\begin{theorem}\label{FYZ}
Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a  closed real (1,1)-form such that the principal argument of  $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$. If there exists a smooth function $\underline{u}$ satisfying \eqref{subsolution},
then the critical LYZ equation \eqref{LYZ1} admits a unique smooth solution u with $\sup_M u=0$.
\end{theorem}

\begin{align}\label{kHessian}
    \sigma_{k}(\alpha) := C_{n}^{k} \frac{\alpha^k \wedge \omega^{n-k}}{\omega^n}.
\end{align}

\begin{remark}\label{vlaplacebound}
    The condition $\Delta v \le C$ in the weak sense means that there exists a constant $C > 0$ such that for every nonnegative test function $f \in C_c^{\infty}(\mathbb{C}^n)$ with $\int_{\mathbb{C}^n} f(z) \, \mathcal{E}^n(z) = 1$, we have
    \[
    \int_{\mathbb{C}^n} v \, \sqrt{-1}\partial\bar{\partial} f \wedge \mathcal{E}^{n-1} \le C.
    \]
    From this condition, it follows immediately that the regularization $[v]_r$ has bounded Laplacian and hence bounded complex Hessian.
\end{remark}

In K\"ahler geometry, the LYZ (Leung-Yau-Zaslow) equation, also known as the dHYM (deformed Hermitian Yang-Mills) equation, is a fully nonlinear elliptic PDE that arises from mirror symmetry \cite{LYZ1999,MMMS2000}. This equation is of current interest in both mathematics and physics.

Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a closed real $(1,1)$-form.
Suppose that $\mathcal{Z}=\int_{M}(\chi+\sqrt{-1}\omega)^n\neq 0$ and let $\hat\theta$ be the principal argument of $\mathcal{Z}$.
 The LYZ equation seeks a smooth function $u: M\rightarrow\mathbb{R}$ such that
 \begin{equation}\label{LYZe}
 \begin{aligned}
 \mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 \end{aligned}
 \end{equation}
 Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be the eigenvalues of $\chi_u$ with respect to $\omega$ and define the special Lagrangian operator as
\[
\theta_{\omega}(\chi_u):=\sum\limits_{i=1}^n\arctan \lambda_i.
\]
 Then equation (\ref{LYZe}) can be rewritten in the form
 \begin{equation}\label{LYZ2}
 \begin{aligned}
 \theta_{\omega}(\chi_u)=\theta
 \end{aligned}
 \end{equation}
 where $\theta=n\frac{\pi}{2}-\hat \theta$. We refer to (\ref{LYZ2}) as the \emph{supercritical} LYZ equation if $\theta\in\big( (n-2)\frac{\pi}{2}, n\frac{\pi}{2}\big)$, and as the \emph{critical} LYZ equation if $\theta=(n-2)\frac{\pi}{2}$, i.e., $\hat \theta=\pi$.

A key requirement for solving the LYZ equation is the existence of a {\sl subsolution}:
A smooth function $\underline{u}$ satisfying
 \begin{align}\label{subsolutiontheta}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>\theta-\frac{\pi}{2}.
 \end{align}
This notion of subsolution is equivalent to that introduced by Guan \cite{guan2014} and Sz\'ekelyhidi \cite{szek2018}.
\subsection{Previous Results}

The 2-dimensional case of the LYZ equation was solved by Jacob-Yau \cite{jacobyau2017},
who also obtained partial results in higher dimensions. Collins-Jacob-Yau \cite{cjy2020} then solved the supercritical case under the assumption of a subsolution $\underline{u}$ and an additional condition $\theta_{\omega}(\chi_{\underline{u}})>(n-2)\frac{\pi}{2}$. The authors \cite{fyz2024} introduced a geometric flow to solve the equation under the same conditions. The extra condition was
subsequently removed by Pingali \cite{Pingali2022APDE} for $n=3$, by Lin \cite{lin2023adv} for $n=3,4$, and finally
by Lin \cite{lin2023} in full generality.
An alternative proof, based on Chen's significant result \cite{Chen2021Invent}, was given by Sun \cite{sun2024}.
For further recent developments, we refer to
\cite{ChanJacob2023arxiv, ChuCollinsLee2021GT,ChuLee2023Crelle, clt2024jdg, CollinsXieYau2018,CollinsYau2021APDE,HanJin2021CVPDE,HanJin2023Tran,HanJin2024Manu,HuangZhangZhang2022SCM,Jacob2021PAMQ, jacob2022arxiv,JacobSheu2022AsianJM,KhalidDyrefelt2024imrn,Lin2024MRL,Takahashi2021ijm}.

\subsection{Our main results}
Collins-Jacob-Yau \cite{cjy2020} suggested studying the LYZ equation when $\theta\le (n-2)\frac{\pi}{2}$. This was later explicitly posed as a question by Li \cite{YangLi2022arxiv}, as resolving it is essential for the LYZ equation to induce a Bridgeland stability condition on $D^b \text{Coh}(M)$, the bounded derived category of coherent sheaves on $M$.

In this paper, we solve the critical case; that is,
we establish the existence of smooth solutions for the \textit{critical} LYZ equation:
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
In this case, the principal argument of $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$,
and the subsolution condition becomes
\begin{align}\label{subsolution}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{equation}\label{LYZ2}
 	\begin{aligned}
 		\theta_{\omega}(\chi_u)=\theta
 	\end{aligned}
 \end{equation}

\begin{equation}\label{LYZe}
 	\begin{aligned}
 		\mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 	\end{aligned}
 \end{equation}

\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a closed real $(1,1)$-form.
Suppose that $\mathcal{Z}=\int_{M}(\chi+\sqrt{-1}\omega)^n\neq 0$ and let $\hat\theta$ be the principal argument of $\mathcal{Z}$.
 The LYZ equation seeks a smooth function $u: M\rightarrow\mathbb{R}$ such that
 \begin{equation}\label{LYZe}
 \begin{aligned}
 \mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 \end{aligned}
 \end{equation}
 Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be the eigenvalues of $\chi_u$ with respect to $\omega$ and define the special Lagrangian operator as
\[
\theta_{\omega}(\chi_u):=\sum\limits_{i=1}^n\arctan \lambda_i.
\]
 Then equation (\ref{LYZe}) can be rewritten in the form
 \begin{equation}\label{LYZ2}
 \begin{aligned}
 \theta_{\omega}(\chi_u)=\theta
 \end{aligned}
 \end{equation}
 where $\theta=n\frac{\pi}{2}-\hat \theta$. We refer to (\ref{LYZ2}) as the \emph{supercritical} LYZ equation if $\theta\in\big( (n-2)\frac{\pi}{2}, n\frac{\pi}{2}\big)$, and as the \emph{critical} LYZ equation if $\theta=(n-2)\frac{\pi}{2}$, i.e., $\hat \theta=\pi$.

In this paper, we solve the critical case; that is,
we establish the existence of smooth solutions for the \textit{critical} LYZ equation:
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
In this case, the principal argument of $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$,
and the subsolution condition becomes
\begin{align}\label{subsolution}
 \min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{remark}
For the special Lagrangian equation in $\mathbb{R}^n$, interior Hessian estimates for critical and supercritical cases were proved by
Warren-Yuan \cite{WarrenYuan2010} and Wang-Yuan \cite{WangYuan2014}.
For $\theta<(n-2)\frac{\pi}{2}$,
$C^{1,\alpha}$ solutions were constructed by Nadirashvili-Vl\u{a}du{t} \cite{NV2010IHP} and Wang-Yuan \cite{wangyuan2013ajm}. Very recently, Mooney-Savin \cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$.
\end{remark}

To solve the critical LYZ equation \eqref{LYZ1},
we consider the family of equations:
\begin{align}\label{LYZt}
 \theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
We verify that there exists a uniform constant $C>0$ such that
$$\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,
(n-2)\frac{\pi}{2}+2Ct)$$ for small $t>0$ (see Lemma \ref{thetatrange} in Section 2). Hence equations \eqref{LYZt} are the supercritical LYZ equations. Then we prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$. Thus we can apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024} ) to obtain the smooth solution $u^t$ to equation \eqref{LYZt} with $\sup_M u^{t}=0$.

