\section{Introduction}
This paper is a continuation of a previous work \cite{Huang-Wang} in which we proved the uniqueness of complex geodesics with prescribed boundary data on bounded strongly linearly convex domains with $C^3$-smooth boundary and solved a homogeneous complex Monge-Amp\`{e}re (HCMA for short) equation with prescribed {\it boundary singularity} on such domains. The purpose of the present paper is to investigate the parameter dependence of such complex geodesics and the higher regularity of solutions to the aforementioned complex Monge-Amp\`{e}re equation.

Our motivation for this work stems mainly from the following considerations. On the one hand, complex geodesics have proved to be very useful in the study of several complex variables and CR geometry (see \cite{Lempert81, Lempert86, Abatebook, Burns-Krantz, Huang-Pisa94, Bracci-JEMS}, to name only a few), and further exploration may lead to new applications. On the other hand, although it has long been known that the regularity of solutions to HCMA equations is generally at most $C^{1,\,1}$  (see \cite{Gam-Sibony} for a well-known example, and also \cite{BF-counter}), higher regularity than $C^{1,\,1}$---which is quite important in applications---is still not well understood, even in certain special contexts with rich geometric structure. Encouragingly, much progress has been made in this regard over the last few decades, among which we mention particularly \cite{Lempert81, Lempert83, Lempert86, Donaldson, Bracci-MathAnn, Chen-Tian, LV13}. The higher regularity of solutions to the HCMA equation that we consider in this paper will be established via a detailed study of the associated complex foliation whose leaves are complex geodesic discs initiated from a fixed boundary point, where the equation has a simple singularity. Our approach is largely inspired by the work of Lempert \cite{Lempert81, Lempert86} and Chang-Hu-Lee \cite{Chang-Hu-Lee88}. In connection with the subject of the present paper and the approach we adopt here, we also mention that complex (or rather Monge-Amp\`{e}re) foliations and HCMA equations are closely related and they have played a significant role in proving many fascinating results that illustrate the interplay between pluripotential theory, several complex variables and complex geometry. For a historical overview and references in this regard we refer the reader to \cite{BK-CPAM, Patriziobook, RWN_Survey}.

We now elaborate on our main results. To begin with, we fix some necessary notation and terminology. Let $\Delta$ be the open unit disc in $\mathbb C$ and $\Omega$ be a bounded strongly linearly convex domain in $\mathbb C^n\, (n>1)$ with $C^3$-smooth boundary. A complex geodesic of $\Omega$ is by definition a holomorphic mapping from $\Delta$ to $\Omega$ that realizes the Kobayashi distance between any two distinct points in its image. Significantly different from real geodesics in Riemannian geometry, complex geodesics are global analytic objects whose existence is a very subtle problem. In his celebrated paper \cite{Lempert81} and later work \cite{Lempert84}, Lempert showed that complex geodesics exist in great abundance on $\Omega$ and enjoy sufficiently high boundary regularity and a number of other good properties.

To proceed let us further assume that $\partial \Omega$ is $C^{m,\,\alpha}$-smooth, where $m\geq 4$ is an integer and $0<\alpha<1$. And throughout the paper we always make this assumption unless otherwise specified. Set
\begin{equation}\label{defn:fiberbundle}
   S_{\partial\Omega}:=\big\{(p, v)\in\mathbb C^n\times\mathbb C^n\!: p\in\partial\Omega,\ v\in L_p\big\},
\end{equation}
where
   $$
   L_p:=\big\{v\in\mathbb C^n\!: |v|=1,\ \langle v, \nu_p\rangle>0\big\},
   $$
$\nu_p$ is the unit outward normal to $\partial\Omega$ at $p$, and  $\langle\:,\,\rangle$ denotes the standard Hermitian inner product on $\mathbb C^n$. Then $S_{\partial\Omega}$ is a $C^{m-1,\,\alpha}$-smooth real submanifold of $\mathbb C^{2n}$ with dimension $4n-3$.
According to \cite{Huang-Pisa94, Huang-Wang} (see also \cite{Chang-Hu-Lee88} for prior partial results), for every $(p, v)\in S_{\partial\Omega}$ there exists a unique\footnote{In fact, the $C^3$-regularity of $\partial\Omega$ is sufficient for the purpose here, as shown in \cite{Huang-Wang}.} complex geodesic $\varphi_{p,\, v}$ of $\Omega $ such that $\varphi_{p,\, v}(1)=p$, $\varphi_{p,\, v}'(1)=\langle v, \nu_p\rangle v$, and
   $$
   \left.\frac{d}{d\theta}\right|_{\theta=0}|\varphi^{\ast}_{p,\, v}(e^{i\theta})|=0,
   $$
where  $\varphi^{\ast}_{p,\, v}$ is the dual mapping of $\varphi_{p,\, v}$ (see Subsection \ref{Kobayashi-geodesic} below for a precise definition). Following \cite{Huang-Wang}, we refer to such a unique $\varphi_{p,\, v}$ as the {\it preferred complex geodesic of $\Omega$ associated to  $(p, v)\in S_{\partial\Omega}$}.

We can now state our first main result.

\begin{thmx}\label{thm:parameter dependence}
Let $\Omega$ and $S_{\partial\Omega}$ be as above. Then for every $k\in\{2,\ldots,m-2\}$ and $\varepsilon\in (0,\alpha)$, the mapping
   $$
   \Phi\!: S_{\partial\Omega}\to\mathcal{O}(\Delta,\,\mathbb C^n)
   \cap C^{k,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n), \quad (p, v)\mapsto \varphi_{p,\, v}
   $$
is locally $C^{m-k-1,\,\alpha-\varepsilon}$-smooth.
\end{thmx}

Here $\mathcal{O}(\Delta,\,\mathbb C^n)$ denotes the set of all holomorphic mappings from $\Delta$ to $\mathbb C^n$, and $C^{k,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n)$ is the usual H\"{o}lder space on $\overline{\Delta}$ with norm defined in a standard way.


Combining Theorem \ref{thm:parameter dependence} with some fundamentally critical results for extremal mappings (with respect to the Kobayashi-Royden metric) proved by Huang \cite{Huang-Illinois94, Huang-Pisa94}, we then obtain the following result concerning the differentiable dependence of preferred complex geodesics on their parameters.

\begin{thmx}\label{thm:joint-regularity}
The mapping
   $$
   \widetilde{\Phi}\!: S_{\partial\Omega}\times\overline{\Delta}\to \mathbb C^n,
   \quad (p, v, \zeta)\mapsto \varphi_{p,\,v}(\zeta)
   $$
is locally $C^{m-3,\,\alpha-\varepsilon}$-smooth for all $\varepsilon\in (0,\alpha)$.
\end{thmx}

Next, we turn to some consequences of Theorem \ref{thm:joint-regularity}, particularly an application to the higher regularity of solutions to the HCMA equation considered in \cite{Bracci-MathAnn, Bracci-Trans, Bracci-Adv, Huang-Wang}. For this we need to recall the so-called boundary spherical representation for $\Omega$,  constructed by Huang and the author in \cite{Huang-Wang}. Denote by $\mathbb B^n$ the open unit ball in $\mathbb C^n$. It is easy to check that for every $(q, v)\in S_{\partial \mathbb B^n}$,  the mapping
\begin{equation*}\label{ball-CG}
  \eta_{q,\, v}\!:\Delta \to\mathbb B^n, \quad \zeta \mapsto q+(\zeta-1)\langle v, q\rangle v
\end{equation*}
is the associated preferred complex geodesic of $\mathbb B^n$. The boundary spherical representation $\Psi_p\!:\overline{\Omega}\to\overline{\mathbb B}^n$ is then defined as $\Psi_p(p):=\nu_p$ and
\begin{equation}\label{defn:BSR}
   \Psi_p(z):=\nu_p+(\zeta_{z,\,p}-1)\langle v_{z,\,p},\, \nu_p\rangle v_{z,\,p},\quad
   z\in\overline{\Omega}\!\setminus\!\{p\},
\end{equation}
where $\zeta_{z,\,p}:=\varphi_{p,\, v_{z,\,p}}^{-1}(z)$ and $v_{z,\,p}\in L_p$ is the only data such that the associated preferred complex geodesic $\varphi_{p,\, v_{z,\,p}}$ of $\Omega$ passes through $z$, i.e.,  $z\in \varphi_{p,\, v_{z,\,p}}(\overline{\Delta})$. In this way $\Psi_p$ is a bijection with inverse $\Psi_p^{-1}$ given by $\Psi^{-1}_p(\nu_p)=p$ and
\begin{equation}\label{defn:inv-BSR}
   \Psi^{-1}_p(z)=\varphi_{p,\,v_{w,\,\nu_p}}(\zeta_{w,\,\nu_p}),\quad w\in\overline{\mathbb B}^n\!\setminus\!\{\nu_p\},
\end{equation}
where $(v_{w,\,\nu_p},\,\zeta_{w,\,\nu_p})\in L_p\times\overline{\Delta}$ is the only data such that $\eta_{\nu_p,\,v_{w,\,\nu_p}}(\zeta_{w,\,\nu_p})=w$; more explicitly,
\begin{equation}\label{ball-splitting}
   v_{w,\,\nu_p}=-\frac{1-\langle \nu_p, w\rangle}{|1-\langle \nu_p, w\rangle|}\frac{w-\nu_p}{|w-\nu_p|},
   \quad \zeta_{w,\,\nu_p}=1-\frac{|w-\nu_p|^2}{|1-\langle \nu_p, w\rangle|^2}(1-\langle w, \nu_p\rangle).
\end{equation}
It was also shown in \cite{Huang-Wang} that $\Psi_p\!:\overline{\Omega}\to\overline{\mathbb B}^n$ is a homeomorphism and, among other things,
   $$
   \Psi(z,\,p):=\Psi_p(z),\quad (z,\,p)\in\overline{\Omega}\times\partial\Omega,
   $$
is continuous (provided $\partial\Omega$ is only $C^3$-smooth; see \cite[Section 3]{Huang-Wang} for details). To get a better understanding of the relationship between the regularity of $\Psi$ and that of the boundary $\partial\Omega$, we prove in this paper the following

\begin{thmx}\label{thm:regularity-BSR}
For every $p\in \partial\Omega$ and sufficiently small $\delta>0$, the mapping
   $$
   \Psi_p\!:\overline{\Omega}\!\setminus\!B(p, \delta)\to
            \overline{\mathbb B}^n\!\setminus\!\Psi_p\big(B(p, \delta)\big)
   $$
is a $C^{m-3,\,\alpha-\varepsilon}$-smooth diffeomorphism for all $\varepsilon\in (0,\alpha)$, where
$B(p, \delta)$  denotes the open ball of radius $\delta$ centered at $p$ in $\mathbb C^n$. Moreover,
   $$
   \Psi\in C^{m-3,\,\alpha-\varepsilon}\big((\overline{\Omega}\times\partial\Omega)
   \!\setminus\!{\rm dist}_{\delta}\, (\overline{\Omega}\times\partial\Omega)\big)
   $$
for all $\varepsilon\in (0,\alpha)$, where
   $$
   {\rm dist}_{\delta}\, (\overline{\Omega}\times\partial\Omega):=\big\{(z,\, p)\in \overline{\Omega}\times\partial\Omega\!: |z-p|<\delta\big\}.
   $$
\end{thmx}

This result strengthens a similar (and deep) result of Chang-Hu-Lee \cite{Chang-Hu-Lee88}, drawing a stronger conclusion under a much weaker condition, and thus may lead to more applications. Relying heavily on Lempert's deformation theory (see \cite{Lempert81}) and the Chern-Moser-Vitushkin normal form theory, the original argument of Chang-Hu-Lee requires $\partial\Omega$ to be sufficiently smooth and is also technically involved (and quite sketchy at some points). In contrast, with the great help of Theorem \ref{thm:joint-regularity} and some new observations (see Theorems \ref{thm:RH-prob2} and \ref{thmA:normalization} below) we can largely follow the strategy of Lempert \cite{Lempert86}, for which the $C^{4,\,\alpha}$-regularity of $\partial\Omega$ turns out to be sufficient. In addition to this, the treatment in this paper is much simpler, as we shall see later.


