=&i^\ell(\Phi_{\lambda',\lambda''}^{\lambda'''}h,F)_{L^2(\R_+^2,x^{1-\lambda'}y^{1-\lambda''}dxdy)}.
\end{eqnarray*}
Here, in the second equality we have used \eqref{eqn:imeasure}
and \eqref{eqn:iholograph}. Thus Proposition \ref{lem:2} is
proved.
\end{proof}

\begin{prop}\label{prop:PhiPsi}
With the notation \eqref{eqn:FPsi}, we have
$$
\widehat\Psi_{\lambda',\lambda''}^{\lambda'''}
=i ^\ell
\Phi_{\lambda',\lambda''}^{\lambda'''}.
$$
\end{prop}
Before giving a proof of Proposition \ref{prop:PhiPsi}, we need the following.

\begin{lem}\label{lem:3}
For any $g\in \mathcal H^2(\Pi)_\lambda$, we have
$$
\left(\frac d{dt}\right)^\ell\int_0^\infty z^{-\ell}(\mathcal F_1^{-1}g)(z) e^{izt}dz=i^\ell g(t).
$$
\end{lem}

\begin{proof}
The statement follows from the (classical) Fourier inversion formula
 and the Paley--Wiener theorem for $g$.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:PhiPsi}]
It suffices to show
$$
\mathcal F_2\circ\Phi_{\lambda',\lambda''}^{\lambda'''}\circ\mathcal F_1^{-1}
=(-i) ^\ell
\Psi_{\lambda',\lambda''}^{\lambda'''}.
$$

We set
$$
t(v):=\frac12 \left((\zeta_2-\zeta_1)v+(\zeta_1+\zeta_2)\right).
$$
By the definitions \eqref{eqn:Azv}
 and \eqref{eqn:iholograph} of $M$ and $\Phi_{\lambda',\lambda''}^{\lambda'''}$, we have,

\begin{eqnarray*}
&&2^{\alpha+\beta+1}(\mathcal F_2\circ\Phi_{\lambda',\lambda''}^{\lambda'''}h)(\zeta_1,\zeta_2)\\
&=&
\int_0^\infty \int_{-1}^1 z^{-\ell} (1-v)^\alpha (1+v)^\beta P^{\alpha,\beta}_\ell(v) h(z) e^{iz t(v)}dvdz
\\
&=&
\frac{(-1)^\ell}{2^{\ell}\ell!}
\int_0^\infty \int_{-1}^1
z^{-\ell}h(z) e^{izt(v)} \left(\frac d{dv}\right)^\ell \left((1-v)^{\alpha+\ell} (1+v)^{\beta+\ell}\right) dvdz
\\
&=&
\frac{i^\ell}{2^{\ell}\ell!}
 \int_{-1}^1
(1-v)^{\alpha+\ell} (1+v)^{\beta+\ell}\left(\frac{dt(v)}{dv}\right)^\ell
\left(\mathcal F_1 h\right)(t(v))
dv,
\end{eqnarray*}
where the second equality follows from the Rodrigues formula \eqref{eqn:JRodrigues} for the Jacobi polynomials, and
the third one from integration by parts and Lemma \ref{lem:3}.
Putting $g=\mathcal F_1 h$, we obtain
\begin{eqnarray*}
&&(\mathcal F_2\circ\Phi_{\lambda',\lambda''}^{\lambda'''}\circ\mathcal F_1^{-1}g)(\zeta_1,\zeta_2)\\
&=&
\frac{(\zeta_1-\zeta_2)^\ell (-i)^\ell}{2^{\alpha+\beta+2\ell+1}\ell!}
\int_{-1}^1 g(t(v))(1-v)^{\alpha+\ell} (1+v)^{\beta+\ell}dv\\
&=&
(-i)^\ell\left(\Psi_{\lambda',\lambda''}^{\lambda'''}g\right)(\zeta_1,\zeta_2).
\end{eqnarray*}

Hence Proposition
 \ref{prop:PhiPsi} is proved.
\end{proof}

\subsubsection{Proof of Theorem \ref{thm:phi}}\label{subsec:pfthmphi}

In this subsection we give a proof of Theorem \ref{thm:phi}.

\begin{proof}[Proof of Theorem \ref{thm:phi}]
As the Rankin--Cohen bidifferential operator ${\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ intertwines the tensor product
$\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$ and $\pi_{\lambda'''}$, so does its Fourier transform $\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}$ (see \eqref{eqn:RCFourier}), and in turn its adjoint operator $\left(\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''}\right)^*$
because $\pi_{\lambda'},\pi_{\lambda''}$, and $\pi_{\lambda'''}$ are unitary representations.
Hence Theorem \ref{thm:phi} follows from Proposition \ref{lem:2}.
\end{proof}

\subsubsection{Proof of Theorem \ref{thm:psi}}\label{subsec:pfthmpsi}
Theorem \ref{thm:phi} together with an argument of holomorphic continuation on parameters
completes the proof of Theorem \ref{thm:psi} as follows.

\begin{proof}[Proof of Theorem \ref{thm:psi}]

The second assertion follows from Proposition \ref{prop:PhiPsi} because $\Phi_{\lambda',\lambda''}^{\lambda'''}$
is an intertwining operator as it was shown in Theorem \ref{thm:phi}.

Let $\ell\in\N$ and $\lambda'''=\lambda'+\lambda''+2\ell$. If $(\lambda',\lambda'')\in\C^2$ satisfies \eqref{eqn:HOconverge} then the
 integral \eqref{eqn:psi} converges for all $g\in\mathcal O(\Pi)$, and $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is continuous viewed
as a map from the Montel
space $\mathcal O(\Pi)$ to the one $\mathcal O(\Pi\times\Pi)$.

On the other hand, if furthermore $\lambda',\lambda''$ are real and $\lambda',\lambda''>1$,
then $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is a $G$-homomorphism on $\mathcal H^2(\Pi)_{\lambda'''}$ by the second statement.
Since $\mathcal H^2(\Pi)_{\lambda'''}$
 is dense in the Montel space $\mathcal O(\Pi)$ as its subspace of $K$-finite functions is already
 dense in
 $\mathcal O(\Pi)$, the continuous map
$\Psi_{\lambda',\lambda''}^{\lambda'''} \colon \mathcal O(\Pi)\longrightarrow\mathcal O(\Pi\times\Pi)$ intertwines
$\pi_{\lambda'''}$ and the tensor product representation $\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$
if $\lambda',\lambda''>1$. Since $\Psi_{\lambda',\lambda''}^{\lambda'''}g\in \mathcal O(\Pi\times\Pi)$
 depends holomorphically on $(\lambda',\lambda'')\in\C^2$ subject to \eqref{eqn:HOconverge} and since the actions
 $\pi_{\lambda'}, \pi_{\lambda''}$ and $\pi_{\lambda'+\lambda''+2\ell}$ of ${SL(2,\R)}{\;}\widetilde{}$ also depend holomorphically on  $(\lambda',\lambda'')\in\C^2$,
 the first statement
is shown.
\end{proof}

\subsubsection{Adjoint of the Rankin--Cohen operator}

As the last part of the diagram \eqref{diagram:squig}, we show that $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is the adjoint of the Rankin--Cohen operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ up to scalar multiplication.

Suppose $\lambda',\lambda'',\lambda'''>1$ and $\lambda'''-\lambda'-\lambda''\in2\N$. We regard
the Rankin--Cohen operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ as a continuous map between
Hilbert spaces
$$
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}
\colon
\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}\longrightarrow \mathcal H^2(\Pi)_{\lambda'''}.
$$
By \eqref{eqn:3holograph} and Lemma \ref{lem:opFourier} below, we obtain
\begin{prop}\label{prop:PsiRC}
Let $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$. The adjoint of $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ is given by
$$
\left(\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\right)^*=
r_\ell(\lambda',\lambda'')\Psi_{\lambda',\lambda''}^{\lambda'''}.
$$
\end{prop}

\subsection{Proof of the Parseval--Plancherel type theorem for the symmetry breaking transform and the holographic transform}

In this section we complete the proof  of Theorems \ref{thm:RCinv} and \ref{thm:PPTT4RCB} in the holomorphic model and Theorems
\ref{thm:FRCinv} and \ref{thm:CL2SL2} in the $L^2$-model for
the Parseval--Plancherel type results
for the symmetry breaking and holographic transforms.
Our strategy consists in applying the $F$-method,
 and then in reducing the proof of these theorems
 to the fact that the Jacobi polynomials $\left\{P_\ell^{\alpha,\beta}\right\}_{\ell\in\N}$ form an orthogonal basis of the Hilbert space $L^2([-1,1], (1-v)^{\alpha}(1+v)^{\beta}dv)$.

\subsubsection{Some properties of operators on Hilbert spaces}

We review a general fact on operators on Hilbert spaces. Suppose a Hilbert space $V$ is decomposed into a Hilbert direct sum of closed subspaces
$\{V_\ell\}_{\ell\in\N}$, that is, $V\simeq\sum_{\ell\in\N}^{\oplus} V_\ell$, where the inner product on $V_\ell$ is induced from that of $V$.
Let $\mathrm{pr}_{V\to V_\ell} \colon V\longrightarrow V_\ell$ be the projection operator.
 Let $\{W_\ell\}_{\ell\in\N}$ be another family of Hilbert spaces. Suppose that we are given a continuous map $R_\ell \colon V\longrightarrow W_\ell$ such that the
restriction $R_\ell\vert_{V_\ell}:V_\ell\longrightarrow W_\ell$ is a unitary operator up to scalar multiplication and $R_\ell\vert_{V_\ell^\perp}\equiv 0$ for every $\ell\in\N$.
Then the adjoint operator
$R_\ell^* \colon W_\ell\longrightarrow V$ is an isometry (up to scalar) onto $V_\ell$.
We write $\Vert R_\ell\Vert_{\mathrm{op}}$ for the operator norm of $R_\ell$ and set
$$
C_\ell:=\Vert R_\ell\Vert_{\mathrm{op}}^2.
$$
 The following two lemmas are elementary.
\begin{lem}\label{lem:Hilbertdeco}
{}
\begin{itemize}
\item[(1)] The linear map
$\displaystyle R:=\bigoplus\limits_{\ell\in\N}R_\ell \colon V\to\bigoplus\limits_{\ell\in\N} W_\ell$ satisfies
\begin{eqnarray}
R_\ell^*R_\ell&=&C_\ell\mathrm{pr}_{V\to V_\ell}, \label{eqn:RR1}\\
\Vert F\Vert^2_V&=&\sum_{\ell\in\N}\frac1{C_\ell}\Vert R_\ell F\Vert^2_{W_\ell}
\quad\mathrm{for\,\, all}\quad F\in V. \label{eqn:Rexpansion}
\end{eqnarray}
In particular, we have the following inversion formula and the unitarity of the map $R$:
\begin{itemize}
\item[(inversion)] $\qquad F=\sum_{\ell\in\N}\frac1{C_\ell}R_\ell^*(R_\ell F)$,
\item[(unitarity)] $\qquad R$ extends to a unitary operator $\displaystyle V\stackrel
{\sim}{\longrightarrow}{\sum\limits_{\ell\in\N}}^\oplus W_\ell$,
where $\displaystyle{\sum_{\ell\in\N}}^\oplus W_\ell$ is the Hilbert sum associated to the weights $\{C_\ell^{-1}\}_{\ell\in\N}$ (see Definition \ref{def:compsum}).
\end{itemize}
\item[(2)] The linear map $R^*:=\bigoplus\limits_{\ell\in\N}R_\ell^* \colon \bigoplus\limits_{\ell\in\N} W_\ell\to V$ satisfies
\begin{eqnarray}
R_\ell R_\ell^*&=&C_\ell\mathrm{id}_{W_\ell}, \label{eqn:RR2}\\
\Vert R_\ell^*w_\ell\Vert^2_V&=&{C_\ell}\Vert w_\ell\Vert^2_{W_\ell}
\quad\mathrm{for\, \, all}\quad w_\ell\in W_\ell. \nonumber
\end{eqnarray}
In particular $R^*$ extends to a unitary operator $\displaystyle{\sum\limits_{\ell\in\N}}^\oplus W_\ell\stackrel
{\sim}{\longrightarrow} V$,
where $\displaystyle{\sum_{\ell\in\N}}^\oplus W_\ell$ is the Hilbert sum associated to the weights $\{C_\ell\}_{\ell\in\N}$.
\end{itemize}
\end{lem}

\begin{lem}\label{lem:opFourier}
Suppose that $H_j$ and $L_j$ $(j=1,2)$ are Hilbert spaces
 and that $\mathcal F_j \colon L_j\longrightarrow H_j$ are
unitary operators up to scalar multiple. Let $b_j$ be positive numbers such that
$$
\Vert \mathcal F_j(F)\Vert^2_{H_j}=b_j\Vert F\Vert^2_{L_j}\qquad \mathrm{for\,\, all}\quad F\in L_j.
$$
Let $\Psi \colon H_1\longrightarrow H_2$
 and $D \colon H_2\longrightarrow H_1$ be continuous linear maps, and
  we define
 $\widehat \Psi\colon L_1\longrightarrow L_2$ and $\widehat D\colon L_2\longrightarrow L_1$ by
$$
\widehat \Psi:=\mathcal F_2^{-1}\circ \Psi\circ\mathcal F_1,\qquad
\widehat D:=\mathcal F_1^{-1}\circ D\circ\mathcal F_2.
$$
We set $r:=\frac{b_1}{b_2}$. Then,
\begin{itemize}
\item[(1)] the operator norms of these operators satisfy
$$
\Vert \widehat D\Vert^2_{\mathrm{op}}=\frac 1r \Vert D\Vert^2_{\mathrm{op}}, \qquad
\Vert \widehat \Psi\Vert^2_{\mathrm{op}}=r \Vert \Psi\Vert^2_{\mathrm{op}};
$$
\item[(2)]
the adjoint operators of $D$ and $\widehat D$ are related as
$$
\widehat{D^*}= r\left(\widehat D\right)^*.
$$
\end{itemize}
\end{lem}

\subsubsection{Parseval--Plancherel type theorem for $\Phi_{\lambda',\lambda''}^{\lambda'''}$}\label{subsec:PPTPHI}

In this subsection, we prove Theorem \ref{thm:CL2SL2}.
By the (abstract) branching law \eqref{eqn:absbra},
Theorem \ref{thm:CL2SL2} is deduced from the following proposition.

\begin{prop}\label{prop:phinorm}
Suppose $\lambda',\lambda'',\lambda'''>1$
 satisfy $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$. Then,
$$
\Vert \Phi_{\lambda',\lambda''}^{\lambda'''} h\Vert^2_{L^2(\R_+^2)_{\lambda',\lambda''}}=
c_\ell (\lambda',\lambda'')\Vert h\Vert^2_{L^2(\R_+)_{\lambda'''}},
$$
for all $h\in L^2(\R_+)_{\lambda'''}$.
Here we recall \eqref{eqn:v-ell}
 for the definition of $c_{\ell}(\lambda',\lambda'')$.
\end{prop}

We reduce Proposition \ref{prop:phinorm} to
the fact that the Jacobi polynomials are orthogonal polynomials, see \eqref{eqn:Pnorm} in Appendix.

\begin{proof}[Proof of Proposition \ref{prop:phinorm}]
Via the diffeomorphism \eqref{eqn:SLiota}, we get
from the formul{\ae} \eqref{eqn:imeasure} for the measure and \eqref{eqn:iholograph}  for the holographic operator $\Phi_{\lambda',\lambda''}^{\lambda'''}$:
\begin{eqnarray*}
&&\Vert \Phi_{\lambda',\lambda''}^{\lambda'''} h\Vert^2_{L^2(\R_+^2)_{\lambda',\lambda''}}
=
\frac1{2^{\lambda'+\lambda''-1}} \times\\
&&
\int_0^\infty\int_{-1}^1\vert h(z)\vert^2
\left\vert P_\ell^{\lambda'-1,\lambda''-1}(v)\right\vert^2 z^{1-\lambda'''}
\left({1-v}\right)^{\lambda'-1} \left({1+v}\right)^{\lambda''-1}dvdz.
\end{eqnarray*}
By the $L^2$-norm \eqref{eqn:Pnorm} of the Jacobi polynomials, we conclude the proposition.
\end{proof}