When $n=4$, the critical LYZ equation is the Hessian quotient equation
$$
\sigma_3(\chi_u)=\sigma_1(\chi_u).
$$
Another consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018}.
 \begin{corollary}\label{4dHQ}
 Let $(M, \omega)$ be a compact K\"ahler manifold with dimension $n=4$.
 Let $\chi$ be a closed real $(1,1)$-form satisfying $3\chi^2\wedge\omega-\omega^3>0\ \text{as a positive $(3,3)$-form}$ and
 \begin{align*}
 \int_M\chi^3\wedge\omega=&\int_M\chi\wedge\omega^3,\\
 \int_M\chi\wedge\omega^{3}>&0,\
 \mathrm{Re}\int_M (\chi+\sqrt{-1}\omega)^4<0.\label{4dRe}\notag
 \end{align*}
 Then there exists a unique
 smooth function $u$ solving
 \begin{align}
 \sigma_3(\chi_u)=\sigma_1(\chi_u),\\
 \chi_u\in \Gamma_3(M), \ \sup_{M} u=0.\notag
 \end{align}
\end{corollary}

\subsection{$C^0$-estimates}
Let $\hat \theta(t)$ denote the principal argument of the complex number $\int_M (\chi + t\omega + \sqrt{-1}\omega)^n$, and set $\theta(t) = (n-2)\frac{\pi}{2} - \hat \theta(t)$. We first show that $\hat \theta(t)$ is close to $\pi$ when $t>0$ is sufficiently small.
\begin{lemma}\label{thetatrange}
 Assume that $\underline u$ is a subsolution of the critical LYZ equation \eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\chi$ and $\omega$, such that for all $t \in (0, t_0)$,
 \[
 \hat \theta(t) \in \bigl( \pi - 2Ct,\; \pi - Ct \bigr).
 \]
\end{lemma}

\begin{theorem}\label{thm: Hessian estimate}
 Let $\underline{u}$ be a subsolution of the critical LYZ equation \eqref{LYZ1} and let $u^t$ be the solution of the LYZ equation \eqref{LYZt} with $\sup_M u^t = 0$.
 Then there exist uniform positive constants $t_0$ and $C$, independent of $\bigl(\theta(t)-(n-2)\frac{\pi}{2}\bigr)^{-1}$, such that for any $t \in (0, t_0)$,
 \begin{align}\label{eq: Hessian estimate}
 \sup_M |\sqrt{-1}\partial\bar{\partial} u^t|_{\omega} \le C \bigl(1 + \sup_M |\nabla u^t|_{\omega}^2 \bigr).
 \end{align}
\end{theorem}

We consider a family of LYZ equations \eqref{LYZt}
 \begin{align*}
 3(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)^2\wedge\omega=&
 \tan\hat\theta(t)(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)^3\\
 &-3\tan\hat\theta(t)(\chi+t\omega+\sqrt{-1}\partial\bar\partial u^t)\wedge\omega^2+\omega^3.
 \end{align*}
 As a corollary of our theorem, we solve the 2-Hessian equation under weaker conditions instead of the usual condition $\chi\in \Gamma_2(M)$.
 \begin{corollary}
 Let $(M,\omega)$ be a compact K\"ahler manifold with dimension $n=3$ and $\chi$ be a real $(1,1)$-form satisfying the following conditions:
 \begin{enumerate}[(i)]
 \item
 $3\int_M\chi^2\wedge\omega=\int_M\omega^3$,
 $\int_M\chi^3<3\int_M \chi\wedge\omega^2\label{re<0}$;
 \item
 the subsolution condition
 \begin{align}
 \chi\wedge\omega>0\ \ \text{as a $(2,2)$-form}.
 \end{align}
 \end{enumerate}
 Then there exists a unique smooth solution $u$ solving
 \begin{align}
 \sigma_2(\chi_u)=1,
 \\ \chi_u\in \Gamma_{2}(M), \ \sup_M u=0.
 \end{align}
 \end{corollary}

\begin{equation}\label{LYZ2}
 	\begin{aligned}
 		\theta_{\omega}(\chi_u)=\theta
 	\end{aligned}
 \end{equation}

\begin{equation}\label{LYZe}
 	\begin{aligned}
 		\mathrm{Im}\big(e^{-\sqrt{-1}\hat\theta}(\chi_u+\sqrt{-1}\omega)^n\big)=0.
 	\end{aligned}
 \end{equation}

\begin{align}\label{LYZt}
	\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}

\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}

\begin{lemma}\label{thetatrange}
    Assume that $\underline u$ is a subsolution of the critical LYZ equation \eqref{LYZ1}. Then there exist uniform positive constants $t_0$ and $C$, depending only on $\chi$ and $\omega$, such that for all $t \in (0, t_0)$,
    \[
    \hat \theta(t) \in \bigl( \pi - 2Ct,\; \pi - Ct \bigr).
    \]
\end{lemma}

\begin{remark}
For the special Lagrangian equation in $\mathbb{R}^n$, interior Hessian estimates for  critical and supercritical cases were proved by
Warren-Yuan \cite{WarrenYuan2010} and Wang-Yuan \cite{WangYuan2014}.
For $\theta<(n-2)\frac{\pi}{2}$,
$C^{1,\alpha}$ solutions were constructed by Nadirashvili-Vl\u{a}du{t} \cite{NV2010IHP} and Wang-Yuan \cite{wangyuan2013ajm}. Very recently, Mooney-Savin \cite{mooneysavin2023duke} obtained a Lipschitz solution that is not $C^1$.
\end{remark}

To solve the critical LYZ equation \eqref{LYZ1},
we  consider the family of  equations:
\begin{align}\label{LYZt}
	\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
We  verify that there exists a uniform constant $C>0$ such that
$$\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,
(n-2)\frac{\pi}{2}+2Ct)$$ for small $t>0$ (see  Lemma \ref{thetatrange}  in Section 2). Hence equations \eqref{LYZt} are the supercritical LYZ equations.  Then we  prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$.  Thus we can apply  Lin's result \cite{lin2023} (see also Sun \cite{sun2024} ) to obtain  the  smooth solution $u^t$ to  equation \eqref{LYZt} with $\sup_M u^{t}=0$.

To prove Theorem \ref{FYZ},
we establish uniform $C^{2,\alpha}$ estimates  for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$.

\begin{theorem}\label{FYZfamilly}
    Assume the same conditions as in Theorem \ref{FYZ}.
    Let $u^t$ be the smooth solution of \eqref{LYZt} with $\sup_M u^t=0$.
    Then there exist uniform constants $C$ and $t_0$, depending only on $(M, \omega, \chi)$ but independent of $t$, such that for any $t \in (0, t_0)$,
    \begin{align}
        |u^t|_{C^{2,\alpha}(M)} \le C.
    \end{align}
\end{theorem}

According to Collins--Jacob--Yau \cite{cjy2020}, $\sup_{M} |u^t|$ is uniformly bounded.
Their results imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.
Our main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type,
\begin{align}\label{HMW}
    \sup\limits_{M} |\sqrt{-1}\partial\bar{\partial} u|_{\omega} \le C\bigl(1 + \sup\limits_{M} |\partial u|_{\omega}^2 \bigr),
\end{align}
and the uniform gradient estimate via the Dinew--Kolodziej blow-up argument \cite{dk2017ajm}.

However, the situation here is more subtle.
If the gradient estimate fails, we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$ that, in the sense of currents, is a weak solution to one of the following equations:
\[
\sigma_n = 0, \quad
\sigma_{n-1} = 0, \quad \text{or} \quad
\sigma_{n-1} + \sigma_n = 0.
\]
The last case is new and requires a corresponding Liouville-type theorem.
Note that $v + |z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense (see Remark \ref{vlaplacebound}).
When following the argument in \cite{dk2017ajm,szek2018}, we find that the first case can be handled similarly.
For the second case, we need to show that the standard mollification $[v]_r$ of $v$, as well as $\frac{1}{3} [(v^\epsilon)^2]_r - \tau_0 |z|^2$, are subsolutions, where $v^\epsilon = [v]_\epsilon$ and $\tau_0 > 0$ is sufficiently small.
To address this issue, our starting observation is that a smooth function $v$ satisfies $\sigma_{n-1} + \sigma_n = 0$ (resp. $\ge 0$) on $\mathbb{C}^n$ if and only if the function $v + |z_{n+1}|^2$ satisfies $\sigma_n = 0$ (resp. $\ge 0$) on $\mathbb{C}^{n+1}$, where $z_{n+1}$ is a new complex variable. Based on this observation, we use potential theory to  obtain the required Liouville-type theorem (see Theorem \ref{LiouTh}).