We are now in a position to consider the following HCMA equation on $\Omega$ with $p\in\partial\Omega$:
\begin{equation}\label{eq:Bdry-MA}
\begin{cases}
u\in {\rm Psh}(\Omega)\cap
L_{{\rm loc}}^{\infty}(\Omega), \\
(dd^c u)^n=0 \quad \quad  \;\;\; \ \ \,  {\rm on}\,\ \Omega, \\
\lim_{z\to x}u(z)=0 \;\;\; \!\: \quad {\rm for}\,\ x\in\partial\Omega\!\setminus\!\{p\}, \\
u(z)\approx -|z-p|^{-1} \:\:\,\,\,\!\: {\rm as}
\ \ z\to p\, \ {\rm nontangentially},
\end{cases}
\end{equation}
which was first studied by Bracci-Patrizio \cite{Bracci-MathAnn} when $\Omega$ is further assumed to be strongly convex and to have $C^m$-smooth boundary for $m\ge 14$ (here ${\rm Psh}(\Omega)$  denotes the set of all plurisubharmonic functions on $\Omega$).
Put
  $$
  P_{\Omega}(z, p)=-\frac{1-|\Psi_p(z)|^2}{|1-\langle \Psi_p(z), \nu_p\rangle|^2},
  $$
which is continuous on $(\overline{\Omega}\times\partial\Omega)\!\setminus\!{\rm diag }\, \partial\Omega$,
where
  $$
  {\rm diag }\, \partial\Omega:=\big\{(z, z)\in \mathbb C^{2n}\!: z\in\partial\Omega\big\}.
  $$
By virtue of \cite[Theorem 1.3]{Huang-Wang} (and its proof) we know that for every $p\in\partial\Omega$ the function $P_{\Omega}(\,\cdot\,,\, p)$ solves the above equation. Now as an immediate consequence of Theorem \ref{thm:regularity-BSR} we obtain the following

\begin{thmx}\label{thm:regularity-HCMA}
For every sufficiently small $\delta>0$,
   $$
   P_{\Omega}\in C^{m-3,\,\alpha-\varepsilon}\big((\overline{\Omega}\times\partial\Omega)
   \!\setminus\!{\rm dist}_{\delta}\, (\overline{\Omega}\times\partial\Omega)\big)
   $$
for all $0<\varepsilon<\alpha$.
\end{thmx}

The particular interest of studying equation \eqref{eq:Bdry-MA} lies in the fact that it is a boundary analogue of the following interior one on $\Omega$ with $w\in\Omega$:
\begin{equation}\label{eq:Inter-MA}
  \begin{cases}
   u\in {\rm Psh}(\Omega)\cap
   L_{{\rm loc}}^{\infty}(\Omega\!\setminus\!\{w\}), \\
   (dd^c u)^n=0 \qquad \qquad  \quad \!\quad\ \ \,\, {\rm on}\,\ \Omega \!\setminus\! \{w\}, \\
   \lim_{z\to x}u(z)=0  \qquad \qquad \quad\!  \: {\rm for}\,\ x\in\partial\Omega, \\
   u(z)-\log|z-w|=O(1)\ \ \;  {\rm as} \ \ z\to w,\\
  \end{cases}
\end{equation}
which has been extensively studied in the literature; see \cite{Lempert81, Klimek-Green, Demailly87, BD-counter, GuanMAeqn, Blocki00, Blocki05} among others. The solution to equation \eqref{eq:Inter-MA} can easily be shown to be unique and is now known as the pluricomplex Green function. It shares many properties with the classical Green function in one complex variable and has profound applications, especially when it appears as a weight function in H\"{o}rmander's $L^2$ theory of the $\overline{\partial}$-operator. We note that there has been a great deal of work along this line since the work of Diederich-Ohsawa \cite{Die-Ohsawa95}, and here we refer the reader only to \cite{Blocki-survey, Chen-Survey} for historical background and to \cite{Chen-APDE} for recent developments.

The study of equation \eqref{eq:Bdry-MA} was initiated in \cite{Bracci-MathAnn, Bracci-Trans} and continued very recently in \cite{Huang-Wang, Bracci-Adv}. It turns out that the function $P_{\Omega}(\,\cdot\,,\, p)$ constructed as above bears a strong resemblance to the classical Poisson kernel in real potential theory (see \cite{Bracci-MathAnn, Bracci-Trans}) and has already found interesting applications in the theory of holomorphic semigroups (see \cite{Bracci-JEMS}) and others. Nevertheless, it is still an open question, quite interesting at least from the point of view of PDEs, whether $P_{\Omega}(\,\cdot\,,\, p)$ is the only solution to equation \eqref{eq:Bdry-MA} up to a positive constant multiple. Under certain analytic and/or geometric conditions, a partial answer to this question can be found in \cite{Bracci-Trans}.



At present it seems difficult to answer the above question in full generality. As far as we know, the well-known comparison principle for the complex Monge-Amp\`{e}re operator due to Bedford-Taylor \cite{Bedford82}, as well as its variants in different contexts, remains the only available tool for proving the uniqueness of solutions to complex Monge-Amp\`{e}re equations of various types (for more information on this see, for example, the nice monograph \cite{GZ-MAeqnbook} and the references therein). But unfortunately, such a key tool fails to work in the present context because of the presence of {\it boundary singularity} in equation \eqref{eq:Bdry-MA}. It seems likely that Theorem \ref{thm:regularity-HCMA} could be useful in seeking an affirmative answer to the above question. We hope to focus on this in future work and expect to find further applications of the results obtained in this paper.

We now conclude this introduction with a description of the organization of the paper. In Section \ref{sect:preliminaries} we first collect necessary background material for complex geodesics on bounded strongly linearly convex domains and then establish some preliminary results that we need in the subsequent sections of the paper. Theorems \ref{thm:parameter dependence} and \ref{thm:joint-regularity} are proved in Sections \ref{sect:proof of thm A} and \ref{sect:proof of thm B}, respectively. Section \ref{sect:proof of thm C} is devoted to the proof of Theorem \ref{thm:regularity-BSR}. In the course of the proof we need a result of Lempert on performing a canonical change of coordinates for strongly linearly convex domains along a given complex geodesic disc. We include a detailed proof of this result in Appendix \ref{supplement} for the reader's convenience.

\medskip
\noindent {\bf Acknowledgements.} The author is greatly indebted to Xiaojun Huang for many enlightening discussions concerning this work and for his constant support and encouragement over the years. The author also wishes to express his gratitude to L\'{a}szl\'{o} Lempert, whose landmark work \cite{Lempert81} on complex geodesics has always been an important source of ideas and motivation for this study. Thanks also go to W{\l}odzimierz Zwonek for commenting on the proof of Theorem \ref{thmA:normalization} and for bring the interesting work \cite{KZ16} to the author's attention.




\section{Preliminaries}\label{sect:preliminaries}
\subsection{The Kobayashi distance and complex geodesics}\label{Kobayashi-geodesic}
We briefly recall here the definitions of the Kobayashi-Royden metric and the Kobayashi distance, as well as some well-known results concerning complex geodesics on bounded strongly linearly convex domains in $\mathbb C^n$. For a complete picture of the subject,  the interested reader may consult \cite{Abatebook, Lempert81, Lempert84, KW13}.

Let $\Delta$ be the open unit disc in $\mathbb C$. The {\it Kobayashi-Royden metric} $\kappa_{\Omega}$ on a domain $\Omega\subset\mathbb C^n$ is  the pseudo-Finsler metric defined by
$$
\kappa_{\Omega}(z, v):=\inf\big\{\lambda>0\,|\;\exists \;\varphi\in \mathcal{O}(\Delta,\, \Omega)\!: \varphi(0)=z,\, \varphi'(0)=\lambda^{-1}v \big\}, \quad (z, v)\in \Omega\times\mathbb C^n,
$$
where $\mathcal{O}(\Delta,\, \Omega)$ denotes the set of all holomorphic mappings from $\Delta$ to $\Omega$.
The {\it Kobayashi distance} $k_{\Omega}$ on $\Omega$ is then defined to be the integrated form of $\kappa_{\Omega}$, that is
$$
k_{\Omega}(z, w)=\inf_{\gamma\in \Gamma}\int_{0}^{1}\kappa_{\Omega}(\gamma(t), \,\gamma'(t))dt,
\quad (z, w)\in \Omega\times \Omega,
$$
where $\Gamma$ is the set of all $C^1$-smooth curves $\gamma\!:[0, 1] \rightarrow \Omega$ such that $\gamma(0)=z$ and $\gamma(1)=w$. For the open unit disc  $\Delta\subset\mathbb C$, $k_{\Delta}$ coincides with the classical \textit{Poincar\'{e} distance}, i.e.,
$$
k_{\Delta} (\zeta_1, \zeta_2)=\tanh^{-1}\bigg|\frac{\zeta_1-\zeta_2}{1-\zeta_1\overline{\zeta}_2}\bigg|,
\quad (\zeta_1, \zeta_2)\in \Delta\times\Delta.
$$
A holomorphic mapping $\varphi\!:\Delta\to\Omega$ is called a {\it complex geodesic} of $\Omega$ if it is an isometry between $k_{\Delta}$ and $k_{\Omega}$, i.e.,
\begin{equation*}\label{isometry}
k_{\Omega}(\varphi(\zeta_1),\,\varphi(\zeta_2))=k_{\Delta}(\zeta_1, \zeta_2)
\end{equation*}
for all $\zeta_1$, $\zeta_2\in\Delta$.

We proceed with one more definition.

\begin{definition}
A domain $\Omega$ in $\mathbb C^n\, (n>1)$ is called  {\it strongly linearly convex} if it has $C^2$-smooth boundary and admits a $C^2$-defining function $\rho\!:\mathbb C^n\to\mathbb R$ whose real Hessian is positive definite on the complex tangent space of $\partial\Omega$, i.e.,
  $$
  \sum_{j,\, k=1}^n\frac{\partial^2 \rho}{\partial z_j\partial\overline
  z_k}(p)v_j\overline{v}_k
  >\bigg|\sum_{j,\, k=1}^n\frac{\partial^2 \rho}{\partial z_j\partial z_k}(p)v_jv_k\bigg|
  $$
for all $p\in\partial\Omega$ and non-zero $v=(v_1,\ldots,v_n)\in T_p^{1,\, 0}\partial\Omega$.
\end{definition}

At first sight, the class of strongly linearly convex domains seems to be rather restrictive. However, as recently discovered, this is not the case. There are bounded strongly linearly convex domains with real-analytic boundary, which are not biholomorphic to convex ones and even cannot be exhausted by biholomorphic images of such domains; see \cite{Pflug-Zwonek}.

We also record the following intrinsic characterization of complex geodesics on strongly linearly convex domains, which is due to Lempert \cite{Lempert84} (see also \cite{KW13} for a very clear exposition) and will be used in this paper in a crucial way.

\begin{theorem}\label{thm:Lempert1}
Let $\Omega$ be a bounded strongly linearly convex domain in $\mathbb C^n\, (n>1)$ with $C^2$-smooth boundary  and $\nu$ be the unit  outward normal vector field of $\partial\Omega$. Then a holomorphic mapping $\varphi\!:\Delta\to\Omega$ is a complex geodesic of $\Omega$ if and only if it satisfies the following conditions:
\begin{enumerate}[leftmargin=2.0pc, parsep=4pt]
\item [{\rm(i)}]
$\varphi\in C^{1/2}(\overline{\Delta},\,\mathbb C^n)$;
\item [{\rm(ii)}]
$\varphi$ is proper, i.e., $\varphi(\partial\Delta)\subset\partial\Omega$;
\item [{\rm(iii)}]
there exists a positive function $\mu\in C^{1/2}(\partial\Delta,\,\mathbb R)$ such that the mapping
$$
\partial\Delta\ni \zeta\mapsto \zeta\mu(\zeta)\overline{\nu\circ\varphi(\zeta)}\in\mathbb C^n
$$
extends holomorphically to $\Delta$; and
\item [{\rm(iv)}]
the winding number of the function
$$
\partial\Delta\ni \zeta\mapsto \langle z-\varphi(\zeta),\, \nu\circ\varphi(\zeta)\rangle
$$
is zero for some {\rm(}and hence all\,{\rm)} $z\in\Omega$.
\end{enumerate}
\end{theorem}

Regarding the last condition, observe that  the (strong) linear convexity of $\Omega$ implies that the function under consideration does not vanish  anywhere, so its winding number is well-defined.

For a complex geodesic $\varphi$ of $\Omega$, the resulting holomorphic mapping as in {\rm(iii)}, which we denote by $\varphi^{\ast}$, can be normalized so that $\langle\varphi', \, \overline{\varphi^{\ast}}\rangle=1$ on $\overline{\Delta}$. The mapping $\varphi^{\ast}$ satisfying such a normalization condition is unique and usually called the {\it dual mapping} of $\varphi$. Lempert also proved that if $\partial\Omega$ is further $C^{m,\,\alpha}$-smooth for some $m\ge 2$ and $\alpha\in(0, 1)$, then both $\varphi$ and $\varphi^{\ast}$ are  $C^{m-1,\,\alpha}$-smooth up to the boundary.


\subsection{Riemann-Hilbert problems}\label{Lem:RH-problem}
Given $m\in\mathbb N$ and $\alpha\in(0, 1)$, we denote by $\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ the subspace of $C^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ consisting of mappings that admit holomorphic extension to $\Delta$. Throughout the paper we shall always not distinguish between a mapping $\varphi\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ and its holomorphic extension to $\Delta$, which allows us to speak of its interior values.

