\label{thm: main-ucp}
Let $T>0$ and $G\subset [0,T]\times {\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.
\begin{itemize}
\item [(1)] UCP holds on $G$.

\item [(2)] $G$ satisfies \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-trivial pair $(A, B)$\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.
\end{itemize}

\begin{equation}\label{eq: wave-eq-0}
(\partial_t^2-\partial_x^2)u=0,\quad (u,\partial_tu)\big|_{t=0}=(u_0,u_1)\in \dot H^1(\T)\times L^2(\T),
\end{equation}

\begin{definition}[Symmetric function pair]\label{def:classSc}
We say
$(f,g)$ is a symmetric  function pair if  there exists a decomposition pair $\{(A_k, B_k),\ k = 1,\dots, K\}$, a sequence of different numbers $\{s_k\}_{k= 1}^K$ such that
\begin{equation*}
 f=\sum_{1\leq k\leq K}s_k\chi_{A_k}, \quad g=\sum_{1\leq k\leq K}s_k\chi_{B_k},
\end{equation*}
and
\begin{equation*}
    \int_{\T} (f+ g)(x) \d x= 0.
\end{equation*}
The set of all symmetric function pairs is denoted by $\mathcal{S}^2_c$.
\end{definition}

\begin{equation}\label{eq: wave-ob}
\|u_0\|^2_{\dot H^1(\T)}+\|u_1\|^2_{L^2(\T)}\leq C\int_G|\partial_tu(t,x)|^2\d t\d x.
\end{equation}

Let $T > 0$. Let $G \subset [0,T] \times \mathbb{T}$ be a spacetime measurable set with positive measure.
Consider the {\it observability problem} of the wave equation: whether there exists some $C= C(G)>0$ such that every solution $u$ to
\begin{equation}\label{eq: wave-eq-0}
(\partial_t^2-\partial_x^2)u=0,\quad (u,\partial_tu)\big|_{t=0}=(u_0,u_1)\in \dot H^1({\mathbb T})\times L^2({\mathbb T}),
\end{equation}
satisfies
\begin{equation}\label{eq: wave-ob}
\|u_0\|^2_{\dot H^1({\mathbb T})}+\|u_1\|^2_{L^2({\mathbb T})}\leq C\int_G|\partial_tu(t,x)|^2{\,\rm d} t{\,\rm d} x.
\end{equation}

Thanks to the classical Hilbert uniqueness method and an argument to raise regularity (see Appendix \ref{sec: HUM-app} for details on this reduction), the observability of \eqref{eq: wave-ob} is equivalent to the {\it exact controllability} of
the controlled wave equation on $[0,T]\times{\mathbb T}$ with control force $f\in L^2(G)$:
\begin{equation}\label{eq: wave-eq}
(\partial_t^2-\partial_x^2)u =f\mathbf{1}_G,\quad (u, \partial_t u)\big|_{t=0}=(u_0,u_1)\in \dot H^1({\mathbb T})\times L^2({\mathbb T}).
\end{equation}

Our primary objective is to seek the sufficient and necessary geometric conditions on $G$ such that the unique continuation and observability hold.

\subsection{History and setting}
From the early work of Russell \cite{Russell-1}, the studies of the controllability and observability for wave equations have been central topics in control theory.

\subsubsection{Geometric control condition}
In the pioneering work of Rauch and Taylor \cite{RT}, they first related the observability to a geometric condition on the damped region $\omega$ and the rays of geometric optics in the boundaryless case.
Later, in another pioneering work of Bardos-Lebeau-Rauch \cite{BLR-gcc}, the well-known \textit{Geometric control condition} is introduced: for an open observed region $\omega$, every generalized ray should meet $\omega$ in a finite time.

Since then, it has become one of the most natural assumptions for the controlled waves. In the existing literature, the observation is most often made on cylindrical domains $G=(0,T)\times \omega$, with $\omega$ being an open subset. Under suitable smooth conditions, it is well-known that GCC is sufficient, and depending on the domain, necessary for the observability in the cylindrical domains $(0,T)\times \omega$, see \cite{BLR-gcc,BG-97}. When considering stabilization problems, GCC is also a useful condition for the exponential decay of energy. We refer to \cite{RT,Haraux}. Otherwise, one may have logarithmic type of energy decay results \cite{LR-97, Burq-98}. GCC also plays a role in practical issues, such as sensor designs, tomography techniques used for imaging bodies (see \cite{RouLebeauAnalPDE2017} for example), etc. For a comprehensive reference of the numerical study, we refer to \cite{zuazua-review} and its references therein.

Finally, we give a very brief overview of the boundary control case. There are also fruitful results in this direction. For related GCC, we refer to \cite{Lebeau-boundary}. In particular, for 1D wave equations, one can find many nonlinear results \cite{Li-1,LY-2} and the references therein.

\subsubsection{Unique continuation}
A qualitative version of observability is the unique continuation property.
The easiest way to ensure this property is to apply the analyticity based on Holmgren's theorem. Besides, H\"ormander's pseudo-convexity condition and Carleman estimates are powerful tools dealing with the unique continuation problem. In this direction, there is a large literature such as \cite{RZ-98,Tataru,Hormander-92,Hormander-96} and more recent work includes \cite{Laurent-Leautaud-2019, MS-21,FCL,Shao}. Here we point out that to construct the appropriate weight to apply Carleman estimates, it is crucial to understand the behavior across the suitable spacetime surfaces, which is more delicate than considering a cylindrical region.

\subsubsection{Our setting: spacetime measurable observable region}
Recently, researchers started to focus on
 the spacetime setting \cite{Castro-Cindea-Munch-2014, RouLebeauAnalPDE2017,Shao,PK}. Meanwhile, the study of the case where $G$ is a spacetime region is far from complete, even for open regions.

 In this paper, we focus on the setting that $G\subset[0,T]\times{\mathbb T}$ is measurable of positive measure. The following condition may be viewed as the natural analogue of the standard geometric control condition in this spacetime measurable setting.

\vspace{2mm}
\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}
\vspace{2mm}

From now on, in this paper, \blackhyperref{def:GCC}{({\rm GCC})}  refers to the preceding definition with respect to \eqref{eq:GCC-wave:0}.
It is well-known that under the standard setting, namely the observable region is a cylinder $[0, T]\times \omega$, GCC yields several important properties: weak observability and the necessity of observability. These results can be generalized to the spacetime \blackhyperref{def:GCC}{({\rm GCC})} for a spacetime measurable observable region. We put the former result in Section \ref{sec: weak-ob-sec} and the later in Appendix \ref{sec:app:3}.

\subsection{A new symmetry condition}

Extend the system $2\pi-$periodically to $x\in \mathbb{R}$,  and introduce the null coordinate:
\begin{equation}
    \xi=x+t \; \textrm{ and } \; \eta=x-t. \notag
\end{equation}
Under this new coordinate,
\begin{equation*}
  2  u_{\xi}= u_{x}+ u_t \; \textrm{ and } \;  2  u_{\eta}= u_{x}- u_t
\end{equation*}
and the wave equation \eqref{eq: wave-eq} becomes\footnote{We considered the equation on $U(\xi, \eta):= u(t, x)$ and $F(\xi, \eta)= f(t, x)$. For ease of notation, we still denote $U$ by $u$ and $F$ by $f$.  }
\begin{equation}
    4 \partial_{\xi} \partial_{\eta} u=  f \mathbf{1}_G. \notag
\end{equation}

For any $\xi_0\in\mathbb{R}$, we denote the  line $\{(t,x):x+t=\xi_0\}$ by $L_{\xi=\xi_0}$,  and call it the $\xi$-characteristic. Similarly, we denote the  line $\{(t,x):x- t=\eta_0\}$ by $L_{\eta=\eta_0}$ and  call it $L_{\eta=\eta_0}$ the $\eta$-characteristic. Thus the \blackhyperref{def:GCC}{({\rm GCC})} is equivalent to
\begin{equation}\label{eq:GCC:measure}
      \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})\geq c_0, \;\;
  \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})\geq c_0.
\end{equation}
We further define measurable cylinders for any measurable sets $A, B\subset [0, 2\pi]$ as
\begin{equation}\label{eq: cylinder}
L_{\xi\in A}:=\bigcup_{\xi_0\in A}L_{\xi=\xi_0} \; \textrm{ and } \; L_{\eta\in B}:=\bigcup_{\eta_0\in B}L_{\eta=\eta_0}.
\end{equation}

Introduce the following symmetry condition:

\vspace{2mm}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset {\mathbb T}$ be two measurable subsets. $G\in [0, T]\times {\mathbb T}$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\mathbb{R}^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\mathbb{R}^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}

\vspace{2mm}

We call a pair $(A, B)$ trivial, if $|A|= |B|= 0$ or $|A|= |B|= 2\pi$.  In principle, we are interested in {\it non-trivial} pairs $(A, B)$. Since the observable symmetry condition is automatically satisfied for trivial pairs ($A, B$).

The observable symmetry condition is very important in our work, see Fig. \ref{fig:OSC-role}.