Finally, we find some applications of the main result.
	In general, the \emph{critical} LYZ equation has the form
$$
\sigma_{n-1}(\chi_u)=\sigma_{n-3}(\chi_u)-\sigma_{n-5}(\chi_u)+\cdots,
$$
where $\sigma_{k}(\chi_u)$  is the $k$-Hessian operator of $\chi_u$ with respect to the metric $\omega$ (see \eqref{kHessian}).

When $n=3$, the critical  LYZ equation is just the 2-Hessian equation
$$\sigma_2(\chi_u)=1.$$
As a consequence of our result, we can solve the 2-Hessian equation in dimension 3 under weaker conditions than those by Sun \cite{sun2017cpam}.
	\begin{corollary}
	Let $(M,\omega)$ be a compact K\"ahler manifolds with dimension $n=3$ and $\chi$ be a closed real  $(1,1)$-form satisfying  $\chi\wedge\omega>0$\  \text{as a positive $(2,2)$-form} and
		\begin{align*}
			3\int_M\chi^2\wedge\omega=\int_M\omega^3,\ \
			\int_M\chi^3<3\int_M \chi\wedge\omega^2.
		\end{align*}
	Then there exists a unique smooth  solution $u$ solving
\[
	\begin{aligned}
		\sigma_2(\chi_u)=1,
		\\ \chi_u\in \Gamma_{2}(M), \ \sup_M u=0.
	\end{aligned}
	\]
\end{corollary}
Note that Hou-Ma-Wu \cite{hmw2010} proved the complex Hessian estimate of Hou-Ma-Wu type for the $k$-Hessian equation $\sigma_k(\omega_u)=f$ on a compact K\"{a}hler manifold.
Then Dinew-Kolodziej \cite{dk2017ajm}
obtained the gradient estimate by a Liouville  theorem for $\sigma_{k}=0$ in $\mathbb{C}^n$ and thus  solved the equation. The Hermitian case was solved independently  by Sz\'ekelyhidi \cite{szek2018} and Zhang \cite{zhang2017pjm}.

When $n=4$, the critical LYZ equation is the Hessian quotient equation
$$
\sigma_3(\chi_u)=\sigma_1(\chi_u).
$$
Another consequence of our result is to solve the above Hessian quotient equation in dimension 4 under weaker conditions than those by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018}.
	\begin{corollary}\label{4dHQ}
	Let $(M, \omega)$ be a compact K\"ahler manifold with dimension $n=4$.
	Let   $\chi$ be a closed real $(1,1)$-form satisfying 	$3\chi^2\wedge\omega-\omega^3>0\  \text{as a positive $(3,3)$-form}$ and
		\begin{align*}
			\int_M\chi^3\wedge\omega=&\int_M\chi\wedge\omega^3,\\
			\int_M\chi\wedge\omega^{3}>&0,\
			\mathrm{Re}\int_M (\chi+\sqrt{-1}\omega)^4<0.\label{4dRe}\notag
		\end{align*}
	Then there exists a unique
	smooth function $u$ solving
	\begin{align}
		\sigma_3(\chi_u)=\sigma_1(\chi_u),\\
		\chi_u\in \Gamma_3(M), \ \sup_{M} u=0.\notag
	\end{align}
\end{corollary}

{We remark that the Hessian quotient equation is of the form $\sigma_k(\chi_u)=c\sigma_{l}(\chi_u)$ with $0\le l<k\le n$. It reduces to the $J$-equation when $k=n$ and $l=n-1$ which was solved by Song-Weinkove \cite{songweinkove2008} using the flow method, and  the case $k=n, 1\le l\le n-1$ was solved by Fang-Lai-Ma \cite{flm2011}. The general case was subsequently solved by Sun \cite{sun2017cpam} and Sz\'{e}kelyhidi \cite{szek2018} via different continuity methods.}

	The organization of this paper is as follows. In Section 2, we  provide preliminaries on $k$-Hessian operators, the special Lagrangian operator $\theta_{\omega}$, and the approximating family $\theta(t)$. We also give the $C^0$-estimate for $u^t$ using results from \cite{cjy2020}. In Section 3, we first prove a key inequality implied by the subsolution condition, then we establish the uniform  complex Hessian estimate  of Hou-Ma-Wu-type which is independent of $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$. In Section 4, based on the complex Hessian estimate,  we prove the gradient estimate by a blow up argument. Then we prove the main theorems. In Section 5, we apply the main results to the 3-dimensional 2-Hessian equation and 4-dimensional Hessian quotient equation.

To solve the critical LYZ equation \eqref{LYZ1}, the authors “consider the family of equations”
\begin{align}\label{LYZt}
\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
They “verify that there exists a uniform constant $C>0$ such that $\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,(n-2)\frac{\pi}{2}+2Ct)$ for small $t>0$,” so “equations \eqref{LYZt} are the supercritical LYZ equations.” Next they “prove the subsolution $\underline{u}$ of the critical LYZ equation \eqref{LYZ1} is also a subsolution of equations \eqref{LYZt} for small $t>0$,” and then “apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024}) to obtain the smooth solution $u^t$ to equation \eqref{LYZt}$ with $\sup_M u^{t}=0$.”

“To prove Theorem \ref{FYZ}, we establish uniform $C^{2,\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \ref{FYZfamilly}. Using Collins--Jacob--Yau \cite{cjy2020}, “$\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”
\begin{align}\label{HMW}
\sup_{M} |\sqrt{-1}\partial\bar{\partial}u|_{\omega}\le C\bigl(1+\sup_{M}|\partial u|_{\omega}^2\bigr),
\end{align}
“and the uniform gradient estimate via the Dinew--Ko\l{}odziej blow-up argument \cite{dk2017ajm}.”

They explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$” which is a weak solution of “$\sigma_n=0$, $\sigma_{n-1}=0$, or $\sigma_{n-1}+\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\sigma_{n-1}+\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^{n+1}$.” “Based on this observation, we use potential theory to obtain the required Liouville-type theorem (see Theorem \ref{LiouTh}).”

With the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in Theorem \ref{FYZ}.

To solve the critical LYZ equation
\begin{align}\label{LYZ1}
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}.
\end{align}
, the authors “consider the family of equations”
\begin{align}\label{LYZt}
\t\theta_{\omega}(\chi_{u^{t}}+t\omega)=\theta(t).
\end{align}
They “verify that there exists a uniform constant $C>0$ such that $\theta(t)\in ((n-2)\frac{\pi}{2}+Ct,(n-2)\frac{\pi}{2}+2Ct)$ for small $t>0$,” so “the equation above is the supercritical LYZ equation.” Next they “prove the subsolution $\underline{u}$ of the critical LYZ equation above is also a subsolution of the equation above for small $t>0$,” and then “apply Lin's result \cite{lin2023} (see also Sun \cite{sun2024}) to obtain the smooth solution $u^t$ to the equation above with $\sup_M u^{t}=0$.”

To prove the main theorem, “we establish uniform $C^{2,\alpha}$ estimates for $u^{t}$ that are independent of $t$ and $(\theta(t)-(n-2)\frac{\pi}{2})^{-1}$,” i.e. the uniform a priori estimate stated as Theorem \ref{FYZfamilly}. Using Collins--Jacob--Yau \cite{cjy2020}, “$\sup_{M} |u^t|$ is uniformly bounded,” and their results “imply that the $C^{2,\alpha}$ estimates can be reduced to proving uniform complex Hessian estimates.” The “main contribution is the proof of the uniform complex Hessian estimate of Hou--Ma--Wu type”
\begin{align}\label{HMW}
\sup_{M} |\sqrt{-1}\partial\bar{\partial}u|_{\omega}\le C\bigl(1+\sup_{M}|\partial u|_{\omega}^2\bigr),
\end{align}
“and the uniform gradient estimate via the Dinew--Ko\l{}odziej blow-up argument \cite{dk2017ajm}.”

They explain the blow-up subtlety: if the gradient estimate fails, “we obtain a bounded non-constant $C^1$ function $v$ on $\mathbb{C}^n$” which is a weak solution of “$\sigma_n=0$, $\sigma_{n-1}=0$, or $\sigma_{n-1}+\sigma_n=0$,” where “the last case is new and requires a corresponding Liouville-type theorem.” They note that “$v+|z|^2$ is plurisubharmonic and $\Delta v$ is uniformly bounded in the weak sense,” and that while “the first case can be handled similarly,” for the second/third cases they need subsolution properties for mollifications and then use a key observation: “a smooth function $v$ satisfies $\sigma_{n-1}+\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^n$ if and only if the function $v+|z_{n+1}|^2$ satisfies $\sigma_n=0$ (resp. $\ge0$) on $\mathbb{C}^{n+1}$.”