\subsubsection{Operator norm of the holographic operator $\Psi_{\lambda',\lambda''}^{\lambda'''}$ in the holomorphic model}\label{subsec:PPTPSY}

\begin{prop}\label{prop:psinorm}
Suppose $\lambda',\lambda'',\lambda'''>1$ and $\ell:=\frac12(\lambda'''-\lambda'-\lambda'')\in\N$. Then
$$
\Vert \Psi_{\lambda',\lambda''}^{\lambda'''}g\Vert^2_{\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}}
= \frac{c_\ell(\lambda',\lambda'')}
{r_\ell(\lambda',\lambda'')}
\Vert g\Vert^2_{\mathcal H^2(\Pi)_{\lambda'''}}\,\mathrm{for\,\,all}\,\, g\in\mathcal H^2(\Pi)_{\lambda'''}.
$$
\end{prop}

\begin{proof
By Proposition \ref{prop:PhiPsi} and Fact \ref{fact:Laplace}, we have
$$
\Vert \Psi_{\lambda',\lambda''}^{\lambda'''}g\Vert^2_{\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}}=
b(\lambda')b(\lambda'')\Vert
\Phi_{\lambda',\lambda''}^{\lambda'''}\mathcal F_1^{-1}g\Vert^2_{L^2(\R_+^2)_{\lambda',\lambda''}}.
$$

By Proposition \ref{prop:phinorm} and Fact \ref{fact:Laplace} again, the right-hand side of the above equality amounts to
$$
\frac{b(\lambda')b(\lambda'')}{b(\lambda''')}c_\ell(\lambda', \lambda'')
\Vert g\Vert^2_{\mathcal H^2(\Pi)_{\lambda'''}}.
$$
Now the proposition follows from the definition \eqref{eqn:b-ell} of $r_\ell(\lambda',\lambda'')$.
\end{proof}

\subsubsection{Norm of the Rankin--Cohen bidifferential operators}

 We find the operator norm of $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
as below.
\begin{prop}\label{prop:RCopnorm}
Suppose that $\lambda',\lambda''> 1$ and  $\lambda'''=\lambda'+\lambda''+2\ell \,(\ell\in\N)$.
Then the operator norm of the Rankin--Cohen bidifferential operator
$\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ seen as a map from the weighted Bergman
space $\mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}$ to $\mathcal H^2(\Pi)_{\lambda'''}$ is given by
$$
\Vert \mathcal{RC}_{\lambda',\lambda''}^{\lambda'''} \Vert_{\mathrm{op}}^2=r_\ell(\lambda',\lambda'')
c_\ell(\lambda',\lambda'').
$$
\end{prop}

\begin{proof}
By Lemma \ref{lem:opFourier} (1), we have
$$
\Vert \mathcal{RC}_{\lambda',\lambda''}^{\lambda'''} \Vert_{\mathrm{op}}^2=r_\ell(\lambda',\lambda'')
\Vert \widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'''} \Vert_{\mathrm{op}}^2,
$$
which equals $r_\ell(\lambda',\lambda'')c_\ell(\lambda',\lambda'')$ by Propositions \ref{lem:2} and
\ref{prop:phinorm}.
\end{proof}

\subsubsection{Proof of Theorem \ref{thm:PPTT4RCB}}\label{subsec:pfthmPPTT4RCB}

Let us complete the proof of the Parseval--Plancherel type theorem for the Rankin--Cohen transform
$\mathcal{RC}_{\lambda',\lambda''}$
 and the holographic transform $\Psi_{\lambda',\lambda''}$.

\begin{proof}[Proof of Theorem \ref{thm:PPTT4RCB}]

(1) We apply Lemma \ref{lem:Hilbertdeco} with
$R_\ell={\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$.
By Proposition \ref{prop:RCopnorm}, we have
$$
\Vert R_\ell\Vert^2_{\mathrm{op}}=r_\ell({\lambda',\lambda''})c_\ell({\lambda',\lambda''}),
$$
hence the first statement follows from Lemma \ref{lem:Hilbertdeco} (1).

(2) We apply Lemma \ref{lem:Hilbertdeco} with
$R_\ell=\frac1{r_\ell({\lambda',\lambda''})}{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$.
By Proposition \ref{prop:RCopnorm}, we have
$$
\Vert R_\ell\Vert^2_{\mathrm{op}}=\frac{c_\ell({\lambda',\lambda''})}{r_\ell({\lambda',\lambda''})}.
$$
Since $\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}=R_\ell^*$ (see Proposition \ref{prop:PsiRC}), we get the second statement by
Lemma \ref{lem:Hilbertdeco} (2).
\end{proof}

\subsubsection{Proof of Theorem \ref{thm:RCinv}}\label{subsec:pfthmRCinv}

We are ready to complete the proof of Theorem \ref{thm:RCinv}.
\begin{proof}[Proof of Theorem \ref{thm:RCinv}]

By Lemma \ref{lem:Hilbertdeco} (1)
applied to $R_\ell=\mathcal{RC}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$, the above proof of Theorem \ref{thm:PPTT4RCB} (1)
implies
$$
f=\sum_{\ell=0}^\infty\frac1{r_\ell({\lambda',\lambda''})c_\ell({\lambda',\lambda''})} R_\ell^*R_\ell f
$$
for any $f\in \mathcal H^2(\Pi)_{\lambda'}\widehat\otimes \mathcal H^2(\Pi)_{\lambda''}$. Now Theorem \ref{thm:RCinv}
follows from the equation $R_\ell^*=r_\ell(\lambda',\lambda'')\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$ (see Proposition \ref{prop:PsiRC}).
\end{proof}

\subsubsection{Proof of Theorem \ref{thm:FRCinv}}\label{subsec:pfthmFRCinv}

Finally, we show Theorem \ref{thm:FRCinv}.
\begin{proof}[Proof of Theorem \ref{thm:FRCinv}]
We apply Lemma \ref{lem:Hilbertdeco} (1) with $R_\ell=\widehat{\mathcal{RC}}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$.
By Lemma \ref{lem:opFourier} and Proposition \ref{prop:RCopnorm}, we obtain
$$
\Vert R_\ell\Vert^2_{\mathrm{op}}=\frac1{r_\ell(\lambda',\lambda'')}\left\Vert
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}
\right\Vert^2_{\mathrm{op}}= c_\ell(\lambda',\lambda'').
$$
Since
$R_\ell^*=i^\ell\; \Phi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$ by Proposition \ref{lem:2},
Theorem \ref{thm:FRCinv} follows from Lemma \ref{lem:Hilbertdeco} (1).
\end{proof}

\subsection{Some applications of symmetry breaking and holographic transforms}\label{sec:applications}

We point out two applications of the symmetry breaking and holographic transforms introduced in the previous
section. First, we provide explicit description of the minimal $K$-types of the ${SL(2,\R)}{\;}\widetilde{}$-module
$(\pi_\lambda,\mathcal O(\Pi))$ in both holomorphic
model $\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$
 and $L^2$-model $L^2(\R_+^2)_{\lambda',\lambda''}$ (see Propositions
\ref{cor:Ktypeholom} and \ref{cor:KtypeL2}). Second, we find in Theorem \ref{thm:180297} an integral expression of any eigenfunction
for a specific second-order holomorphic partial differential operator arising from the diagonal action of the Casimir in the enveloping algebra.

\subsubsection{Minimal $K$-types}

The minimal $K$-type of the ${SL(2,\R)}{\;}\widetilde{}$-module $(\pi_\lambda,\mathcal O(\Pi))$
is given by $\C (\zeta+i)^{-\lambda}$, see \eqref{eqn:piKtype} for the whole set of $K$-types.
 As an application of the integral formula \eqref{eqn:psi} we find an explicit expression for the minimal $K$-types of submodules in the tensor product $\pi_{\lambda'}\widehat\otimes\pi_{\lambda''}$ as follows.

\begin{prop}\label{cor:Ktypeholom}
Suppose ${\mathrm{Re}\;\lambda',\mathrm{Re}\;\lambda''}>0$ and $\lambda'''=\lambda'+\lambda''+2\ell\,(\ell\in\N)$. Then the holomorphic function
$$
(\zeta_1,\zeta_2)\mapsto (\zeta_1-\zeta_2)^\ell(\zeta_1+i)^{-\lambda'-\ell}(\zeta_2+i)^{-\lambda''-\ell}
$$
is a minimal $K$-type in the submodule $\Psi_{\lambda',\lambda''}^{\lambda'''}(\mathcal O(\Pi))$ in $\pi_{\lambda'} \widehat\otimes\pi_{\lambda''}$.
\end{prop}

\begin{proof}
We set $g(\zeta):= (\zeta+i)^{-\lambda'''}$. By the change of variables $t=\frac12(1+v)$, the definition \eqref{eqn:psi} shows
$$
\left(\Psi_{\lambda',\lambda''}^{\lambda'''} g\right) (\zeta_1,\zeta_2)
=\frac1{\ell!} (\zeta_1-\zeta_2)^\ell (\zeta_1+i)^{-\lambda'''}
\int_0^1t^{a-1}(1-t)^{c-a-1}(1-tz)^{-b}dt,
$$
where $a=\lambda''+\ell, b=c=\lambda'''$, and $z=\frac{\zeta_1-\zeta_2}{\zeta_1+i}$.

By the Euler integral  representation of the hypergeometric function ${}_2F_1$, and by the fact that
${}_2F_1(a,b;b;z)=(1-z)^{-a}$, we obtain
\begin{equation}\label{eqn:181223}
\left(\Psi_{\lambda',\lambda''}^{\lambda'''} g\right) (\zeta_1,\zeta_2)
=\frac1{\ell!}
B(\lambda'+\ell,\lambda''+\ell) (\zeta_1-\zeta_2)^\ell
(\zeta_1+i)^{-\lambda'-\ell} (\zeta_2+i)^{-\lambda''-\ell},
\end{equation}
where $B(\cdot,\cdot)$ stands for the Euler beta function.
\end{proof}

\begin{prop}\label{cor:KtypeL2}
Suppose $\lambda',\lambda''>1$ and $\ell\in\N$. Then
the function
$$
(x,y)\mapsto (x^{\lambda'-1}e^{-x})(y^{\lambda''-1}e^{-y})(x+y)^\ell P_\ell^{\lambda'-1,\lambda''-1}\left(\frac{y-x}{x+y}\right)
$$
belongs to $L^2(\R_+^2)_{\lambda',\lambda''}$, and gives a minimal $K$-type in the
image of the holographic operator $\Phi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$.
\end{prop}
\begin{proof}
Since $z^{\lambda'''-1}e^{-z}$ belongs to the minimal $K$-type in the irreducible re\-presentation $L^2(\R_+)_{\lambda'''}$, so does
$\Phi_{\lambda',\lambda''}^{\lambda'''}(z^{\lambda'''-1}e^{-z})$ in the irreducible
representation $\Phi_{\lambda',\lambda''}^{\lambda'''}(L^2(\R_+^2)_{\lambda',\lambda''})$. Then
  the formula \eqref{eqn:defPhi}
of the holographic operator $\Phi_{\lambda',\lambda''}^{\lambda'''}$ with
$\lambda'''=\lambda'+\lambda''+2\ell$ shows Proposition \ref{cor:KtypeL2}.
\end{proof}

\subsubsection{An application of the integral formula}

Fix $\lambda',\lambda''\in\C$ and consider eigenfunctions of the following holomorphic differential operator on $\Pi\times\Pi$:
\begin{equation}\label{eqn:diagCa}
P_{\lambda',\lambda''}:=(\zeta_1-\zeta_2)^2\frac{\partial^2}{\partial\zeta_1\partial\zeta_2}+
(\lambda''\zeta_2+\lambda'-\lambda'')\zeta_1\frac{\partial}{\partial\zeta_1}+
(\lambda'\zeta_1-\lambda'+\lambda'')\zeta_2\frac{\partial}{\partial\zeta_2},
\end{equation}
and define for $\mu\in\C$
$$
\mathcal{S}o\ell(\Pi\times\Pi, \mathcal M_{\lambda',\lambda'',\;\mu}):=\left\{f\in\mathcal O(\Pi\times\Pi)\colon P_{\lambda',\lambda''}f=\mu f\right\}.
$$
The integral transform \eqref{eqn:psi} constructs all eigenfunctions of $P_{\lambda',\lambda''}$ as follows.

\begin{thm}\label{thm:180297}
Suppose $\lambda',\lambda''\in\C$. Then, the following hold.
\begin{itemize}
\item[(1)] $\mathcal{S}o\ell(\Pi\times\Pi, \mathcal M_{\lambda',\lambda'',\;\mu})\neq\{0\}$ if and only if $\mu$ is of the form
$$
\mu=-\ell(\lambda'+\lambda''+\ell-1) \quad\mathrm{for}\,\,\mathrm{some}\,\,\ell\in\N.
$$
\item[(2)] For any $\lambda', \lambda''\in\C$ and $\ell\in\N$,
$$
(\zeta_1-\zeta_2)^\ell (\zeta_1+i)^{-\lambda'-\ell}(\zeta_2+i)^{-\lambda''-\ell}
\in
\mathcal{S}o\ell (\Pi\times\Pi, \mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)}).
$$

\item[(3)] If ${\mathrm{Re}\;\lambda',\mathrm{Re}\;\lambda''}>0$ and $\ell\in\N$, then
 the integral transform \eqref{eqn:psi} gives a bijection
$$
\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}
\colon
\mathcal O(\Pi)
\stackrel
{\sim}{\longrightarrow}
\mathcal{S}o\ell(\Pi\times\Pi, \mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)}).
$$
The inverse map is proportional to the Rankin--Cohen bidifferential operator, namely,
$$
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}
\circ  \Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}= c_\ell(\lambda',\lambda'')\mathrm{id}\quad
\mathrm{on}\,\, \mathcal O(\Pi),
$$
where $c_\ell(\lambda',\lambda'')$ is defined as in \eqref{eqn:v-ell}.
\end{itemize}
\end{thm}

\subsubsection{Quick review of representations of the universal covering group $SL(2,\R){\;}\widetilde{}$}

In order to prove Theorem \ref{thm:180297} we recall some properties of representations of ${SL(2,\R)}{\;}\widetilde{}$.
The universal covering group ${SO(2)}{\;}\widetilde{}$ of the maximal compact subgroup $K=SO(2)$ is isomorphic to $\R$.
We parametrize its characters $\chi_{\lambda}$
 by $\lambda \in \C$
 as an extension of the following group homomorphisms
 originally defined for $\lambda\in\Z$:
$$
\R\simeq{SO(2)}{\;}\widetilde{}\longrightarrow SO(2)\longrightarrow\C^\times,\quad \theta\mapsto
\left(\begin{matrix}
\cos\theta&-\sin\theta\\\sin\theta&\cos\theta
\end{matrix}\right)
\mapsto e^{i\lambda\theta}.
$$
The representation $\pi_\lambda$ on $\mathcal O(\Pi)$ given in Section \ref{subsec:pilmd} is a highest weight module with highest weight $-\lambda$ because it has the following $K$-types:
\begin{equation}\label{eqn:piKtype}
-\lambda,-\lambda-2,-\lambda-4,\ldots.
\end{equation}
Choose the standard basis of the Lie algebra $\mathfrak{sl}(2,\R)$:
$$
H:=
\left(\begin{matrix} 1&0\\0&-1\end{matrix}\right),\quad
X:=\left(\begin{matrix} 0&1\\0&0\end{matrix}\right),\quad
Y:=\left(\begin{matrix} 0&0\\1&0\end{matrix}\right).
$$
Then the Casimir element $C$ is expressed as
$
C=\frac18( H^2+2XY+2YX).
$

The infinitesimal action $d\pi_\lambda$ is given by holomorphic differential operators:
\begin{equation}\label{eqn:sl2vect}
d\pi_\lambda(H)=-\lambda-2z\frac{d}{dz},\quad
d\pi_\lambda(X)=-\frac{d}{dz},\quad
d\pi_\lambda(Y)=\lambda z +z^2\frac{d^2}{dz^2},
\end{equation}
and the Casimir element $C$ acts on $(d\pi_\lambda,\mathcal O(\Pi))$ as $d\pi_\lambda(C)=\frac18\lambda(\lambda-2)\mathrm{id}$.
In general,
 if $\pi$ is a highest weight module of ${SL(2,\R)}{\;}\widetilde{}\,\,$ with highest weight $\nu$ ($\nu\in\C$), then the
Casimir element is given via $d\pi$ as the scalar
multiplication $\frac18\nu(\nu+2)\mathrm{id}$.