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

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\draw    (329,82.5) -- (329,134.5) ;
\draw [shift={(329,136.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (430,82.5) -- (509.44,146.25) ;
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\draw (219,47) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observable symmetry condition};
\draw (81,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Conservation law};
\draw (282,146.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {\begin{minipage}[lt]{66.78pt}\setlength\topsep{0pt}
\begin{center}
Controllability \\Observability \\UCP
\end{center}

\end{minipage}};
\draw (455,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Symetric \ \ \  class };
\draw (515,167.9) node [anchor=north west][inner sep=0.75pt]  [font=\small]  {$\mathcal{S}^{2}$};
\draw (128,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Noether}};
\draw (333,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize necessary}};
\draw (486,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize generate}};

\end{tikzpicture}

    \caption{The role of (OSC) in this paper}
    \label{fig:OSC-role}
\end{figure}

\vspace{3mm}
The first result is a new conservation law of the forced equation \eqref{eq: wave-eq}.
\begin{proposition}[Conservation law]
\label{prop-sym-conser}
Let $A,B\subset {\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\it observable symmetry condition}.
Let $(u,u_t)\in C([0,T];\dot{H}^1({\mathbb T})\times L^2({\mathbb T}))$ be a solution to the forced equation \eqref{eq: wave-eq} and define the energy \footnote{Indeed, under the $(\xi, \eta)-$coordinate this conservation is write as
\begin{equation*}
 \frac{1}{2} I(t)=   \int_{A-t}u_{\xi}(t,x){\,\rm d} x+\int_{B+t}u_{\eta}(t,x){\,\rm d} x.
\end{equation*}
}
$$
I(t)= \int_{A-t}(u_x+u_t)(t,x){\,\rm d} x+\int_{B+t}(u_x-u_t)(t,x){\,\rm d} x.
$$
Then we have
$$
I(t)=I(0), \;  \forall t\in [0, T].
$$
In the case $|A|= |B|= 2\pi$, this conservation law is exactly the condition $\int_{{\mathbb T}} u_x (t, x) {\,\rm d} x= 0$.
\end{proposition}

\begin{remark}
  This proposition can be understood as a variant of Noether’s theorem\footnote{Noether’s theorem: ``Every continuous symmetry of the action (or the equations of motion, in a suitable sense) implies a conserved quantity.''} adapted to a controlled wave equation: a geometric symmetry of the forcing region yields a conserved functional of the solution. Indeed,
the key hypothesis is that the region $G$ satisfies the  {\it observable symmetry condition} for ($A, B$). The conclusion shows that the energy $I(t)$ is invariant in time, regardless of the force term.
\end{remark}

This new type of symmetry provides a {\it necessary condition} for the controllability and observability of \eqref{eq: wave-eq}, and even a {\it necessary condition} for the unique continuation of  \eqref{eq: wave-ob}. See Section \ref{sec:symmetry} for the detailed construction of counterexamples.  Moreover, this construction gives rise to two classes of function pairs satisfying the symmetry condition, $\mathcal{S}^2_c$ and $\mathcal{S}^2$, in Definitions \ref{def:classSc} and \ref{def:classS}.

\begin{figure}[htp]
    \centering
\begin{tikzpicture}[scale=1.0]

  \draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};
  \draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};

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  \draw (0,6) node[left] {$2\pi$} -- (0.1,6);

  \draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;

ll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;

\node at (barycentric cs:A1=1,B1=1,C1=1) {\textcolor{white}{-1}};

ll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;

\node at (barycentric cs:A2=1,B2=1,C2=1) {\textcolor{white}{1}};

ll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;

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\node at (barycentric cs:A5=1,B5=1,C5=1) {\textcolor{white}{-1}};

ll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;

\node at (barycentric cs:A6=1,B6=1,C6=1) {\textcolor{white}{1}};

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\node at (10,4) {$G=G_{0}\cup G_1$};

\node at (10, 3) {$G_{0}=\{(x,t): u_\xi=u_\eta=-1\}$: blue part};

\node at (10, 2) {$G_1=\{(x,t): u_\xi=u_\eta=1\}$: red part};

\end{tikzpicture}
    \caption{Observability fails on $G$, though $G$ satisfies GCC. }
    \label{fig:GCCfail1}
\end{figure}

\vspace{2mm}
\noindent {\bf The counterexample.}
 Let $T= 2\pi$. Let $G\subset [0,2\pi]\times {\mathbb T}$ be given in Fig. \ref{fig:GCCfail1}. Clearly $G$ satisfies \blackhyperref{def:GCC}{({\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\pi)$ or $A=B=(\pi,2\pi)$. Then one can show that the observability inequality \eqref{eq: wave-ob}  fails on $G$, see Section \ref{sec-osc-nece}.

\subsection{Necessary and sufficient unique continuation conditon}
The second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\in [0, T]\times {\mathbb T}$ be a measurable set.  UCP is the qualitative version of the observability and asks:
$$
\mbox{(UCP) \;\;\;\;\;\;\;\;\;\;\;\;\;\; Let $u$ be a solution of \eqref{eq: wave-eq-0} and }   u_t =0 \mbox{ a.e. in } G \Longrightarrow u \equiv 0.  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
$$

Motivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.

\vspace{2mm}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s>0.   \notag
\end{equation}
\end{enumerate}

\vspace{2mm}

This condition is equivalent to
\begin{equation}
      \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})>0, \;\;
  \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})>0. \notag
\end{equation}
Compared to the standard \blackhyperref{def:GCC}{({\rm GCC})}, it is not required a uniform positive lower bound.

\vspace{2mm}
\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}
\vspace{2mm}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset {\mathbb T}$ be two measurable subsets. $G\in [0, T]\times {\mathbb T}$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

\vspace{3mm}
The first result is a new conservation law of the forced equation \eqref{eq: wave-eq}.
\begin{proposition}[Conservation law]
\label{prop-sym-conser}
Let $A,B\subset {\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\it observable symmetry condition}.
Let $(u,u_t)\in C([0,T];\dot{H}^1({\mathbb T})\times L^2({\mathbb T}))$ be a solution to the forced equation \eqref{eq: wave-eq} and define the energy \footnote{Indeed, under the $(\xi, \eta)-$coordinate this conservation is write as
\begin{equation*}
 \frac{1}{2} I(t)= \int_{A-t}u_{\xi}(t,x){\,\rm d} x+\int_{B+t}u_{\eta}(t,x){\,\rm d} x.
\end{equation*}
}
$$
I(t)= \int_{A-t}(u_x+u_t)(t,x){\,\rm d} x+\int_{B+t}(u_x-u_t)(t,x){\,\rm d} x.
$$
Then we have
$$
I(t)=I(0), \; \forall t\in [0, T].
$$
In the case $|A|= |B|= 2\pi$, this conservation law is exactly the condition $\int_{{\mathbb T}} u_x (t, x) {\,\rm d} x= 0$.
\end{proposition}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s>0. \notag
\end{equation}
\end{enumerate}

\vspace{2mm}

This condition is equivalent to
\begin{equation}
 \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})>0, \;\;
 \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})>0. \notag
\end{equation}
Compared to the standard \blackhyperref{def:GCC}{({\rm GCC})}, it is not required a uniform positive lower bound.

\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}

\begin{equation}\label{eq: wave-eq}
(\partial_t^2-\partial_x^2)u =f\mathbf{1}_G,\quad (u, \partial_t u)\big|_{t=0}=(u_0,u_1)\in \dot H^1(\T)\times L^2(\T).
\end{equation}

\begin{equation}\label{eq:GCC:measure}
      \textrm{ for } a.e. \; x\in \T, \mbox{meas}_\R(G\cap L_{\eta= x})\geq c_0, \;\;
  \mbox{meas}_\R(G\cap L_{\xi=x})\geq c_0.
\end{equation}

\begin{figure}[htp]
 \centering

\subsection{Necessary and sufficient observability/controllability condition}
Finally, we obtain the following sharp and complete geometric characterization of observable regions.
\begin{theorem}\label{thm: main-ob}
Let $T>0$ and let $G\subset [0, T]\times \T$ be a measurable set. Then the following two statements are equivalent.
\begin{itemize}
 \item[(1)] The observability inequality \eqref{eq: wave-ob} holds on $G$;
 \item [(2)] $G$ satisfies \blackhyperref{def:GCC}{{\rm (GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-tirvial pair ($A, B$).
\end{itemize}
\end{theorem}

\begin{proposition}[Unique continuation implies weak GCC]
Let $T>0$ and $G\in [0, T]\times \T$. Assume that the following unique continuation property holds
$$
\mbox{ Let u be a solution of \eqref{eq: wave-eq-0} and } u_t =0 \mbox{ a.e. in } G \Longrightarrow u \equiv 0.
$$
Then $G$ satisfies the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}}.
\end{proposition}
\begin{proof}
We argue by contradiction. Suppose that $G$ does not satisfy the \blackhyperref{def:weakGCC}{({\rm weak GCC})}, there exists a subset $A\subset \T$ such that $\mbox{meas}_\R(A)>0$ and
\begin{align*}
\mbox{meas}_\R(G\cap L_{\eta\in A})=0 \quad \mbox{ or } \quad
 \mbox{meas}_\R(G\cap L_{\xi\in A})=0.
\end{align*}
Without loss of generality, we only consider the case
\begin{align}\label{equ-ucp-wgcc-1}
\mbox{meas}_\R(G\cap L_{\eta\in A})=0.
\end{align}
Since $A$ has positive measure, one can find a non-zero function $\phi$ such that
\begin{align}\label{equ-ucp-wgcc-2}
 \int_A \phi(x)\d x=0, \quad \int_A|\phi| \d x=\mbox{meas}_{\R}(A)>0.
\end{align}
Set
$$
u_{0x}=-u_1=\chi_A \phi.
$$
This is possible since $\int_\T u_{0,x}\d x =\int_A\phi \d x =0$. Then we have
$$
\partial_\xi u|_{t=0}=u_{0x}+u_1=0, \quad \partial_\eta u|_{t=0}=u_{0x}-u_1=2\chi_A\phi.
$$
It follows that
$$
\partial_\xi u=0 \quad \mbox{ on } [0,T]\times \T
$$
and
$$
\partial_\eta u= 0 \quad \mbox{ on } L_{\eta\in \T\backslash A}.
$$
Thus we have
$$
u_t=\frac{1}{2}(\partial_\xi u-\partial_\eta u)=0 \quad \mbox{ on } L_{\eta\in \T\backslash A}.
$$
Thanks to \eqref{equ-ucp-wgcc-1}, we see $G\subset L_{\eta\in \T\backslash A}$, and thus $u_t=0$ on $G$. By the UCP, we must have $\partial_\eta u|_{t=0}=2\chi_A\phi\equiv 0$, which leads a contradiction with \eqref{equ-ucp-wgcc-2}.
\end{proof}