We next use the following theorem.
\begin{theorem}\label{LiouTh}
    Let $v: \mathbb{C}^n \rightarrow \mathbb{R}$ be a $C^1$ function that is $(n-1)$-subharmonic and such that $v + |z|^2$ is plurisubharmonic. Assume there exists a positive constant $C$ such that $\|v\|_{C^1(\mathbb{C}^n)} \le C$ and $\Delta v \le C$ in the weak sense (see Remark~\ref{vlaplacebound} for details). If $v$ is a weak solution in the sense of currents to the equation
    \begin{align}\label{nn-1equation}
        \sigma_{n-1}(\sqrt{-1}\partial\bar{\partial} v) + \sigma_{n}(\sqrt{-1}\partial\bar{\partial} v) = 0,
    \end{align}
    then $v$ must be constant.
\end{theorem}

With the uniform complex Hessian and gradient estimates, they obtain uniform $C^{2,\alpha}$ bounds for $u^t$ independent of $t$ (Theorem \ref{FYZfamilly}), and then “prove the main theorems,” yielding existence/uniqueness in establishing the main theorem.

\label{FYZ}
Let $(M,\omega)$ be a compact K\"ahler manifold and $\chi$ a  closed real (1,1)-form such that the principal argument of  $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$. If there exists a smooth function $\underline{u}$ satisfying
\begin{align}\label{subsolution}
	\min\limits_{1\le j\le n}\sum\limits_{\substack{i=1 \\ i\neq j}}^n\mathrm{arctan}\lambda_i(\chi_{\underline {u}})>{(n-3)\frac \pi 2}.
\end{align}
then the critical LYZ equation (in establishing the main theorem) admits a unique smooth solution u with $\sup_M u=0$.

Question:
Let $(M,\omega)$ be a compact K\"ahler manifold, let $\chi$ be a closed real $(1,1)$-form, and for a smooth real-valued function $v$ write $\chi_v:=\chi+\sqrt{-1}\,\partial\bar\partial v$. If $\lambda_1(\alpha),\dots,\lambda_n(\alpha)$ are the eigenvalues of a real $(1,1)$-form $\alpha$ with respect to $\omega$, define
\[
\theta_{\omega}(\alpha):=\sum_{i=1}^n \arctan \lambda_i(\alpha).
\]
Assume that the principal argument of
\[
\int_M (\chi+\sqrt{-1}\,\omega)^n
\]
is $\pi$, and suppose there exists a smooth function $\underline u$ such that
\[
\min_{1\le j\le n}\sum_{\substack{i=1\\ i\neq j}}^n \arctan \lambda_i(\chi_{\underline u})>(n-3)\frac{\pi}{2}.
\]
Which statement holds for the critical LYZ equation
\[
\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}?
\]

There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ and the normalization $\sup_M u=0$.

Choices:
B. There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$, and moreover the solution is obtained with uniform a priori bounds independent of $\bigl(\theta-(n-2)\frac{\pi}{2}\bigr)^{-1}$.
C. There exists a smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$.
D. There exists a unique smooth function $u:M\to\mathbb{R}$ satisfying $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ provided one additionally assumes the stronger phase condition $\theta_{\omega}(\chi_{\underline u})>(n-2)\frac{\pi}{2}$ for the same subsolution $\underline u$, together with $\sup_M u=0$.
E. For every sufficiently small $t>0$, there exists a unique smooth function $u^t:M\to\mathbb{R}$ with $\sup_M u^t=0$ solving the perturbed supercritical equation $\theta_{\omega}(\chi_{u^t}+t\omega)=\theta(t)$, and hence the critical equation $\theta_{\omega}(\chi_u)=(n-2)\frac{\pi}{2}$ need not itself admit a smooth solution under the stated hypotheses.

uniform-estimate statement for approximating family incorrectly inserted into theorem conclusion

dropped uniqueness and normalization $\sup_M u=0$

reintroduced extra supercritical subsolution phase assumption removed in the critical theorem

approximation-by-$t$ family mistaken for nonexistence of the limiting critical solution

The stem does not explicitly reveal the correct conclusion. It gives hypotheses and asks for the resulting theorem statement, but it does not directly state existence/uniqueness.

This is essentially a theorem-recall item: the hypotheses are stated almost verbatim and the correct option is the theorem's conclusion. It tests recognition of the statement more than mathematical inference.

Some reasoning is needed to distinguish the exact conclusion from nearby variants (existence vs uniqueness, extra assumptions, false strengthening). However, the item primarily rewards memory of the theorem rather than genuine generative reasoning.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, others add illegitimate strengthening or confuse approximation/uniform estimates with the actual conclusion. They reflect realistic failure modes.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is largely tautological and only moderately probes reasoning.


When $d\leq4$, for any open subset $\omega\subset M$ satisfying Geometric Control Condition (GCC) and for any $R_{0}>0$, there exist $T(R_{0})>0$ and $C>0$ such that for every $u_{0}$ and $u_{1}$ in $H^{2}(M)$ with
	\begin{equation}
		\|u_{0}\|_{H^{2}(M)}\leq R_{0}~~ \textit{and}~~\|u_{1}\|_{H^{2}(M)}\leq R_{0}
	\end{equation}
	there exists a control $h\in L^{2}([0,T]; H^{2}(M))$ with $\|h\|_{L^{2}([0,T]; H^{2}(M))}\leq C$ supported in $[0,T]\times\bar{\omega}$ such that the unique solution $u$ in $E_{T}$ of the Cauchy problem
	\begin{equation*}
		\begin{cases}
			i\partial_{t}u+\Delta_{g}^{2}u-\beta\Delta_{g}u+|u|^{2k}u=1_{\omega}h(t,x),&(t,x)\in[0,T]\times M\\ u(0,x)=u_{0}(x)\in H^{2}(M)
		\end{cases}
	\end{equation*}
	satisfies $u(T)=u_{1}$.

	Moreover, when $d=5$, the controllability result also holds for  the Cauchy problem
	\begin{equation*}
		\begin{cases}
			i\partial_{t}u+\Delta_{g}^{2}u-\beta\Delta_{g}u+|u|^{2}u=1_{\omega}h(t,x),&(t,x)\in[0,T]\times M\\ u(0,x)=u_{0}(x)\in H^{2}(M)
		\end{cases}
	\end{equation*}
	with initial data
	\begin{equation}
		\|u_{0}\|_{H^{2}(\mathbb{S}^{5})}\leq R_{0}~~ \textit{and}~~\|u_{1}\|_{H^{2}(\mathbb{S}^{5})}\leq R_{0}
	\end{equation}
	such that the unique solution $u\in X^{2,b}_{T}$
	satisfies $u(T)=u_{1}$.

In this paper, we study the global stabilization and exact controllability of the fourth-order nonlinear Schr\"odinger equation
\begin{equation}\label{4NLS}
	\begin{cases}
		i\partial_tu+(\Delta_g^2-\beta\Delta_g)u=-|u|^{\alpha-1}u,&(t,x)\in\R\times M,\\
		u(0,x)=u_0(x),& ~~~~~~x\in M,
	\end{cases}
\end{equation}
posed on a smooth compact Riemannian manifold without boundary.
where $u:I\times M\to\mathbb{C}$ is the unknown function  with $1\leq d\leq5$. Here $\Delta_{g}$ denotes the Laplace-Beltrami operator on $M$ associate with the Riemannian metric $g$.  We write $\Delta_{g}$ as follows:
\begin{equation}
	\Delta_{g}=\frac{1}{\sqrt{|g|}}\sum_{i,j=1}^{d}\frac{\partial}{\partial x_{i}}\Big(g^{ij}\sqrt{|g|}\frac{\partial}{\partial x_{j}}\Big)
\end{equation}
The  equation \eqref{4NLS} possesses a Hamiltonian structure and preserves both the mass and the energy
\begin{equation}
	\operatorname{Mass}: M(u)(t)=\int_{M}|u|^{2}\,\dd x
\end{equation}
\begin{equation}
	\operatorname{Energy}: E(u)(t)=\int_{M}|\Delta_{g}u|^{2}+\beta|\nabla_{g} u|_{g}^{2}+\frac{1}{2(\alpha+1)}|u|^{\alpha+1}\,\dd x.
\end{equation}

In the past few decades, there has been a surge of interest in applying the fourth order Schr\"odinger equation to characterize physical phenomena and has been extensively studied in the literature.  It is known that fourth order Schr\"odinger equations  are not merely formal generalizations of second order counterparts, but also arise naturally from intrinsic physical mechanisms, {for example, relativistic quantum corrections, dipolar interactions in Bose-Einstein condensations, and high-order dispersion in nonlinear optics, }and describe specific phenomena that second order equations cannot capture, for instance, {asymmetric spectral broadening of ultrashort pulses, enhanced quantum fluctuations in dipolar condensates}. On the other hand, many physicists and
mathematicians are devoted to studying the Gross-Pitaevskii equation which the confining potential is added to the fractional Laplacian operator. The confining effect
brought by the potential is close to the localization effect on the compact manifold.
The fractional quantum mechanics in the spatial-inhomogeneous media motivates us
to consider defocusing fourth order nonlinear Schr?dinger equation(FNLS) on compact
manifold $M$.