Of crucial importance in the proofs of Theorems \ref{thm:parameter dependence} and \ref{thm:regularity-BSR} are the following two results, which can be thought of as the generalizations of a result of Lempert  \cite{Lempert86} from the $0$-jet (i.e., one-point) case to the $1$-jet and two-point cases, respectively.

\begin{theorem}\label{thm:RH-prob1}
Suppose that $m\in\mathbb N^{\ast},\, \alpha\in(0, 1)$, and $H,\, S\in C^{m,\,\alpha}(\partial\Delta,\,\mathbb C^{n\times n})$ are such that $H(\zeta)$ is Hermitian,  $S(\zeta)$ is symmetric, and
\begin{equation}\label{eq:cond-RH}
   v^tH(\zeta)\bar{v}>|v^tS(\zeta)v|
\end{equation}
for all $\zeta\in\partial\Delta$ and $v\in\mathbb C^n\!\setminus\!\{0\}$. Then for every $f\in C^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ and $(\zeta_0,\, z_0,\,v_0)\in \overline{\Delta}\times\mathbb C^n\times\mathbb C^n$, there exists a unique $g\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ such that $g(\zeta_0)=z_0$, $g'(\zeta_0)=v_0$, and
   $$
   H\overline{(g/{\rm Id_{\partial\Delta}})}+Sg/{\rm Id_{\partial\Delta}}+f\in
   \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n).
   $$
\end{theorem}


\begin{proof}
We argue by considering the following two cases:

\medskip
\noindent  {\bf Case 1:} $\zeta_0\in \Delta$.
\smallskip

If $\zeta_0=0$, the result is an immediate consequence of \cite[Lemma 4.2]{Lempert86}. Otherwise, we take a $\sigma\in {\rm Aut}(\Delta)$ such that $\sigma(0)=\zeta_0$. Then there exists a positive smooth function $\mu\!:\partial\Delta\to \mathbb R$ such that
$\partial\Delta\ni\zeta\mapsto\zeta\mu(\zeta)/\sigma(\zeta)$ extends to a nowhere vanishing holomorphic function $h$ on $\overline{\Delta}$. In fact, it suffices to take
   $$
   \mu(\zeta):=\frac{\sigma(\zeta)}{\zeta\sigma'(\zeta)},\quad \zeta\in \partial\Delta.
   $$
Now the desired result follows by applying the one in the case of $\zeta_0=0$ to
the pair $(\mu^{-1}H\circ\sigma,\, \mu^{-1}S\circ\sigma)$ instead of $(H,\,S)$.


\medskip
\noindent  {\bf Case 2:} $\zeta_0\in \partial\Delta$.
\smallskip

Let $f$ be as described. According to \cite[Lemma 4.2]{Lempert86}, there exists a unique $g^{\ast}\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ such that $g^{\ast}(0)=0$ and
   $$
   H\overline{g^{\ast}}+Sg^{\ast}+f\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n).
   $$
Similarly, there exist unique $g^r_k,\,g^i_k\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ such that
   $$
   g^r_k(0)=e_k,\quad  H\overline{g^r_k}+Sg^r_k\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n),
   $$
and
   $$
   g^i_k(0)=ie_k,\quad H\overline{g^i_k}+Sg^i_k\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n),
   $$
where $k=1,\ldots,n$, and $\{e_1,\ldots, e_n\}$ denotes the standard basis of $\mathbb C^n$. In the same fashion, one can also find  unique $h^r_k,\,h^i_k\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ with $h^r_k(0)=h^i_k(0)=0$ such that
\begin{equation*}
\begin{split}
   H\overline{(h^r_k+e_k/{\rm Id_{\partial\Delta}})}+S(h^r_k+e_k/{\rm Id_{\partial\Delta}})
   &=H\overline{h^r_k}+Sh^r_k+{\rm Id_{\partial\Delta}}He_k+Se_k/{\rm Id_{\partial\Delta}}\\
   &\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
   H\overline{(h^i_k+ie_k/{\rm Id_{\partial\Delta}})}+S(h^i_k+ie_k/{\rm Id_{\partial\Delta}})
   &=H\overline{h^i_k}+Sh^i_k-i{\rm Id_{\partial\Delta}}He_k+iSe_k/{\rm Id_{\partial\Delta}}\\
   &\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n),
\end{split}
\end{equation*}
where $k=1,\ldots,n$. Now a mapping $g:=(g_1, \ldots,g_n)^t
\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ satisfies
   $$
   H\overline{(g/{\rm Id_{\partial\Delta}})}+Sg/{\rm Id_{\partial\Delta}}+f\in
   \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)
   $$
if and only if the mapping
   $$
   h:=\frac{g-g(0)}{{\rm Id_{\partial\Delta}}}-g^{\ast}-
      \sum_{k=1}^{n}\Big(\big({\rm Re}\,g_k(0)\big)h^r_k+\big({\rm Im}\,g_k(0)\big)h^i_k\Big)
     \in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)
   $$
satisfies
   $$
   H\overline{h}+Sh\in\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n).
   $$
As $h(0)=g'(0)$, it follows from the uniqueness part of \cite[Lemma 4.2]{Lempert86} that
   $$
   h=\sum_{k=1}^{n}\Big(\big({\rm Re}\,g'_k(0)\big)g^r_k+\big({\rm Im}\,g'_k(0)\big)g^i_k\Big).
   $$
Consequently, $g$ takes the form
\begin{equation}\label{eq:solutionformula}
   g=g(0)+{\rm Id_{\partial\Delta}}\bigg\{g^{\ast}+\sum_{k=1}^{n}\Big(\big({\rm Re}\,g'_k(0)\big)g^r_k
     +\big({\rm Im}\,g'_k(0)\big)g^i_k+\big({\rm Re}\,g_k(0)\big)h^r_k+
      \big({\rm Im}\,g_k(0)\big)h^i_k\Big)\bigg\}.
\end{equation}
To prove the existence and uniqueness of $g$ with the desired property as in the theorem, it now suffices to show that $g(0)$ and $g'(0)$ are uniquely determined by the prescribed boundary data.

In view of  equality \eqref{eq:solutionformula}, it suffices to show that the  vectors
$$
\begin{pmatrix}
    g^r_k(\zeta_0)\\
    (g^r_k)'(\zeta_0)\\
\end{pmatrix},
\quad
\begin{pmatrix}
    g^i_k(\zeta_0)\\
    (g^i_k)'(\zeta_0)\\
\end{pmatrix},
\quad
\begin{pmatrix}
    h^r_k(\zeta_0)+\zeta^{-1}_0e_k\\
    (h^r_k)'(\zeta_0)-\zeta^{-2}_0e_k\\
\end{pmatrix},
\quad
\begin{pmatrix}
   h^i_k(\zeta_0)+i\zeta^{-1}_0e_k\\
   (h^i_k)'(\zeta_0)-i\zeta^{-2}_0e_k\\
\end{pmatrix},
$$
$k=1,\ldots, n$, are linearly independent over $\mathbb R$. This amounts to proving that $\varphi=0$ is the only element of $\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ that satisfies
\begin{equation*}\label{eq:solution-unique1}
   H\overline{(\varphi/{\rm Id_{\partial\Delta}})}+S\varphi/{\rm Id_{\partial\Delta}}\in
   \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n),\quad \varphi(\zeta_0)=\varphi'(\zeta_0)=0.
\end{equation*}
To prove this, it is clearly enough to consider the case when $m=1$. Taking into account the invariance of the last condition under rotations we may assume that $\zeta_0=1$. Also by virtue of \cite[Th\'{e}or\`{e}me B]{Lempert81}, we may further assume that $H=I_n$. Then condition \eqref{eq:cond-RH} means precisely that for every $\zeta\in\partial\Delta$ the operator norm $\|S(\zeta)\|$ of the matrix $S(\zeta)$ is less than one.

Now suppose $\varphi\in \mathcal{O}^{1,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ satisfies $\varphi(1)=\varphi'(1)=0$ and
\begin{equation}\label{eq:solution-varphi}
   \widetilde{\varphi}:=\overline{\varphi/{\rm Id_{\partial\Delta}}}+S\varphi/{\rm Id_{\partial\Delta}}
   \in\mathcal{O}^{1,\,\alpha}(\partial\Delta,\,\mathbb C^n).
\end{equation}
Set
   $$
   q:=\sup\big\{p\geq 0\!: ({\rm Id_{\partial\Delta}}-1)^{-2}\varphi
      \in L^p(\partial\Delta,\, \mathbb C^n)\big\}.
   $$
Clearly $q\geq(1-\alpha)^{-1}>1$. We claim that $q=\infty$. If this were not the case, there would exist a $Q>q$ such that $(1-\alpha)Q<q$. Then
\begin{align}\label{ineq:L^Q-intergability1}
\begin{split}
 \int_{0}^{2\pi}&\bigg|\frac{(S(e^{i\theta})-S(1))\varphi(e^{i\theta})}{(e^{i\theta}-1)^2}\bigg|^Qd\theta
   \leq \int_{0}^{2\pi}\bigg(\frac{\|S(e^{i\theta})-S(1)\||\varphi(e^{i\theta})|}
        {|e^{i\theta}-1|^2}\bigg)^Qd\theta\\
     &=\int_{0}^{2\pi}\bigg(\frac{\|S(e^{i\theta})-S(1)\|}{|e^{i\theta}-1|^{\alpha}}\bigg)^Q
     \bigg|\frac{\varphi(e^{i\theta})}{e^{i\theta}-1}\bigg|^{\alpha Q}
     \bigg|\frac{\varphi(e^{i\theta})}{(e^{i\theta}-1)^2}\bigg|^{(1-\alpha)Q}d\theta<\infty,
\end{split}
\end{align}
since the first two factors of the integrand in the last integral are bounded and the last one is integrable.
Note that the function ${\rm Id_{\partial\Delta}}/({\rm Id_{\partial\Delta}}-1)^2$ takes values in $\mathbb R$. By \eqref{eq:solution-varphi} we then have the  decomposition
\begin{equation}\label{eq:L^Q-decomposition1}
   \frac{\,(S-S(1))\varphi\,}{({\rm Id_{\partial\Delta}}-1)^2}
   =\frac{\,{\rm Id_{\partial\Delta}}\widetilde{\varphi}-S(1)\varphi\,}{({\rm Id_{\partial\Delta}}-1)^2}
    -\overline{\, \frac{\varphi}{({\rm Id_{\partial\Delta}}-1)^2}\,}.
\end{equation}
In view of this,  \eqref{ineq:L^Q-intergability1}, and a classical result of Riesz that the Hilbert transform maps $L^Q(\partial\Delta, \,\mathbb C)$ into itself, we conclude that both terms on the right-hand side of \eqref{eq:L^Q-decomposition1} belong to $L^Q(\partial\Delta, \,\mathbb C^n)$. This contradicts the fact that $Q>q$, and the claim follows.

Now we see in particular that both $({\rm Id_{\partial\Delta}}-1)^{-2}\varphi$ and
$({\rm Id_{\partial\Delta}}-1)^{-2}S\varphi$ belong to $L^2(\partial\Delta,\, \mathbb C^n)$. Let
  $$
  \sum_{k=0}^{\infty}a_k\zeta^k,\quad\sum_{k=-\infty}^{\infty}b_k\zeta^k
  $$
be their Fourier expansions, respectively. By \eqref{eq:solution-varphi} we have $b_{-k}=-\overline{a}_k$ for all $k\geq0$. If $\varphi\neq 0$, an application of the Parseval identity and the fact that $\sup_{\partial\Delta}\|S\|<1$ would give
\begin{align*}\label{ineq:RH-contradiction1}
\begin{split}
    \frac{1}{2\pi}&\int_{0}^{2\pi}\bigg|\frac{\varphi(e^{i\theta})}{(e^{i\theta}-1)^2}\bigg|^2d\theta
    =\sum_{k=0}^{\infty}|a_k|^2\leq\sum_{k=-\infty}^{\infty}|b_k|^2\\
    &=\frac{1}{2\pi}\int_{0}^{2\pi}\bigg|S(e^{i\theta})\frac{\varphi(e^{i\theta})}{(e^{i\theta}-1)^2}\bigg|^2
     d\theta<\frac{1}{2\pi}\int_{0}^{2\pi}\bigg|\frac{\varphi(e^{i\theta})}{(e^{i\theta}-1)^2}\bigg|^2d\theta,
\end{split}
\end{align*}
yielding a contradiction. It follows therefore that $\varphi=0$, as desired. This proves the theorem.
\end{proof}

Similar to the $1$-jet case, we also have the following 2-point case.