\subsection{OSC and weak GCC imply unique continuation }\label{sec: OSC+WGCC>UCP}
In this sequel, we generalize the result in Proposition \ref{prop-ucp-S2}, which forms the following proposition.
\begin{proposition}[Weak GCC implies UCP up to $\mathcal{S}^2$]\label{prop-weakGCC-ucp-S2}
 Let $T>0$. Let $G\in [0, T]\times \T$ satisfy the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}}.
 If $u$ solves the wave equation \eqref{eq: wave-eq-0} and $u_t = 0$ on $G$, then its initial state $(\partial_\xi u, \partial_\eta u)|_{t=0}$ belongs to the symmetry function pair class $\mathcal{S}^2$.
\end{proposition}
\begin{proof}
We use the same notation in the proof of Proposition \ref{prop-ucp-S2}. Recall that we have the relation \eqref{equ-xi-eta-exchange}, namely
\begin{equation*} \partial_{\eta}u|_{\{t= 0\}\cap L_{\eta= \eta_0}}=\partial_{\eta}u(\xi_0,\eta_0)= \partial_{\xi}u|_{\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}}}
\end{equation*}
up to sets with zero measure.
Namely, the value of $\partial_{\eta}u(\xi_0,\eta_0)$ determines almost every value of $\partial_{\xi}u|_{\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}}}$. Since $G$ satisfies the \blackhyperref{def:weakGCC}{({\rm weak GCC})}, we know
$$
\mbox{meas}_\R(\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}})>0.
$$
In other words, $\partial_{\xi}u|_{\{t= 0\}}$ equals to $\partial_{\eta}u(\xi_0,\eta_0)$ at least on a positive measure subset set of $\T$. Thus the set
$$
\mathcal{U}_{(\xi_0,\eta_0)}:=\{x_0\in \T: \partial_{\xi}u|_{\{t= 0,x=x_0\}}=\partial_{\eta}u(\xi_0,\eta_0)\}
$$
has positive measure. Repeating this process for other point $(\xi_0',\eta_0')\in [0,T]\times \T$, we shall find that, there exists a family sets $\mathcal{U}_\ell (\ell\in I)$, $I$ is a index set, such that
\begin{align}\label{equ-Ul-129}
 \mathcal{U}_\ell\cap \mathcal{U}_{\ell'}=\emptyset \mbox{ for } \ell\neq \ell', \quad \mbox{meas}_\R(\mathcal{U}_\ell)>0 \mbox{ for } \ell\in I , \quad \T=\cup_{\ell\in I}\mathcal{U}_\ell
\end{align}
and $\partial_{\xi}u|_{\{t= 0\}}$ is a constant on each $\mathcal{U}_\ell$.
One can show that, under the restrictions \eqref{equ-Ul-129}, the index set $I$ is at most countable, thus, after relabeling if necessary, $\{\mathcal{U}_\ell, \ell\ge 1\}$ is a weak decomposition of $\T$. Moreover, there exists a sequence of different numbers $\{s_\ell\}$ such that
$$
\partial_{\xi}u|_{\{t= 0\}}=s_\ell \mbox{ on } \mathcal{U}_\ell.
$$
Note that $\partial_{\xi}u|_{\{t= 0\}}\in L^2(\T)$, and $\mathcal{U}_\ell$ are disjoint for different $\ell$, we find
$$
\|\partial_{\xi}u|_{\{t= 0\}}\|_{L^2(\T)}^2=\sum_\ell |s_\ell|^2\mbox{meas}_\R(\mathcal{U}_\ell)<\infty.
$$
Thus we obtain
$$
\partial_{\xi}u|_{\{t= 0\}} = \sum_{\ell\geq 1} s_\ell \chi_{\mathcal{U}_\ell} \quad \mbox{ in } L^2(\T).
$$
Similarly, we can find a weak decomposition $\mathcal{V}_{\ell}(\ell\geq 1)$ such that
$$
\partial_{\eta}u|_{\{t= 0\}} = \sum_{\ell\geq 1} s_\ell \chi_{\mathcal{V}_\ell} \quad \mbox{ in } L^2(\T).
$$
In summary, we find that $(\partial_{\xi}u|_{\{t= 0\}}, \partial_{\eta}u|_{\{t= 0\}})\in \mathcal{S}^2$.
\end{proof}

The following theorem gives a sufficient and necessary condition of the observable set for the transport equation. We only state the result for \eqref{eq: transport-eq}, the reader easily figures out the the necessary modifications for \eqref{eq: transport-eq-2}.
\begin{theorem}\label{thm-tran-suff-nece}
Let $T>0$ and $G$ be a measurable subset of $[0,T]\times \T$ with positive measure. Then the observability inequality
\begin{align}\label{equ-tran-suff-necc-1}
 \|u_0\|^2_{L^2(\T)}\leq C\int_G|u(t,x)|^2\d x \d t
\end{align}
holds with a positive constant $C>0$ for all solutions to the transport equation \eqref{eq: transport-eq} if and only if there exists a constant $c_0>0$ such that
\begin{equation}\label{equ-tran-suff-necc-2}
\int_0^T\mathbf{1}_G(s,x- s)\d s\geq c_0 \quad \mbox{ for a.e. } x\in\T
\end{equation}
\end{theorem}
\begin{proof}
The direction $\eqref{equ-tran-suff-necc-2}\Longrightarrow \eqref{equ-tran-suff-necc-1}$ follows clearly from Proposition \ref{prop: tran-eq-ob}. To show the inverse direction, namely $\eqref{equ-tran-suff-necc-1}\Longrightarrow \eqref{equ-tran-suff-necc-2}$, we use the contradiction argument. Suppose that \eqref{equ-tran-suff-necc-2} is not true, then for any $\varepsilon>0$, there exists a set $E_\varepsilon\subset \T$ with positive measure $|E_\varepsilon|>0$ such that
\begin{align}\label{equ-tran-suff-necc-3}
\int_0^T\mathbf{1}_G(s,x- s)\d s<\varepsilon \quad \mbox{ for a.e. } x\in E_\varepsilon.
\end{align}
Let $u_{0\varepsilon}(x)=1_{E_\varepsilon}(x)$. Then $u_{0\varepsilon}\in L^2(\T)$ and its support is contained in $E_\varepsilon$. Then, similar to the proof in Proposition \ref{prop: tran-eq-ob}, we have
\begin{align*}
\int_G|u(t,x)|^2\d x \d t =\int_{\T}|u_{0\varepsilon}(x)|^2\int_{\T}\mathbf{1}_G(t,x-t)\d t\d x\leq \varepsilon \|u_{0\varepsilon}\|^2_{L^2(\T)}.
\end{align*}
Taking $\varepsilon>0$ small enough, say $\varepsilon=C/2$, we obtain a contradiction with \eqref{equ-tran-suff-necc-1}.
\end{proof}
\subsection{GCC is necessary for the wave observability}\label{sec:app:3}
\begin{corollary}\label{wave-GCC-nece}
If the observability \eqref{eq: wave-ob} holds on some observation set $G\subset [0,T]\times \T$, then $G$ needs to satisfies \blackhyperref{def:GCC}{{\rm (GCC)}}.
\end{corollary}
\begin{proof}

\begin{remark}
    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\ref{def:classS}). In other words, the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.
\end{remark}

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;
\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;
\draw    (303,146.5) -- (415,146.5) ;
\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (416,154.5) -- (305,153.52) ;
\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\blackhyperref{def:OSC}{${\rm OSC}$} \ + \ \blackhyperref{def:weakGCC}{{\rm weak GCC}}};
\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};
\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec: OSC+WGCC>UCP}}};
\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec:neosctoucp}, Sec \ref{sec: ucp-wgcc}}};

\end{tikzpicture}

    \caption{Outline of the proof to the equivalence.}
    \label{fig:OSC+WGCC=UCP}
\end{figure}

\subsection{Necessary and sufficient observability/controllability condition}
Finally, we obtain the following sharp and complete geometric characterization of observable regions.
\begin{theorem}\label{thm: main-ob}
Let $T>0$ and let $G\subset [0, T]\times {\mathbb T}$ be a measurable set. Then the following two statements are equivalent.
\begin{itemize}
    \item[(1)] The observability inequality \eqref{eq: wave-ob}  holds on $G$;
    \item [(2)] $G$ satisfies \blackhyperref{def:GCC}{{\rm (GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-tirvial pair ($A, B$).
\end{itemize}
\end{theorem}

To the best of our knowledge, this work provides the first necessary and sufficient geometric condition for observability of the wave equation on spacetime region. The observable symmetry condition plays a central role: it completes the classical GCC by capturing the additional geometric structure. This viewpoint also leads to necessary and sufficient conditions in a variety of other important geometric configurations, such as $[0, T]\times \omega$,  measurable Cartesian products  $E_t\times F_{x}$, and general spacetime open sets.  A detailed comparison is presented in Section~\ref{sec:example}.