In this article, we focus on the controllability and stabilization associated with the Cauchy problem  \eqref{4NLS}.
\begin{itemize}
	\item[$\bullet$]The controllability problem asks whether one can steer the system to a prescribed terminal state $y(T)=y_{1}$ by means of an appropriate control input $h$ belonging to a control space $U_{T}$. Exactly speaking, suppose that $H$ is a Hilbert space, and linear abstract Cauchy problem
	\begin{equation}\label{Abstract Cauchy problem}
		\begin{cases}
			\frac{\dd}{\dd t}y=Ay+Bh, &t\in[0,T]\\ y(0)=y_{0}\in H
		\end{cases}
	\end{equation}
	is wellposed on $H$.
	Given a goal state $y_{1}\in H$, can we find a control function $h\in U_{T}$, where $U_{T}$ is another Hilbert space, called control space, such that the solution of Cauchy problem\eqref{Abstract Cauchy problem} attains the goal $y(T)=y_{1}$. A powerful approach to this problem is the Hilbert Uniqueness Method (HUM) introduced by J. Lions, which converts controllability of \eqref{Abstract Cauchy problem} into an observability inequality for the adjoint system.
	This method has become a central tool in control theory for dispersive equations.
	In the context of nonlinear Schr?dinger-type equations, the interaction between high-order dispersion, geometry of the underlying manifold, and nonlinear effects presents substantial analytical challenges.

	\item[$\bullet$]The stabilization problem, tracing back to Lyapunov's fundamental work, may be viewed as a particular manifestation of controllability.
	Its objective is to design a feedback mechanism ensuring that the trajectory of a dynamical system, typically an evolutionary PDE converges asymptotically to a designated equilibrium, most often the zero solution. That is to say, we search a control function $h$ such that solution $y(t)\to 0$ in $H$ as $t\to\infty$. A classical example arises in the study of the wave equation, where internal damping of the form $u_{t}$ is employed as a feedback term.
	Under appropriate geometric conditions, this yields a decay of the associated energy, as established in the seminal work of Burq and G\'erard \cite{Burqdamping}.
	Such results illustrate the general mechanism by which suitable dissipation induces stabilization, a principle that extends to a broad class of dispersive and hyperbolic equations.
\end{itemize}

It is well-known that  establishing the well-posedness constitutes the fundamental step to the control and stabilization,
\begin{equation}
	\begin{cases}
		i\partial_{t}u+(\Delta_{g}^{2}-\beta\Delta_{g})u=-|u|^{\alpha-1}u+1_{\omega}h(t,x),&(t,x)\in\mathbb{R}\times M,\\ u(0,x)=u_{0}(x),& x\in M
	\end{cases}
\end{equation}

In recent years, dispersive equations on non-flat geometries have attracted substantial attention, accompanied by the development of a range of sophisticated analytical tools. The key contribution in the area comes from Kapitanski's seminal work, where the WKB construction was initially proposed to establish Strichartz estimates for the wave equation. In contrast, establishing Strichartz estimates for the linear Schr\"odinger equation in such settings proves considerably more delicate.
A central difficulty is the inevitable loss of derivatives which is influenced by  the curvature of manifold. This novel and nontrivial behavior was identified by Burq, G\'erard, and Tzvetkov\cite{Burq-AJM} through the framework of semiclassical analysis. Their work revealed the intrinsic geometric sensitivity of Schr\"odinger dynamics and provided the foundational Strichartz estimates with derivative loss, laying the groundwork for subsequent investigations into this sensitivity for the linear Schr\"odinger equation. Specifically, they established the following inequality:
\begin{equation}\label{strichartz on manifolds}
	\|e^{it\Delta_{g}}f\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\leq C(I)\|f\|_{H^{\frac{1}{p}}},~\frac{2}{p}=d\big(\frac{1}{2}-\frac{1}{q}\big),~p\geq2,~q<\infty
\end{equation}

As an application of Strichartz estimates, Burq-G\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\"odinger equation with initial data $u_0\in H^{s}$, with $s>\frac{d-1}{2}$. Later, Dinh \cite{Dinh} generalized  \eqref{strichartz on manifolds} to fractional case:
\begin{equation}
	\|e^{it(-\Delta_{g})^{\frac{\sigma}{2}}}f\|_{L^{p}_{t}L^{q}_{x}(I\times M)}\leq C(I)\|f\|_{H^{\frac{1}{p}}},~ \frac{2}{p}\leq d\big(\frac{1}{2}-\frac{1}{q}\big),~ p\geq2, ~q<\infty.
\end{equation}
Then, he obtained the well-posedness of cubic fractional Schr\"odinger equation for $s>\frac{d-1}{2}$.

Building on the above well-posedness results established in \cite{Burq-AJM}, Dehman-G\'erard-Lebeau \cite{Dehman} were the first to obtain exact controllability and stabilization results for the Schr\"odinger equation on compact manifolds without boundary.Their analysis proceeds by reducing the local exact controllability problem to observable inequality and stabilization problem to the energy decay estimate for damped Schr\"odinger equation. Combining these ingredients, they derived global exact controllability. Later, Laurent utilized the multilinear Strichartz estimate established in \cite{Burq-Multilinear} to prove stabilization for cubic NLS on several three-dimensional manifolds including $\Bbb T^3$, $\Bbb S^3$ and $\Bbb S^2\times\Bbb S^1$. Very recently, applying the partial Floquet transformation, Niu-Zhao \cite{Niu-Zhao} proved the controllability for cubic NLS on semi-periodic space. Following the approach of \cite{Dehman}, R. Capistrano-Filho and A. Pampu\cite{Robertomathz} derived  the controllability and stabilization results by employing  Dinh's Strichartz estimates and strategies from \cite{Dehman}.  For fourth order case, R. Capistrano-Filho and M. Cavalcante \cite{Robertoamo} established the controllability and stabilization in $L^{2}(\mathbb{T})$ for equation
\begin{equation*}
	i\partial_{t}u+\partial_{x}^{2}u-\partial_{x}^{4}u=\lambda |u|^{2}u.
\end{equation*}
We also refer to \cite{Cavalcanti,Dehman-Lebeau-Zuazua,Laurent-JFA,Laurent-Joly} for the results on controllability of nonlinear wave/Klein-Gordon equation.
\subsection{Main results}
Our goal in this paper concerns controllability and stabilization problem for \eqref{4NLS}.  More precise,  the  exact controllability and stabilization   theorem.

In this article, we focus on the controllability and stabilization associated with the Cauchy problem \eqref{4NLS}.
\begin{itemize}
 \item[$\bullet$]The controllability problem asks whether one can steer the system to a prescribed terminal state $y(T)=y_{1}$ by means of an appropriate control input $h$ belonging to a control space $U_{T}$. Exactly speaking, suppose that $H$ is a Hilbert space, and linear abstract Cauchy problem
 \begin{equation}\label{Abstract Cauchy problem}
 \begin{cases}
 \frac{\dd}{\dd t}y=Ay+Bh, &t\in[0,T]\\ y(0)=y_{0}\in H
 \end{cases}
 \end{equation}
 is wellposed on $H$.
 Given a goal state $y_{1}\in H$, can we find a control function $h\in U_{T}$, where $U_{T}$ is another Hilbert space, called control space, such that the solution of Cauchy problem\eqref{Abstract Cauchy problem} attains the goal $y(T)=y_{1}$. A powerful approach to this problem is the Hilbert Uniqueness Method (HUM) introduced by J. Lions, which converts controllability of \eqref{Abstract Cauchy problem} into an observability inequality for the adjoint system.
 This method has become a central tool in control theory for dispersive equations.
 In the context of nonlinear Schr?dinger-type equations, the interaction between high-order dispersion, geometry of the underlying manifold, and nonlinear effects presents substantial analytical challenges.