\subsubsection{Proof of Theorem \ref{thm:180297}}\label{subsec:intsol}
\begin{lem}\label{lem:diagCasimir}
The Casimir element $C$ of $\mathfrak{sl}(2,\R)$ acts
 on $\mathcal O(\Pi \times \Pi)$ via $d\pi_{\lambda'}\otimes d\pi_{\lambda''}$
 by
$$
\left(d\pi_{\lambda'}\otimes d\pi_{\lambda''}\right)(\mathrm{diag(C)})=-\frac12 P_{\lambda',\lambda''}
+\frac18(\lambda'+\lambda'')(\lambda'+\lambda''-2),
$$
where the holomorphic differential operator $P_{\lambda',\lambda''}$ is defined in \eqref{eqn:diagCa}.
\end{lem}
\begin{proof}
By \eqref{eqn:sl2vect} and the Leibniz rule,
 we have
\begin{eqnarray*}
\left(d\pi_{\lambda'}\otimes d\pi_{\lambda''}\right)(\mathrm{diag(H)})
&=&
-\lambda'-\lambda''-2\left(\zeta_1\frac{\partial}{\partial\zeta_1}+\zeta_2\frac{\partial}{\partial\zeta_2}\right),\\
\left(d\pi_{\lambda'}\otimes d\pi_{\lambda''}\right)(\mathrm{diag(X)})
&=&
-\frac{\partial}{\partial\zeta_1}-\frac{\partial}{\partial\zeta_2},\\
\left(d\pi_{\lambda'}\otimes d\pi_{\lambda''}\right)(\mathrm{diag(Y)})
&=&
\lambda'\zeta_1+\lambda''\zeta_2+\left(\zeta_1^2\frac{\partial^2}{\partial\zeta_1^2}+\zeta_2^2\frac{\partial^2}{\partial\zeta_2^2}\right).
\end{eqnarray*}
Now the lemma follows by a direct computation.
\end{proof}
The tensor product representation $\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}$ does not always split into a direct sum of irreducible representations in the nonunitary case when $\lambda',\lambda''\in\C$, see \cite{KP16b} for instance.
We determine the set of possible infinitesimal characters
 of subrepresentations of the tensor product $\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}$ in this case.

\begin{lem}\label{lem:subtensor}
Let $\lambda',\lambda''\in\C$.
Suppose $\pi$ is a subrepresentation  of $\pi_{\lambda'} \widehat\otimes \pi_{\lambda''}$ such that the Casimir element $C$ acts as scalar multiplication via $d\pi$.
Then this scalar must be of the form
$$
\frac18 (\lambda'+\lambda''+2\ell)(\lambda'+\lambda''+2\ell-2)\quad\mathrm{for}\,\mathrm{some}\,\,\ell\in\N.
$$
\end{lem}

\begin{proof}
We use the general theory of discretely decomposable restrictions
 of (nonunitary) representations \cite{kdeco98}.
First we observe from \eqref{eqn:piKtype}
 that the $K$-types of the tensor product representation $\pi_{\lambda'} \widehat\otimes \pi_{\lambda''}$ are of the form
$$
-\lambda'-\lambda''-2(\ell'+\ell'') \quad\mathrm{for}\,\,\mathrm{some}\,\,\ell',\ell''\in\N.
$$
Thus the tensor product representation $\pi_{\lambda'}\widehat\otimes \pi_{\lambda''}$ on $\mathcal O(\Pi\times\Pi)$
contains the direct sum of $K$-isotypic spaces
\begin{equation}\label{eqn:Ktensor}
\bigoplus_{\ell\in\N}(\ell+1)\chi_{-\lambda'-\lambda''-2\ell}
\end{equation}
as a dense subset,
 where $(\ell +1)$ stands for the multiplicity.

In particular, each $K$-type occurs in $\pi_{\lambda'} \widehat\otimes \pi_{\lambda''}$ with at most finite multiplicities.
Hence,
 any subrepresentation $\pi$ is admissible,
 and of highest weight $-\lambda'-\lambda''-2\ell$ for some $\ell\in\N$.
Therefore,
 if the Casimir element acts as a scalar via $d\pi$,
 then this scalar must coincide with
 $\frac18 (\lambda'+\lambda''+2\ell)(\lambda'+\lambda''+2\ell-2)$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:180297}]
(1) By Lemma \ref{lem:diagCasimir}, $\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',\mu})$ is characterized as the eigen\-space of the Casimir operator $C$ as follows:
\begin{multline}\label{eqn:SolCas}
\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',\mu})\cr
=\{f\in\mathcal O(\Pi\times\Pi)\colon
\left(d\pi_{\lambda'}\otimes d\pi_{\lambda''}\right)(\mathrm{diag}(C))=e(\lambda',\lambda'',\mu) f\},
\end{multline}
where we set
$e(\lambda',\lambda'',\mu)=-\frac12\mu+\frac18(\lambda'+\lambda'')(\lambda'+\lambda''-2)$.

On the other hand, Lemma \ref{lem:subtensor} tells that $e(\lambda',\lambda'',\mu)=
\frac18 (\lambda'+\lambda''+2\ell)(\lambda'+\lambda''+2\ell-2)$ for some $\ell\in\N$.
This gives the desired formula for $\mu$, showing the ``if\," part of the first statement. The ``only if\," part
follows from the second statement.
\vskip5pt

(2) By the assumption $\mathrm{Re}\,\lambda', \mathrm{Re}\,\lambda''>0$, the integral \eqref{eqn:psi}
converges for any $\ell\in\N$.
Since the Casimir element $C$ acts on $\mathcal O(\Pi)$
 as the scalar
$\frac18\lambda'''(\lambda'''-2)$ via $d\pi_{\lambda'''}$,
 and since $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is an intertwining operator,
the Casimir element $C$ acts on the image of $\Psi_{\lambda',\lambda''}^{\lambda'''}$ by the same scalar.
Therefore $\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}(\mathcal O(\Pi))\subset\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)})$ by \eqref{eqn:SolCas}. In turn, Proposition \ref{cor:Ktypeholom} and Lemma \ref{lem:diagCasimir} imply
that
$$
\left(
P_{\lambda',\lambda''} +\ell(\lambda'+\lambda''+\ell-1)
\right)
\left(
(\zeta_1-\zeta_2)^\ell (\zeta_1+i)^{-\lambda'-\ell}(\zeta_2+i)^{-\lambda''-\ell}
\right)
=0
$$
as far as $\mathrm{Re}\,\lambda', \mathrm{Re}\,\lambda''>0$. By the analytic continuation, the equation holds for all $\lambda',\lambda''\in\C$.

(3) We begin with the case where
$\lambda'$ and $\lambda''$ are real and $\lambda',\lambda''>1$. Then it follows from Proposition
\ref{prop:RCopnorm}, Lemma \ref{lem:Hilbertdeco} (2) and Proposition \ref{prop:PsiRC} that
$$
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\circ\Psi_{\lambda',\lambda''}^{\lambda'''}=
c_\ell({\lambda',\lambda''})\mathrm{id}\quad\mathrm{on}\;{\mathcal H^2(\Pi)_{\lambda'''}}.
$$
Since ${\mathcal{H}}^2(\Pi)_{\lambda'''}$ is dense in ${\mathcal{O}}(\Pi)$,
 the equality holds on
 the whole $\mathcal O(\Pi)$ by continuity.

Moreover,
 since the operators $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ and $\Psi_{\lambda',\lambda''}^{\lambda'''}$ depend
holomorphically on $(\lambda',\lambda'')\in\C^2$ satisfying \eqref{eqn:HOconverge}, we conclude that
\begin{equation}\label{eqn:RCPsic}
\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\circ\Psi_{\lambda',\lambda''}^{\lambda'''}=
c_\ell({\lambda',\lambda''})\mathrm{id}\quad\mathrm{on}\;{\mathcal O(\Pi)}
\end{equation}
for any $(\lambda',\lambda'',\lambda''')$
 subject to \eqref{eqn:HOconverge}
 by analytic continuation.
In particular, $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is injective and $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ is surjective because $c_\ell(\lambda',\lambda'')\neq0$ in this case.

Let us prove that $ \Psi_{\lambda',\lambda''}^{\lambda'''}\colon \mathcal O(\Pi)\longrightarrow
\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)})$
is surjective. We let ${SL(2,\R)}{\;}\widetilde{}$ act on $\mathcal O(\Pi\times\Pi)$ via $\pi_{\lambda'}\widehat\otimes
\pi_{\lambda''}$.
 Since the map $\N\longrightarrow\C,\, \ell\mapsto -\ell(\lambda'+\lambda''+\ell-1)$ is injective
if  $\mathrm{Re}\,\lambda', \mathrm{Re}\,\lambda''>0$,
 we have the following inclusion of ${SL(2,\R)}{\;}\widetilde{}$ -submodules of $\mathcal O(\Pi\times\Pi)$:
$$
\bigoplus_{\ell\in\N} \Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}
\left(\mathcal O(\Pi) \right)\subset
\bigoplus_{\ell\in\N}\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)}).
$$

We observe that $K$-multiplicities coincide by \eqref{eqn:Ktensor} because the holographic operator $\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}$ is injective for any $\ell\in\N$.
Therefore,
$$\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell} \colon \mathcal O(\Pi)\longrightarrow
\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)})$$
is surjective on the level of $(\mathfrak g, K)$-modules.

Since $c_\ell(\lambda',\lambda'')\neq0$,  the symmetry breaking operator $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$
is injective on $\Psi_{\lambda',\lambda''}^{\lambda'+\lambda''+2\ell}\left(\mathcal O(\Pi)\right)$
by \eqref{eqn:RCPsic}, and therefore the underlying  $(\mathfrak g, K)$-module of
$\mathrm{Ker}\left(\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}\right)\cap
\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)})$ must be zero for any $\lambda'''=\lambda'+\lambda''+2\ell$
($\ell\in\N$). Hence $\mathcal{RC}_{\lambda',\lambda''}^{\lambda'''}$ is injective
when restricted to $\mathcal So\ell(\Pi\times\Pi,\mathcal M_{\lambda',\lambda'',-\ell(\lambda'+\lambda''+\ell-1)})$. Now we conclude that $\Psi_{\lambda',\lambda''}^{\lambda'''}$ is surjective.
Thus the theorem is proved.
\end{proof}

\section{Holomorphic Juhl transform and its holographic transform}
\label{sec:HJT}

Another remarkable family of differential operators is provided by conformally covariant differential operators for the pair $S^n\supset S^{n-1}$ of Riemannian manifolds, introduced by Juhl \cite{Juhl}.
These operators are symmetry breaking operators from spherical principal series representations
of the Lorentz group $O(1,n+1)$ to those of the subgroup $O(1,n)$.
This setting is intimately related to the holographic or AdS/CFT correspondence in string theory (see \emph{e.g}. \cite{Ma,W1}).

The \emph{holomorphic Juhl operators} are the holomorphic continuation of Juhl's operators, which map holomorphic functions on the $n$-dimensional Lie ball to those on the $(n-1)$-dimensional Lie ball,
intertwining (relative) discrete series representations of $G=SO_o(2,n)$
with those of the subgroup $G'=SO_o(2,n-1)$, see \cite{KP16b}.

In this section we solve Problems A and \ref{prob:B} stated in Section \ref{sec:Intro}
 for the symmetry breaking transform
 associated with the holomorphic Juhl operators.
We assume $n\geq 3$ throughout this section. The case
$n=2$ can be recovered from the previous section with an appropriate change of parameters.

\subsection{Holomorphic Juhl operators}
\subsubsection{Holomorphic discrete series of $SO_o(2,n)$}
Let $Q_{p,q}$ be the standard quadratic form of signature $(p,q)$ on $\R^{p+q}$.
The indefinite orthogonal group
$$
O(p,q):=\left\{ g\in GL(p+q,\R) : Q_{p,q}(gx)=Q_{p,q}(x)\,\mathrm{for\,all}\,x\in\R^{p+q}
             \right\}
$$
has four connected components when $p,q>0$.
Let $G=SO_o(2,n)$ be the identity component of $O(2,n)$ and $K$ a maximal compact subgroup of $G$.
We write $\mathfrak c(\mathfrak k)$ for the first factor of the Lie algebra
$\mathfrak k\simeq \R\oplus \mathfrak{so}(n)$, and
fix a characteristic element $H_0\in \mathfrak c(\mathfrak k)$ such that ad$(H_0)$ gives
the eigenspace decomposition of $\mathfrak g_\C= \mathrm {Lie}(G)\otimes_\R\C$ as
\begin{equation}\label{eqn:gknn}
\mathfrak g_\C=\mathfrak k_\C+\mathfrak n_++\mathfrak n_-
\end{equation}
for eigenvalues $0,-i$ and $i$, respectively.
The complex structure
 of the homogeneous space $G/K$ is
 given by the $G$-translation of $\exp\left(\mathrm{ad}\left(\frac\pi2 H_0\right)\right)\in GL_\R(T_o(G/K))$, or
 equivalently, it is
  induced from the Borel embedding $G/K\subset
 G_\C/K_\C\exp\mathfrak n_+$.

Let $\widetilde G$ be the universal covering of $G=SO_o(2,n)$, $p:\widetilde G\longrightarrow G$ the covering homomorphism, and set $\widetilde K:=p^{-1}(K)$.
For $\lambda\in\C$, we define a character of  $\mathfrak c(\mathfrak k)$ by
$tH_0\mapsto\lambda t$, which lifts  to a character $\C_\lambda$ of $\widetilde K$.

 Then one can define a $\widetilde G$-equivariant holomorphic line bundle $\mathcal L_\lambda=\widetilde G\times_{\widetilde K}\C_\lambda$ over $X:=\widetilde G/\widetilde K\simeq G/K$ for all $\lambda\in\C$, and
obtain representations $\pi_\lambda^{(n)}$ of $\widetilde G$ on the space $\mathcal O(X,\mathcal L_\lambda)$ of holomorphic sections. The representation $\pi_\lambda^{(n)}$ descends to $G$ if $\lambda\in\Z$.

If $\lambda\in\R$, then the line bundle $\mathcal L_\lambda\longrightarrow X$ carries a $\widetilde G$-invariant Hermitian metric, and therefore we can define a Hilbert space $\left( \mathcal O\cap L^2\right)(X,\mathcal L_\lambda)$. This Hilbert space  is nonzero if and only if $\lambda>n-1$, and the resulting unitary representation of $\widetilde G$, to be denoted by the same symbol $\pi_\lambda^{(n)}$, is called a \emph{(relative) holomorphic discrete series representation} of $\widetilde G$. For
actual computations we use its realization in the weighted Bergman space as below.