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw   (178,130) -- (292,130) -- (292,170) -- (178,170) -- cycle ;
\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;
\draw    (303,146.5) -- (415,146.5) ;
\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (416,154.5) -- (305,153.52) ;
\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

\draw (188,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\blackhyperref{def:OSC}{${\rm OSC}$} \ + \ \blackhyperref{def:GCC}{{\rm GCC}}};
\draw (439,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observability};
\draw (343,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec:sharp:obser}}};
\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec-osc-nece}, Sec \ref{sec:app:3}}};

\end{tikzpicture}
    \caption{Outline of the proof to the equivalence.}
    \label{fig:OSC+GCC=OB}
\end{figure}

\begin{remark}
The main theorem reveals two structural features of observability that distinguish the wave equation from other important models, for example, the heat equation,  \cite{AEWZ,WWZZ}. First, the family of observable regions $\mathcal{O}(T)$ depends intrinsically on $T$; non-observable sets arise from geometric obstructions that vary with $T$, leading to a genuinely time-dependent structure for $\{\mathcal{O}(T)\}_{T>0}$.
Second, our characterization also suggests that no waiting time is needed for observability, indicating that, for wave equations, the spacetime geometry of the observation region may play a more decisive role than the duration of control. See Section~\ref{sec:furthercomment}.
\end{remark}

Moreover, we believe that the symmetry mechanism uncovered in this work may extend to other instances of wave equations and to a broader range of models, including coupled systems, semilinear equations, and geometric wave equations. These ideas may also provide insight for higher-dimensional problems, where geometric propagation and microlocal structures are considerably more intricate. In addition, the geometric perspectives here may prove useful in questions of sensor placement, optimization of observation regions, and numerical implementations of control and observability.

\subsection*{Acknowledgements}
The authors would like to thank Nicolas Burq for valuable and useful discussions during the preparation of this manuscript.  Shengquan Xiang is partially supported by the NSFC under grants 12571474 and 12301562.   Ming Wang was partially supported by the NSFC under grants 12571260, 12171442 and 12171178.

A proof sketch for Theorem~\ref{thm: main-ucp} is indicated as follows.

- The key mechanism is the \blackhyperref{def:OSC}{(OSC)} (\emph{observable symmetry condition}): “Under the minimal \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\ref{def:classS}).”

- Conceptual structure: “the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.”

- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “\blackhyperref{def:OSC}{{\rm OSC}} + \blackhyperref{def:weakGCC}{{\rm weak GCC}}” and “UCP,” with the relevant parts located in “Sec \ref{sec: OSC+WGCC>UCP}” and “Sec \ref{sec:neosctoucp}, Sec \ref{sec: ucp-wgcc}.”

A proof sketch for the main theorem is indicated as follows.

- The key mechanism is the following observable symmetry condition:
\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}
Under the minimal assumption
\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}
unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, where
\begin{definition}[Weak symmetric function pair]\label{def:classS}
We say
$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\{(A_k, B_k), k\in I\}$, a sequence of different  numbers $\{s_k\}_{k\in I}$ such that
\begin{equation*}
 f=\sum_{k\in I}s_k\chi_{A_k}, \quad g=\sum_{k\in I}s_k\chi_{B_k} \quad \mbox{ in } L^2(\T)
\end{equation*}
and
\begin{equation*}
    \int_{\T} (f+ g)(x) \d x= 0.
\end{equation*}
The set of all weak symmetric function pairs is denoted by $\mathcal{S}^2$.
\end{definition}
In particular, this class is generated precisely by the observable symmetry condition above.

- Conceptual structure: the weak GCC condition stated above eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.

- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “the observable symmetry condition above + the weak GCC condition above” and “UCP,” with the relevant parts located later and in \ref{sec: OSC+WGCC>UCP} and \ref{sec:neosctoucp}, \ref{sec: ucp-wgcc}.

\label{thm: main-ucp}
Let $T>0$ and $G\subset [0,T]\times {\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.
\begin{itemize}
\item [(1)] UCP holds on $G$.

\item [(2)] $G$ satisfies
\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}
and does not obey
\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}
for any non-trivial pair $(A, B)$\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.
\end{itemize}

Biconditional or Equivalence
Classification or Bijection

Question: Let $T>0$ and let $G\subset [0,T]\times \mathbb T$ be a measurable spacetime set, where $\mathbb T$ is the one-dimensional torus. Which explicit geometric condition on $G$ is equivalent to the unique continuation property (UCP) on $G$ for the wave setting?

$G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0,$$ where $\Delta$ denotes symmetric difference.

Choices:
B. $G$ satisfies the geometric control condition in the strong sense that there exists a constant $c_0>0$ such that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds\ge c_0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds\ge c_0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0.$$
C. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$.
D. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any pair $(A,B)$ of measurable subsets of $\mathbb T$, including the trivial pair; equivalently, for every measurable pair $(A,B)$ one has $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)>0.$$
E. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ at least one of the two integrals $\int_0^T \mathbf 1_G(s,x+s)\,ds$ or $\int_0^T \mathbf 1_G(s,x-s)\,ds$ is strictly positive, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0.$$

restriction to non-trivial symmetry pairs

both characteristic directions required in weak GCC


\label{thm:charac-Markov}
  Let $X_0$ be a smooth affine surface over an algebraically closed field $K$ with a non-elementary automorphism group.
  If $X_0$ is completable by a cycle of rational curves, then we have two mutually exclusive possibilities.
  \begin{enumerate}
    \item $X_0 = \G_m^2$.
    \item $X_0$ is a cubic affine surface of Markov type.
  \end{enumerate}
  The distinction between the two cases comes from whether $X_0$ admits non-constant invertible regular functions.

\label{sec:intro}

Let $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \emph{completion} of $X_0$ is a
smooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the
\emph{dynamical degree} $\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,
then
\begin{equation}
 \lambda (f) = \lim_n \left( (f^n)^* H \cdot H \right)^{1/n}.
 \label{}
\end{equation}
The limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\lambda
(f) \geq 1$.
we say that an automorphism of $X_0$ is \emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's
work in \cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\mathbf{C}$ the topological entropy of $f$ is equal to $\log
\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\Aut (X_0)$ is
\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.

In \cite{abboudDynamicsEndomorphismsAffine2023}, the author studied the dynamics of loxodromic automorphisms of normal
affine surfaces. Using Gizatullin's work on affine surfaces, we showed that there is a dichotomy. If $X_0$ is a normal
affine surface with a loxodromic automorphism, then either $X_0$ is
completable by a zigzag of rational curves or by a cycle of rational curves. We have the following equivalent
conditions. We denote by $\overline \kappa (X_0)$ the log Kodaira dimension of $X_0$.

\begin{prop}\label{prop:charac-cycle}
 Let $X_0$ be a normal affine surface with a loxodromic automorphism, then we have the following dichotomy: either
 $X_0$ is completable by a tree of rational curves or by a cycle of rational curves and
 the following are equivalent.
 \begin{enumerate}
 \item $X_0$ is completable by a cycle of rational curves.
 \item $\overline \kappa (X_0) = 0$.
 \item For every loxodromic automorphism $f$ of $X_0$, $\lambda (f)$ is a quadratic integer.
 \end{enumerate}
 Or
 \begin{enumerate}
 \item $X_0$ is completable by a tree of rational curves.
 \item $\overline \kappa (X_0) = - \infty$.
 \item For every loxodromic automorphism $f$ of $X_0$, $\lambda (f)$ is an integer.
 \end{enumerate}
\end{prop}
The case of a tree of rational curve is quite rich, the affine plane $\mathbf{A}^2$ is the main example of such an affine surface but there are
many other examples of non isomorphic affine surfaces. In \cite{blancAffineSurfacesHuge2013}, Blanc and Dubouloz showed
that there are affine surfaces completable by a zigzag with a huge automorphism group. In
\cite{botSmoothComplexRational2023}, Bot showed that there are moduli spaces of such surfaces which gives a smooth rational
complex affine surface with uncountably many non-isomorphic real forms.

For the cycle case, the main example is the algebraic torus $\G_m^2$ and cubic affine surfaces of Markov type, i.e the
complement of a triangle of lines $\Delta$ in a smooth cubic projective surface $S$ in $\mathbf{P}^3$, (see
\cite{el-hutiCubicSurfacesMarkov1974}). If $q$ is one of the three intersection points of the triangle of lines in
$S$, it defines an involution $\sigma_p$ as follows: if $p \in S \setminus \Delta$, the line between $q$ and $p$
intersects $S \setminus \Delta$ in one other point which is $\sigma_q (p)$. Up to finite index the automorphism group of
such an affine surface is generated
by the three involutions $\sigma_{q_1}, \sigma_{q_2}, \sigma_{q_3}$ where $q_1, q_2, q_3$ are the three intersection
points in $\Delta$. These families of cubic affine surfaces have been studied extensively as they appear in different
areas of mathematics. They are related to the Painlevé Equation, to the character varieties of the 4-punctured sphere or
the once punctured torus, see \cite{cantatHolomorphicDynamicsPainleve2007}.
The purpose of this paper is to prove the following theorem which states that these are the only smooth examples.