It is well-known that establishing the well-posedness constitutes the fundamental step to the control and stabilization,
\begin{equation}
 \begin{cases}
 i\partial_{t}u+(\Delta_{g}^{2}-\beta\Delta_{g})u=-|u|^{\alpha-1}u+1_{\omega}h(t,x),&(t,x)\in\mathbb{R}\times M,\\ u(0,x)=u_{0}(x),& x\in M
 \end{cases}
\end{equation}

In recent years, dispersive equations on non-flat geometries have attracted substantial attention, accompanied by the development of a range of sophisticated analytical tools. The key contribution in the area comes from Kapitanski's seminal work, where the WKB construction was initially proposed to establish Strichartz estimates for the wave equation. In contrast, establishing Strichartz estimates for the linear Schr\"odinger equation in such settings proves considerably more delicate.
A central difficulty is the inevitable loss of derivatives which is influenced by the curvature of manifold. This novel and nontrivial behavior was identified by Burq, G\'erard, and Tzvetkov\cite{Burq-AJM} through the framework of semiclassical analysis. Their work revealed the intrinsic geometric sensitivity of Schr\"odinger dynamics and provided the foundational Strichartz estimates with derivative loss, laying the groundwork for subsequent investigations into this sensitivity for the linear Schr\"odinger equation. Specifically, they established the following inequality:
\begin{equation}\label{strichartz on manifolds}
 \|e^{it\Delta_{g}}f\|_{L_{t}^{p}(I,L_{x}^{q}(M))}\leq C(I)\|f\|_{H^{\frac{1}{p}}},~\frac{2}{p}=d\big(\frac{1}{2}-\frac{1}{q}\big),~p\geq2,~q<\infty
\end{equation}

As an application of Strichartz estimates, Burq-G\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\"odinger equation with initial data $u_0\in H^{s}$, with $s>\frac{d-1}{2}$. Later, Dinh \cite{Dinh} generalized \eqref{strichartz on manifolds} to fractional case:
\begin{equation}
 \|e^{it(-\Delta_{g})^{\frac{\sigma}{2}}}f\|_{L^{p}_{t}L^{q}_{x}(I\times M)}\leq C(I)\|f\|_{H^{\frac{1}{p}}},~ \frac{2}{p}\leq d\big(\frac{1}{2}-\frac{1}{q}\big),~ p\geq2, ~q<\infty.
\end{equation}
Then, he obtained the well-posedness of cubic fractional Schr\"odinger equation for $s>\frac{d-1}{2}$.

Building on the above well-posedness results established in \cite{Burq-AJM}, Dehman-G\'erard-Lebeau \cite{Dehman} were the first to obtain exact controllability and stabilization results for the Schr\"odinger equation on compact manifolds without boundary.Their analysis proceeds by reducing the local exact controllability problem to observable inequality and stabilization problem to the energy decay estimate for damped Schr\"odinger equation. Combining these ingredients, they derived global exact controllability. Later, Laurent utilized the multilinear Strichartz estimate established in \cite{Burq-Multilinear} to prove stabilization for cubic NLS on several three-dimensional manifolds including $\Bbb T^3$, $\Bbb S^3$ and $\Bbb S^2\times\Bbb S^1$. Very recently, applying the partial Floquet transformation, Niu-Zhao \cite{Niu-Zhao} proved the controllability for cubic NLS on semi-periodic space. Following the approach of \cite{Dehman}, R. Capistrano-Filho and A. Pampu\cite{Robertomathz} derived the controllability and stabilization results by employing Dinh's Strichartz estimates and strategies from \cite{Dehman}. For fourth order case, R. Capistrano-Filho and M. Cavalcante \cite{Robertoamo} established the controllability and stabilization in $L^{2}(\mathbb{T})$ for equation
\begin{equation*}
 i\partial_{t}u+\partial_{x}^{2}u-\partial_{x}^{4}u=\lambda |u|^{2}u.
\end{equation*}
We also refer to \cite{Cavalcanti,Dehman-Lebeau-Zuazua,Laurent-JFA,Laurent-Joly} for the results on controllability of nonlinear wave/Klein-Gordon equation.
\subsection{Main results}
Our goal in this paper concerns controllability and stabilization problem for \eqref{4NLS}. More precise, the exact controllability and stabilization theorem.

As an application of Strichartz estimates, Burq-G\'erard-Tzvetkov established the local well-posedness for the cubic nonlinear Schr\"odinger equation with initial data $u_0\in H^{s}$, with $s>\frac{d-1}{2}$. Later, Dinh \cite{Dinh} generalized \eqref{strichartz on manifolds} to fractional case:
\begin{equation}
 \|e^{it(-\Delta_{g})^{\frac{\sigma}{2}}}f\|_{L^{p}_{t}L^{q}_{x}(I\times M)}\leq C(I)\|f\|_{H^{\frac{1}{p}}},~ \frac{2}{p}\leq d\big(\frac{1}{2}-\frac{1}{q}\big),~ p\geq2, ~q<\infty.
\end{equation}
Then, he obtained the well-posedness of cubic fractional Schr\"odinger equation for $s>\frac{d-1}{2}$.

Now, we prove an observability inequality uniform with respect to the potential for solutions at high frequency.
 \begin{proposition}
 Let $T>0$, $M>0$, $d\in\mathbb{N}$ and $s>\frac{d}{2}$ be fixed. Let $\omega$ be an open set satisfying the GCC and let $\mathcal{C}u=b_{\omega}u$. Then there exist $n_{0}\in\mathbb{N}$ and $C>0$ such that for any $V_{1}$, $V_{2}\in \mathcal{B}_{M}^{[0,T]}(H^{s}(M))$ and $n\geq n_0$, the following observability inequality holds
 \begin{equation}
 \|w_{0}\|_{H^{s}}^{2}\leq C\int_{0}^{T}\|\mathcal{C}w(t)\|_{H^{s}}^{2}\,\dd t
 \end{equation}
 for any $w_0\in\mathcal{Q}_{n}H^{s}(M)$, where $w\in C^{0}([0,T],\mathcal{Q}_{n}H^{s}(M))$ is solution to
 \begin{equation}\label{fourth order with potential}
 \begin{cases}
 i\partial_{t}w+\Delta_{g}^{2}w-\beta\Delta_{g}w=\mathcal{Q}_{n}(V_{1}w+V_{2}\bar{w}),&(t,x)\in (0,T)\times M,\\ w(0)=w_{0},& x\in M.
 \end{cases}
 \end{equation}
 \end{proposition}
 \begin{proof}
 We proceed by contradiction. Suppose, for contradiction, there exist a sequence $\{n_{j}\}_{j\in\mathbb{N}}$ with $n_{j}\to\infty$ as $j\to\infty$, a sequence of potentials $\{
 (V_{1,j},V_{2,j})\}_{j\in\mathbb{N}}$ belonging to $\mathcal{B}_{M}^{[0,T]}(H^{s}(M))$ and a sequence of solutions $(w_{j})_{j}$ to \eqref{fourth order with potential} in $C([0,T],\mathcal{Q}_{n_{j}}H^{s}(M))$ associated with $(V_{1,j})$ and $(V_{2,j})$ with $\|w_{0,j}\|_{H^{s}}=1$ such that
 \begin{equation}
 \int_{0}^{T}\|\mathcal{C}w_{j}(t)\|^{2}_{H^{s}(M)}\,\dd t\to0,~~j\to\infty.
 \end{equation}
 The uniform bounds on $(w_{0,j})$, $(V_{1,j})$ and $(V_{2,j})$ implies that $w_{j}$ is a uniformly bounded sequence in $C([0,T],H^{s}(M))$. Moreover, since $w_{j}\in \mathcal{Q}_{n_{j}}H^s(M)$ for almost every $t\in[0,T]$, by spectral decomposition, we have \begin{equation}
 \|w_{j}\|_{L^{\infty}([0,T],H^{s-1}(M))}\leq \lambda_{n_{j}}^{-1/2}\|w_{j}\|_{L^{\infty}([0,T],H^{s}(M))}
 \end{equation}
 and thus infer $w_{j}\to0$ in $L^{\infty}([0,T],H^{s-1}(M))$ as $j\to\infty$.