\begin{theorem}\label{thm:RH-prob2}
Suppose that $m\in\mathbb N,\, \alpha\in(0, 1)$, and $H,\, S\in C^{m,\,\alpha}(\partial\Delta,\,\mathbb C^{n\times n})$ are such that $H(\zeta)$ is Hermitian, $S(\zeta)$ is symmetric, and
  $$
   v^tH(\zeta)\bar{v}>|v^tS(\zeta)v|
  $$
for all $\zeta\in\partial\Delta$ and $v\in\mathbb C^n\!\setminus\!\{0\}$. Then for every $f\in C^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ and $(\zeta_0,\, z_0),\, (\xi_0,\, w_0)\in \overline{\Delta}\times\mathbb C^n$ with $\zeta_0\neq\xi_0$, there exists a unique $g\in \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ such that $g(\zeta_0)=z_0$, $g(\xi_0)=w_0$, and
   $$
   H\overline{(g/{\rm Id_{\partial\Delta}})}+Sg/{\rm Id_{\partial\Delta}}+f\in
   \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n).
   $$
\end{theorem}

\begin{proof}
The argument is essentially identical to the one in the proof of Theorem \ref{thm:RH-prob1}. We only consider here the case when $\zeta_0,\,\xi_0\in \partial\Delta$ (the remaining follows as easily as Case 1 in the proof of Theorem \ref{thm:RH-prob1}). In this case the mapping $g$ we are looking for must be of the form \eqref{eq:solutionformula}. Consequently, the problem shifts to proving that $g(0)$ and $g'(0)$ are uniquely determined by the prescribed boundary data. In view of equality \eqref{eq:solutionformula} again, it suffices to show that the  vectors
$$
\begin{pmatrix}
    g^r_k(\zeta_0)\\
    g^r_k(\xi_0)\\
\end{pmatrix},
\quad
\begin{pmatrix}
    g^i_k(\zeta_0)\\
    g^i_k(\xi_0)\\
\end{pmatrix},
\quad
\begin{pmatrix}
    h^r_k(\zeta_0)+\zeta_0^{-1}e_k\\
    h^r_k(\xi_0)+\xi_0^{-1}e_k\\
\end{pmatrix},
\quad
\begin{pmatrix}
   h^i_k(\zeta_0)+i\zeta^{-1}_0e_k\\
   h^i_k(\xi_0)+i\xi^{-1}_0e_k\\
\end{pmatrix},
$$
$k=1,\ldots, n$, are linearly independent over $\mathbb R$. This really amounts to proving that $\psi=0$ is the only element of $\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ that satisfies
\begin{equation*}\label{eq:solution-unique2}
   H\overline{(\psi/{\rm Id_{\partial\Delta}})}+S\psi/{\rm Id_{\partial\Delta}}\in
   \mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n),\quad \psi(\zeta_0)=\psi(\xi_0)=0.
\end{equation*}
It suffices to consider the case when $m=0$.

For simplicity we may assume as before that $H=I_n$. Suppose
$\psi\in \mathcal{O}^{\alpha}(\partial\Delta,\,\mathbb C^n)$ satisfies $\psi(\zeta_0)=\psi(\xi_0)=0$ and
\begin{equation}\label{eq:solution-psi}
   \widetilde{\psi}:=\overline{\psi/{\rm Id_{\partial\Delta}}}+S\psi/{\rm Id_{\partial\Delta}}
   \in\mathcal{O}^{\alpha}(\partial\Delta,\,\mathbb C^n).
\end{equation}
Set
   $$
   q:=\sup\Big\{p\geq 0\!: ({\rm Id_{\partial\Delta}}-\zeta_0)^{-1}
       ({\rm Id_{\partial\Delta}}-\xi_0)^{-1}\psi\in L^p(\partial\Delta,\,\mathbb C^n)\Big\}.
   $$
Clearly $q\geq(1-\alpha)^{-1}>1$. We claim that $q=\infty$. Suppose on the contrary that $q<\infty$ and choose a $Q>q$ such that $(1-\alpha)Q<q$. Then
\begin{align}\label{ineq:L^Q-intergability2}
\begin{split}
     \int_{0}^{2\pi}&\bigg|\bigg(\frac{e^{i\theta}-\xi_0}{\zeta_0-\xi_0}\big(S(e^{i\theta})-S(\zeta_0)\big)
          +\frac{\zeta_0-e^{i\theta}}{\zeta_0-\xi_0}\big(S(e^{i\theta})-S(\xi_0)\big)\bigg)
          \frac{\psi(e^{i\theta})}{(e^{i\theta}-\zeta_0)(e^{i\theta}-\xi_0)}\bigg|^Qd\theta\\
     \leq\, & \frac{1}{|\zeta_0-\xi_0|}\int_{0}^{2\pi}
          \bigg(\frac{\|S(e^{i\theta})-S(\zeta_0)\|}{|e^{i\theta}-\zeta_0|}+
          \frac{\|S(e^{i\theta})-S(\xi_0)\|}{|e^{i\theta}-\xi_0|}
          \bigg)^Q|\psi(e^{i\theta})|^Qd\theta\\
     \leq\, &\frac{2^{Q-1}}{|\zeta_0-\xi_0|}\int_{0}^{2\pi}
          \bigg\{\bigg(\frac{\|S(e^{i\theta})-S(\zeta_0)\|}{|e^{i\theta}-\zeta_0|}\bigg)^Q+
          \bigg(\frac{\|S(e^{i\theta})-S(\xi_0)\|}{|e^{i\theta}-\xi_0|}\bigg)^Q
          \bigg\}|\psi(e^{i\theta})|^Qd\theta\\
     =\, &\frac{2^{Q-1}}{|\zeta_0-\xi_0|}\int_{0}^{2\pi}
          \bigg(\frac{\|S(e^{i\theta})-S(\zeta_0)\|}{|e^{i\theta}-\zeta_0|^{\alpha}}\bigg)^Q|
          \psi(e^{i\theta})|^{\alpha Q}
          \bigg|\frac{\psi(e^{i\theta})}{e^{i\theta}-\zeta_0}\bigg|^{(1-\alpha)Q}d\theta\\
         &+\frac{2^{Q-1}}{|\zeta_0-\xi_0|}\int_{0}^{2\pi}
          \bigg(\frac{\|S(e^{i\theta})-S(\xi_0)\|}{|e^{i\theta}-\xi_0|^{\alpha}}\bigg)^Q|
          \psi(e^{i\theta})|^{\alpha Q}
          \bigg|\frac{\psi(e^{i\theta})}{e^{i\theta}-\xi_0}\bigg|^{(1-\alpha)Q}d\theta\\
      <\, &\infty.
\end{split}
\end{align}
On the other hand, by \eqref{eq:solution-psi} we have the  decomposition
\begin{equation}\label{eq:L^Q-decomposition2}
\begin{split}
   \bigg(&\frac{{\rm Id_{\partial\Delta}}-\xi_0}{\zeta_0-\xi_0}\big(S-S(\zeta_0)\big)
        +\frac{\zeta_0-{\rm Id_{\partial\Delta}}}{\zeta_0-\xi_0}\big(S-S(\xi_0)\big)
   \bigg)\frac{\psi}{({\rm Id_{\partial\Delta}}-\zeta_0)({\rm Id_{\partial\Delta}}-\xi_0)}\\
   =\,&\frac1{({\rm Id_{\partial\Delta}}-\zeta_0)({\rm Id_{\partial\Delta}}-\xi_0)}
     \bigg\{{\rm Id_{\partial\Delta}}\widetilde{\psi}-
     \bigg(\frac{{\rm Id_{\partial\Delta}}-\xi_0}{\zeta_0-\xi_0}S(\zeta_0)
        +\frac{\zeta_0-{\rm Id_{\partial\Delta}}}{\zeta_0-\xi_0}S(\xi_0)\bigg)\psi\bigg\}\\
     &-\overline{\,\frac{\zeta_0\xi_0\psi}{({\rm Id_{\partial\Delta}}-\zeta_0)
      ({\rm Id_{\partial\Delta}}-\xi_0)}\,},
\end{split}
\end{equation}
from which together with \eqref{ineq:L^Q-intergability2} and the aforementioned Riesz theorem we conclude that both terms on the right-hand side of \eqref{eq:L^Q-decomposition2} belong to $L^Q(\partial\Delta, \,\mathbb C^n)$. This contradicts the fact that $Q>q$, and the claim follows.

We now obtain in particular
   $$
   \frac{\psi}{({\rm Id_{\partial\Delta}}-\zeta_0)
      ({\rm Id_{\partial\Delta}}-\xi_0)}\in L^2(\partial\Delta,\,\mathbb C^n).
   $$
Also, by \eqref{eq:solution-psi} we have
   $$
   \overline{\,\frac{\zeta_0\xi_0\psi}{({\rm Id_{\partial\Delta}}-\zeta_0)
   ({\rm Id_{\partial\Delta}}-\xi_0)}\,}+S\frac{\psi}{({\rm Id_{\partial\Delta}}-\zeta_0)
   ({\rm Id_{\partial\Delta}}-\xi_0)}
   =\frac{{\rm Id_{\partial\Delta}}\widetilde{\psi}}{({\rm Id_{\partial\Delta}}-\zeta_0)
   ({\rm Id_{\partial\Delta}}-\xi_0)},
   $$
which combined with a similar argument at the end of the proof of Theorem~\ref{thm:RH-prob1} implies  $\psi=0$.  This completes the proof.
\end{proof}



\section{Proof of Theorem \ref{thm:parameter dependence}}\label{sect:proof of thm A}

Let us start with the following

\begin{remark}\label{rem:observations}
We make here some observations to be used in the proof of the theorem, all of which are very simple, but clarify matters greatly.

\begin{enumerate}[leftmargin=1.8pc, parsep=4pt]
\item  [{\rm (i)}]
Let $\Omega$ be a domain in $\mathbb C^n\, (n>1)$ with $C^{m,\,\alpha}$-smooth boundary, where $m\in\mathbb N^{\ast}$ and $0<\alpha<1$. Then the set $S_{\partial\Omega}$ given as in \eqref{defn:fiberbundle} is a $C^{m-1,\,\alpha}$-smooth real submanifold of $\mathbb C^{2n}$ with dimension $4n-3$. In fact, $S_{\partial\Omega}$ is even a fiber bundle over $\partial\Omega$ with fiber
   $$
   \mathbb S^{2n-2}_+:=\big\{(\sqrt{1-|\hat{v}|^2},\, \hat{v})\in\mathbb R\times\mathbb C^{n-1}\!: |\hat{v}|<1\big\}.
   $$
To see this we denote by $U(n)$ the unitary group of degree $n$ and by $^t$ the transpose of matrices. For every $p_0\in\partial\Omega$ we can find a small neighborhood $U_{\partial\mathbb B^n}\subset \partial\mathbb B^n$ of $\nu_{p_0}$  and a smooth  mapping
   $$
   U_{\partial\mathbb B^n}\to U(n),\quad v\mapsto\gamma_v
   $$
such that $\gamma_v(e_1)=v$ for all $v\in U_{\partial\mathbb B^n}$, where $e_1:=(1,0,\ldots, 0)^t$. Set
   $$
   U_{\partial\Omega}:=\nu^{-1}(U_{\partial\mathbb B^n})\ni p_0\quad \mbox{and}\quad
   U_{S_{\partial\Omega}}:=\big\{(p, v)\in S_{\partial\Omega}\!: \ p\in U_{\partial\Omega}\big\},
   $$
where $\nu\!:\partial\Omega\to\partial\mathbb B^n$ denotes the unit outward normal vector field of $\partial\Omega$. Then it is evident that the mapping
\begin{equation}\label{defn:local-trivialization}
  U_{S_{\partial\Omega}}\to U_{\partial\Omega}\times \mathbb S^{2n-2}_+,\quad
  (p, v)\mapsto (p,\, \gamma^{-1}_{\nu_p}(v))
\end{equation}
gives a locally $C^{m-1,\,\alpha}$-smooth local trivialization of $S_{\partial\Omega}$ over  $U_{\partial\Omega}$.

\item  [{\rm (ii)}]
Let $\Omega$, $\widetilde{\Omega}$ be bounded strongly linearly convex domains in $\mathbb C^n\, (n>1)$ with $C^3$-smooth boundary, and $f\!: \Omega\to\widetilde{\Omega}$ be a biholomorphic mapping. If $\varphi$ is a complex geodesic of $\Omega$ with $\varphi^{\ast}$ as its dual mapping, then $f\circ\varphi$ is a complex geodesic of $\widetilde{\Omega}$ whose dual mapping $(f\circ\varphi)^{\ast}$ is related to $\varphi^{\ast}$ via the formula
   $$
   \varphi^{\ast}=(f'\circ\varphi)^t(f\circ\varphi)^{\ast}.
   $$
In particular if $\varphi_{p,\, v}$ is a preferred complex geodesic of $\Omega$ associated to some $(p, v)\in S_{\partial\Omega}$, then $U\circ \varphi_{p,\, v}$ is the preferred complex geodesic of $\widetilde{\Omega}:=U(\Omega)$ associated to $(Up,\, Uv)\in S_{\partial\widetilde{\Omega}}$ for each $U\in U(n)$. Roughly speaking, unitary transformations preserve preferred complex geodesics.