Let $K$ be an algebraically closed field and let $X_0$ be an irreducible smooth affine surface. A \emph{completion} of $X_0$ is a
smooth projective surface $X$ that contains $X_0$ a Zariski dense open subset. If $f$ is an automorphism of $X_0$, the
\emph{dynamical degree} $\lambda (f)$ of $f$ is defined as follows: take a completion $X$ of $X_0$ and $H$ an ample divisor over $X$,
then
\begin{equation}
 \lambda (f) = \lim_n \left( (f^n)^* H \cdot H \right)^{1/n}.
 \label{}
\end{equation}
The limit exists, it does not depend on $X$ or on the choice of the ample divisor $H$. We always have that $\lambda
(f) \geq 1$.
we say that an automorphism of $X_0$ is \emph{loxodromic} if its dynamical degree is $>1$. It follows from the author's
work in \cite{abboudDynamicsEndomorphismsAffine2023} that if $K=\C$ the topological entropy of $f$ is equal to $\log
\lambda(f)$ so loxodromic automorphisms are exactly the ones with positive entropy. We say that $\Aut (X_0)$ is
\emph{non-elementary} if it contains two loxodromic automorphisms with no common iterates.

\begin{rmq}\label{rmq:}
 If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$
 is a curve of degree 3 in $\P^2$ with a nodal singularity, then $X_0 := \P^2 \setminus C$ is a smooth affine surface
 completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of
 $X_0$ such that $\Delta_X = C_1 \cup C_2$ which are two smooth rational curves that meet at two points. Here
 $\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is
 elementary.
\end{rmq}

Let $X_0$ be a normal affine surface, we write $K[X_0]$ for its ring of regular functions. A $K$-valuation over
$K[X_0]$ is a map $v : K[X_0] \rightarrow \R \cup \left\{ \infty \right\}$ such that
\begin{enumerate}
 \item $v_{|K^\times} = 0$.
 \item $v(0) = \infty$.
 \item $\forall P,Q \in K[X_0], \quad v(PQ) = v(P) + v (Q)$.
 \item $\forall P,Q \in K[X_0], \quad v(P+Q) \geq \min \left( v(P), v(Q) \right)$.
\end{enumerate}
We give two examples that will be useful for this note. First we have \emph{divisorial} valuations. Let $X$ be a
completion of $X_0$ and $E$ a prime divisor at infinity, then $\ord_E$ the order of vanishing along $E$ is a valuations
over $K[X_0]$. The second example is as follows. Let $E,F$ be two divisors at infinity in $X$ intersecting at a point
$p$, then we can find local coordinates $(x,y)$ at $p$ such that $x = 0 $ is a local equation of $E$ and $y = 0$ is a
local equation of $F$. For $\alpha, \beta > 0$ we define $v_{\alpha,\beta}$ as follows. The completion of the local ring
at $p$ with respect to its maximal ideal is $K \left[ [ x,y ] \right]$ the ring of formal power series in $x,y$. Define
\begin{equation}
 v_{\alpha,\beta} \left( \sum_{i,j} a_{ij} x^i y^j \right) = \min \left( \alpha i + \beta j : a_{ij} \neq 0 \right).
 \label{<+label+>}
\end{equation}
This defines a valuation over $K[X_0]$ because every $P \in K[X_0]$ is in the fraction field of $\OO_{X,p}$ the local
ring at $p$ and this fraction field embeds into $K [ [ x ,y ] ]$. We have two possibilities, if $\alpha / \beta \in \Q$,
then $v_{\alpha,\beta} = \ord_G$ is also divisorial but we need to blow up $p$ and infinitely near points to make $G$
appear. Otherwise $v_{\alpha,\beta}$ is \emph{irrational}.

\section{Proof of Theorem \ref{thm:charac-Markov}}\label{sec:}
Let $X_0$ be a smooth affine surface completable by a cycle of rational curves and assume that $\Aut (X_0)$ is not elementary.
\begin{prop}\label{prop:vanishing-self-intersection}
 For any $[v] \in \cC_\infty$, we have $Z_v^2 = 0$.
\end{prop}
\begin{proof}
 Let $f,g $ be two loxodromic automorphisms generating a non-elementary subgroup. Write $v_f^\pm, v_g^\pm$ for the
 eigenvaluations of $f$ and $g$ respectively. Recall that they are all different by Proposition
 \ref{prop:different-eigenvaluations}. From Proposition \ref{prop:eigenvaluations-divisor}, we have that
 \begin{equation}
 Z_{v^\pm_f}^2 = Z_{v^\pm_g}^2 = 0.
 \label{<+label+>}
 \end{equation}
 Let $Y$ be a cyclic completion of $X_0$ such that the centers of these four eigenvaluations are distinct. Recall that
 they are irrational valuations so their centers is always a satellite point. Let
 $p = E \cap F = c_Y (v_f^+)$. We look at the monomial valuations centered at $p$. The function
 \begin{equation}
 \phi: s \in [0, +\infty) \mapsto Z_{v_{1,s}}^2
 \label{<+label+>}
 \end{equation}
 is a polynomial map of $s$ of degree $\leq 2$ by Lemma \ref{lemme:self-intersection-irrational}. We have that $v_f^+ = v_{1, s_0}$ for some $s_0 > 0$. Now, we know
 that $f^n_* [v_g^+] \rightarrow [v_f^+]$ by Theorem \ref{thm:dynamics-loxodromic-automorphisms}. This implies that after a finite number of steps we have up to
 renormalisation
 \begin{equation}
 f^n_* v_g^+ = v_{1, s_n}
 \label{<+label+>}
 \end{equation}
for some $s_n > 0$ with $s_n \neq s_0$ and $s_n \rightarrow s_0$. But since $Z_{v_g^+}^2 = 0$ and this is invariant by the action of $f$
we have that $Z_{v_{1,s_n}}^2 = 0$ so the function $\phi$ is zero. Since for any valuation $[v] \in \cC_\infty \setminus
\left\{ [v_f^-] \right\}, f^n_* [v] \rightarrow [v_f^+]$ we have that $Z_v^2 = 0$.
\end{proof}
This implies by Proposition \ref{prop:nef-valuation} that for every $v \in \cC_\infty, Z_v$ is nef.
\begin{cor}\label{cor:intersection-positive}
 If $E,F$ are different prime divisors at infinity of a cyclic completion $X$ of $X_0$, then
 \begin{equation}
 \hat E \cdot \hat F > 0.
 \label{<+label+>}
 \end{equation}
\end{cor}
\begin{proof}
 We have by Proposition \ref{prop:vanishing-self-intersection} that $(\hat E)^2 = (\hat F)^2 = 0$ so that they are nef
 and effective divisors by Proposition \ref{prop:nef-valuation}. This implies that $\hat E \cdot \hat F \geq 0$.
 If the intersection number is zero, then by the Hodge index theorem we must have that $\hat E = \hat F$ but this is a
 contradiction.
\end{proof}

\begin{prop}\label{prop:trivial-k-plus-delta}
 Let $X_0$ be a smooth affine surface completable by a cycle of rational curves with a non-elementary automorphism
 group, then the class $K+\Delta \in \cNS_{cyc} (X_0)$ is equal to $0$.
\end{prop}
\begin{proof}
Recall by Proposition \ref{prop:Cartier-canonical-class} that $(K+ \Delta) \in \Cartier_{cyc}(X_0)_\R$ and it is defined
 by $K_X + \Delta_X$ for any cyclic completion $X$ of $X_0$. Furthermore, it is fixed by $\Aut (X_0)$. We show that
 \begin{equation}
 \forall v \in \cC_\infty, \quad Z_v \cdot (K+\Delta) = 0.
 \end{equation}
 Let $f,g \in \Aut (X_0)$ be two loxodromic automorphisms not sharing a common iterate with their eigenvaluations
 $v_h^\pm$ for $h = f,g$. We have that
 \begin{equation}
 Z_{v_h^\pm} \cdot (K+\Delta) = (h^{\pm 1})^* \left( Z_{v_h^\pm} \cdot (K+\Delta) \right) = \lambda \left(h^{\pm 1}\right) Z_{v_h^\pm}
 \cdot (K+\Delta).
 \label{<+label+>}
 \end{equation}
 This implies that $Z_{v_h^\pm} \cdot (K+\Delta) = 0$. Now, let $X$ be a cyclic completion of $X_0$ and let $p = E \cap F = c_X (v_f^+)$. The function
 \begin{equation}
 \phi: s > 0 \mapsto Z_{v_{1,s}} \cdot (K+\Delta)
 \label{<+label+>}
 \end{equation}
 is a polynomial of degree at most 1 because we have
 \begin{equation}
 Z_{v_{1,s}} \cdot (K+\Delta) = (Z_{v_{1,s}})_X \cdot (K+\Delta)_X = (\hat E + s \hat F) \cdot (K_X + \Delta_X).
 \label{<+label+>}
 \end{equation}
 Now we have that $v_f^+ = v_{1,s_0}$ for some $s_0 > 0$. As in the proof of Proposition
 \ref{prop:vanishing-self-intersection}, for $n$ large enough, $f^n_* v_g^+ = v_{1,s_n}$ and we have $\phi(s_n) = 0$.
 So that the function $\phi$ is 0. Now, for any $[v] \in \cC_\infty \setminus \left\{ [v_{-,f}] \right\}$ we have that
 $f^n_* [v] \rightarrow [v_{+, f}]$ so it will belong to $[v_E, v_F[$ after finitely many iterations. So that we get
 $Z_v \cdot (K+\Delta) = (f^n)_* (Z_v \cdot (K+\Delta)) = Z_{f^n_* v} \cdot (K+\Delta) = 0 $.

\begin{rmq}\label{rmq:}
  If we do not suppose that the automorphism group is non-elementary then other examples can arise. For example, if $C$
  is a curve of degree 3 in $\mathbf{P}^2$ with a nodal singularity, then $X_0 := \mathbf{P}^2 \setminus C$ is a smooth affine surface
  completable by a cycle of rational curves. Indeed, if we blow up the nodal singularity we get a completion $X$ of
  $X_0$ such that $\Delta_X = C_1 \cup C_2$ which are two smooth rational curves that meet at two points. Here
  $\Aut (X_0)$ is virtually cyclic with the generator being a loxodromic automorphism so the automorphism group is
  elementary.
\end{rmq}

\subsection{Idea of the proof}\label{subsec:}
The proof goes as follows. We use the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}. In the
case of an affine surface completable by a cycle of rational curves, the space of valuations centered at infinity of
$X_0$ contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle. This circle is homeomorphic to the
completion of the inverse limits of the dual graphs of cyclic completions. If $\Aut(X_0)$ is non-elementary, then its
action on this circle has very large orbits. This imposes strong constraints on the intersection form on the space of
divisors supported at infinity. We then conclude that $X_0$ has to be the complement of a triangle of lines in a smooth
cubic surface unless $X_0 = \G_m^2$.