\begin{proof}
 If $u$ is a solution of \eqref{fourth order schrodinger with general f} which belongs to $X^{2,b}_{T}$ with $b>\frac{1}{2}$, due to elliptic regularity arguments, $u$ is smooth in $(0,T)\times\omega$ and then we use the propagation of regularity theorem \ref{propagation nonlinear} to obtain that $u$ belongs to $X_{T}^{1+\nu,b}$ for $\nu\in(1/2,1]$. According to Sobolev embedding, it also belongs to $C^{0}([0,T],H^{1+\nu}(M))$ and hence $t\in(0,T)\mapsto u(t,\cdot)\in H^{1+\nu}(M)$ is analytic by Theorem \ref{analytic propogation theorem}. Let $z=\partial_{t}u$, we can apply Theorem \ref{Tataru-Robbiano-Zuily-Hormander} to obtain $\partial_{t}u=0$ in $(0,T)\times M$. Multiplying $\bar{u}$ and integral by parts gives that $u\equiv0$.
 \end{proof}
 \subsubsection{Propogation of regularity in Bourgain space}
 \begin{proposition}\label{propagation of regularity in Bourgain space}
 Let $T>0$ and $0\leq b<1$, and suppose that $u\in X^{s,b}_{T}$, $s\in\mathbb{R}$, be a solution of
 \begin{equation*}
 i\partial_{t}u+\Delta^{2}_{g}u-\Delta_{g}u=f\in X^{s,-b}_{T}
 \end{equation*}
 Given $\rho_{0}=(x_{0},\xi_{0})\in T^{\star}M\backslash\{0\}$, we assume that there exists a zeroth pseudo-differential operator $\chi(x,D_{x})$, elliptic in $\rho_{0}$ such that
 \begin{equation*}
 \chi(x,D_{x})u\in L^{2}_{\rm loc}((0,T), H^{s+\nu})
 \end{equation*}
 for some $\nu\leq\frac{1-b}{2}$. Then, for every $\rho_{1}\in\Phi_{t}(\rho_{0})$, the geodesic ray starting at $\rho_{0}$, there exists a pseudo-differential operator $\psi(x,D_{x})$ elliptic at $\rho_{1}$ such that
 \begin{equation}
 \psi(x,D_{x})u\in L^{2}_{\rm loc}((0,T), H^{s+\nu}).
 \end{equation}
 Moreover, if an open set $\omega$ satisfies GCC condition and $a(x)u\in L^{2}_{\rm loc}((0,T),H^{s+\nu})$ with $a(x)\in C^{\infty}(M)$, $\operatorname{supp} a(x)\subset\omega$ also satisfies GCC, then $u\in L^{2}_{\rm loc}((0,T),H^{s+\nu})$.
 \end{proposition}
 \begin{proof}
 Firstly, we regularize $u_{n}=e^{\frac{1}{n}\Delta_{g}}u$ with $\|u_{n}\|_{X^{s,b}_{T}}\leq C$. Let $r=s+\nu$ and $B(x,D_{x})$ be a pseudo-differential operator of order $2r-1=2s+2\nu-1$ that will be chosen later. Let $A=A(t,x,D_{x})=\varphi(t)B(x,D_{x})$, where $\varphi\in C_{0}^{\infty}((0,T))$.

We prove a cutoff version of \eqref{obervability inequality} $\forall u_{0}\in L^{2}(M)$ and $0<h<h_{0}$
 \begin{equation}\label{cutoff observability inequality}
 \|\Pi_{h}u_{0}\|^{2}_{L^{2}(M)}\leq C\int_{0}^{T}\int_{\omega}|e^{it(\Delta^{2}_{g}-\beta\Delta_{g})}\Pi_{h}u_0|^2\,\dd x\dd t.
 \end{equation}
 If \eqref{cutoff observability inequality} is not right, there exist a sequence $h_{j}\to0$ and functions $u^{0}_{h_{j}}$ such that
 \begin{equation}
 \|\Pi_{h_{j}}u_{h_{j}}^{0}\|_{L^{2}(M)}=1, ~~~\lim_{j\to\infty}\int_{0}^{T}\int_{\omega}|e^{it(\Delta_{g}^{2}-\Delta_{g})}\Pi_{h_{j}}u^{0}_{h_j}|^2\,\dd x\dd t=0,
 \end{equation}
 the first equality means that $W^{h_{j}}_{u_{h_{j}}}$ converges to a finite measure in $T^{\star}M$, and the second inequality means that $W^{h_{j}}_{u_{h_{j}}}$ converges to zero measure in $T^{\star}\omega\setminus0$.
 Thus, we have
 \begin{equation}\label{contradiction part}
 \int_{0}^{T}\mu_{t}(T^{\star}M\setminus0)\,\dd t=T,~~\int_{0}^{T}\int_{T^{\star}M}b(x)\mu_{t}(\dd x,\dd\xi)\,\dd t=0,~~\forall b\in C^{\infty}_{c}(\omega)
 \end{equation}
 Denote $$F_{\omega}^{T}:=\bigcup_{t\in[0,T]}\{\phi_{t}(x,\xi):(x,\xi)\in T^{\star}\omega\setminus0\}.$$ Since $\omega$ satisfies Geometric Control Condition, there exsits $T_{0}$ such that $F_{\omega}^{T_{0}}=T^{\star}M\setminus0$. Combining the fact that $\mu_{t}$ is invariant by the Hamiltonian flow of $\sigma_{4}(|\xi|_{g}^{2})$, we infer that $\mu_t(T^{\star}M\setminus0)=0$, which is a contradition to the first part of \eqref{contradiction part}.
 \section{Local Controllability}
 Before the proof of nonlinear controllability, we briefly recall the HUM method for linear equation
 \begin{equation}
 \begin{cases}
 i\partial_{t}u+(\Delta_{g}^{2}-\beta\Delta_{g})u=-|u|^{2k}u+h(x,t),\\ u(0,x)=u_{0}(x).
 \end{cases}
 \end{equation}
 When observability inequality is proved, HUM method gives the exact controllability of linear equation
 \begin{equation}\label{linear equation}
 \begin{cases}
 i\partial_{t}u+(\Delta_{g}^{2}-\beta\Delta_{g})u=0,\\ u(0,x)=u_{0}(x).
 \end{cases}
 \end{equation}
 \subsection{Local controllability in energy space}
 Since $\omega$ satisfies the Geometric Control Condition, we can find $\omega'$ satisfying GCC and $\omega'\Subset \omega$, we then introduce $\varphi\in C^{\infty}(M)$ real-valued supported in $\omega$ such that $\varphi=1$ on $\omega'$.
 Consider the systems
 \begin{equation}\label{HUM dual double equation}
 \begin{cases}
 i\partial_{t}u+\Delta^2_g u-\beta\Delta_{g}u=h\in L^{2}(0,T;H^2), u(T)=0,\\ i\partial_{t}v+\Delta^{2}_{g}v-\beta\Delta_{g}v=0, v(0)=v_{0}\in H^{-2}(M),
 \end{cases}
 \end{equation}

Taking control function $h=a(x)(1-\Delta_{g})^{-2}\big(a(x)u_{t}\big)$, the stabilization system is
\begin{equation}\label{damped}
	\begin{cases}
		i\partial_{t}u+\Delta^{2}_{g}u-\beta\Delta_{g}u+|u|^{2k}u+u=a(x)(1-\Delta_{g})^{-2}\big(a(x)u_{t}\big),& (t,x)\in [0,T]\times M,\\ u(0,x)=u_{0}\in H^{2}(M)
	\end{cases}
\end{equation}
By  the direct computation, one can observe that the damped equation does not conserve its energy:
\begin{equation*}
	E(u)(t)-E(u)(0)=-\int_{0}^{t}\|(1-\Delta_{g})^{-1}\big(a(x)\partial_{t}u\big)\|_{L^{2}(M)}^{2}\,\dd t.
\end{equation*}The presence of damping term helps us obtain the exponential decay of the energy, which is stated as follows.