\item  [{\rm (iii)}]
Let $\Omega$, $\varphi$, and $\varphi^{\ast}$ be as in ${\rm (ii)}$. Suppose further that $\partial\Omega$ is $C^{3,\,\alpha}$-smooth and $\rho\!:\mathbb C^n \to \mathbb R$ is a $C^{3,\,\alpha}$-defining function for $\Omega$ whose (real) gradient is of length two on $\partial\Omega$. Then
$\langle\rho_z\circ\varphi(1),\, \overline{\varphi'(1)}\rangle>0$ and
\begin{align*}
   \begin{split}
       &\left.\frac{d}{d\theta}\right|_{\theta=0}
         \big|\langle\rho_z\circ\varphi(e^{i\theta}),\,\overline{\varphi'(e^{i\theta})}\rangle\big|^2
        =2\langle\rho_z\circ\varphi(1),\, \overline{\varphi'(1)}\rangle
         \left.\frac{d}{d\theta}\right|_{\theta=0}{\rm Re}
         \langle\rho_z\circ\varphi(e^{i\theta}),\, \overline{\varphi'(e^{i\theta})}\rangle\\
       &\qquad=-2\langle\rho_z\circ\varphi(1),\, \overline{\varphi'(1)}\rangle
        {\rm Im}\Big(\langle \rho_z\circ\varphi(1),\,\overline{\varphi''(1)}\rangle+
        \big\langle\rho_{zz}\circ\varphi(1)\varphi'(1),\, \overline{\varphi'(1)}\big\rangle\Big),
   \end{split}
\end{align*}
where
   $$
   \rho_z=\bigg(\frac{\partial\rho}{\partial z_1},\ldots, \frac{\partial\rho}{\partial z_n}\bigg)^t
   \quad {\rm and} \quad
   \rho_{zz}=\bigg(\frac{\partial^2\rho}{\partial z_j\partial z_k}\bigg)_{1\leq j,\, k\leq n}.
   $$
Observe also that
$|\varphi^{\ast}(\zeta)|=1/\langle\zeta\rho_z\circ\varphi(\zeta),\,\overline{\varphi'(\zeta)}\rangle$ for all $\zeta\in\partial\Delta$. The above calculation shows that
\begin{equation}\label{dual-nor-cond}
  \left.\frac{d}{d\theta}\right|_{\theta=0}|\varphi^{\ast}(e^{i\theta})|=0
\end{equation}
if and only if
   $$
   {\rm Im}\Big(\langle\rho_z\circ\varphi(1),\,\overline{\varphi''(1)}\rangle+
   \big\langle\rho_{zz}\circ\varphi(1)\varphi'(1),\, \overline{\varphi'(1)}\big\rangle\Big)=0.
   $$

\item  [{\rm (iv)}]
Note that for every $m\in\mathbb N$ and $0<\alpha<1$, the norms $\|\ \|_{C^{m,\,\alpha}(\overline{\Delta})}$ and $\|\ \|_{C^{m,\,\alpha}(\partial\Delta)}$ are equivalent on the space $\mathcal{O}(\Delta,\,\mathbb C^n)\cap C^{m,\,\alpha}(\overline{\Delta},\,\mathbb C^n)$. This enables us to identify mappings in $\mathcal{O}(\Delta,\,\mathbb C^n)\cap C^{m,\,\alpha}(\overline{\Delta},\,\mathbb C^n)$ with their trace on $\partial\Delta$. Recall that in Subsection~\ref{Lem:RH-problem} we have denoted by
$\mathcal{O}^{m,\,\alpha}(\partial\Delta,\,\mathbb C^n)$ the space consisting of these elements. With this notation, the result stated in Theorem \ref{thm:parameter dependence} is equivalent to saying that
    $$
    \Phi\!: S_{\partial\Omega}\to\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n),
            \quad (p, v)\mapsto \varphi_{p,\, v}
    $$
is locally $C^{m-k-1,\,\alpha-\varepsilon}$-smooth for each $2\leq k\leq m-2$ and $0<\varepsilon<\alpha$.
\end{enumerate}
\end{remark}


We are now ready to prove Theorem \ref{thm:parameter dependence}.

\begin{proof}[Proof of Theorem {\rm\ref{thm:parameter dependence}}]
Fix an arbitrary point $(p_0, v_0)\in S_{\partial\Omega}$. Let $\varphi_0:=\varphi_{p_0,\, v_0}$ be the associated preferred complex geodesic of $\Omega$ and $\varphi^{\ast}_0$ be its dual mapping. Then $\varphi_0, \,\varphi^{\ast}_0\in C^{m-1,\,\alpha}(\overline{\Delta})$. With Remark \ref{rem:observations} (ii) in mind, by making a unitary change of coordinates  we may assume that the first two components $\varphi^{\ast}_{0,\, 1},\, \varphi^{\ast}_{0,\, 2}$ of $\varphi^{\ast}_0$ do not vanish simultaneously on $\overline{\Delta}$. Then a standard argument as in Wolff's approach to the corona
theorem (see \cite{Wolff-corona}), together with the well-known regularity property of the Cauchy-Green transform on $\Delta$ (see, e.g., \cite[Theorem 1.32]{Vekua_book}), yields two functions $\phi_1,\, \phi_2\in \mathcal{O}(\Delta)\cap C^{m-1,\,\alpha}(\overline{\Delta})$ satisfying
   $$
   \phi_1\varphi^{\ast}_{0,\, 1}+\phi_2\varphi^{\ast}_{0,\, 2}=1
   \quad {\rm on}\ \, \overline{\Delta}.
   $$

As in \cite{Lempert86}, we next consider the mapping $A_0\!: \Delta\to \mathbb C^{n\times n}$ defined by
\begin{equation}\label{defn:flatenning}
A_0:=\begin{pmatrix}
    \varphi'_{0,\, 1} & -\varphi^{\ast}_{0,\, 2} & -\phi_1\varphi^{\ast}_{0,\, 3} & \cdots & -\phi_1\varphi^{\ast}_{0,\, n} \\
    \varphi'_{0,\, 2} & \varphi^{\ast}_{0,\, 1} & -\phi_2\varphi^{\ast}_{0,\, 3} & \cdots & -\phi_2\varphi^{\ast}_{0,\, n} \\
    \varphi'_{0,\, 3} & 0 & 1 & \cdots & 0 \\
    \vdots & \vdots & \vdots & \ddots &  \vdots \\
    \varphi'_{0,\, n} & 0 & 0 & \cdots & 1 \\
\end{pmatrix}.
\end{equation}
Then $A_0\in C^{m-2,\,\alpha}(\overline{\Delta},\,\mathbb C^{n\times n})$ and ${\rm det}\, A_0\neq 0$ on $\overline{\Delta}$. Moreover,
\begin{equation}\label{eq:A_0-flatenning}
   A^t_0\varphi^{\ast}_0=e_1:=(1,0,\ldots, 0)^t.
\end{equation}
We also choose a $C^{m,\,\alpha}$-defining function $\rho\!:\mathbb C^n \to \mathbb R$ for $\Omega$ with gradient  of length two on $\partial\Omega$, so that the unit outward normal vector field of $\partial\Omega$ is exactly
   $$
   \rho_{\bar{z}}=\bigg(\frac{\partial\rho}{\partial\bar{z}_1},\ldots, \frac{\partial\rho}{\partial\bar{z}_n}\bigg)^t.
   $$
Let $k$ and $\varepsilon$ be as in the theorem. Since the winding number is stable under a small perturbation, Theorem \ref{thm:Lempert1}  implies that a mapping $\varphi$ in $\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)$ that is sufficiently close to $\varphi_0$ is (the boundary value of) a complex geodesic  of $\Omega$ associated to $(p, v)\in S_{\partial\Omega}$ if and only if it satisfies
\begin{equation*}\label{eq:cg-condition}
\left\{
  \begin{aligned}
    &\,\rho\circ\varphi=0, \\
    &\,\exists\, \mu\in C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb R^+),\,
    \mbox{s.t.}\, \, {\rm Id_{\partial\Delta}}\mu\rho_{z}\circ\varphi\in
    \mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n),\\
    &\,\varphi(1)=p, \\
    &\,\varphi'(1)=\langle v, \nu_p\rangle v.
  \end{aligned}
\right.
\end{equation*}
Thanks to \eqref{eq:A_0-flatenning}, for each such $\varphi$ the second of the above conditions can be equivalently reformulated as
   $$
   [A^t_0\,\rho_{z}\circ\varphi]/(A^t_0\,\rho_{z}\circ\varphi)_1\in \mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1}).
   $$
Here and henceforward, the first component of a vector $z\in \mathbb C^n$ is denoted (as before) by $z_1$ or $(z)_1$, whereas the last $(n-1)$-component (considered as a vector in $\mathbb C^{n-1}$) is denoted by $[z]$.

Let $\mathcal{H}\!: C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})\to C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})$ be the standard Hilbert transform  and define
  $$
  \Pi\!: C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})\to
  C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1}),\ \  u\mapsto
  \frac12(u-i\mathcal{H}u)-\frac{1}{4\pi}\int_{0}^{2\pi}u(e^{i\theta})d\theta.
  $$
Then $\Pi$ is continuous and idempotent (i.e., $\Pi^2=\Pi$), and
${\rm Ker}\,\Pi=\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})$. This leads us to consider the mapping
\begin{equation*}
\begin{split}
  \Theta\!:\big(&\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)\cap B_{k,\,
  \varepsilon} (\varphi_0,\,\varepsilon_0)\big)\times\partial\Omega\times\mathbb B^{n-1}\to\\
  &C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb R)\times
  \Pi\big(C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})\big)\times\mathbb
  C^n\times\mathbb C^n\times \mathbb R
\end{split}
\end{equation*}
defined by
\begin{equation*}\label{defn:Theta}
\begin{split}
  \Theta(\varphi,\, p,\, \hat{v})=\bigg(&\rho\circ \varphi,\,
  \Pi\bigg(\frac{[A^t_0\,\rho_{z}\circ\varphi]}{(A^t_0\,\rho_{z}\circ\varphi)_1}\bigg),\,
  \varphi(1)-p,\, \varphi'(1)-\sqrt{1-|\hat{v}|^2}\,\gamma_{\nu_p}\big(\sqrt{1-|\hat{v}|^2},\,
  \hat{v}^t\big)^t,\, \\
  &{\rm Im}\Big(\langle\rho_{z}\circ\varphi(1),\,\overline{\varphi''(1)}\rangle+
   \big\langle\rho_{zz}\circ\varphi(1)\varphi'(1),\, \overline{\varphi'(1)}\big\rangle\Big)\bigg),
\end{split}
\end{equation*}
where $B_{k,\, \varepsilon} (\varphi_0,\,\varepsilon_0)$ denotes the open ball in
$C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)$ with center $\varphi_0$ and radius $\varepsilon_0>0$, which is a sufficiently small constant. By virtue of the preceding argument together with Remark \ref{rem:observations} (iii), we conclude that a mapping $\varphi$ in $\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)$ that is sufficiently close to $\varphi_0$ is the preferred complex geodesic of $\Omega$ associated to $(p, v)\in S_{\partial\Omega}$ close to
$(p_0, v_0)$ if and only if it solves the equation
\begin{equation}\label{eq:CG-equation}
  \Theta(\varphi,\, p,\, \hat{v})=0.
\end{equation}
Here, with a slight abuse of notation we identify $(p,\, \hat{v})$ with
  $$
 (p, v):=\big(p,\, \gamma_{\nu_p}\big(\sqrt{1-|\hat{v}|^2},\, \hat{v}^t\big)^t\big)
  $$
via the local trivialization \eqref{defn:local-trivialization} of $S_{\partial\Omega}$ over  $U_{\partial\Omega}\ni p_0$. Therefore to investigate the regularity of the mapping  $\Phi$ (in the theorem) near $(p_0, v_0)$, it is sufficient to solve equation \eqref{eq:CG-equation} with
$(p, \hat{v})\in\partial\Omega\times\mathbb B^{n-1}$ given near $(p_0, \hat{v}_0)$.