We define the tube domain
$$
T_\Omega\equiv T_{\Omega(n)}:=\R^n+ i\Omega(n)
$$
by taking $\Omega(n)$ to be the time-like cone in the Minkowski space $\R^{1,n-1}$, namely,
$$
\Omega(n):=\left\{ \eta\in\R^n\;:\;Q_{1,n-1}(\eta)>0,\; \eta_1>0\right\}.
$$

Then the tube domain $T_\Omega$ is biholomorphic to the bounded symmetric domain  of type $\mathrm{IV}_n$, sometimes
referred to as the Lie ball. From a group-theoretic viewpoint, $T_\Omega$ is isomorphic
to the Hermitian symmetric space $X=G/K$, which is realized as an open subset of
$\mathfrak n_- (\simeq\C^n)$ via the following maps
\begin{equation}\label{eqn:BorelGK}
\mathfrak n_-\underset{\mathrm{open}}{\hookrightarrow}
G_\C/K_\C\exp(\mathfrak n_+)\underset{\mathrm{open}}{\supset}G/K.
\end{equation}

The homogeneous holomorphic line bundle $\mathcal L_\lambda\longrightarrow X$ is trivialized via
the Bruhat decomposition, and the Hilbert space $(\mathcal O\cap L^2)(X,\mathcal L_\lambda)$
is then identified with the weighted Bergman space
$$
\mathcal H^2(T_{\Omega(n)})_\lambda:=\mathcal O(T_{\Omega(n)})\cap
L^2(T_{\Omega(n)}, Q_{1,n-1}(\eta)^{\lambda-n}d\xi d\eta),
$$
on which $G$ acts as a multiplier representation
by
$$
f(\zeta)\mapsto b_\lambda (g,\zeta)f(g^{-1}.\zeta)
$$
for $g\in G$ and $f(\zeta)\in\mathcal O(T_{\Omega(n)})$. Here the multiplier
$$
b_\lambda: G\times T_{\Omega(n)}\longrightarrow \C^\times
$$
is a 1-cocycle defined by $b_\lambda(g,\zeta):=\chi_{-\lambda}(k(g,\zeta))$, where
$\chi_\lambda \colon K_\C\longrightarrow \C^\times$ is the holomorphic extension of the character $\C_\lambda$ of $K$ and $k(g,\zeta)$ is an element of $K_\C$ determined by
$$
g^{-1}\exp(\zeta)\in \exp(g^{-1}.\zeta)k(g,\zeta)\exp (\mathfrak n_+),
$$
see \eqref{eqn:BorelGK}.

For $\lambda>n-1$ this Hilbert space admits a reproducing kernel $K_\lambda(\zeta,\tau)$ given
by:
\begin{equation}\label{eqn:Klmbd}
K_\lambda(\zeta,\tau)= k_{\lambda,n}Q_{1,n-1}\left({\zeta-\overline \tau}\right)^{-\lambda},
\end{equation}
where $Q_{1,n-1}(\zeta)$ stands for the
holomorphic extension of
$
Q_{1,n-1},$ and
 we set
$$
k_{\lambda,n}:=\frac{(2i)^{2\lambda}}{(4\pi)^{n}}\frac{\left(\lambda-\frac n2\right)\Gamma(\lambda)}{\Gamma(\lambda-n+1)},
$$
see \cite[Prop.~XIII.1.2]{FK}.
We note that $k_{\lambda,n}\neq0$ if $\lambda>n-1$.

We realize $O(2,n-1)$ as the subgroup of $O(2,n)$
 which fixes the $(n+2)$-th coordinate,
 and set $G'=SO_o(2,n-1)$
 as its identity component.
 By abuse of notation, we write $\widetilde G'$ for the connected
 subgroup of $\widetilde G=SO_o(2,n){\;}\widetilde{}$\,
 corresponding to $G'\subset G$. The subgroup $\widetilde G'$ is simply connected if $n\neq4$.
  Similarly, a
 (relative) holomorphic discrete series representation $\pi_\nu^{(n-1)}$
of $\widetilde G'$
 is defined for $\nu >n-2$,
 as an irreducible unitary representation
 on the weighted Bergman space $\mathcal H^2(T_{\Omega(n-1)})_\nu$.
 By abuse of notation, the same symbol $\pi_\nu^{(n-1)}$ will be used to denote a (nonunitary) representation
 on $\mathcal O(T_{\Omega(n-1)},\mathcal L_\nu)$
 for $\nu \in {\mathbb{C}}$.

\subsubsection{Holomorphic Juhl operators}

Let $\Delta_{\C^{1,n-2}}:=\frac{\partial^2}{\partial\zeta_1^2}-\frac{\partial^2}{\partial\zeta_2^2}
 -\cdots-\frac{\partial^2}{\partial\zeta_{n-1}^2}$ be the holomorphic Laplacian on $\C^{n-1}$
 associated to the complexified quadratic form $Q_{1,n-2}$.
For $\alpha \in {\mathbb{C}}$
 and $\ell, k \in {\mathbb{N}}$
 with $\ell \ge 2 k$,
 we define a polynomial of $\alpha$
 of degree $\ell-k$
by
\begin{equation}\label{eqn:alalpha}
a_k(\ell,\alpha):=
\frac{(-1)^k 2^{\ell-2k} \cdot \Gamma\left(\alpha+\ell-k\right)}
{\Gamma\left(\alpha\right) k! (\ell-2k)!}.
\end{equation}
 The coefficients $a_k(\ell,\alpha)$ appear in the definition \eqref{eqn:gegen} of the Gegenbauer
 polynomials $C_\ell^\alpha(x)$, see Appendix.
We define a holomorphic differential operator
$\mathcal D_\ell^\alpha$ on $\C^n$ by
\begin{equation}\label{eqn:JuhlonCn}
\mathcal D_\ell^\alpha:=\sum_{k=0}^{\left[\frac\ell2\right]}
a_k\left(\ell,\alpha\right)
 \left(\frac{\partial}{\partial\zeta_n}\right)^{\ell-2k}\Delta^k_{\C^{1,n-2}}.
\end{equation}

For $\lambda,\nu\in \C$ with $\ell:=\nu-\lambda\in\N$,
 the holomorphic Juhl operator $D_{\lambda\to\nu} \colon \mathcal O(T_{\Omega(n)})\longrightarrow
\mathcal O(T_{\Omega(n-1)})$
 is defined as the composition
\begin{eqnarray}\label{eqn:holoJuhl}
D_{\lambda\to\nu}:=\mathrm{Rest}_{\zeta_n=0}\circ
\mathcal D_\ell^{\lambda-\frac{n-1}2}.
 \end{eqnarray}

The operator $D_{\lambda\to\nu}$ may be viewed as the holomorphic extension
 of the original Juhl operator \cite{Juhl},
 which is a conformally covariant differential operator
 $C^{\infty}(S^{n})\to C^{\infty}(S^{n-1})$.
In our setting,
 the hyperbolic space $H^n$ is realized
 as a totally real submanifold of the tube domain $T_{\Omega(n)}$, and
likewise, $H^{n-1}$ is that of $T_{\Omega(n-1)}$.
The restriction of the holomorphic Juhl operator to these real manifolds
 also yields
a conformally covariant operator
$C^\infty(H^n)\longrightarrow C^\infty(H^{n-1})$,
see \cite[Thm.~E]{KKPAdS}.

The holomorphic Juhl operator $D_{\lambda\to\nu}$ gives yet another symmetry breaking operator,
intertwining the (relative) holomorphic discrete series representation $\pi_\lambda^{(n)}$ of $\widetilde G$ and the one $\pi_{\nu}^{(n-1)}$
of the subgroup $\widetilde G'$ \cite[Thm.~6.3]{KP16b}.
Moreover, the differential operator $D_{\lambda\to\nu}$ induces a \emph{continuous} map
between the Bergman spaces by the general theory \cite[Thm.~5.13]{KP16a}.
Its adjoint is denoted by $D_{\lambda\to\nu}^*$.
We determine the operator norm of $D_{\lambda \to \nu}$
 in Proposition \ref{prop:normJuhl}.

\subsection{Two constants $c_\ell(\lambda)$ and $r_\ell(\lambda)$}\label{sec:const-conf}

Throughout Section \ref{sec:HJT}
the parameter set is $(\lambda,\nu)\in\C^2$ with $\nu-\lambda\in\N$ and $n\geq3$.
We use the following notation:
\begin{equation}
\label{eqn:alphal}
\alpha=\lambda-\frac{n-1}2,\qquad \ell=\nu-\lambda.
\end{equation}

As in Section \ref{sec:constants} devoted to the tensor product case, the main results in this section involve the following two constants:
\begin{eqnarray}\label{eqn:c-ell}
c\equiv c_\ell(\lambda)&:=& \int_{-1}^1\left\vert C_\ell^\alpha(v)\right\vert^2 (1-v^2)^{\alpha-\frac12}dv\nonumber\\
&=&  \frac{\pi 2^{n-2\lambda} \Gamma(2\lambda+\ell-n+1)}{\ell!\left(\lambda+\ell-\frac{n-1}2\right)
\Gamma\left(\lambda-\frac{n-1}2\right)^2},\\
\label{eqn:r-ell}
r\equiv r_\ell(\lambda)&:=&\frac{b_{n-1}(\nu)}{b_n(\lambda)}\nonumber\\
&=&
\frac{\Gamma\left(\lambda+\ell-\frac{n-1}2\right) \Gamma\left(\lambda+\ell-n+2\right)}
{(2\pi)^{\frac32}2^{2\ell+1} \Gamma\left(\lambda-\frac n2\right)\Gamma\left(\lambda-n+1\right)},
\end{eqnarray}
where
$b_n(\lambda)$ is a Plancherel density (see \eqref{eqn:bnlmd} below).

\subsection{Holomorphic Juhl transforms}

\begin{defn}[holomorphic Juhl transform]
For $\lambda\in\C$, the holomorphic Juhl transform $D_\lambda$
 is a collection of the holomorphic Juhl operators
$$
D_\lambda \colon \mathcal O(T_{\Omega(n)})\longrightarrow
\operatorname{Map}(\mathbb N, \mathcal O(T_{\Omega(n-1)})),
\quad
f \mapsto \{(D_{\lambda} f)_{\ell}\}_{\ell \in {\mathbb{N}}},
$$
where $ (D_\lambda f)_\ell:=D_{\lambda\to\lambda+\ell} f$.
\end{defn}
The holomorphic Juhl transform $D_\lambda$ intertwines $(\pi_\lambda^{(n)},\mathcal O(T_{\Omega(n)}))$ with the formal direct sum $\widehat\bigoplus_{\ell\in\N}(\pi_{\lambda+\ell}^{(n-1)},\mathcal O(T_{\Omega(n-1)}))$; its
 inversion formula and the corresponding Parseval--Plancherel type theorems
 are given as follows.

\begin{thm}\label{thm:170887}
Suppose $\lambda>n-1$.
\begin{itemize}
\item[(1)] \emph{(inversion formula).}
 Any $f\in \mathcal H^2(T_{\Omega(n)})_\lambda$ is recovered from
 $D_{\lambda} f$ by
$$
f=\sum_{\ell=0}^\infty \frac1{r_\ell(\lambda)c_\ell(\lambda)}
D_{\lambda\to{\lambda+\ell}}^* \left(D_\lambda f\right)_\ell.
$$
\item[(2)]
\emph{(Parseval--Plancherel type theorem).}
For every $f\in \mathcal H^2(T_{\Omega(n)})_\lambda$, we have
\begin{equation}
\Vert f\Vert^2_{\mathcal H^2_\lambda(T_{\Omega(n)})}=
\sum_{\ell=0}^\infty \frac1{r_\ell(\lambda)c_\ell(\lambda)}
\Vert \left(D_{\lambda}f\right)_\ell\Vert^2_{\mathcal H^2_{\lambda+\ell}(T_{\Omega(n-1)})}.
\end{equation}
\end{itemize}
\end{thm}
Theorem \ref{thm:170887} is proved in Section \ref{subsec:proofTHM32}. It gives an answer to Problems A.2 and B raised in Section 1 for the holomorphic Juhl transform $D_{\lambda}$.
An answer to Problem A.1 (explicit integral formula for holographic transform) will be given in Theorem \ref{thm:CIRM18}.

From a representation-theoretic viewpoint, Theorem \ref{thm:170887} gives quantitative information on the corresponding branching law for the restriction of the (relative) holomorphic discrete series representation $\pi_\lambda^{(n)}$ of $\widetilde G$ to the subgroup $\widetilde G'$, which decomposes the restriction $\pi_\lambda^{(n)}\big\vert_{\widetilde G'}$
 into a multiplicity-free direct Hilbert sum of irreducible representations of the subgroup $\widetilde G'$, see \cite[Thm.~8.3]{K08}:
\begin{equation}\label{eqn:absbraConf}
\pi_\lambda^{(n)}\vert_{\widetilde G'}\simeq {\sum_{\ell\in\N}}^\oplus \pi_{\lambda+\ell}^{(n-1)}.
\end{equation}

\begin{cor}[projection operator]
Suppose $\lambda>n-1$ and $\ell:=\nu-\lambda\in\N$. Then
$$
\frac1{r_\ell(\lambda) c_\ell(\lambda)}D_{\lambda\to\nu}^*D_{\lambda\to\nu}
$$
is the projection operator from the Hilbert space $\mathcal H^2(T_{\Omega(n)})_\lambda$
onto the summand which is unitarily equivalent to the irreducible representation $(\pi_\nu^{(n-1)},\mathcal H^2(T_{\Omega(n-1)})_\nu)$,
see \eqref{eqn:absbraConf}.
\end{cor}