\subsection{Study of endomorphisms of affine surfaces of Markov type}\label{subsec:endomorphisms-intro}
We fully classify the dynamics of cubic affine surfaces of Markov type in characteristic zero by showing that they do not have
non-invertible dominant endomorphisms.

\begin{bigthm}\label{thm:endomorphisms-affine-surface}
  Let $K$ be a field of characteristic zero. If $X_0$ is a smooth cubic affine surface of Markov type over $K$ and $f$ is a
  dominant endomorphism of $X_0$, then $f$ is an automorphism.
\end{bigthm}
To show this result we can assume that $K = \mathbf{C}$, then using valuative techniques and the geometry of the space of
valuations centered at infinity we show that $f$ must be proper and unramified. Thus, it is a covering of the complex
manifold $X_0 (\mathbf{C})$ which is simply connected (see eg \cite[Lemma 3.10]{cantatHolomorphicDynamicsPainleve2007}). Hence
$f$ must be a homeomorphism and therefore an automorphism.

Notice that this is not true for singular cubic affine surfaces. Indeed, let $\sC \subset \mathbf{A}^3$ be the Cayley cubic defined by
\begin{equation}
  x^2 + y^2 + z^2 = xyz + 4.
  \label{<+label+>}
\end{equation}
It is the quotient of $\G_m^2$ by the involution $(u,v) \mapsto (u^{-1}, v^{-1})$. The Cayley cubic has four singular
points which are orbifold singularities of order 2. It has many endomorphisms, namely every monomial endomorphism of
$\G_m^2$ descends to an endomorphism of $\sC$. The proof fails for this surface because it is not simply connected as an
orbifold. Indeed, its orbifold fundamental group is not trivial.

\subsection*{Acknowledgements} I thank Serge Cantat and Matteo Ruggiero for related discussions on this problem.

The proof of Theorem~\ref{thm:charac-Markov} is described in the subsection \emph{Idea of the proof}. One "use[s] the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which "contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle." This circle is identified as "homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions." If $\Aut(X_0)$ is non-elementary, then "its action on this circle has very large orbits," which "imposes strong constraints on the intersection form on the space of divisors supported at infinity." From these constraints one concludes that "$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \G_m^2$."

The proof of the main theorem is described in the subsection \emph{Idea of the proof}. One "use[s] the valuative techniques from \cite{abboudDynamicsEndomorphismsAffine2023}" and, for an affine surface completable by a cycle of rational curves, considers the space of valuations centered at infinity of $X_0$, which "contains an $\Aut(X_0)$-equivariant set which is homeomorphic to a circle." This circle is identified as "homeomorphic to the completion of the inverse limits of the dual graphs of cyclic completions." If $\Aut(X_0)$ is non-elementary, then "its action on this circle has very large orbits," which "imposes strong constraints on the intersection form on the space of divisors supported at infinity." From these constraints one concludes that "$X_0$ has to be the complement of a triangle of lines in a smooth cubic surface unless $X_0 = \G_m^2$."

Question: Let $X_0$ be a smooth affine surface over an algebraically closed field $K$. Assume that $\operatorname{Aut}(X_0)$ is non-elementary, meaning it contains two loxodromic automorphisms with no common iterate, where loxodromic means dynamical degree $>1$. Also assume that $X_0$ admits a smooth projective completion whose boundary divisor is a cycle of rational curves. Which description gives the complete classification of such surfaces?

Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.

Choices:
B. Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is the complement of a cycle of rational curves in an arbitrary smooth del Pezzo surface; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.
C. One always has the following dichotomy: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type.
D. If $X_0$ is completable by a cycle of rational curves and $\operatorname{Aut}(X_0)$ contains a loxodromic automorphism, then exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is a cubic affine surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.
E. Exactly two mutually exclusive possibilities occur: either $X_0\cong \mathbb{G}_m^2$, or $X_0$ is the complement of a triangle of lines in a cubic projective surface of Markov type; the distinction between these two cases is given by whether $X_0$ admits non-constant invertible regular functions.

specific ambient cubic-surface conclusion

criterion via non-constant invertible regular functions

non-elementary hypothesis weakened to existence of one loxodromic automorphism


\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in ${\bf C}^n$ with $\partial M\in C^2$. Assume that   the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\partial M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\overline M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Omega^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\overline M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.

\begin{thm}\label{vbint-}
Let $0<r\leq\infty$.
Let $M$ be a domain in $\cc^n$ with $\pd M\in C^2$. Assume that   the Levi form of $\pd M$ on $T^{(1,0)}\pd M$ has either $n-1$ positive or at least three negative eigenvalues at  $\zeta_0\in\pd M$.
Let $E$ be a complex vector bundle of class $Lip$ $($resp. $\Lambda^{r+1})$ over $\ov M$. Assume that a connection  $D\colon Lip(M,E)\to L_{(1)}^\infty(M,E)$  $($resp. $\Lambda^{r+1}(M,E)\to \Lambda^{r}_{(1)}(M,E))$ satisfies $\Om^{(0,2)}=0$ on $ M$ in the sense of distributions. Then $E|_U$ has a    holomorphic vector bundle structure, where $U$ is some open subset of $\ov M$ and $\zeta_0\in U$. Furthermore, the holomorphic structure is of class $C^{1/2}$ $($resp. $\Lambda^{r+1/2})$ with respect to the original $Lip$ $($resp. $\Lambda^{r+1})$ structure of $E|_U$.
\end{thm}

\label{sec1}
 Let $M\subset{\bf C}^n$ be a relatively compact domain with boundary $\partial M\in C^2$. Let $E$ be a smooth (i.e. $C^\infty$) complex vector bundle over the closure $\overline M$. Denote by $C^\infty(\overline M, E), C_{(k)}^\infty(E,\overline M)$ the spaces of smooth sections of $E$ and $E$-valued smooth $k$-forms on $\overline M$, respectively.
A Koszul {\it connection} $D$ in $E$ of is a ${\bf C}$-linear map
from $
 C_{(k)}^\infty(\overline M,E)$ to $ C^\infty_{(k+1)}(\overline M,E)
$
satisfying, for any $f\in C_{(k)}^\infty(\overline M)$ and $s\in C^\infty(\overline M,E)$,
$$
D(fs)=df\otimes s +(-1)^kf\wedge Ds.
$$ The curvature $R$ of $D$ is defined as
$
R=D\circ D\colon C^\infty_{(k)}(\overline M,E)\to C^\infty_{(k+2)}(\overline M,E).
$
 Let $e=(e_1,\dots, e_{k_0})^t$ be a smooth local frame of $E$. Then
$$
De_i=\sum \omega^j_i\otimes e_j, \ De=\omega e;\quad D^2e_i=\sum \Omega^j_i\otimes e_j, \ D^2e=\Omega e,
$$
where $\omega$ and $\Omega$ are the connection and curvature forms.
If $\hat e=Ae$ is another frame, then
\begin{equation}\label{hatomom}
\hat \omega =(dA)A^{-1} +A\omega A^{-1}, \quad \widehat\Omega=A\Omega A^{-1}.
\end{equation}
Decomposing into types, $\omega=\omega^{(1,0)}+\omega^{(0,1)}$ and
$\Omega=\Omega^{(2,0)}+\Omega^{(1,1)}+\Omega^{(0,2)}$, yields
\begin{equation}\label{om02}
 \Omega^{(0,2)}=\dbar\omega^{(0,1)}-\omega^{(0,1)}\wedge\omega^{(0,1)}.
\end{equation}

The {\it integrability problem} studied in this paper is to find a frame $\hat e=Ae$ for which $\hat\omega^{(0,1)}=0$, which is equivalent to solving
\begin{equation}\label{dbAA}
A^{-1}\dbar A+\omega^{(0,1)}=0.
\end{equation}
If such frames $\hat e_k$ exist on an open covering $\{U_k\}$ of $\overline M$, then on overlap $U_j\cap U_k$ their transition matrices satisfy $\dbar A_{kj}=0$. In this case we say that $(E,\overline M,D)$ carries a holomorphic vector bundle structure. By (\ref{hatomom})-(\ref{om02}), the solvability of (\ref{dbAA}) requires the {\it formal} integrability condition
\begin{equation}\label{Om02=0}
\dbar\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)},\quad \text{i.e.}\quad \Omega^{(0,2)}=0.
\end{equation}
Koszul and Malgrange~~\cite{MR131882} proved that when the boundary is not considered, the formal integrability condition is sufficient for the integrality; see also Kobayashi~~\cite{MR0909698}*{Prop.~3.7, p.~17} for a proof using the Newlander-Nirenberg theorem.