Now it is routine to verify that $\Theta$ is $C^{m-k-1,\,\alpha-\varepsilon}$-smooth provided $\varepsilon_0>0$ is sufficiently small. An application of the implicit function theorem in Banach spaces would yield the desired regularity of $\Phi$, if we could prove that the partial derivative
\begin{equation*}
L:=\big(d\Theta(\,\cdot\,,\, p_0,\, \hat{v}_0)\big)_{\varphi_0}\!:
\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)\to C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb R)\times\Pi\big(C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})\big)
\times\mathbb C^n\times\mathbb C^n\times \mathbb R
\end{equation*}
is invertible. An easy calculation using \eqref{eq:A_0-flatenning} gives
\begin{align*}
 \begin{split}
   L(\varphi)&=\left.\frac{d}{d\tau}\right|_{\tau=0}\Theta(\varphi_0+\tau\varphi,\, p_0,\, \hat{v}_0)\\
     &=\bigg(2{\rm Re}\langle \varphi,\, \rho_{\bar{z}}\circ\varphi_0\rangle,\,
       \Pi\bigg(\frac{[A^t_0(\rho_{zz}\circ\varphi_0)\varphi+A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{\varphi}]}
       {(A^t_0\,\rho_{z}\circ\varphi_0)_1}\bigg),\,
   \varphi(1),\, \varphi'(1),\, \hat{L}(\varphi)\bigg),
\end{split}
\end{align*}
where
\begin{align}\label{eq:partial-der}
\begin{split}
\hat{L}(\varphi):={\rm Im}\bigg(&\langle\rho_z(p_0),\, \overline{\varphi''(1)}\rangle
       +2\big\langle\rho_{zz}(p_0)\varphi'_0(1),\, \overline{\varphi'(1)}\big\rangle\\
      &+\big\langle\rho_{zz}(p_0)\varphi(1)+\rho_{z\bar{z}}(p_0)\overline{\varphi(1)},\,
        \overline{\varphi''_0(1)}\big\rangle\\
      &+\Big\langle\Big(\sum_{j=1}^{n}\frac{\partial \rho_{zz}}{\partial z_j}(p_0)\varphi_j(1)
       +\sum_{j=1}^{n}\frac{\partial \rho_{zz}}{\partial \bar{z}_j}(p_0)\overline{\varphi_j(1)}
           \Big)\varphi'_0(1),\, \overline{\varphi'_0(1)}\Big\rangle\bigg),
\end{split}
\end{align}
and $\varphi=(\varphi_1,\ldots,\varphi_n)^t$. Thus the invertibility of $L$ means that for every
   $$
   (r,\, f,\, w,\,  v,\, \lambda)\in C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb R)\times
   \Pi\big(C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})\big)\times\mathbb C^n
   \times\mathbb C^n\times \mathbb R,
   $$
the following system of equations admits exactly one solution
$\varphi\in \mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^n)$:
\begin{equation*}\label{eq:L-invertibility1}
\left\{
  \begin{aligned}
    &\,{\rm Re}\langle \varphi,\, \rho_{\bar{z}}\circ\varphi_0\rangle=r,\\
    &\,\Pi\bigg(\frac{[A^t_0(\rho_{zz}\circ\varphi_0)\varphi+A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{\varphi}]}
    {(A^t_0\,\rho_{z}\circ\varphi_0)_1}\bigg)=f,\\
    &\,\varphi(1)=w,\\
    &\,\varphi'(1)=v,\\
    &\,\hat{L}(\varphi)=\lambda.
  \end{aligned}
\right.
\end{equation*}

To solve this system we use Theorem \ref{thm:RH-prob1} and the basic properties of the Hilbert transform $\mathcal{H}$. Writing
   $$
   \left.\varphi^{\ast}_0\right|_{\partial\Delta}={\rm Id_{\partial\Delta}}\mu_0\,\rho_{z}\circ\varphi_0
   $$
and using \eqref{eq:A_0-flatenning}, we see that $\varphi$ solves the above system if  and only if $\psi=A^{-1}_0\varphi$ satisfies the following conditions:
\begin{equation}\label{eq:L-invertibility21}
   {\rm Re}\,(\psi_1/{\rm Id_{\partial\Delta}})=\mu_0r,
\end{equation}
\begin{equation}\label{eq:L-invertibility22}
   \Pi\bigg({\rm Id_{\partial\Delta}}\mu_0\big[A^t_0(\rho_{zz}\circ\varphi_0)A_0\psi
            +A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{A_0}\,\overline{\psi}\,\big]\bigg)=f,
\end{equation}
\begin{equation}\label{eq:L-invertibility23}
   \psi(1)=A^{-1}_0(1)w,
\end{equation}
\begin{equation}\label{eq:L-invertibility24}
   \psi'(1)=A^{-1}_0(1)v+(A^{-1}_0)'(1)w,
\end{equation}
and
\begin{equation}\label{eq:L-invertibility25}
   \hat{L}(A_0\psi)=\lambda.
\end{equation}
We first show that $\psi_1$ is uniquely determined by conditions \eqref{eq:L-invertibility21} and \eqref{eq:L-invertibility23}-\eqref{eq:L-invertibility25}. For this we rephrase condition \eqref{eq:L-invertibility21} as
   $$
   {\rm Re}\bigg(\frac{\psi_1-\psi_1(0)}{{\rm Id_{\partial\Delta}}}+
   \overline{\psi_1(0)}\,{\rm Id_{\partial\Delta}}\bigg)=\mu_0r.
   $$
Set
   $$
   f_0:=\mu_0r+i\mathcal{H}(\mu_0r).
   $$
Then by the Privalov theorem $f_0\in \mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C)$, and
${\rm Im}\,f_0(0)=0$ by the definition of $\mathcal{H}$. It follows that
   $$
   \frac{\psi_1-\psi_1(0)}{{\rm Id_{\partial\Delta}}}+
   \overline{\psi_1(0)}\,{\rm Id_{\partial\Delta}}=f_0+i{\rm Im}\,\psi'_1(0),
   $$
i.e.,
\begin{equation}\label{eq:formula-for-psi_1}
\psi_1=\psi_1(0)+{{\rm Id_{\partial\Delta}}}\big(f_0-\overline{\psi_1(0)}\,{\rm Id_{\partial\Delta}}
       +i{\rm Im}\,\psi'_1(0)\big).
\end{equation}
We then need to show that $\psi_1(0)$ and ${\rm Im}\,\psi'_1(0)$ are uniquely determined by conditions \eqref{eq:L-invertibility23}-\eqref{eq:L-invertibility25}. To this end, observe first that  \eqref{eq:L-invertibility23} and \eqref{eq:L-invertibility24} uniquely determine $\psi(1)$ and $\psi'(1)$, which in turn determine
${\rm Im}\langle\varphi''(1),\, \nu_{p_0}\rangle$ (via \eqref{eq:L-invertibility25} and \eqref{eq:partial-der}). On the other hand,  \eqref{eq:formula-for-psi_1} gives
\begin{equation*}\label{eq:psi_1-determination}
\left\{
 \begin{aligned}
     &\,2{\rm Im}\,\psi_1(0)+{\rm Im}\,\psi'_1(0)={\rm Im}\big(\psi_1(1)-f_0(1)\big),\\
     &\,{\rm Re}\,\psi_1(0)=\frac12{\rm Re}\big(f_0(1)+f'_0(1)-\psi'_1(1)\big),\\
     &\,{\rm Im}\,\psi_1(0)=\frac12{\rm Re}\big(\psi''_1(1)-f''_0(1)-2f'_0(1)\big),
 \end{aligned}
\right.
\end{equation*}
indicating that $\psi_1(0)$ and ${\rm Im}\,\psi'_1(0)$ are expressible in terms of $\psi_1(1),\, \psi_1'(1)$ and ${\rm Im}\,\psi''_1(1)$ in a unique way. Thus the problem reduces to proving that  ${\rm Im}\,\psi''_1(1)$ is uniquely determined by ${\rm Im}\langle\varphi''(1),\, \nu_{p_0}\rangle$. But this is clearly true, as can be easily seen from \eqref{eq:A_0-flatenning} and the fact that $A_0\in \mathcal{O}^{2,\,\alpha}(\partial\Delta,\,\mathbb C^{n\times n})$ and ${\rm det}\, A_0\neq 0$ on $\overline{\Delta}$.

Up to now we have proved that $\psi_1$ is uniquely determined by conditions \eqref{eq:L-invertibility21} and \eqref{eq:L-invertibility23}-\eqref{eq:L-invertibility25}. We next consider the last $(n-1)$-component $[\psi]=(\psi_2,\ldots,\psi_n)^t$. Set
   $$
   H_0=(h_{ij})_{1\leq i,\,j\leq n}:=\mu_0A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{A_0},
   $$
   $$
   S_0=(s_{ij})_{1\leq i,\,j\leq n}:=({\rm Id_{\partial\Delta}})^2\mu_0A^t_0(\rho_{zz}\circ\varphi_0)A_0,
   $$
and
\begin{equation*}\label{defn:Her-sym}
H:=(h_{ij})_{2\leq i,\,j\leq n},\quad S:=(s_{ij})_{2\leq i,\,j\leq n}.
\end{equation*}
Also we put $g:=[\psi]$. Then condition \eqref{eq:L-invertibility22} amounts to
\begin{equation}\label{eq:RH-problem}
   H\overline{(g/{\rm Id_{\partial\Delta}})}+Sg/{\rm Id_{\partial\Delta}}+\widetilde{f}
   \in\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1}),
\end{equation}
where
   $$
   \widetilde{f}:=-f+\overline{(\psi_1/{\rm Id_{\partial\Delta}})}(h_{21},\ldots, h_{n1})^t+
                  (\psi_1/{\rm Id_{\partial\Delta}})(s_{21},\ldots, s_{n1})^t
                 \in C^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1}).
   $$
Thus it suffices to show that there exists a unique $g\in\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})$ with $g(1)=[\psi(1)]$ and $g'(1)=[\psi'(1)]$ such that \eqref{eq:RH-problem} holds. Now as $\Omega$ is strongly linearly convex, it follows that
   $$
   \sum_{i,\, j=1}^n\frac{\partial^2\rho}{\partial z_i\partial\bar {z}_j}\circ\varphi_0(\zeta)v_i\bar{v}_j
   >\bigg|\sum_{i,\, j=1}^n\frac{\partial^2\rho}{\partial z_i\partial z_j}\circ\varphi_0(\zeta)v_iv_j\bigg|
   $$
for all $\zeta\in\partial\Delta$ and non-zero $v\in T_{\varphi_0(\zeta)}^{1,\, 0}\partial\Omega$. This means precisely that
   $$
   \hat{v}^tH(\zeta)\bar{\hat{v}}>|\hat{v}^tS(\zeta)\hat{v}|
   $$
for all $\zeta\in\partial\Delta$ and $\hat{v}\in \mathbb C^{n-1}\!\setminus\!\{0\}$. Consequently, Theorem \ref{thm:RH-prob1} guarantees the existence of a unique  $g\in\mathcal{O}^{k,\,\varepsilon}(\partial\Delta,\,\mathbb C^{n-1})$ with the desired property. This proves the invertibility of $L$ and hence the theorem.
\end{proof}


\section{Proof of Theorem \ref{thm:joint-regularity}} \label{sect:proof of thm B}

We argue by considering the following two cases:

\medskip
\noindent  {\bf Case 1:} $m=4$.
\smallskip

By Theorem \ref{thm:parameter dependence} (with $k=2$) we know that
   $$
   \Phi\!: S_{\partial\Omega}\to C^{2,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n),
           \quad (p, v)\mapsto \varphi_{p,\, v}
   $$
is locally $C^{1,\,\alpha-\varepsilon}$-smooth for all $0<\varepsilon<\alpha$. Note also that the evaluation mapping
   $$
   e\!: C^{2,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n)\times \overline{\Delta}\to\mathbb C^n,\quad
       (\varphi,\, \zeta) \mapsto \varphi(\zeta)
   $$
is locally $C^{2,\,\varepsilon}$-smooth. It then follows that
$\widetilde{\Phi}\!=e\circ(\Phi\times {\rm Id}_{\overline{\Delta}})$
is necessarily locally $C^{1,\,\alpha-\varepsilon}$-smooth for all $0<\varepsilon<\alpha$.



\medskip
\noindent  {\bf Case 2:} $m\geq 5$.
\smallskip

Let
   $$
   U_{S_{\partial\Omega}}\to U_{\partial\Omega}\times \mathbb S^{2n-2}_+,\quad
   (p, v)\mapsto (p,\, \gamma^{-1}_{\nu_p}(v))
   $$
be a locally $C^{m-1,\,\alpha}$-smooth local trivialization of $S_{\partial\Omega}$ over  an open subset $U_{\partial\Omega}$ of $\partial\Omega$, which is as in \eqref{defn:local-trivialization}. For every
$(p, \hat{v})\in U_{\partial\Omega}\times \mathbb B^{n-1}$ we write
   $$
   \varphi_{p,\,\hat{v}}:=\varphi_{p,\,\gamma_{\nu_p}(v)}\quad
    {\rm with}\quad v:=(\sqrt{1-|\hat{v}|^2},\, \hat{v}^t)^t
   $$
to simplify the notation. Then what we need to prove is no other than that for every $r\in (0, 1)$ the  mapping
\begin{equation}\label{defn:variation-local-version}
   \widetilde{\Phi}_r\!: U_{\partial\Omega}\times r\mathbb B^{n-1}\times\overline{\Delta}\to \mathbb C^n,
   \quad (p, \hat{v}, \zeta)\mapsto \varphi_{p,\,\hat{v}}(\zeta)
\end{equation}
is $C^{m-3,\,\alpha-\varepsilon}$-smooth for all $0<\varepsilon<\alpha$, after shrinking $U_{\partial\Omega}$ if necessary.