$$
$$

\subsection{Key operators in the proof of Theorem \ref{thm:170887}}
~~~\par
\vskip10pt

Analogously to the case of the tensor product representations (Section \ref{sec:RCT}), we introduce the following continuous operators for the proof of Theorem \ref{thm:170887}:
\begin{alignat*}{4}
D_{\lambda\to\nu}^*& \colon \mathcal H^2(T_{\Omega(n-1)})_\nu && \longrightarrow && \mathcal
 H^2(T_{\Omega(n)})_\lambda
\qquad
&& \mathrm{adjoint\, of}\,\, D_{\lambda\to\nu},\\
\widehat D_{\lambda\to\nu}&\colon L^2(\Omega(n))_\lambda &&\longrightarrow&& L^2(\Omega(n-1))_\nu
\qquad &&\mathrm{Fourier\,transform\, of}\,\, D_{\lambda\to\nu},\\
\Phi_\lambda^\nu&\colon L^2(\Omega(n-1))_\nu &&\longrightarrow&&  L^2(\Omega(n))_\lambda
\qquad &&\mathrm{holographic\, operator}.
\end{alignat*}
See \eqref{eqn:JFourier} and \eqref{eqn:hologSO} below for the definitions of $\widehat D_{\lambda\to\nu}$  and $\Phi_\lambda^\nu$, respectively.
\vskip 7pt

The link between these operators may be summarized in the following diagram:

\begin{equation}\label{diagram:squig2}
\xymatrix{
D_{\lambda\to\nu}
\ar@{~>}[d]_{\mathrm{Proposition}\, \ref{lem:170770}}
\ar@{~>}[rr]^{\mathrm{Theorem}\,\ref{thm:CIRM18}}
&& D_{\lambda\to\nu}^*
 \\
\widehat{D}_{\lambda\to\nu}
\ar@{~>}[rr]_{\mathrm{Proposition}\, \ref{lem:170772}}&&  \Phi_{\lambda}^{\nu}
\ar@{~>}[u]_{\mathrm{Proposition}\,\ref{prop:181248}}
}
\end{equation}
\vskip 7pt
Among them, the holographic operator $\Phi_\lambda^\nu$ in the $L^2$-model will play a crucial role in the proof of Theorem \ref{thm:170887}.
$$
$$

\subsection{Holographic transform in the $L^2$-model on the time-like cone
$\Omega(n)$}
\subsubsection{$L^2$-model of holomorphic discrete series}

For $\lambda>n-1$,
we denote by $L^2(\Omega)_\lambda\equiv L^2(\Omega,m_\lambda(y)dy)$
 the Hilbert space of square integrable functions
 on the time-like cone $\Omega\equiv \Omega(n)$
 with respect to the measure
 $m_{\lambda}(y) d y$,
 where $m_{\lambda}$ is a positive-valued function on $\Omega$
 given by
$$
m_\lambda(y):=Q_{1,n-1}(y)^{\frac n2-\lambda}.
$$
Let $\langle y,\zeta\rangle=\sum_{j=1}^ny_j\zeta_j$.
Since the cone $\Omega$ is self-dual in $\R^n$, the Fourier--Laplace transform
$$
\left(\mathcal F_nF\right)(\zeta):=\int_\Omega F(y) e^{i\langle y,\zeta\rangle}dy
$$
is a holomorphic function of $\zeta\in T_\Omega$ if $F\in C_c(\Omega)$.

\begin{fact}
[Faraut--Koranyi {\cite[Thm.~XIII.1.1]{FK}}]
\label{fact:Fnisom}
For $\lambda >n-1$,
 we set
\begin{equation}\label{eqn:bnlmd}
b_n(\lambda):=\left(2\pi\right)^{\frac{3n}2-1}2^{-2\lambda+n}\Gamma\left(\lambda-\frac n2\right)\Gamma(\lambda-n+1).
\end{equation}
Then the Fourier--Laplace transform
$\mathcal F_n \colon C_c(\Omega)
\longrightarrow \mathcal O(T_\Omega)$
extends to a linear bijection:
\begin{eqnarray}
\label{eqn:Fnisom}
\mathcal F_n \colon L^2(\Omega)_\lambda
{\stackrel{\sim}{\to}}
\mathcal H^2_\lambda(T_{\Omega(n)}).
\end{eqnarray}
with
$$
\Vert\mathcal F_n F\Vert^2_{\mathcal H^2_\lambda(T_{\Omega(n)})}=
b_n(\lambda)\Vert F\Vert^2_{L^2(\Omega(n))_\lambda}
\quad
\text{for all $F \in L^2(\Omega)_{\lambda}$.  }
$$
\end{fact}
Via the isomorphism \eqref{eqn:Fnisom},
 an irreducible unitary representation of $\widetilde G$
 is defined on $L^2(\Omega)_\lambda$ for $\lambda >n-1$,
 to which we refer as the $L^2$-\emph{model}
 of the (relative) holomorphic discrete series representation $\pi_\lambda^{(n)}$.

Similarly,
 we define a positive-valued function
 $$
 m_\nu'(y'):=Q_{1,n-2}(y')^{\frac{n-1}2-\nu}
 $$
 on the time-like cone $\Omega'\equiv\Omega(n-1)$ in $\R^{1,n-2}$, and we set
$L^2(\Omega')_\nu:= L^2(\Omega', m_\nu'(y')dy')$
 on which the $L^2$-model
 of the (relative) holomorphic discrete series representation
 $\pi_{\nu}^{(n-1)}$ of $\widetilde G'$ is defined via the unitary map
$$
\mathcal F_{n-1} \colon L^2(\Omega')_\nu
{\stackrel{\sim}{\to}}
\mathcal H^2_\nu(T_{\Omega(n-1)}) \qquad
\mathrm{for}\quad\nu>n-2
$$
up to rescaling $b_{n-1}(\nu)^{-\frac12}$.

\subsubsection{Gegenbauer polynomial and Juhl's conformally covariant operator}

Let  $a_k(\ell,\alpha)$ be as in \eqref{eqn:alalpha}.
We define a polynomial of two variables by
\begin{equation}\label{eqn:IlCuv}
(I_\ell C_\ell^{\alpha})(u,v):=
\sum_{k=0}^{\left[\frac\ell2\right]}a_k(\ell,\alpha)u^k v^{\ell-2k}.
\end{equation}
For instance, we have
 $(I_0 C_0^{\alpha})(u,v)=1$,
 $(I_1 C_1^{\alpha})(u,v)=2\alpha v$,
 and $(I_2 C_2^{\alpha})(u,v)=\alpha(2(\alpha+1)v^2-u)$.
By definition,
 $(I_\ell C_\ell^{\alpha})(w^2,v)$ is a homogeneous polynomial of degree $\ell$, and
$(I_\ell C_\ell^{\alpha})(1,v)$
coincides with the Gegenbauer polynomial
$C_\ell^{\alpha}(v)$, see \eqref{eqn:gegen} in Appendix. (This is the reason why we adopted
the notation $I_\ell C_\ell^{\alpha}$.)

The F-method \cite[Thm.~6.3]{KP16b} shows
 that the differential operator $\mathcal D_{\ell}^{\alpha}$
 (see \eqref{eqn:JuhlonCn}) is expressed as

\begin{equation}\label{eqn:iJuhl}
\mathcal D^{\alpha}_\ell=i^{-\ell}\left(I_\ell C^{\alpha}_\ell\right)
\left(-\Delta_{\C^{1,n-2}}, i\frac\partial{\partial\zeta_n}\right).
\end{equation}

\subsubsection{Construction of discrete summands in the $L^2$-model}

Following the idea of the F-method \cite{K14}, we introduce the \emph{holographic operator}
$\Phi_\lambda^\nu$ as a multiplication operator like in the tensor product case (\emph{cf.}
Definition \ref{def:Phi}). By the simplicity of its formula, the holographic operator
$\Phi_\lambda^\nu$ in the $L^2$-model plays a crucial role in the proof of Theorem \ref{thm:170887}.

Retain the basic setting \eqref{eqn:alphal} where $\ell=\nu-\lambda\in\N$
and $\alpha=\lambda-\frac{n-1}2$.
For a function $h(y')$ on $\Omega(n-1)$, we define $\left(\Phi_\lambda^\nu h\right)(y)$ on $\Omega(n)$ by
\begin{equation}\label{eqn:hologSO}
\left(\Phi_\lambda^\nu h\right)(y):=Q_{1,n-2}(y')^{-\left(\ell+\frac12\right)}(1-y_n^2)^{\lambda-\frac n2} \left(I_\ell
C^\alpha_\ell\right)(Q_{1,n-2}(y'),-y_n)h(y').
\end{equation}

Then $\Phi_\lambda^\nu$ gives rise to a holographic operator
in the $L^2$-model in the following sense:
\begin{prop}\label{lem:170772}
Suppose that $\lambda>n-1$ and $\nu=\lambda+\ell$ for some $\ell\in\N$.
\begin{itemize}
\item[(1)] The linear map
$\Phi_{\lambda}^{\nu} \colon L^2(\Omega(n-1))_{\nu}\longrightarrow L^2(\Omega(n))_\lambda$ is an isometry up to scalar:
$$
\Vert
\Phi_{\lambda}^{\nu}(h)\Vert^2_{L^2(\Omega(n))_\lambda}=c_\ell(\lambda)\Vert h\Vert^2_{L^2(\Omega(n-1))_\nu}\quad\mathrm{for}\,\,\mathrm{all}\,\, h\in L^2(\Omega(n-1))_\nu,
$$
where we recall that the constant $c_\ell(\lambda)$ is given in \eqref{eqn:c-ell}.
\item[(2)]
$
\Phi_{\lambda}^{\nu}
$
 intertwines the irreducible unitary representation $\pi_{\nu}^{(n-1)}$ of the subgroup
$\widetilde G'$
 with the restriction $\pi_\lambda^{(n)}\big\vert_{\widetilde G'}$.
\end{itemize}
\end{prop}
We also discuss some further basic properties of the holographic operators
$\Phi_\lambda^\nu$ in Proposition \ref{prop:181248}.

\subsubsection{Proof of Theorem \ref{thm:170887}}\label{subsec:proofTHM32}

Postponing the proof of Proposition \ref{lem:170772} until Section
\ref{sec:357} and Proposition \ref{prop:181248} below until Section \ref{sec:358}
 we complete the proof of Theorem  \ref{thm:170887}.
\begin{prop}\label{prop:normJuhl}
Suppose $\lambda>n-1$ and $\nu=\lambda+\ell$ with $\ell\in\N$. Then the differential operator
$D_{\lambda\to\nu}$ extends to a continuous linear map
$D_{\lambda\to\nu} \colon \mathcal H^2(T_{\Omega(n)})_\lambda\longrightarrow \mathcal H^2(T_{\Omega(n-1)})_\nu$ with the following operator norm:
$$
\Vert D_{\lambda\to\nu}\Vert_{\mathrm{op}}^2=r_\ell(\lambda)c_\ell(\lambda).
$$
\end{prop}
\begin{proof}
It follows from Lemma \ref{lem:opFourier} (1) and  Propositions \ref{lem:170772}
and \ref{prop:181248} that
$$
\Vert D_{\lambda\to\nu}\Vert^2_{\mathrm{op}}=
r_\ell(\lambda) \Vert \widehat D_{\lambda\to\nu}\Vert^2_{\mathrm{op}}
=r_\ell(\lambda)\Vert \Phi_{\lambda}^{\nu}\Vert^2_{\mathrm{op}}
=r_\ell(\lambda)c_\ell(\lambda).
$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:170887}]
By Lemma \ref{lem:Hilbertdeco} and by the (abstract) branching law \eqref{eqn:absbraConf}
for the restriction $\widetilde G\downarrow \widetilde G'$,
the theorem follows from the expression of the operator norm of the differential operator $D_{\lambda\to\nu}$ given in Proposition \ref{prop:normJuhl}.
\end{proof}

${}$

The rest of this section is devoted to the proof of Proposition \ref{lem:170772}.

\subsubsection{Coordinate change in the $L^2$-model.}

 As in Section \ref{sec:coord-changeSL2} for the tensor product case, we introduce the following coordinates of the time-like cone $\Omega(n)$ in $\R^{1,n-1}$:
\begin{equation}\label{eqn:fibration}
\iota \colon \Omega(n-1)\times (-1,1)\stackrel
{\sim}{\longrightarrow}
\Omega(n), \quad (y',v)\mapsto (y',-\sqrt{Q_{1,n-2}(y')}v),
\end{equation}
which is a bijection because $(y',y_n)\in\Omega(n)$ if and only if
$y'\in\Omega(n-1)$ and $ y_n^2<Q_{1,n-2}(y')$.

We define a function on $\Omega(n-1) \times (-1,1)$ by
\begin{equation}\label{eqn:Ayv}
M(y',v)\equiv M_{\lambda,\nu}(y',v):= Q_{1,n-2}(y')^{\frac{\ell+1}2}(1-v^2)^{\frac n2-\lambda}.
\end{equation}

Via the isomorphism $\iota$, the ratio of the densities $m_\lambda(y)$ and $m_\nu(y')$, and the holographic
operator $\Phi_\lambda^\nu$ are expressed as follows:
\begin{eqnarray}
\label{eqn:mmAQ-SO}
\frac{m'_\nu(y')}{m_\lambda(y)} &=&M(y',v)^{-1}\;Q_{1,n-2}(y')^{-\frac\ell2},\\
\label{eqn:mmQ}
m_\lambda(y)dy&=&M(y',v)^2\; m'_{\nu}(y')dy'(1-v^2)^{\lambda-\frac n2}dv,\\
\left(\Phi_\lambda^\nu h\right)\circ \iota(y',v)&=&M(y',v)^{-1}\; C_\ell^\alpha(v) h(y'). \label{eqn:holoiota}
\end{eqnarray}

\subsubsection{Fourier transform of the holomorphic Juhl operators}

For $\lambda>n-1$ and $\nu=\lambda+\ell$ ($\ell\in\N$), the holomorphic Juhl operator $D_{\lambda\to\nu}$ gives rise to a continuous operator $\mathcal H^2(T_{\Omega(n)})_\lambda\longrightarrow
\mathcal H^2(T_{\Omega(n-1)})_\nu$ between the weighted Bergman spaces \cite[Thm.~5.13]{KP16a}.

 We define a linear map
$
\widehat D_{\lambda\to\nu}:L^2(\Omega)_\lambda\longrightarrow L^2(\Omega')_{\nu}
$
 by
\begin{equation}\label{eqn:JFourier}
\widehat D_{\lambda\to\nu}:=\mathcal F_{n-1}^{-1}\circ D_{\lambda\to\nu}\circ\mathcal F_n.
\end{equation}

Then the idea of the F-method \cite{KP16a} implies
 that the ``Fourier transform''
$\widehat D_{\lambda\to\nu}$ of the holomorphic Juhl operator
$D_{\lambda\to\nu}$ is given by a Gegenbauer transform (\emph{cf}. \cite[Chap.~15]{DB})
 along the trajectory in \eqref{eqn:fibration}
 where the parameter $v$  moves:

\begin{prop}\label{lem:170770}
The operator $\widehat D_{\lambda\to\nu}$ is given by the following integral transform:
$$
\left(\widehat D_{\lambda\to\nu} F\right)(y')=
 i^{-\ell} Q_{1,n-2}(y')^{\frac{\ell+1}2}\int_{-1}^1F\circ\iota(y',v) C_\ell^\alpha(v)dv \quad \mathrm{for}
 \quad y'\in\Omega',
$$
where we set $\alpha=\lambda-\frac{n-1}2$.
\end{prop}

\begin{proof}
Let $\alpha=\lambda-\frac{n-1}2$.
Then it follows from \eqref{eqn:JuhlonCn} and \eqref{eqn:iJuhl} that
\begin{eqnarray}
\left(i^{\ell}D_{\lambda\to\nu}\circ\mathcal F_n\right)F&=&
\mathrm{Rest}_{\zeta_n=0}\circ \left(I_\ell C_\ell^\alpha\right)
\left( -\Delta_{\C^{1,n-2}},i\frac{\partial}{\partial\zeta_n}\right)\mathcal F_nF
\nonumber\\
&=&\mathrm{Rest}_{\zeta_n=0}\circ\mathcal F_n
\left( I_\ell C_\ell^\alpha(Q_{1,n-2}(y'),-y_n) F\right), \label{eqn:FJuhl}
\end{eqnarray}
for $F\in L^2(\Omega(n))_\lambda$.
Since $(I_\ell C_\ell^\alpha)(u^2,w)=u^\ell C_\ell^\alpha\left(\frac wu\right)$,
 the right-hand side amounts to
\begin{eqnarray*}
&&\int_{\Omega(n-1)}\int_{-1}^1 F\circ\iota(y',v)
Q_{1,n-2}(y')^{\frac{\ell+1}2}
C^\alpha_\ell(v) e^{i\langle y',\zeta'\rangle} dy' dv\\
&=&\mathcal F_{n-1}\left(
Q_{1,n-1}(y')^{\frac{\ell+1}2}\int_{-1}^1F\circ\iota(y',v) C^\alpha_\ell(v)dv\right)
\end{eqnarray*}
via the diffeomorphism \eqref{eqn:fibration}. Since the left-hand side
 of \eqref{eqn:FJuhl} is equal to $i^\ell \mathcal F_{n-1}\circ\widehat D_{\lambda\to\nu}F$
 by the definition \eqref{eqn:JFourier} of $\widehat D_{\lambda\to\nu}$, we proved Proposition
 \ref{lem:170770}.
\end{proof}

\subsubsection{Proof of Proposition \ref{lem:170772}}\label{sec:357}

\begin{proof}[Proof of Proposition \ref{lem:170772}]
Let $\alpha=\lambda-\frac{n-1}2$ as before.
By \eqref{eqn:mmQ} and \eqref{eqn:holoiota} we have
$$
\Vert\Phi_{\lambda}^{\nu} h\Vert^2_{L^2(\Omega(n))_\lambda}=\Vert h\Vert^2_{L^2(\Omega(n-1))_{\lambda+\ell}}