Our main result establishes integrability and optimal boundary regularity for the H\"older-Zygmund class (see Section 3 for definition) under the formal integrability condition on $M$.

\begin{thm}[convex configuration estimate]\label{conv-est} Let $r>0$ and $1\leq q\leq n-1$.
Let $(D^1,D_{r_2}^2)$ be an $(n-q)$-convex configuration. Let $D^{12}_r=D^1\cap D^2_r$.
The homotopy operators $H_q, H_{q+1}$ in \rta{hf-c} satisfies
\aln{}\|H_s\var\|_{\Lambda^{r+1/2}(D^{1 2}_{(1-\theta)r_2})}&\leq \f{C_r( \nabla\rho^1,\nabla^2\rho^1)}{\theta^{3r+2r_0}}\|\var\|_{\Lambda^r(D^{1 2}_{r_2})},\quad \forall\theta\in(0,1).
\end{align*}
Moreover, $C_r(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$.
\end{thm}
\begin{proof}Recall from \re{hq1}-\re{nhq2-} that $H_s =H^{(1)}_s + H_s^{(2)}$ with
\aln{}
 H^{(1)}_s f&:=R_{ D^{2}; s-1}^0 \cL E_{D^{12}} f+R_{U^1;s-1 }^{01}[\db,\cL E_{D^{12}}] f,\\
H^{(2)}_sf&=L_{1^+;{s-1}
}^{01} \cL E_{D^{12}} f +L_{2;s-1}^{02} f+L_{12;s-1}^{012}f.
\end{align*}
Take a cut-off function $\chi\in C^\infty_0(D^{2}_{(1-\theta/3)r_2})$ with $\chi=1$ on $D^{2}_{(1-\theta/2)r_2}$, and decompose
$
 H^{(1)}_s f= H^{(1)}_s(\chi f)+ H^{(1)}_s( (1-\chi)f).
$
Then the estimate in~\cite{gong-shi-nn} says that
$$
\|H_s^{(1)}(\chi \var)\|_{\Lambda^{r+1/2}(D^{1 2}_{r_2})}\leq C_r \|\chi\var\|_{\Lambda^r(D^{1 2}_{r_2})}.
$$
We can choose $\chi$ such that $\|\chi\|_{C^r}\leq {C_r}{\theta^{-r-1}}$. Thus,
$$
\|\chi\var\|_{\Lambda^r(D^{12}_{r_2})}\leq {C_r }{\theta^{-r-r_0}}\|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
Since $(1-\chi)\var$ vanishes on $D_{(1-\theta/2)r}^{12}$ and
\ga{}\label{giest}
|g^i(z,\zeta)\cdot(\zeta-z)|\geq |\zeta-z|^2/C,\quad z\in D^{1}\cap D^2,\zeta\in D^2\setminus D^1, i=0,1,2,
\end{gather}
then we can obtain $$
\|H_s^{(1)}((1-\chi) \var)\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r\theta^{-2r-r_0} \|(1-\chi)\var\|_{\Lambda^r(D^{12}_{r_2})}.
$$
The $H_s^{(2)}$ is given by boundary integrals on $S^1_+, S^2$ and $ S^{12}$ respectively. The estimate \re{giest} also yields
$\|H_s^{(2)}\var\|_{\Lambda^{r+1/2}(D^{12}_{(1-\theta)r_2})}\leq C_r \theta^{-2r-r_0} \|\var\|_{\Lambda^r(D^{12}_{r_2})}.
$
We have obtained the desired estimate of $H_s$.
\end{proof}
We now derive estimates for the concave case. We still have $H_s=H_s^{(1)}+H_s^{(2)}$, where $H_s^{(1)}$ has the same form as in the convex case. However,
$$
H_{s}^{(2)}f=\sum_{i=1}^2L_{i3,s}^{0i3}f- L^{01}_{1^+,s-1}\cL Ef +L_{12;s-1}^{123}f-\hat T_{ D^{23}_{(1-\theta)r}, s}L^{23}_{12;s} f.
$$
By \re{g3theta}, we have $
\RE\{g^3(z,\zeta)\cdot(\zeta-z)\} \geq\theta r_3^2$ for $z\in D^{23}_{(1-\theta)r}$ and $\zeta\in S^{13}\cup S^{23}\cup S^{12}\subset\pd D^{123}_r$. Some kernels in $H^{(2)}_s$ involve first-order $z$-derivatives of $\nabla\rho^1(z)$.
As in the convex case, we can obtain
$$\|H^{(2)}_sf\|_{\Lambda^{r+1/2}(D^{123}_{(1-\theta)r})}\leq {C_r(\nabla\rho^1 )}{\theta^{-r-r_0}}\|\rho^1\|_{\Lambda^{3/2}}\|f\|_{\Lambda^\e}.
$$The kernels of $H^{(1)}f$ involve the second-order $z$-derivatives $ \nabla\rho^2(z)$.
Then by ~\cite{gong-shi-nn}*{Thm. 5.12}, we have
$$\|H^{(1)}_sf
\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq {C_r }{\theta^{-r-r_0}}\left(\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\e}+ \|f\|_{\Lambda^r}\right).
$$Therefore, we have obtained part $(a)$ below.
\begin{thm}[concave configuration estimate]\label{concave-est} Let $\e>0,r>0$ and $1\leq q\leq n-3$.
Assume that $\rho^1\in C^{2+\e}$. The following hold.
\bpp
\item The operators $H_q, H_{q+1}$ in Theorem~$\ref{cchf}$ satisfy
\al{}
\label{hqfr12}
\|H_sf\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}&\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon({D_{r}^{123}})}+ \|f\|_{\Lambda^r(D^{123}_{r})})
\end{align}
for $0\leq\delta\leq1/2$ and $0<\theta<1$. Furthermore, $C_{r,\e}=C_{r,\e}(\nabla\rho^1,\nabla^2\rho^1)$ is stable under small $C^2$ perturbations of $\rho^1$;
 the same estimate holds for the $\db$ solution operator $H_q$ in \rta{cchf-closed}.
 \item
The homotopy formula \rea{tsqf+-cv} in Theorem~$\ref{cchf}$ holds for $f\in \Lambda^{\e}_{(0,q)}(D^{123})$ when $\db f\in \Lambda^{\e}_{(0,q)}(D^{123})$, and the $\db$-solution operator $H_q$ in \rta{cchf-closed} is valid for all $\db$-closed $f\in\Lambda^{\e}_{(0,q)}(D^{123})$ and satisfies \rea{hqfr12}.
\epp
\end{thm}
\begin{proof}We verify $(b)$ using $(a)$. We know that $f=\db H_qf+H_{q+1}\db f$ when $f\in C^{1+\e}$. Let us write $H_q$ as $H_{D^{123}_r}$ to indicate the dependent on the domain. Now assume $f\in \Lambda^r_{(0,q)}(D^{123})$ with $\db f\in\Lambda^r$. We find a sequence $f_j\in C^{\infty}_{(0,q)}(D^{123})$ such that $f_j $ and $\db f_j $ converge to $f,\db f$ in $\Lambda^{r'}(D^{123})$ as $j\to\infty$ for any $r'<r$. Furthermore, $$\|f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|f\|_{\Lambda^{r}(D^{123})}, \quad \|\db f_j\|_{\Lambda^{r}(D^{123})}\leq C_r\|\db f\|_{\Lambda^{r}(D^{123})}.$$
Then $H_qf=\lim H_qf_j$ and $H_{q+1}\db f=\lim H_{q+1}\db f_j$ gives us \rea{tsqf+-cv}.
Assume now $f$ is $\db$-closed. Set $L_t(z)=z+tie_{q+2}$ with $t>0$, where $e_{q+2}$ is the $(q+2)$-th unit vector of $\cc^n$. Fix $\theta\in(0,1/2)$. When $0<t<t_\theta$, we have $L_t(D^{123}_{(1-\theta)r})\subset D^{123}_r$. Using a mollifier, we find $\db$-closed $f_j\in C^\infty(D^{123}_{(1-\theta)r})$ such that $f_j$ converges to $f$ in $\Lambda^{r'}(D^{123}_{(1-\theta)r})$-norm for any $r'<r$, whereas $\|f_j\|_{\Lambda^{r}(D^{123}_{(1-\theta)r})}\leq C_r\|f\|_{\Lambda^{r}(D^{123}_{r})}$. We have $f_j=\db H_{D^{123}_{(1-\theta')r},q}f_j$ on $D^{123}_{(1-\theta)r}$ for $0<\theta'<\theta/2$.
Letting $j\to\infty$, we obtain $f=\db H_{D^{123}_{(1-\theta')r},q}f$ on $D^{123}_{(1-\theta)r}$ and
\eq{Hth''}
\|H_{D^{123}_{(1-\theta')r},q}f\|_{\Lambda^{r+\del}(D^{123}_{(1-\theta)r})}\leq \f{C_{r,\e}}{\theta^{3r+r_0}} (\|\rho^1\|_{\Lambda^{r+2+\del}}\|f\|_{\Lambda^\epsilon }+ \|f\|_{\Lambda^r }).
\eeq
By the Arzel\`{a}-Ascoli theorem, $H_{D^{123}_{(1-\theta'_j)r},q}f$ converges to $H_{D^{123}_{r},q}f$ in $\Lambda^{r'+\delta}(D^{123}_{(1-\theta)r})$ norm for a sequence $\theta'_j\to0$ for any $r'<r$. Then \re{Hth''} yield
\re{hqfr12} for $H_q$.
\end{proof}

\setcounter{thm}{0}\setcounter{equation}{0}
\section{Integrability of complex vector bundles}\label{sec:NM}
In Section 2, we have seen that \rt{vbint-} follows from the following result.
\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$. Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$ of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}
\begin{proof}The proof is based on a KAM-type iteration method. Such an approach was developed by Webster~\ci{MR999729} for an interior version of Newlander-Nirenberg theorem. It was also used in~\cites{MR2742034,MR2829316} to establish the integrability of CR vector bundles on strictly pseudoconvex hypersurfaces in $\cc^n$ for $n\geq4$.