First of all, a similar argument as in the proof of \cite[Theorem 3.1]{Huang-Wang} shows
   $$
   \big\{\varphi_{p,\,\hat{v}}(0)\!: (p, \hat{v})\in  U_{\partial\Omega}\times
   r\mathbb B^{n-1}\big\}\subset\subset\Omega,
   $$
whence
   $$
   \sup\left\{\|\varphi_{p,\,\hat{v}}\|_{C^{m-1,\,\min\{\alpha,\, 1/2\}}(\overline{\Delta})}\!:
       (p, \hat{v})\in  U_{\partial\Omega}\times r\mathbb B^{n-1}\right\}<\infty
   $$
for all $r\in(0, 1)$ (see \cite[Proposition 1]{Huang-Illinois94}).
Combining this with Theorem \ref{thm:parameter dependence} then yields
\begin{align}\label{ineq:Holder-estimate1}
\begin{split}
\big|\varphi^{(k)}_{p_1,\,\hat{v}_1}(\zeta_1)-\varphi^{(k)}_{p_2,\,\hat{v}_2}(\zeta_2)\big|
&\leq\big|\varphi^{(k)}_{p_1,\,\hat{v}_1}(\zeta_1)-\varphi^{(k)}_{p_1,\,\hat{v}_1}(\zeta_2)\big|
    +\big|\varphi^{(k)}_{p_1,\,\hat{v}_1}(\zeta_2)-\varphi^{(k)}_{p_2,\,\hat{v}_2}(\zeta_2)\big|\\
&\leq\|\varphi_{p_1,\,\hat{v}_1}\|_{C^{m-1}(\overline{\Delta})}|\zeta_1-\zeta_2|
    +\|\varphi_{p_1,\,\hat{v}_1}-\varphi_{p_2,\,\hat{v}_2}\|_{C^{m-2}(\overline{\Delta})}\\
&\leq {\rm const}_{r,\, \Omega}|(p_1, \hat{v}_1, \zeta_1)-(p_2, \hat{v}_2, \zeta_2)|
\end{split}
\end{align}
for $k=1,\ldots, m-2$, and $(p_1, \hat{v}_1, \zeta_1),\, (p_2, \hat{v}_2, \zeta_2)\in U_{\partial\Omega}\times r\mathbb B^{n-1}\times\overline{\Delta}$.

Before proceeding with further estimates, let us write
   $$
   \Phi(p, \hat{v}):=\Phi\big(p,\, \gamma_{\nu_p}\big(\sqrt{1-|\hat{v}|^2},\, \hat{v}^t\big)^t\big)
   $$
for all $(p, \hat{v})\in U_{\partial\Omega}\times \mathbb B^{n-1}$ and identify $p$ with its local coordinates via a coordinate chart. Consider all multi-indices $(\lambda, \mu)\in \mathbb N^{2n-1}\times\mathbb N^{2n-2}$ with
   $$
   1\leq|\lambda|+|\mu|\leq m-3.
   $$
When $|\lambda|+|\mu|=m-3$, we conclude from Theorem \ref{thm:parameter dependence} (with $k=2$) that
   $$
   \frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}(p, \hat{v})
   \in\mathcal{O}(\Delta,\,\mathbb C^n)\cap C^{2,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n),\quad \forall\,(p, \hat{v})\in U_{\partial\Omega}\times \mathbb B^{n-1}.
   $$
Moreover,
\begin{align}\label{ineq:Holder-estimate2}
 \begin{split}
  &\bigg|\bigg(\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
                (p_1, \hat{v}_1)\bigg)(\zeta_1)
               -\bigg(\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
                (p_2, \hat{v}_2)\bigg)(\zeta_2)
          \bigg|\\
  &\leq \bigg|\bigg(\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)\bigg)(\zeta_1)
       -\bigg(\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)\bigg)(\zeta_2)
        \bigg|\\
  &\quad\, +\bigg|\bigg(\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)-\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_2, \hat{v}_2)\bigg)(\zeta_2)\bigg|\\
  &\leq \bigg\|\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1,\hat{v}_1)\bigg\|_{C^{1}(\overline{\Delta})}|\zeta_1-\zeta_2|
           +\bigg\|\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)-\frac{\partial^{m-3}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_2, \hat{v}_2)\bigg\|_{C^{0}(\overline{\Delta})}\\
  &\leq {\rm const}_{r,\,\alpha,\,\varepsilon,\,\Omega}
   \big(|\zeta_1-\zeta_2|+|(p_1, \hat{v}_1)-(p_2, \hat{v}_2)|^{\alpha-\varepsilon}\big)
 \end{split}
\end{align}
for all $(p_1, \hat{v}_1, \zeta_1),\, (p_2, \hat{v}_2, \zeta_2)\in U_{\partial\Omega}\times r\mathbb B^{n-1}\times\overline{\Delta}$. We now turn to the case when $1\leq |\lambda|+|\mu|<m-3$. Note that
   $$
   2\leq m-|\lambda|-|\mu|-2\leq m-3.
   $$
Applying Theorem \ref{thm:parameter dependence} with $k:=m-|\lambda|-|\mu|-2$ we obtain
   $$
   \frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}(p, \hat{v})
   \in\mathcal{O}(\Delta,\,\mathbb C^n)\cap C^{m-|\lambda|-|\mu|-2,\,\varepsilon}(\overline{\Delta},\,\mathbb C^n), \quad \forall\,(p, \hat{v})\in U_{\partial\Omega}\times \mathbb B^{n-1},
   $$
and
\begin{align}\label{ineq:Holder-estimate3}
 \begin{split}
  \bigg|&\frac{\partial^{j+|\lambda|+|\mu|}\widetilde{\Phi}_r}{\partial \zeta^j\partial p^{\lambda}
                \partial \hat{v}^{\mu}}(p_1, \hat{v}_1, \zeta_1)
                -\frac{\partial^{j+|\lambda|+|\mu|}\widetilde{\Phi}_r}{\partial \zeta^j\partial p^{\lambda}
                \partial \hat{v}^{\mu}}(p_2, \hat{v}_2, \zeta_2)
          \bigg|\\
  &=\bigg|\frac{\partial^j}{\partial \zeta^j}\bigg(\frac{\partial^{|\lambda|+|\mu|}\Phi}
          {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_1, \hat{v}_1)\bigg)(\zeta_1)
          -\frac{\partial^j}{\partial \zeta^j}\bigg(\frac{\partial^{|\lambda|+|\mu|}\Phi}
          {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_2, \hat{v}_2)\bigg)(\zeta_2)\bigg|\\
  &\leq\bigg|\frac{\partial^j}{\partial \zeta^j}\bigg(\frac{\partial^{|\lambda|+|\mu|}\Phi}
            {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_1, \hat{v}_1)\bigg)(\zeta_1)
            -\frac{\partial^j}{\partial \zeta^j}\bigg(\frac{\partial^{|\lambda|+|\mu|}\Phi}
            {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_1, \hat{v}_1)\bigg)(\zeta_2)\bigg|\\
  &\quad\, +\bigg|\frac{\partial^j}{\partial \zeta^j}
            \bigg(\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
                  (p_1, \hat{v}_1)
                 -\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
                  (p_2, \hat{v}_2)\bigg)(\zeta_2)
            \bigg|\\
  &\leq\bigg\|\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1,\hat{v}_1)\bigg\|_{C^{m-|\lambda|-|\mu|-3}(\overline{\Delta})}|\zeta_1-\zeta_2|\\
  &\quad\, +\bigg\|\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)-\frac{\partial^{|\lambda|+|\mu|}\Phi}
              {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_2, \hat{v}_2)
       \bigg\|_{C^{m-|\lambda|-|\mu|-4}(\overline{\Delta})}\\
  &\leq{\rm const}_{r,\, \Omega}|(p_1, \hat{v}_1, \zeta_1)-(p_2, \hat{v}_2, \zeta_2)|
 \end{split}
\end{align}
for all $(p_1, \hat{v}_1, \zeta_1),\, (p_2, \hat{v}_2, \zeta_2)\in U_{\partial\Omega}\times r\mathbb B^{n-1}\times\overline{\Delta}$ and all $j\in \mathbb N$ with
   $$
  j<m-|\lambda|-|\mu|-3.
   $$
Similarly, for $j=m-|\lambda|-|\mu|-3$ an application of  Theorem \ref{thm:parameter dependence} with
   $$
  k:=m-|\lambda|-|\mu|-1
   $$
gives
\begin{align}\label{ineq:Holder-estimate4}
 \begin{split}
   \bigg|&\frac{\partial^{m-3}\widetilde{\Phi}_r}{\partial \zeta^{m-|\lambda|-|\mu|-3}
                 \partial p^{\lambda}\partial \hat{v}^{\mu}}(p_1, \hat{v}_1, \zeta_1)
                -\frac{\partial^{m-3}\widetilde{\Phi}_r}{\partial \zeta^{m-|\lambda|-|\mu|-3}
                 \partial p^{\lambda}\partial \hat{v}^{\mu}}(p_2, \hat{v}_2, \zeta_2)
   \bigg|\\
   &\leq\bigg\|\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1,\hat{v}_1)\bigg\|_{C^{m-|\lambda|-|\mu|-2}(\overline{\Delta})}|\zeta_1-\zeta_2|\\
   &\quad\, +\bigg\|\frac{\partial^{|\lambda|+|\mu|}\Phi}{\partial p^{\lambda}\partial \hat{v}^{\mu}}
              (p_1, \hat{v}_1)-\frac{\partial^{|\lambda|+|\mu|}\Phi}
              {\partial p^{\lambda}\partial \hat{v}^{\mu}}(p_2, \hat{v}_2)
        \bigg\|_{C^{m-|\lambda|-|\mu|-3}(\overline{\Delta})}\\
  &\leq {\rm const}_{r,\,\alpha,\,\varepsilon,\,\Omega}
   \big(|\zeta_1-\zeta_2|+|(p_1, \hat{v}_1)-(p_2, \hat{v}_2)|^{\alpha-\varepsilon}\big)
 \end{split}
\end{align}
for all $(p_1, \hat{v}_1, \zeta_1),\, (p_2, \hat{v}_2, \zeta_2)\in U_{\partial\Omega}\times r\mathbb B^{n-1}\times\overline{\Delta}$.

Now putting \eqref{ineq:Holder-estimate1}-\eqref{ineq:Holder-estimate4} together, we conclude that for every $r\in (0, 1)$ the mapping $\widetilde{\Phi}_r$ in \eqref{defn:variation-local-version} is indeed  $C^{m-3,\,\alpha-\varepsilon}$-smooth for all $0<\varepsilon<\alpha$, as desired. This completes the proof of the theorem.

\section{Proof of Theorem \ref{thm:regularity-BSR}}\label{sect:proof of thm C}
In light of Theorem \ref{thm:joint-regularity} together with \eqref{defn:inv-BSR} and \eqref{ball-splitting}, it is evident that each $\Psi_p^{-1}$ has the desired regularity.  To establish the same regularity for $\Psi$, by definition it suffices to show that for every sufficiently small $\delta>0$ the mapping
  $$
  (\overline{\Omega}\times\partial\Omega)\!\setminus\!{\rm dist}_{\delta}\, (\overline{\Omega}\times\partial\Omega)
  \to \mathbb C^n\times (\overline{\Delta}\!\setminus\!\{1\}),\quad (z,\, p)\mapsto (v_{z,\,p},\,\zeta_{z,\,p})
  $$
determined in \eqref{defn:BSR} is $C^{m-3,\,\alpha-\varepsilon}$-smooth for all $0<\varepsilon<\alpha$. Observe also that for every
$(z, p)\in\overline{\Omega}\times\partial\Omega$ with $z\neq p$, $(v_{z,\,p},\,\zeta_{z,\,p})$ is the only solution to the equation $\widetilde{\Phi}(p, v, \zeta)-z=0$. It therefore suffices to verify the injectivity of the (real) differential
   $$
   (d\widetilde{\Phi}(p,\,\cdot\,,\,\cdot\,))_{(v,\,\zeta)}\!: \mathbb R^{2n-2}\times\mathbb R^2
   \to \mathbb R^{2n}
   $$
for every $(p, v,\zeta)\in S_{\partial\Omega}\times(\overline{\Delta}\!\setminus\!\{1\})$, for then the desired regularity follows immediately from Theorem \ref{thm:joint-regularity} and  the implicit function theorem.