\int_{-1}^1\left\vert C^{\alpha}_\ell(v)\right\vert^2 (1-v^2)^{\alpha-\frac 12} dv.
$$
Thus the first assertion holds by the definition \eqref{eqn:c-ell} of $c_\ell(\lambda)$.

The second assertion follows readily from Proposition \ref{prop:181248}
 below.
\end{proof}

\subsubsection{Holographic operators and the adjoint of symmetry breaking operators}
\label{sec:358}

We have constructed operators $\Phi_\lambda^\nu$ in the
$L^2$-model and $\widehat\Phi_\lambda^\nu$ in the holomorphic model using the F-method. On the other hand, the adjoint of symmetry
 breaking operators between unitary representations yield holographic operators in general (\emph{cf}. Theorem \ref{thm:SBTinj} (1) below).
In our setting, these operators are proportional to each other because the branching law \eqref{eqn:absbraConf} is multiplicity-free. We determine the proportionality constants:
\begin{prop}\label{prop:181248}
Suppose $\lambda> n-1$ and $\ell=\nu-\lambda\in\N$. Then we have
\begin{align*}
\left(\widehat D_{\lambda\to\nu}\right)^* &=i^\ell \Phi_{\lambda}^{\nu},
\\
D_{\lambda\to\nu}^* &=i^\ell r_\ell(\lambda) \widehat{\Phi_\lambda^\nu}.
\end{align*}
\end{prop}
\begin{proof}
Take  any $h\in L^2(\Omega(n-1))_\nu$ and $F\in L^2(\Omega(n))_\lambda$ with $\nu-\lambda=\ell\in\N$.
Since $\alpha=\lambda-\frac{n-1}2$ is real, $C_\ell^\alpha(v)$ is real-valued for $-1< v< 1$, hence we have from Proposition
\ref{lem:170770}
\begin{eqnarray*}
&&\left( h,\widehat D_{\lambda\to\nu}F\right)_{L^2(\Omega(n-1))_\nu}\\&=&i^\ell
\int_{\Omega(n-1)}h(y')\left(\int_{-1}^1 Q_{1,n-2}(y')^{\frac{\ell+1}2}\overline{F\circ\iota(y',v )}
 C_\ell^\alpha(v)dv\right) m'_\nu(y')dy'\\
&=& i^\ell \int_{\Omega(n)}h(y')\overline{F(y)}(I_\ell C_\ell^\alpha)(Q_{1,n-2}(y'),-y_n)m'_\nu(y')dy\\
&=& i^\ell \int_{\Omega(n)}\left(\Phi_{\lambda}^{\nu}h\right)(y)\overline{F(y)} m_\lambda(y)dy\\
&=& i^\ell\left(\Phi_{\lambda}^{\nu}h,F\right)_{L^2(\Omega(n))_\lambda}.
\end{eqnarray*}
Here we have used \eqref{eqn:mmAQ-SO} and \eqref{eqn:holoiota} in the third identity.
Hence the first equality is shown.
By Lemma \ref{lem:opFourier},
 the second equality follows from the first one.
\end{proof}

\subsection{Explicit integral formula for the holographic operator}

In this section, we give an integral formula for the holographic operator $D_{\lambda\to\nu}^*$
in the holomorphic model, giving thus an answer to Problem A.1 in Section 1, see Theorem
\ref{thm:CIRM18} below.

\subsubsection{Construction of discrete summands in the holomorphic model}

In contrast to the holographic operators $\Psi_{\lambda',\lambda''}^{\lambda'''}$ (Definition \ref{def:HOtensor}) in the tensor product case in Section \ref{sec:RCT}, we do not have a simple integral
expression for a holographic operator like \eqref{eqn:psi} in the present setting.
Instead, we adopt an alternative approach to construct a holographic operator
 by introducing a {\it{relative reproducing kernel}} $K_{\lambda,\nu}(\zeta,\tau')$ as below.

For $\lambda,\nu\in\C$ with $\ell:=\nu-\lambda\in\N$, we set
\begin{equation}\label{eqn:holoKlmdnu}
K_{\lambda,\nu}(\zeta,\tau'):=\frac{
\left((\zeta_1-\overline \tau_1)^2
-(\zeta_2-\overline \tau_2)^2-\cdots - (\zeta_{n-1}-\overline \tau_{n-1})^2-\zeta_n^2\right)^{-\nu}}
{\zeta_n^{\lambda-\nu}},
\end{equation}
where $\zeta=(\zeta_1,\cdots,\zeta_n)\in T_{\Omega(n)}$ and $\tau'=(\tau_1,\cdots,\tau_{n-1})\in T_{\Omega(n-1)}$.

\begin{rem}
$K_{\lambda,\nu}(\zeta,\tau')$ may be viewed as the holomorphic counterpart of the distribution kernel  of a conformally covariant \emph{integral} symmetry breaking operator for the pair of Riemannian
manifolds $(S^n, S^{n-1})$, see \cite{KS}. See also \eqref{eqn:Klmdnu} for the case $n=2$.
\end{rem}

Let $d\mu_\nu$ be a measure on $T_{\Omega(n-1)}$ given
by
$$d\mu_\nu(\tau'):=\left(\frac i2\right)^{n-1}Q_{1,n-2}(\mathrm{Im}\; \tau')^{\nu-n+1}d\tau'd\overline \tau'.$$

\begin{thm}[holographic operator]\label{thm:CIRM18}
Let $n\geq3$.
 Suppose $\lambda>n-1$ and
 $\nu=\lambda+\ell$ with $\ell\in\N$.
 Then the integral
 $$
 \int_{T_{\Omega(n-1)}} K_{\lambda,\nu} (\zeta,\tau') g(\tau') d\mu_\nu(\tau')
 $$
 converges for all
  $g\in \mathcal H^2(T_{\Omega(n-1)})_\nu$ and $\zeta\in T_{\Omega(n)}$.
  Moreover, it gives the adjoint  operator $D^*_{\lambda\to\nu}$ up to scalar multiplication:
\begin{eqnarray*}
\left(D^*_{\lambda\to\nu} g\right)(\zeta)=C\int_{T_{\Omega(n-1)}} K_{\lambda,\nu}(\zeta,\tau') g(\tau') d\mu_\nu(\tau'),
\end{eqnarray*}
where the constant $C$ is given by
\begin{equation}\label{eqn:CIRM18}
C=
\frac{2^{2\lambda-2n+\ell-1}(\lambda-n+1)_{n+\ell-1}(2\lambda-n)_{\ell+1}}{i^{2\lambda+2\ell} \pi^n \ell !}.
\end{equation}
In particular, it
 yields an injective continuous $\widetilde G'$-intertwining operator between weighted Bergman spaces,
$
 \mathcal H^2(T_{\Omega(n-1)})_\nu
\hookrightarrow \mathcal H^2(T_{\Omega(n)})_\lambda.
$
\end{thm}

We first show that Theorem \ref{thm:CIRM18} is derived from the following
Bernstein--Sato type
identity for the holomorphic Juhl operator.

\begin{thm}\label{thm:Cassis}
Let $\mathcal D_\ell^\alpha$ be the differential operator as in \eqref{eqn:iJuhl}.
We set
\begin{eqnarray}\label{eqn:qnl}
q(n,\ell;\lambda):=\frac{2^{\ell}}{\ell!}\,(2\lambda-n+1)_\ell(\lambda)_\ell.
\end{eqnarray}
Then,
$$
\zeta_n^{-\ell}\,\mathcal D_{\ell}^{\lambda-\frac{n-1}2} Q_{1,n-1}(\zeta)^{-\lambda}= q(n,\ell;\lambda)\, Q_{1,n-1}(\zeta)^{-\lambda-\ell}.
$$
\end{thm}

\begin{rem}\label{rem:Cassis}
Theorem \ref{thm:Cassis} shows that the complex power of the quadratic form
$Q_{1,n-1}$ satisfies the Bernstein--Sato type identity not only for the power of the Laplacian (see \eqref{eqn:LapQpower} below) but also for another operator closely
related to the holomorphic Juhl operator.
\end{rem}
Postponing the proof of Theorem \ref{thm:Cassis}, we complete the proof of Theorem \ref{thm:CIRM18}. For this, we also
 use the following lemma.
\begin{lem}\label{lem:adker}
Let $D_j$ ($j=1,2$) be complex manifolds, and $\mathcal H_j$ Hilbert spaces contained in $\mathcal O(D_j)$ with
reproducing kernels $K^{(j)}(\cdot,\cdot)$.
Suppose that $\mathcal R \colon \mathcal H_1\to\mathcal H_2$ is a continuous linear map, and
$\mathcal R^* \colon \mathcal H_2\to\mathcal H_1$ is its adjoint operator.
Then,
\begin{itemize}
\item[(1)] $\overline{\mathcal R K^{(1)}(\,\cdot, \zeta)(\tau')}=\left(\mathcal R^*K^{(2)}(\,\cdot,\tau')\right)(\zeta)$ for $\zeta\in D_1, \tau'\in D_2$;
\item[(2)] $\left(\mathcal R^*g\right)(\zeta)=\left( g, \mathcal R K^{(1)}(\,\cdot, \zeta)\right)_{\mathcal H_2}$ for $g\in\mathcal H_2$ and $\zeta\in D_1$.
\end{itemize}
\end{lem}
\begin{proof}
(1)\enspace The first assertion results from the reproducing property of $K^{(j)}(\cdot,\cdot)$ applied to the following identity:
$$
(\mathcal R^* K^{(2)}(\,\cdot,\tau'), K^{(1)}(\,\cdot,\zeta))_{\mathcal H_1}
=( K^{(2)}(\,\cdot,\tau'),\mathcal R K^{(1)}(\,\cdot,\zeta))_{\mathcal H_2}.
$$
\par\noindent
(2)\enspace  The statement is immediate from the following:
$$\left(\mathcal R^*g\right)(\zeta)= \left(\mathcal R^*g, K^{(1)}(\,\cdot,\zeta)\right)_{\mathcal H_1}= (g,\mathcal R K^{(1)}(\,\cdot,\zeta))_{\mathcal H_2}.
$$
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:CIRM18}]
Applying Lemma \ref{lem:adker} to the triple
$$
(\mathcal R, \mathcal H_1,\mathcal H_2)=( D_{\lambda\to\nu}, \mathcal H^2(T_{\Omega(n)})_\lambda, \mathcal H^2(T_{\Omega(n-1)})_\nu),
$$
we obtain the following integral expression of the adjoint operator $D_{\lambda\to\nu}^*$:
\begin{equation}\label{eqn:DKint}
\left(D_{\lambda\to\nu}^*g\right)(\zeta)=
\int_{T_{\Omega(n-1)}} g(\tau')\overline{D_{\lambda\to\nu} K_\lambda(\,\cdot,\zeta)(\tau')}
d\mu_\nu(\tau').
\end{equation}
Here, we have viewed the reproducing kernel $K_\lambda(\tau,\zeta)=k_{\lambda,n} Q_{1,n-1}(\tau-\overline{\zeta})^{-\lambda}$ defined in \eqref{eqn:Klmbd} as a function of
$\tau\in T_{\Omega(n)}$ with parameter $\zeta\in T_{\Omega(n)}$ and applied
the holomorphic Juhl operator $D_{\lambda\to\nu}$.
Writing $\tau$ as $\tau =(\tau', \tau_n)$, we get from Theorem \ref{thm:Cassis}:
\begin{eqnarray*}
&&D_{\lambda\to\nu} K_\lambda(\tau,\zeta)\\
&=&
k_{\lambda,n}\, q(n,\ell;\lambda)\,
\mathrm{Rest}_{\tau_n=0}\circ
(\tau_n-\overline\zeta_n)^\ell
Q_{1,n-1}(\tau-\overline{\zeta})^{-\lambda-\ell}\\
&=&
(-1)^\ell{k_{\lambda,n}} q(n,\ell;\lambda) \overline{K_{\lambda,\nu}(\zeta, \tau')}
 \end{eqnarray*}
for $\tau'\in T_{\Omega(n-1)},
$
by the definition \eqref{eqn:holoKlmdnu} of the relative reproducing kernel $K_{\lambda,\nu}(\zeta,\tau')$. Since $q(n,\ell;\lambda)\in\R$ when $\lambda\in\R$, the integral formula of
Theorem
\ref{thm:CIRM18} is shown with the constant $C=(-1)^\ell\overline{k_{\lambda,n}} q(n,\ell;\lambda)$.
A short computation shows the formula  \eqref{eqn:CIRM18}.
\end{proof}
The rest of this section is devoted to the proof of Theorem \ref{thm:Cassis}.

\subsubsection{Proof of Theorem \ref{thm:Cassis}}

\begin{lem}\label{lem:cirm180318}
Suppose $\ell\in\N$.
Then there exist  $q_j\equiv q_j(n,\ell;\lambda)$ ($0\leq 2j\leq\ell)$ such that
$$
\mathcal D_\ell^\alpha Q_{1,n-1}(\zeta)^{-\lambda}=\sum_{j=0}^{\left[\frac\ell2\right]}
q_j\zeta_n^{\ell-2j} Q_{1,n-1}(\zeta)^{-\lambda-\ell+j}.
$$
\end{lem}

\begin{proof
It is easy to see that an analogous statement holds for $\left(\frac{\partial}{\partial\zeta_n}\right)^\ell$ instead of $\mathcal D_\ell^\alpha$, namely, there
exist $q_j'\equiv q_j'(n,\ell;\lambda)$ ($0\leq 2j\leq\ell$) such that
\begin{equation}\label{eqn:derQ}
\left(\frac{\partial}{\partial\zeta_n}\right)^\ell
Q_{1,n-1}(\zeta)^{-\lambda}=\sum_{j=0}^{\left[\frac\ell2\right]}
q_j'\zeta_n^{\ell-2j} Q_{1,n-1}(\zeta)^{-\lambda-\ell+j}.
\end{equation}

We rewrite $\mathcal D_\ell^\alpha$ as a polynomial of $\Delta_{\C^{1,n-1}}=\frac{\partial^2}{\partial\zeta_1^2}-\frac{\partial^2}{\partial\zeta_2^2}-\cdots-\frac{\partial^2}{\partial\zeta_n^2}$
 and $\frac{\partial}{\partial\zeta_n}$ by substituting $\Delta_{\C^{1,n-2}}=\Delta_{\C^{1,n-1}}+\frac{\partial^2}{\partial\zeta_n^2}$ into
\eqref{eqn:JuhlonCn}:
\begin{equation}\label{eqn:Dshiftp}
\mathcal D_\ell^\alpha =
\sum_{k=0}^{\left[\frac\ell2\right]} p_k(n,\ell;\alpha)\left(\frac{\partial}{\partial\zeta_n}\right)^{\ell-2k}
\left(\Delta_{\C^{1,n-1}}\right)^k,
\end{equation}
where the first coefficient is given by
\begin{equation}\label{eqn:p0}
p_0(n,\ell;\alpha)=\sum_{k=0}^{\left[\frac\ell2\right]}a_k(\ell,\alpha)=
 C_\ell^\alpha(1)=\frac{(2\alpha)_\ell}{\ell!}.
\end{equation}

An iterated use of the formula
$$
\Delta_{\C^{1,n-1}} Q_{1,n-1}(\zeta)^{-\lambda}=2\lambda(2\lambda-n+2)Q_{1,n-1}(\zeta)^{-\lambda-1},
$$
leads us to
\begin{equation}\label{eqn:LapQpower}
\left(\Delta_{\C^{1,n-1}}\right)^kQ_{1,n-1}(\zeta)^{-\lambda}=s_k(n,\lambda) Q_{1,n-1}(\zeta)^{-\lambda-k},
\end{equation}
for some polynomials $s_k(n,\lambda)$ of $\lambda$ of degree $2k$.
We note that $s_0(n,\lambda)=1$.
 Now the lemma follows from \eqref{eqn:derQ}.
\end{proof}

Clearly the coefficients
$q_j=q_j(n,\ell; \lambda)$ in Lemma \ref{lem:cirm180318}
are unique.
The proof of Theorem \ref{thm:Cassis} is reduced to the following proposition
on these coefficients $q_j(n,\ell; \lambda)$.

\begin{prop}\label{prop:DQtop}
${}$
\begin{itemize}
\item[(1)]
{\rm{(the first term)}}. Recall that
$q(n,\ell;\lambda)$ is defined in \eqref{eqn:qnl}. Then,
$$
q_0\left(n,\ell;\lambda\right)=q(n,\ell;\lambda).
$$
\item[(2)]
{\rm{(vanishing of higher terms)}}.
$$
q_j\left(n,\ell;\lambda\right)=0\quad\mathrm{for}\,\mathrm{all}\,\, j\geq1.
$$
\end{itemize}
\end{prop}

\begin{proof}[Proof of Proposition \ref{prop:DQtop} (1)]
In the expression
$$
\left(\frac{\partial}{\partial\zeta_n}\right)^{\ell-2k} \left(\Delta_{\C^{1,n-1}}\right)^k
Q_{1,n-1}(\zeta)^{-\lambda}=s_k(n,\lambda)\left(\frac{\partial}{\partial\zeta_n}\right)^{\ell-2k}
Q_{1,n-1}(\zeta)^{-\lambda-k},
$$
the term $\zeta_n^\ell \;Q_{1, n-1}(\zeta)^{-\lambda-\ell}$ occurs
only when $k=0$, and its coefficient is given by
$
s_0(n,\lambda) 2^\ell (\lambda)_\ell=2^\ell(\lambda)_\ell.
$
By \eqref{eqn:Dshiftp}, we get
$$
q_0(n,\ell;\lambda)=p_0(n,\ell;\alpha)\cdot 2^\ell(\lambda)_\ell.
$$
Now the first assertion of Proposition \ref{prop:DQtop}
follows from \eqref{eqn:p0}.
\end{proof}

In order to prove the second assertion of Proposition \ref{prop:DQtop}, we discuss the kernel of the holomorphic Juhl operator
 $D_{\lambda\to\nu} \colon \mathcal O(T_{\Omega(n)})\longrightarrow
\mathcal O(T_{\Omega(n-1)})$.

\begin{prop}\label{prop:KerJuhl}