Our theorem concerns the integrability of $(E,D)$ in the H\"older-Zygmund spaces $\Lambda^r$, which is the standard H\"older spaces $C^r$ when $r$ is not an integer; see Section 5 for definition. The theorem also treats  a Koszul connection $D$ that maps the space $ Lip(M,E)$ of Lipschitz sections
 into  $ L^\infty_{(1)}( M,E)$, the space  of $E$-valued $1$-forms of class $L^\infty$ on $M$. Such a connection extends trivially to a Koszul connection on $E$ over $\overline M$.   The theory of $L^\infty$ connections, which is of independent interest, is required  even when the original connections are $C^\infty$ for the (strictly) $3$-concave case. Theorem~\ref{vbint-}  is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of   $L^\infty$ connections on Lipschitz bundles  and of  {\it admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

\medskip

We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear.
 We   apply homotopy formulas to the study of (\ref{dbAA}).
The use of integral representations to establish regularity of $\overline\partial$-solutions on strongly pseudoconvex domains in ${\bf C}^n$ has a long history. Sup-norm estimate for $\dbar$-solutions to $(0,1)$-forms was proved by  Grauert and Lieb~\cite{MR273057} and  Henkin~\cite{MR0249660}.   Kerzman~\cite{MR0281944} obtained  $L^p$ and $C^{\beta}$ estimates of $\dbar$-solutions for $(0,1)$-forms and all $\beta<1/2$.
  Lieb~~\cite{MR283235} obtained the $L^\infty$ and the $C^\beta$ estimates of $\dbar$-solutions for $(0,q)$-forms.
Henkin and Romanov~\cite{MR0293121}  proved the sharp $C^{1/2}$ estimate of
$\dbar$-solutions for continuous $(0,1)$-forms.

We recall several important developments on integral representation formulas for  $\overline\partial$   that lead to derivative estimates on strongly pseudoconvex domains $D\subset{\bf C}^n$. When $\partial D$ is of class $ C^{k+2}$,   Siu~\cite{MR330515} showed that the Henkin solution operator satisfies a $C^{k+1/2}$ estimate  for  $\dbar$-closed $(0,1)$ forms of class $C^k$, and for  $\dbar$-closed $(0,q)$-forms   with $q\geq1$,  Lieb and Range~\cite{MR597825} constructed a new  $\dbar$ solution operator   and established  the same estimate. When $\partial D$ is only $ C^2$,  a homotopy formula was constructed  in~\cite{MR3961327} using the Stein extension, producing   homotopy operators with $\Lambda^{r+1/2}$ estimate for $r>1$.    Shi and Yao~\cites{MR4688544,MR4861589} later developed   a homotopy formula   employing the Rychkov extension and obtained $\Lambda^{r+1/2}$ estimates for all $r>0$, as well as $H^{s+1/2,p}$ estimates  for $1<p<\infty$ and $s>1/p$ when $\partial D\in C^2$; they also showed that the  estimates hold  for all $s\in{\bf R}$ when $\partial D$ is sufficiently smooth.  Very recently, Yao~~\cite{yaoc2} further reduced the boundary smoothness assumption to $ C^2$ for all $s\in{\bf R}$. We also refer the reader to references cited in~\cites{MR3961327,MR4688544,MR4861589,yaoc2}.

We also note that   Range and Siu~~\cite{MR338450} proved the $C^\beta$ estimate for all $\beta<1/2$ for $\dbar$-solutions for continuous $(0,q)$-forms on the (real) transversal intersection of
  strictly pseudoconvex domains. Whether the optimal gain of $1/2$  derivative   estimates  holds for $\dbar$-solutions on such transversal intersections remains an open problem.
  Higher-order derivative estimates in this setting were obtained by Brinkmann~~\cite{Br84}, Michel~~\cite{MR928297}, and Michel-Perotti~~\cite{MR1038709}.
 Peters~~\cite{MR1135535}  constructed a homotopy formula for the weakly transversal intersection of strictly pseudoconvex domains and established higher-order estimates, though with a loss of derivatives.
 It would be interesting to determine whether the results mentioned above can be applied to the integrability problem for intersections of strictly pseudoconvex domains. We also remark that the integrability of almost CR vector bundles on strongly pseudoconvex hypersurfaces in $\mathbb C^n$ with $n\geq4$ was proved by Webster~~\cite{MR1128608}, and sharper regularity results in the finitely smooth case were obtained by Gong-Webster~\cites{MR2742034,MR2829316}.
It remains an open question whether Theorem~\ref{vbint-} holds if $\partial M$ has two negative Levi eigenvalues. This appears to be related to the unsettled embedding problem for strongly pseudoconvex local CR structures in ${\bf R}^5$, investigated by Webster~\cite{MR0995504} and Gong-Webster~~\cite{MR2868966}.

We organize the paper as follows. In Section 2, we examine the formal integrability condition in detail for connections on Lipschitz complex vector bundles and introduce the notion of admissible frame changes. Sections 3 and 4 reformulate the local homotopy formulas in~\cites{MR986248,MR4866351}
 for shrinking domains. In Section 5, we recall  estimates established in \cites{MR986248,gong-shi-nn} and derive  new estimates on shrinking domains. Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-}  by establishing Theorem~\ref{vbint}.

\setcounter{thm}{0}\setcounter{equation}{0}

Theorem~\ref{vbint-} is reduced to Theorem~\ref{vbint}, and the reduction requires a careful discussion in Section~\ref{sec:int-formal} of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; see Proposition~\ref{d2=0}.

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and “Finally, in Section 6, we complete the proof of Theorem~\ref{vbint-} by establishing Theorem~\ref{vbint}.”

To prove the main theorem, we reduce it to the following theorem.

\begin{thm}\label{vbint}
Let $M$ be a domain in $\mathbb C^n$ with $\pd M\in C^2$. Assume that    the Levi-form of $M$ has
either $(n-1)$ positive or at least $3$ negative Levi eigenvalues at $\zeta_0\in\pd M$. Let $U$ be an open set in $\ov M$ and $\zeta_0\in U$.    Assume that $\om$ is a $k_0\times k_0$ matrix of $(0,1)$ forms on $U$ such that $\db\om=\om\wedge\om$ holds on $U\setminus\pd M$ in the sense of distributions. Assume that $\om$ is in $L^\infty(U)$ $($resp. $\Lambda^r(U)$ with $r>0)$. Then there is an invertible matrix $A$   of class $\Lambda^{1/2}$ $($resp. $\Lambda^{r+1/2})$ on some neighborhood $U'$ of $\zeta_0$ in $\ov M$ such that
$
\db A+A\om=0
$
in the sense of distributions on $U'\setminus\pd M$.
\end{thm}

The reduction requires a careful discussion of $L^\infty$ connections on Lipschitz bundles and of \textit{admissible} frame changes for which the transformation law (\ref{hatomom}) remains valid; for this we use the following proposition.

\begin{prop}\label{d2=0}
Let $D$ be a connection for a complex vector bundle $E$. Let $e=(e_1,\dots, e_{k_0})^t$ be a local frame field of $E$ over $U$. Assume that $De=\om \otimes e$ and $\om\in L^2_{loc}(U)$ and $d\om\in L^1_{loc}(U)$. If $A$ is admissible, then $\hat\om=(dA)A^{-1}+A\om A^{-1}$ is still in $L^2_{loc}(U)$ and  the curvature  forms    satisfy $A\Om A^{-1}=\hat\Om$.
\end{prop}

“We now outline our approach. Equation (\ref{dbAA}) may be viewed as a nonlinear $\dbar$-equation, and the integrability condition (\ref{Om02=0}) is also nonlinear. We apply homotopy formulas to the study of (\ref{dbAA}).”

The proof is then completed as follows: Sections 3 and 4 “reformulate the local homotopy formulas … for shrinking domains,” Section 5 recalls and derives “estimates … on shrinking domains,” and finally we complete the proof of the main theorem by establishing the theorem above.

Question: Let $0<r\le \infty$. Let $M\subset \mathbb C^n$ be a domain with $\partial M\in C^2$, and let $\zeta_0\in \partial M$. Assume that the Levi form of $\partial M$ on $T^{(1,0)}\partial M$ has either $n-1$ positive eigenvalues or at least three negative eigenvalues at $\zeta_0$. Let $E$ be a complex vector bundle over $\overline M$ of class $Lip$ (respectively of Hölder-Zygmund class $\Lambda^{r+1}$). Suppose $D$ is a connection on $E$ of the corresponding regularity, namely $D\colon Lip(M,E)\to L^{\infty}_{(1)}(M,E)$ (respectively $D\colon \Lambda^{r+1}(M,E)\to \Lambda^r_{(1)}(M,E)$), and that its curvature has vanishing $(0,2)$-part on $M$ in the sense of distributions. In a local frame $e$, if $De=\omega e$ and $D^2e=\Omega e$, this means $\Omega^{(0,2)}=0$, equivalently $\bar\partial\omega^{(0,1)}=\omega^{(0,1)}\wedge \omega^{(0,1)}$. A holomorphic vector bundle structure on $E|_U$ means local frames on an open cover of $U$ whose transition matrices are holomorphic (equivalently, local frame changes can be made so that the $(0,1)$-part of the connection vanishes). Under these assumptions, which existence statement holds?

There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1/2}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1/2}$.

Choices:
B. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure. Moreover, relative to the original $Lip$ structure on $E|_U$ this holomorphic structure is of class $C^{1}$, and relative to the original $\Lambda^{r+1}$ structure on $E|_U$ it is of class $\Lambda^{r+1}$.
C. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure.
D. There exists an open subset $U\subset \overline M$ with $\zeta_0\in U$ such that $E|_U$ admits a holomorphic vector bundle structure, provided the identity $\bar\partial\omega^{(0,1)}=\omega^{(0,1)}\wedge\omega^{(0,1)}$ holds on all of $U$, including across $\partial M$, in the sense of distributions.
E. For every neighborhood $U$ of $\zeta_0$ in $\overline M$, the restriction $E|_U$ admits a holomorphic vector bundle structure. Moreover, one can choose a local frame change $A$ of the same regularity as the original bundle structure, namely $A\in Lip(U)$ in the $Lip$ case and $A\in \Lambda^{r+1}(U)$ in the $\Lambda^{r+1}$ case, such that $\bar\partial A+A\omega^{(0,1)}=0$ on $U\setminus\partial M$.

equation required only on interior/distribution on $U\setminus\partial M$

local existence on some neighborhood and reduced regularity of frame change