Fix an arbitrary point $p_0\in \partial\Omega$. Up to a unitary transformation of $\mathbb C^n$ which preserves the strong linear convexity of $\Omega$ and preferred complex geodesics (see Remark \ref{rem:observations} (ii)), we may assume that $\nu_{p_0}=e_1$. Set $\varphi_{\hat{v}}:=\varphi_{p_0,\,v}$ for $v:=(\sqrt{1-|\hat{v}|^2},\, \hat{v}^t)^t$ and consider the mapping
   $$
   \hat{\Phi}\!: \mathbb B^{n-1}\times(\overline{\Delta}\!\setminus\!\{1\})
   \to\overline{\Omega}\!\setminus\!\{p_0\}
   $$
defined by $\hat{\Phi}(\hat{v},\, \zeta)=\varphi_{\hat{v}}(\zeta)$. By the symmetry of $\mathbb B^{n-1}$ we now only need to verify
\begin{equation*}\label{eq:trival-kernel}
   {\rm Ker}\,d\hat{\Phi}_{(0,\,\zeta)}=\{0\}, \quad \forall\,\zeta\in\overline{\Delta}\!\setminus\!\{1\}.
\end{equation*}
With (the uniqueness part of) Theorem \ref{thm:RH-prob2} in hand, we will prove this by first performing for $\Omega$ a holomorphic coordinate transformation as given in Theorem \ref{thmA:normalization} and then carefully analyzing the behavior of the normalization condition \eqref{dual-nor-cond} with respect to such a transformation.

Fix $\zeta_0\in\overline{\Delta}\!\setminus\!\{1\}$ and suppose $(\hat{v},\, \lambda)\in \mathbb C^{n-1}\times\mathbb C$ satisfies
\begin{equation}\label{vanishing}
   d\hat{\Phi}_{(0,\,\zeta_0)}(\hat{v},\,\lambda)=0.
\end{equation}
To show that $\hat{v}=0$ and $\lambda=0$, we note that
$\hat{\Phi}(0,\,\cdot\,)=\varphi_0:=\varphi_{p_0,\, e_1}$, and hence
\begin{equation*}\label{eq:horizontal-var}
   d\hat{\Phi}_{(0,\,\zeta_0)}(0, 1)=(d\hat{\Phi}(0,\,\cdot\,))_{\zeta_0}(1)=\varphi'_0(\zeta_0).
\end{equation*}
Setting
\begin{equation*}\label{defn:v-variation}
   \varphi:=\left.\frac{d}{d\tau}\right|_{\tau=0}\varphi_{\tau\hat{v}}\in\mathcal{O}(\Delta,\,\mathbb C^n)
   \cap C^2(\overline{\Delta},\,\mathbb C^n),
\end{equation*}
which makes sense in view of Theorem \ref{thm:parameter dependence}, we obtain
\begin{equation*}\label{eq:vertical-var}
d\hat{\Phi}_{(0,\,\zeta_0)}(\hat{v},\,0)=(d\hat{\Phi}(\,\cdot\,,\zeta_0))_0(\hat{v})=\varphi(\zeta_0).
\end{equation*}
Consequently, \eqref{vanishing} can be rephrased as
\begin{equation}\label{equi-vanishing}
   \varphi(\zeta_0)+\lambda\varphi'_0(\zeta_0)=0.
\end{equation}

To proceed further, we make use of Theorem \ref{thmA:normalization} in Appendix \ref{supplement}. Choose a biholomorphism $F\!:D\to\Omega$ which extends to a $C^2$-smooth diffeomorphism between $\overline{D}$ and $\overline{\Omega}$, and a $C^2$-defining function $\rho$ for $\Omega$ defined on a neighborhood of $\varphi_0(\overline{\Delta})$ in $\mathbb C^n$  such that
\begin{equation}\label{Straightening1}
   F(\,\cdot\,,0)=\varphi_0 \quad {\rm on}\ \, \overline{\Delta},
\end{equation}
\begin{equation}\label{Straightening2}
{\rm Id_{\partial\Delta}}A^t_0\,\rho_{z}\circ\varphi_0=e_1 \quad {\rm on}\ \, \partial\Delta,
\end{equation}
and
\begin{equation}\label{Straightening3}
A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{A_0}=I_n,
   \quad A^t_0(\rho_{zz}\circ\varphi_0)A_0=
   \begin{pmatrix}
    0 & 0 \\
    0 & S_0\\
    \end{pmatrix} \quad {\rm on}\ \, \partial\Delta,
\end{equation}
where $D\subset\mathbb C^n$ is a bounded strongly pseudoconvex domain  with $C^2$-smooth boundary such that
   $$
   \Delta\times\{0\}\subset D\subset\Delta\times\mathbb C^{n-1},
   $$
$A_0:=F'(\,\cdot\,,0)$ on $\overline{\Delta}$, and  $S_0\!:\partial\Delta\to {\rm Sym}(n-1,\,\mathbb C)
\footnote{${\rm Sym}(n, \mathbb C)$ denotes the set of all complex symmetric  $n\times n$ matrices.}$
is $C^2$-smooth with operator norm less than one. Then we have
\begin{equation}\label{eq:geodesic-family}
\left\{
  \begin{aligned}
    &\,\rho\circ\varphi_{\tau\hat{v}}=0,\\
    &\,\Pi\bigg(\frac{[A^t_0\,\rho_{z}\circ\varphi_{\tau\hat{v}}]}
                     {(A^t_0\,\rho_{z}\circ\varphi_{\tau\hat{v}})_1}\bigg)=0,\\
    &\,\varphi_{\tau\hat{v}}(1)=p_0,\\ &\,\varphi_{\tau\hat{v}}'(1)=
        \sqrt{1-\tau^2|\hat{v}|^2}(\sqrt{1-\tau^2|\hat{v}|^2},\,\tau \hat{v}^t)^t,\\
    &\,\left.\frac{d}{d\theta}\right|_{\theta=0}|\varphi^{\ast}_{\tau\hat{v}}(e^{i\theta})|=0
  \end{aligned}
\right.
\end{equation}
for all $|\tau|<<1$. Recall that, as we assumed before,  $\nu_{p_0}=e_1$. Combining this with \eqref{Straightening1} and \eqref{Straightening2}, we conclude that $\rho_z(p_0)=e_1$ and both the first column of $A_0(1)$ and that of its transpose $A_0^t(1)$ are equal to $e_1$. Note also that $\varphi_{\tau\hat{v}}(1)=p_0$ and
  \begin{equation*}
   |\rho_z\circ\varphi_{\tau\hat{v}}|=|\varphi^{\ast}_{\tau\hat{v}}|
   |\langle\rho_z\circ\varphi_{\tau\hat{v}},\, \overline{\varphi'_{\tau\hat{v}}}\rangle| \quad {\rm on}\ \, \partial\Delta.
  \end{equation*}
A straightforward calculation then shows that the last identity in \eqref{eq:geodesic-family} is equivalent to
\begin{equation*}
\begin{split}
\langle \varphi'_{\tau\hat{v}}(1)&,\, e_1\rangle{\rm Im}\big\langle \rho_{zz}(p_0)\varphi'_{\tau\hat{v}}(1)-\rho_{z\bar{z}}(p_0)\overline{\varphi'_{\tau\hat{v}}(1)},\, e_1\big\rangle\\
&={\rm Im}\big(\langle \varphi''_{\tau\hat{v}}(1),\, e_1\rangle+ \langle\rho_{zz}(p_0)\varphi'_{\tau\hat{v}}(1),\, \overline{\varphi'_{\tau\hat{v}}(1)}\rangle\big).
\end{split}
\end{equation*}
Now differentiating the preceding identities with  respect to $\tau$  and evaluating at $\tau=0$ yield
\begin{equation}\label{eq:injectivity1}
   {\rm Re}\,(\psi_1/{\rm Id_{\partial\Delta}})=0,
\end{equation}
\begin{equation}\label{eq:injectivity2}
   \Pi\bigg({\rm Id_{\partial\Delta}}\big[A^t_0(\rho_{zz}\circ\varphi_0)A_0\psi
            +A^t_0(\rho_{z\bar{z}}\circ\varphi_0)\overline{A_0}\,\overline{\psi}\,\big]\bigg)=0,
\end{equation}
\begin{equation}\label{eq:injectivity3}
   \psi(1)=0,
\end{equation}
\begin{equation}\label{eq:injectivity4}
   \psi'(1)=A_0^{-1}(1)\begin{pmatrix}
    \,0\,\\
    \,\hat{v}\,
\end{pmatrix},
\end{equation}
and
\begin{equation}\label{eq:injectivity5}
{\rm Im}\Big(\langle\varphi''(1),\, e_1\rangle+\big\langle \rho_{zz}(p_0)\varphi'(1)+\rho_{z\bar{z}}(p_0)\overline{\varphi'(1)},\, e_1\big\rangle\Big)=0,
\end{equation}
where $\psi:=A^{-1}_0\varphi$.



We are now in a position to show that $\psi\equiv0$. Consider first the last $(n-1)$-component $[\psi]$ of $\psi$. In view of \eqref{Straightening3}, \eqref{eq:injectivity2} means more explicitly that
   $$
   \overline{[\psi]/{\rm Id_{\partial\Delta}}}+({\rm Id^2_{\partial\Delta}}S_0)[\psi]/{\rm Id_{\partial\Delta}}\in\mathcal{O}^2(\partial\Delta,\,\mathbb C^{n-1}).
   $$
Clearly $[\psi](1)=0$. Observing also that $\varphi'_0$ is the first column of $A_0=F'(\,\cdot\,,0)$ (see \eqref{Straightening1}), we see that \eqref{equi-vanishing} is equivalent to
 \begin{equation}\label{equi-vanishing1}
   \psi(\zeta_0)=-\lambda e_1,
 \end{equation}
and hence $[\psi](\zeta_0)=0$ as well. Then the uniqueness part of Theorem \ref{thm:RH-prob2} implies $[\psi]\equiv 0$, as desired.  We next show that the same is true for the first component $\psi_1$ of $\psi$. Observe that  \eqref{eq:injectivity1}, \eqref{eq:injectivity3}, and \eqref{eq:injectivity4} together imply
  \begin{equation}\label{first-comp}
     \psi_1=ic\,(1-{\rm Id}_{\overline{\Delta}})^2,
  \end{equation}
where $c$ is a real constant (depending on $\hat{v}$). Indeed, \eqref{eq:injectivity1} is equivalent to
\begin{equation*}\label{eq:injectivity1'}
  {\rm Re}\big(\psi_1/(1-{\rm Id}_{\overline{\Delta}})^2\big)=0
  \quad {\rm on}\ \, \partial\Delta\!\setminus\!\{1\}.
\end{equation*}
Since also $\psi_1(1)=\psi'_1(1)=0$, the function $\psi_1/(1-{\rm Id}_{\overline{\Delta}})^2$ belongs to the Hardy space $H^1(\Delta)$ and  \eqref{first-comp} then follows immediately. It remains to show that the constant in \eqref{first-comp} is actually zero.  For this we can first deduce from \eqref{Straightening1} and \eqref{Straightening2} that
   \begin{equation}\label{eq:dual-formula}
     \left.\varphi^{\ast}_0\right|_{\partial\Delta}={\rm Id_{\partial\Delta}}\,\rho_{z}\circ\varphi_0,
   \end{equation}
and hence $A_0^t\varphi^{\ast}_0=e_1$ (i.e., $\varphi^{\ast}_0$ is precisely the first column of $(A^t_0)^{-1}$). Then, by differentiating the identity $\varphi=A_0\psi$ twice and evaluating at $1$ we get
  $$
  \varphi''(1)=A_0(1)\big(\psi''(1)-2(A_0^{-1})'(1)\varphi'(1)\big)
  $$
and consequently
  $$
  \varphi_1''(1)=
  \big\langle\psi''(1)-2(A_0^{-1})'(1)\varphi'(1),\, e_1\big\rangle
  =\psi''_1(1)-2\langle (\varphi^{\ast}_0)'(1),\, \overline{\varphi'(1)}\rangle.
  $$
On the other hand, by \eqref{eq:dual-formula} we have
   $$
   (\varphi^{\ast}_0)'(1)=-i\left.\frac{d}{d\theta}\right|_{\theta=0}\varphi^{\ast}_0(e^{i\theta})
   =e_1+\rho_{zz}(p_0)e_1-\rho_{z\bar{z}}(p_0)e_1.
   $$
Combining the last two equalities with \eqref{eq:injectivity5} and \eqref{first-comp}, we arrive at
\begin{equation*}\label{eq:second-der1}
  c=\frac12{\rm Im}\big\langle \rho_{zz}(p_0)e_1-\rho_{z\bar{z}}(p_0)e_1,\,
   \overline{\varphi'(1)}\big\rangle=-\frac12{\rm Im}\langle e_1, (0, \hat{v}^t)^t
   \rangle=0.
\end{equation*}
Here the second equality follows from \eqref{Straightening3}, \eqref{eq:injectivity4}, and the aforementioned fact that both the first column of $A_0(1)$ and that of its transpose $A_0^t(1)$ are equal to $e_1$.

Now we have $\psi\equiv0$. Together with \eqref{eq:injectivity4} and \eqref{equi-vanishing1}, this in turn implies $\hat{v}=0$ and $\lambda=0$, thereby completing the proof.