Suppose $\lambda-\frac{n-1}2\not\in\{0,-1,-2,\cdots\}$. Then for any $N\in\N$ we have
\begin{multline}
\bigcap_{j=0}^N\mathrm{Ker}\left(D_{\lambda\to\lambda+j}\right)\\=
\left\{f\in\mathcal O(T_{\Omega(n)})\colon \mathrm{Rest}_{\zeta_n=0}\circ \left(\frac{\partial}{\partial\zeta_n}\right)^j f=0\,\,\mathrm{for}\,\mathrm{all}\,0\leq j\leq N\right\}. \nonumber
\end{multline}
\end{prop}

\begin{proof}
By the definition \eqref{eqn:holoJuhl} of $D_{\lambda\to\nu}$,
the right-hand side is clearly contained in the left-hand side. To see the opposite inclusion,
we recall from the definition \eqref{eqn:holoJuhl} that the symmetry breaking operator $D_{\lambda\to\lambda+j}$
is of the form
$$
D_{\lambda\to\lambda+j}=\mathrm{Rest}_{\zeta_n=0}\circ\left(a_0 \left(\frac{\partial}{\partial\zeta_n}\right)^j+
\sum_{k=1}^{\left[\frac j2\right]} a_k
\left(\frac{\partial}{\partial\zeta_n}\right)^{j-2k}
\Delta^k_{\C^{1,n-2}}\right),
$$
where the first coefficient $a_0\equiv a_0\left(j,\alpha\right)$ is given by
$$
 a_0\left(j,\alpha\right)=\frac{2^j}{j!}(\alpha)_\ell \qquad \mathrm{with}\quad\alpha=\lambda-\frac{n-1}2.
$$

We now prove the proposition by induction on $N$. The statement is clear for $N=0$ and $1$ because
$D_{\lambda\to\lambda}= \mathrm{Rest}_{\zeta_n=0}$ and $D_{\lambda\to\lambda+1}= \mathrm{Rest}_{\zeta_n=0}\circ \frac{\partial}{\partial\zeta_n}$.

Suppose that $f\in\mathcal O(T_{\Omega(n)})$
satisfies $D_{\lambda\to\lambda+j}f=0$ for $0\leq j\leq N+1$. By the inductive assumption, we get
$\mathrm{Rest}_{\zeta_n=0}\circ \left(\frac{\partial}{\partial\zeta_n}\right)^jf=0$ ($0\leq j\leq N)$. Since $a_0\equiv a_0(j,\alpha)$ is nonzero for any $j$ by the assumption on $\lambda$, $D_{\lambda\to\lambda+N+1}f=0$ implies
$\mathrm{Rest}_{\zeta_n=0}\circ \left(\frac{\partial}{\partial\zeta_n}\right)^{N+1}f=0$.
Thus the proposition is proved by induction.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:DQtop} (2)]
Let $\nu=\lambda+\ell$. By \eqref{eqn:DKint} and Lemma \ref{lem:cirm180318}, we obtain
\begin{eqnarray}\label{eqn:Dstarsum}
&&\left( D_{\lambda\to\nu}^* g\right)(\zeta)
\\&=&(-1)^\ell \overline{k_{\lambda,n}}\sum_{j=0}^{\left[\frac\ell2\right]}q_j\zeta_n^{\ell-2j}
\int_{T_{\Omega(n-1)}}
g(\tau') Q_{1,n-1}(\zeta-\overline{\tau})^{-\lambda-\ell+j}d\mu_\nu(\tau'), \nonumber
\end{eqnarray}
where we write $\tau = (\tau',0)= (\tau_1',\cdots, \tau_{n-1}',0)$ by abuse of notations.

On the other hand, the composition map
$$
D_{\lambda\to\lambda+j}\circ D_{\lambda\to\lambda+\ell}^* \colon
\mathcal H^2(T_{\Omega(n-1)})_{\lambda+\ell}\longrightarrow \mathcal H^2(T_{\Omega(n-1)})_{\lambda+j}
$$
is a $G'$-intertwining operator for any $j$.
Since the $G'$-modules $\mathcal H^2(T_{\Omega(n-1)})_{\lambda+j}$ ($j\in\N$) are irreducible and mutually inequivalent if $\lambda>n-1$, such an intertwining operator
must be zero unless $j=\ell$. Therefore
$$
\mathrm{Image}\left(D_{\lambda\to\lambda+\ell}^*\right)\subset
\bigcap_{j=0}^{\ell-1}\mathrm{Ker}\left(D_{\lambda\to\lambda+j}\right).
$$
We now prove that $q_j\equiv q_j(n,\ell;\lambda)$ vanishes for all $1\leq j\leq\left[\frac\ell2\right]$
by downward induction on $j$. For simplicity, we treat the case where $\ell$ is even, say $\ell=2m$. The case where $\ell$ is odd can be dealt with similarly.

By Proposition \ref{prop:KerJuhl}, we have
$$
\mathrm{Rest}_{\zeta_n=0}\circ D_{\lambda\to\nu}^*g=0 \quad
\text{for all}\quad g\in
\mathcal H^2(T_{\Omega(n-1)})_\nu.
$$
Then it follows from \eqref{eqn:Dstarsum} that
$$
 (-1)^\ell\, \overline{k_{\lambda,n}} q_m
\int_{T_{\Omega(n-1)}}
g(\tau') Q_{1,n-2}(\zeta'-\overline{\tau})^{-\lambda-m}d\mu_\nu(\tau')=0.
$$
Thus we conclude that $q_m=0$ because
$k_{\lambda,n}\neq0$.

Suppose that we have shown $q_j=0$ for $j=m, m-1,\cdots, m+1-s$ for some $s\geq1$.
 If $s\leq m-1$,
then $2s\leq\ell-1 (=2m-1)$ and we can proceed by
applying Proposition \ref{prop:KerJuhl} with $N=\ell-1$, hence
$$
\mathrm{Rest}_{\zeta_n=0}\circ \left(\frac{\partial}{\partial\zeta_n}\right)^{2s}
\circ D_{\lambda\to\nu}^*g=0.
$$
By the inductive assumption, we obtain
$$
 (-1)^\ell \overline{k_{\lambda,n}} q_{m-s}
\int_{T_{\Omega(n-1)}}
g(\tau') Q_{1,n-2}(\zeta'-\overline{\tau})^{-\lambda-m-s}d\mu_\nu(\tau')=0
$$
for all $g\in
\mathcal H^2(T_{\Omega(n-1)})_\nu$. Thus we conclude that $q_{m-s}=0$
as far as $s\leq m-1$. Hence
 we have shown $q_j=0$ for all $j\geq1$.
 \end{proof}
 Thus the proof of Theorem \ref{thm:Cassis} (hence, also the one of Theorem \ref{thm:CIRM18}) is completed.

\section{Perspectives of symmetry breaking and holographic transforms}
\label{sec:conclusion}

We end this article with discussion
 on a representation-theoretic background of Problems A and B
 in a broader framework.

In Section \ref{sec:concl1},
 we consider these problems from the viewpoint of the branching laws of unitary representations of locally compact groups.
In Section \ref{sec:concl2},
 we investigate Problems A and B for triples $(G,G',\pi)$
 such that

\begin{itemize}
\item[] $\bullet\,(G,G')$ is a reductive symmetric pair of holomorphic type;
\item[] $\bullet\,\pi$ is a unitary highest weight module
 of $G$ of scalar type,
\end{itemize}
generalizing the settings for the main results in Sections \ref{sec:RCT} and \ref{sec:HJT}.
The role of special orthogonal polynomials in these cases
 is clarified in Section  \ref{sec:concl3}.

\subsection{Branching laws, symmetry breaking transform and holographic transform}\label{sec:concl1}

Let $G\supset G'$ be a pair of groups,
 $\pi$ an irreducible $G$-module,
 and $\rho$ an irreducible $G'$-module.
We recall from Section \ref{sec:Intro}
 that an element in $\mathrm{Hom}_{G'}\left(\pi\vert_{G'},\rho\right)$
 (resp. in
$\mathrm{Hom}_{G'}\left(\rho,\pi\vert_{G'}\right)$) is said to be a symmetry breaking operator (resp. a holographic operator).
We also recall that a symmetry breaking transform (resp. a holographic transform) is a collection of symmetry breaking operators (resp. holographic operators)
 where $(\rho,W)$ runs over a certain set $\Lambda$ of irreducible representations of the subgroup $G'$.

If $\pi$ is a unitary representation of a locally compact group $G$
 on a Hilbert space $V$,
 then Mautner's theorem guarantees that the restriction ($\pi\vert_{G'},V$)
 is unitarily equivalent to the direct integral of irreducible unitary representations of the subgroup $G'$:
\begin{equation}\label{eqn:BL}
\pi\vert_{G'}\simeq\int_{\widehat{G'}}^\oplus m_\pi(\rho)\rho\, d\mu(\rho),
\end{equation}
where $\widehat{G'}$ is the set of equivalence classes of irreducible unitary representations of $G'$
(\emph{unitary dual}), $\mu$ is a Borel measure on $\widehat{G'}$ endowed with the Fell topology, and $m_\pi \colon \widehat{G'}\longrightarrow \N\cup\{\infty\}$ is a measurable function (\emph{multiplicity}).
The irreducible decomposition \eqref{eqn:BL} is called branching law
 of the restriction $\pi\vert_{G'}$, which is unique up to isomorphism if $G'$ is a type $I$ group, in particular, if $G'$ is a real reductive group by a theorem of Harish-Chandra \cite{HC}.

The (abstract) branching law \eqref{eqn:BL}
 would be enriched through Problems A and \ref{prob:B}
 by geometric realizations of irreducible representations
 and explicit intertwining operators:
\begin{align*}
&&\text{from (LHS) to (RHS)}  \qquad   & \text\it{symmetry\,\, breaking\,\, transform};\\
&&\text{from (RHS) to (LHS)}  \qquad  & \text{\it{holographic\, transform}}.
\end{align*}

In the unitary case, it is natural to take $\Lambda$ to be the support of the measure $\mu$ in \eqref{eqn:BL}. If $\Lambda$ is a countable set, then the branching law \eqref{eqn:BL} is \emph{discretely decomposable}
 without continuous spectrum.
A criterion for the triple $(G,G',\pi)$ to admit a discretely decomposable restriction $\pi\vert_{G'}$ was studied in \cite{kdeco98,kdeco98Inv} when $G\supset G'$ is a pair of real reductive groups.

On the other hand,
 the multiplicity $m_{\pi}(\rho)$ in \eqref{eqn:BL} is not always finite
 when $\pi$ and $\rho$ are infinite-dimensional.
A geometric criterion for the pair $(G,G')$
 to assure that $\operatorname{Hom}_{G'}(\pi^{\infty}|_{G'}, \rho^{\infty})$
 is finite-dimensional
 for all smooth irreducible $G$-modules $\pi^{\infty}$ and $G'$-modules $\rho^{\infty}$ was established
 in \cite{xKOfm}.

\vskip 1pc

If the branching law \eqref{eqn:BL} is discretely decomposable and multiplicity free,
 then we could expect a simple and detailed study of symmetry breaking transform
 and holographic transform.
In this case,
 since the vector space $\mathrm{Hom}_{G'}(\pi\vert_{G'},\rho)$
 is one-dimensional,
 symmetry breaking operator is unique up to scaling
 for every $\rho$,
 and the symmetry breaking transform is defined
 as the collection of countably many such operators.

\subsection{Symmetric pairs of holomorphic type}\label{sec:concl2}

In this section, we provide a geometric condition (see Setting \ref{setting:takagi18} below) that assures the branching law $\pi\vert_{G'}$ to be discretely decomposable and multiplicity free.
In this case, we see that every
symmetry breaking operator is a
 differential operator (\emph{e.g.}  the Rankin--Cohen transforms studied in Section \ref{sec:RCT} and the holomorphic Juhl transforms in Section \ref{sec:HJT}),
 and that our symmetry breaking transform $D$
is injective, hence giving an affirmative
answer to Problem A.0 in Section \ref{sec:Intro}.
The main results in Sections \ref{sec:RCT} and \ref{sec:HJT}
 are built on special cases of this general setting.

Let us fix some notations. Let $G$ be a connected reductive Lie group, $\theta$ a Cartan
involution, $K=\{g\in G : \theta g=g\}$, $\mathfrak g=\mathfrak k+\mathfrak p$ the corresponding Cartan decomposition, and $\mathfrak g_\C=\mathfrak k_\C+\mathfrak p_\C$ its complexification.
Assume that there exists a central element $Z$ of $\mathfrak k_\C$ such that
$$
\mathfrak g_\C=\mathfrak k_\C+\mathfrak n_++\mathfrak n_-
$$
is the eigenspace decomposition of $\operatorname{ad}(Z)$
 with eigenvalues $0,1$, and $-1$, respectively.
This assumption is satisfied
 if and only if $G$ is locally isomorphic to a direct product of compact Lie groups
 (with $Z=0$) and noncompact Lie groups of Hermitian type.
Then the associated Riemannian symmetric space $X=G/K$
 becomes a Hermitian symmetric space
 with complex structure induced from the Borel embedding $G/K\subset G_\C/K_\C\exp(\mathfrak n_+)$.
Take a Cartan subalgebra $\mathfrak t$ of $\mathfrak k$, and write
$\rho(\mathfrak n_+)$ for half the sum of roots in $\Delta(\mathfrak n_+,
\mathfrak t_\C)$.

\begin{sett}\label{setting:takagi18}
Let $(G,G')$ be a reductive symmetric pair of holomorphic type,
 that is,
 $X=G/K$ and $Y=G'/K'$ are both Hermitian symmetric spaces and the natural embedding $\iota \colon Y\hookrightarrow X$ is holomorphic.
Let $\mathcal L=G\times_K\C_{\lambda}$ be
 a $G$-equivariant holomorphic line bundle over $X$
 associated to a unitary character $\C_\lambda$ of $K$,
 and we set
$\mathcal H^2(X,\mathcal L):=\left(\mathcal O\cap L^2\right)(X,\mathcal L)$.
Assume $\lambda$ satisfies the following condition:
\begin{equation}\label{eqn:lmdpos}
\begin{cases}
\langle \lambda,\alpha\rangle=0
 \quad &\forall\alpha\in\Delta
(\mathfrak k_\C,\mathfrak t_\C),
\\
\langle \lambda-\rho(\mathfrak n_+),\alpha\rangle>0 \quad &\forall\alpha\in\Delta(\mathfrak n_+,\mathfrak t_\C).
\end{cases}
\end{equation}
\end{sett}

 The Hilbert space $\mathcal H^2(X,\mathcal L)$ is naturally identified with a weighted Bergman space, which is nonzero if $\lambda$ satisfies the
 condition \eqref{eqn:lmdpos}.
We denote by
$\pi$ the representation of $G$ on the Hilbert space $\mathcal H^2(X,\mathcal L)$, which is irreducible and unitary, and is
 called a {\it{holomorphic discrete series representation}} of $G$.
The list of irreducible symmetric pairs $(G,G')$
 of holomorphic type
 may be found in \cite[Table 3.4.1]{K08}.
\begin{fact}
[{see \cite[Thm.~B]{K08}}]
In Setting \ref{setting:takagi18}, the restriction $\pi\vert_{G'}$ is discretely decomposable and multiplicity free.
\end{fact}

Any irreducible $G'$-module
 that occurs in the branching law \eqref{eqn:BL}
 for the unitary representation $(\pi,{\mathcal{H}}^2(X,{\mathcal{L}}))$
 is of the form
 $\mathcal H^2(Y,\mathcal W)$
 for some $G'$-equivariant holomorphic vector bundle $\mathcal W$ over $Y$
 associated to an irreducible finite-dimensional unitary representation $W$
 of $K'$,
 and such bundles ${\mathcal{W}}$ are classified.
 Thus $\Lambda$ is parametrized by a subset of $\widehat{K'}$, or by a
 subset of dominant integral weights which can be
  described in terms of the root data (see \cite[Thm.~8.3]{K08}).
 We write $\rho_\ell$ for the irreducible unitary representation
 of $G'$ corresponding to $\ell\in\Lambda$, and identify $\Lambda$ as a subset of $\widehat{G'}$
  by $\ell\mapsto\rho_\ell$.
Here is a summary on general results
 about symmetry breaking operators
 in this setting:
\begin{fact}
\label{fact:4.3}
In Setting \ref{setting:takagi18}, let $\mathcal W$ be the $G'$-equivariant holomorphic vector bundle corresponding to $\ell\in\Lambda$.
\begin{itemize}
\item[(1)] Any continuous $G'$-homomorphism $\mathcal O(X,\mathcal L)\longrightarrow\mathcal O(Y,\mathcal W)$
is given as a holomorphic differential operator, and induces a continuous $G'$-homomorphism between
the Hilbert spaces $\mathcal H^2(X,\mathcal L)\longrightarrow\mathcal H^2(Y,\mathcal W)$.
\item[(2)] Any continuous $G'$-homomorphism $\mathcal H^2(X,\mathcal L)\longrightarrow \mathcal H^2(Y,\mathcal W)$ extends to a continuous $G'$-homomorphism $\mathcal O(X,\mathcal L)\longrightarrow\mathcal O(Y,\mathcal W)$.
\end{itemize}
\end{fact}

\begin{proof}
(1)\enspace The first statement is proved in \cite[Thm.~5.3]{KP16a} (\emph{localness theorem}), and the second one is in \cite[Thm.~5.13]{KP16a}.
\par\noindent
(2)\enspace
By (1) there is a natural injective map
\begin{equation}\label{eqn:OH2}
   \operatorname{Hom}_{G'}(\mathcal O(X,\mathcal L)\big\vert_{G'}, \mathcal O(Y,\mathcal W)) \hookrightarrow
   \operatorname{Hom}_{G'}(\mathcal H^2(X,\mathcal L)\big\vert_{G'}, \mathcal H^2(Y,\mathcal W)).
\end{equation}
To prove that \eqref{eqn:OH2} is surjective, we observe that the left-hand side of
\eqref{eqn:OH2} is understood by the branching law
 for the generalized Verma module
 \cite[Thm.~5.2]{K12} via the duality theorem \cite[Thm.~A]{KP16a},
 whereas the right-hand side of \eqref{eqn:OH2} is given by
the
 branching law
 of the unitary representation $\mathcal H^2(X,\mathcal L)|_{G'}$
 (\cite[Thm.~8.3]{K08}), and that they coincide
 under the condition \eqref{eqn:lmdpos}. Hence \eqref{eqn:OH2} is bijective.
 \end{proof}

In order to clarify the dependence of the parameter $\ell$,
 we write ${\mathcal{W}}_{\ell}$
 for the $G'$-equivariant vector bundle corresponding to $\ell \in \Lambda$ from now.
Then Fact \ref{fact:4.3} tells that the one-dimensional vector space
\[
   \mathrm{Hom}_{G'}\left( \mathcal O(X,\mathcal L)\big\vert_{G'},\mathcal O(Y,\mathcal W_\ell)\right)
\simeq
 \mathrm{Hom}_{G'}\left( \mathcal H^2(X,\mathcal L)\big\vert_{G'},\mathcal H^2(Y,\mathcal W_{\ell})\right)
\]
 is spanned by a
\emph{differential} symmetry breaking operator. We fix such a generator $D_\ell$ for every $\ell\in\Lambda$.

Since $D_\ell \colon \mathcal H^2(X,\mathcal L) \to \mathcal H^2(Y,\mathcal W_\ell)$ is a continuous operator
between the Hilbert spaces,
its operator norm $\Vert D_\ell\Vert_{\mathrm{op}}$ is finite and its adjoint
$D_\ell^*$ is a continuous linear operator. Set
$$
 C_\ell:=\Vert D_\ell\Vert_{\mathrm{op}}^2, \quad \Psi_\ell:=\frac1{C_\ell} D_\ell^*.
$$
Let $D=(D_{\ell})_{\ell \in \Lambda}$ be the symmetry breaking transform.
Then we have the following:
\begin{thm}\label{thm:SBTinj}
Suppose we are in Setting \ref{setting:takagi18}.
\begin{itemize}
\item[(1)] $\Psi_\ell\colon \mathcal H^2(Y,\mathcal W_\ell)\longrightarrow\mathcal H^2(X,\mathcal L)$ is a holographic operator. Moreover, it is an isometry up to renormalization.

\item[(2)] The symmetry breaking transform $D$ is injective on $\mathcal H^2(X,\mathcal L)$.

\item[(3)]
Any $f\in\mathcal H^2(X,\mathcal L)$ is recovered from its symmetry breaking transform $Df$ by
$$
f=\sum_{\ell\in\Lambda}\Psi_\ell \left( Df\right)_{\ell}.
$$
\item[(4)] The norm $\Vert f\Vert_{\mathcal H^2(X,\mathcal L)}$ is recovered from the sequence of
norms $\Vert \left( Df\right)_{\ell}\Vert_{\mathcal H^2(Y,\mathcal W_\ell)}$ by
$$
\Vert f\Vert_{\mathcal H^2(X,\mathcal L)}^2=
\sum_{\ell\in\Lambda}\frac1{C_\ell}
\Vert \left( Df\right)_{\ell}\Vert_{\mathcal H^2(Y,\mathcal W_\ell)}^2.
$$
\end{itemize}
\end{thm}
\begin{proof}
The unitary representation of $G$ on the Hilbert space $\mathcal H^2(X,\mathcal L)$ is decomposed discretely and multiplicity freely into the Hilbert
direct sum:
\begin{equation}\label{eqn:HKSK}
\mathcal H^2(X,\mathcal L)|_{G'}
 \simeq \sum_{\ell\in\Lambda}{}^{\oplus}\mathcal H^2(Y,\mathcal W_\ell)
\end{equation}
as unitary representations of the subgroup $G'$
by Fact \ref{fact:4.3}.
\par\noindent
(1)\enspace
 The adjoint operator $D_\ell^*$ is a $G'$-homomorphism because
 $\mathcal H^2(X,\mathcal L)$ and both $\mathcal H^2(Y,\mathcal W_\ell)$ are unitary representations. The second assertion follows from Schur's lemma
 because $\mathrm{Hom}_{G'}(\mathcal H^2(Y,\mathcal W_\ell), \mathcal H^2(X,\mathcal L)\big\vert_{G'})$ is one-dimensional.
\par\noindent
(2)\enspace
Expand $f\in\mathcal H^2(X,\mathcal L)$ as $f=\sum_{\ell\in\Lambda}f_\ell$
according to the decomposition \eqref{eqn:HKSK}.
Then $(Df)_\ell$ is a nonzero multiple of $f_\ell$ by Schur's lemma since the decomposition \eqref{eqn:HKSK}
is multiplicity free. Hence, if $Df=0$, then $f_\ell=0$ for all $\ell\in\Lambda$, and therefore $f=0$.
\par\noindent
Statements (3) and (4) are direct consequences of Lemma \ref{lem:Hilbertdeco}.
\end{proof}

\subsection{Role of orthogonal polynomials}\label{sec:concl3}
In this section we investigate Problems A and B in Setting \ref{setting:takagi18}, and clarify
the role of the F-method and special orthogonal polynomials for the $L^2$-theory of symmetry breaking transforms consisting of holomorphic differential operators.

Suppose we are in  Setting \ref{setting:takagi18}.
As we have seen in Section \ref{sec:concl2},
Theorem \ref{thm:SBTinj} (2) solves Problem A.0, whereas
Theorem \ref{thm:SBTinj} (1), (3), and (4) give a framework for Problems A.1, A.2, and B, respectively,
 for the symmetry breaking
transform $D=(D_\ell)_{\ell \in \Lambda}$.
Thus the solution to Problems A and B is reduced to the following four
questions of finding explicit description and closed formul{\ae} of
\begin{itemize}
\item the support of $\Lambda$;
\item holomorphic differential operators $D_\ell$;
\item the operator norm $\| D_\ell \|_{\operatorname{op}}$;
\item the adjoint operator $D_\ell^*$.
\end{itemize}

Let us summarize briefly what was known, what has been proved in this article,
and what looks promising.

As we mentioned in Section \ref{sec:concl2},
the explicit description of $\Lambda$, equivalently, the
branching law \eqref{eqn:HKSK} for the restriction $\pi\vert_{G'}$ in Section \ref{sec:concl1} was proved  in
\cite[Thm. 8.3]{K08}, which gives a generalization of the Hua--Kostant--Schmid formula in the case when $G'=K$.
Denote by $\mathrm{rank}_\R G/G'$ the split rank of the reductive symmetric space
$G/G'$.
Then it
turns out
 that $\Lambda$ is a free abelian semigroup generated by
 $\mathrm{rank}_\R G/G'$ elements,
 see \cite{K08}.

It is more involved to construct symmetry breaking operators $D_{\ell}$
 explicitly
than determining $\Lambda$, namely, the
branching law of the restriction $\pi\vert_{G'}$. As of now, an explicit construction of $D_\ell$ for all $\ell\in\Lambda$ with exhaustion theorem is known when $\operatorname{rank}_\R G/G'=1$,
see \cite{KP16b}. There are
six families of such symmetric pairs $(G,G')$,
 and the resulting symmetry breaking operators include
 the Rankin--Cohen operators and the holomorphic Juhl operators.

In order to obtain the operator norm $\Vert D_\ell\Vert_{\mathrm{op}}$
 of such holomorphic differential operators $D_{\ell}$, we
 have developed the idea of the F-method to connect $\Vert D_\ell\Vert_{\mathrm{op}}$ with the $L^2$-norm
of special polynomials $P_\ell$ in the following two cases in this article.
\vskip7pt
{\small{\centerline{
\begin{tabular}{c|c}
$\qquad D_\ell$ & $\qquad P_\ell$\\
  \hline
  &\\
  Rankin--Cohen operators & Jacobi polynomials \\&\\
  \hline
  &\\
  Juhl operators & Gegenbauer polynomials
\end{tabular}}}}
\vskip7pt
\noindent
The relationship between $D_\ell$ and $P_\ell$ follows from the fact that the $G'$-equivariance condition on the operator $D_\ell$ is transformed into a certain differential equation
 (\emph{e.g.}\ Jacobi differential equation \eqref{eqn:JacobiDE})
 for the polynomial $P_\ell$.
It is plausible that this idea would work in the full generality of Setting~\ref{setting:takagi18}.

Concerning the adjoint operator $D_\ell^*$, this article has provided two kinds of integral formul{\ae}, that is, by the line integral (Definition \ref{def:HOtensor}), see
Proposition \ref{prop:PsiRC}, and by the integral over the tube domain (Theorem
\ref{thm:CIRM18}). The former has an advantage that the formula is simple and does not require the unitarity of representations, whilst the latter uses a natural idea of the ``relative reproducing kernel" $K_{\lambda,\nu}(\zeta,\tau')$, see
\eqref{eqn:holoKlmdnu}.

\section{Appendix: Jacobi polynomials and Gegenbauer polynomials}\label{sec:appendixJacob}
\subsection{The Jacobi polynomials}
Suppose $\alpha,\beta\in\C$ and $\ell\in\N$.
The Jacobi polynomial $P^{\alpha,\beta}_\ell(t)$ is a polynomial solution to the
Jacobi differential equation
\begin{equation}\label{eqn:JacobiDE}
\left((1-t^2)\frac{d^2}{dt^2}+(\beta-\alpha-(\alpha+\beta+2)t)\frac d{dt}+\ell(\ell+\alpha+\beta+1)\right)y=0,
\end{equation}
which is normalized  by
 $P^{\alpha,\beta}_\ell(1)=\frac{\Gamma(\alpha+\ell+1)}{\Gamma(\alpha+1)\ell!}=
 \frac{(\alpha+1)_\ell}{\ell!}$. Then it satisfies the Rodrigues formula
\begin{equation}\label{eqn:JRodrigues}
(1-t)^\alpha(1+t)^\beta P_\ell^{\alpha,\beta}(t)=\frac{(-1)^\ell}{2^\ell\ell!}\left(\frac d{dt}\right)^\ell\left((1-t)^{\ell+\alpha}(1+t)^{\ell+\beta} \right).
\end{equation}

The Jacobi polynomial
 $P^{\alpha,\beta}_\ell(t)$ is
nonzero and is
 a polynomial of degree $\ell$ for generic parameters (see \cite[Thm. 11.2]{KP16b} for the precise condition).
Explicitly, one has
\begin{eqnarray}\label{eqn:Jacobi}
 P^{\alpha,\beta}_\ell(t)&=&
\frac{(\alpha+1)_\ell}{\ell!}
 {}_2F_1\left(-\ell,\alpha+\beta+\ell+1;\alpha+1;\frac{1-t}2\right) \nonumber
 \\
 &=&
\sum_{j=0}^\ell
\frac{(\alpha+\beta+\ell+1)_j(\alpha+j+1)_{\ell-j}}{(\ell-j)!j!}\left(\frac{t-1}2\right)^j.
\end{eqnarray}

The first Jacobi polynomials are
\begin{itemize}
\item $P_0^{\alpha,\beta}(t)= 1,$
\item
$P_1^{\alpha,\beta}(t)=\frac{1}{2} (\alpha-\beta+(2+\alpha+\beta) t)$.
\end{itemize}
For real $\alpha,\beta$ with $\alpha,\beta>-1$, the Jacobi polynomials $\left\{P_\ell^{\alpha,\beta}\right\}_{\ell\in\N}$ form an orthogonal
basis in the Hilbert space $L^2\left((-1,1),(1-x)^\alpha(1+x)^\beta dx\right)$ with the following
norm (see \cite[page 301]{AAR} for example):
\begin{equation}\label{eqn:Pnorm}
\int_{-1}^1\left\vert P_\ell^{\alpha,\beta}(x)\right\vert^2(1-x)^\alpha(1+x)^\beta dx=
\frac{2^{\alpha+\beta+1}\Gamma(\ell+\alpha+1)\Gamma(\ell+\beta+1)}{(2\ell+\alpha+\beta+1)\Gamma(\ell+\alpha+\beta+1)\ell!}.
\end{equation}

When $\alpha=\beta$ these polynomials yield Gegenbauer polynomials (see
\eqref{eqn:PtoC} below), and they further reduce to Legendre polynomials in the case when $\alpha=\beta=0$.

\subsection{The Gegenbauer polynomials}
For $\alpha\in\C$ and $\ell\in\N$, the Gegenbauer polynomial
 (or ultraspherical polynomial) $C^\alpha_\ell(t)$
 is defined by
 \begin{equation}\label{eqn:gegen}
C^\alpha_\ell(t)=\sum_{k=0}^{\left[\frac\ell2\right]} a_k(\ell,\alpha) t^{\ell-2k},
\end{equation}
where $ a_k(\ell,\alpha)$ is given in \eqref{eqn:alalpha}. The Gegenbauer polynomials are special cases of the Jacobi polynomials by
\begin{equation}\label{eqn:PtoC}
\frac{(2\alpha+1)_\ell}{(\alpha+1)_\ell} P_\ell^{\alpha,\alpha}(x)= C_\ell^{\alpha+\frac12}(x),
\end{equation}
and have
 the generating function:
$$
(1-2t r+r^2)^{-\alpha}=\sum_{\ell\in\N} C_\ell^\alpha(t) r^\ell.
$$
We note that $C_\ell^\alpha(1)=\frac{(2\alpha)_\ell}{\ell!}$. If $\alpha>-\frac12$, then the Gegenbauer polynomials $\left\{ C_\ell^\alpha(v)\right\}_{\ell\in\N}$ form an
orthonormal basis in the Hilbert space $L^2\left((-1,1), (1-v^2)^{\alpha-\frac12}dv\right)$ with the
following
 $L^2$-norm (see \cite[7.313]{GR}):
\begin{equation}\label{eqn:L2Gegen}
\int_{-1}^1 \left\vert C_\ell^{\alpha}(v)\right\vert^2(1-v^2)^{\alpha-\frac{1}{2}}\,dv = \frac{\pi 2^{1-2\alpha}\Gamma(\ell+2\alpha)}{\ell!(\ell+\alpha)\Gamma\left(\alpha\right)^2}.
\end{equation}

redpajama_set_name: RedPajamaArXiv