\begin{theorem}
	Let $(M,\omega)$ satisfy GCC and $a\in C^{\infty}(M)$, then for every $R_{0}>0$, there exist $C(R_{0})$ and $\gamma(R_{0})$ such that
	\begin{equation}
		\|u(t)\|_{H^{2}(M)}\leq Ce^{-\gamma t}\|u_{0}\|_{H^{2}(M)}, ~t>0,
	\end{equation}
	holds for every solution $u$ of \eqref{damped} with initial data $u_{0}\in H^{2}(M)$ obeying $\|u_{0}\|_{H^{2}(M)}\leq R_{0}$.
\end{theorem}
Recall that in Dehman-G\'erard-Lebeau \cite{Dehman} and Capistrano-Filho-Pampu \cite{Robertomathz}, they studied the control and stabilization issues under the following two assumptions:
\begin{itemize}
	\item Assumption 1. Geometric control condition(GCC): There exists $T_{0}>0$ such that every geodesic of $M$, travelling with speed $1$ and issued at $t=0$, enters the set $\omega$ in a time $t<T_{0}$.
	\item  Assumption 2. Unique continuation principle(UCP): For every $T>0$, the only solution $u$ in $C^{\infty}([0,T]\times M)$ to the system
	\begin{equation}
		\begin{cases}
			i\partial_{t}u+\Delta^{2}_{g}u-\Delta_{g}u+b_{1}(t,x)u+b_{2}(t,x)\overline{u}=0, &(t,x)\in [0,T]\times M,\\ u=0,&(t,x)\in [0,T]\times\omega,
		\end{cases}
	\end{equation}
	where $b_{1}(t,x)$ and $b_{2}(t,x)$ are both in $C^{\infty}([0,T]\times M)$, is the trivial one $u\equiv0$.
\end{itemize}
The geometric control condition (GCC) is a classical sufficient condition for linear controllability, which was introduced in Bardos-Lebeau-Rauch \cite{Bardos}.
It is noteworthy, however, that GCC is not necessary in certain geometries, e.g., on the flat three dimensional torus $\mathbb{T}^{3}$, Jaffard \cite{Jaffard} proved that any open subset can be the control domain. For more information about this, we refer to the  monograph \cite{Komornik} and \cite{Tucsnack} for a comprehensive account of controllability in periodic domains and related geometric phenomena. In contrast, many works rely on the unique continuation property (UCP), which often serves as a technical cornerstone in the analysis of Schr\"odinger-type equations.
For the second-order Schr\"odinger equation, UCP can be derived via Carleman estimates, as shown in Laurent's appendix \cite{Laurent}.
However, extending such arguments to the fourth-order case is substantially more delicate.
The primary difficulty stems from the intricate structure of the biharmonic operator: repeated integrations by parts introduce higher-order  mixed derivatives that are significantly more challenging to control. Consequently, establishing UCP for the fourth-order Schr\"odinger equation requires alternative microlocal methods or refined semiclassical arguments, rather than a straightforward generalization of the second-order case. Fortunately, Laurent-Loyola \cite{LaurentLoyola} and Loyola \cite{Loyola2025} found that the series of unique continuation results for a broad class of Schr\"odinger operators with partially analytic coefficients proved by Tataru \cite{Tataru1999jmpa}, Robbiano-Zuily and H\"ormander will help us to deduce the unique continuation for the linearized equation, which avoids the use of Carleman estimate, and provide a new perspective in proving this property.
\subsection{Outline of the proof}
We adopt the framework of global stabilization combined with local controllability  to establish the main results of this paper, a strategy previously employed in \cite{Dehman}.

\textbf{Step 1: Establish Strichartz estimates and well-posedness}. Using the Kato-Rellich theorem, Helffer-Sj\"ostrand formula, and the analytical arguments in \cite{Burq-AJM}, we derive the Strichartz estimates for fourth-order Schr\"odinger operator $L$ on manifolds with loss of regularity. From these estimates, we further deduce the well-posedness of system in energy space $H^{2}(M)$ for $d\leq4$. For $d=5$ (with $M=\mathbb{S}^{5}$), we follow an analogous strategy from \cite{Burq-Multilinear} to prove multilinear Strichartz estimates in the Bourgain space, thereby establishing the well-posedness of the system in this space.

\textbf{Step 2: Prove observability}. We establish the observability inequality for fourth-order Schr\"odinger operator $i\partial_{t}+\Delta^{2}_{g}-\beta\Delta_{g}$. Notably, this inequality is equivalent to linear controllability of the system via the Hilbert Uniqueness Method(HUM), a classic tool introduced by J.-L. Lions. To verify the observability inequality, we use techniques from semiclassical analysis: specifically, semiclassical microlocal defect measures and the Wigner distribution.

\textbf{Step 3: Obtain local controllability of the nonlinear system}. Using the Banach fixed point theorem, we prove the local controllability of nonlinear equation. The analysis is split by dimension:
\begin{itemize}
	\item For $d\leq4$, we reformulate the nonlinear controllability problem as solving the of nonlinear HUM operator, where all the estimates are conducted in the energy space $H^{2}(M)$.
	\item  For $d=5$, the core process is similar to $d\leq4$. The key distinction lies in replacing all estimates in the energy space with those in the Bourgain space.
\end{itemize}
\textbf{Step 4: Prove stabilization via contradiction argument}. We establish the stabilization result, which reduces to show that the following energy decay inequality $$E(0)\leq C\int_{0}^{T}\|(1-\Delta_{g})^{-1}(a(x)\partial_{t}u)\|^{2}_{L^{2}}\,\dd t.$$  The proof relies on a contradiction argument.
\begin{itemize}
	\item In the case $d\leq4$, we establish the propagation of compactness property in energy space $H^{2}(M)$.
	\item  In the case $d=5$, we extend this analysis to prove the propagation of compactness in the Bourgain space.
\end{itemize}
Subsequently, the entire proof reduces to addressing the unique continuation problem. For this purpose, we employ the methods developed in \cite{LaurentLoyola} and \cite{Loyola2025} and refer to Section $3$ for more details.

\subsection{Structure of the article}
The paper is organized as follows. In Section 2, we collect some basic properties of harmonic analysis tools and functional spaces on $M$. We also prove some Strichartz estimate in various settings and show the local well-posedness for NLS with forced term and damping term. In Section 3, we introduce the abstract analytical framework, which will help us to transfer the linearized equation to linear Schr\"odinger equation with potentials under  partial analytical coefficients. In Section 4, we prove the propagation of singularities, including the regularity and compactness in Sobolev and Bourgain spaces. In Section 5 and 6, we prove the exact controllability  and stabilization respectively.

We “adopt the framework of global stabilization combined with local controllability to establish the main results of this paper.” The outline given is:

\textbf{Step 1: Establish Strichartz estimates and well-posedness}. “Using the Kato-Rellich theorem, Helffer-Sj\"ostrand formula, and the analytical arguments in \cite{Burq-AJM}, we derive the Strichartz estimates for fourth-order Schr\"odinger operator $L$ on manifolds with loss of regularity.” From these, “we further deduce the well-posedness of system in energy space $H^{2}(M)$ for $d\leq4$.” For $d=5$ with $M=\mathbb{S}^{5}$, they “prove multilinear Strichartz estimates in the Bourgain space,” yielding well-posedness there.

\textbf{Step 2: Prove observability}. They “establish the observability inequality for fourth-order Schr\"odinger operator $i\partial_{t}+\Delta^{2}_{g}-\beta\Delta_{g}$,” noting it is “equivalent to linear controllability … via the Hilbert Uniqueness Method(HUM).” The observability proof uses “semiclassical microlocal defect measures and the Wigner distribution.”

\textbf{Step 3: Obtain local controllability of the nonlinear system}. “Using the Banach fixed point theorem, we prove the local controllability of nonlinear equation,” by casting it as “solving … nonlinear HUM operator,” with estimates in $H^{2}(M)$ for $d\le4$, and “replacing all estimates in the energy space with those in the Bourgain space” for $d=5$.

\textbf{Step 4: Prove stabilization via contradiction argument}. They “establish the stabilization result,” reducing it to the energy decay inequality
\[
E(0)\leq C\int_{0}^{T}\|(1-\Delta_{g})^{-1}(a(x)\partial_{t}u)\|^{2}_{L^{2}}\,\dd t.
\]
“The proof relies on a contradiction argument,” proving a “propagation of compactness” property (in $H^{2}(M)$ for $d\le4$; in Bourgain space for $d=5$). “Subsequently, the entire proof reduces to addressing the unique continuation problem,” handled using methods of \cite{LaurentLoyola} and \cite{Loyola2025}.

Question:
Let \((M,g)\) be the underlying compact Riemannian manifold of dimension \(d\), let \(\Delta_g\) be its Laplace--Beltrami operator, and let \(\omega\subset M\) be an open set. Assume \(\omega\) satisfies the geometric control condition (GCC), meaning that there exists \(T_0>0\) such that every nonzero Hamiltonian trajectory in \(T^*M\setminus 0\) meets \(T^*\omega\) within time \(T_0\). For the controlled fourth-order nonlinear Schr\"odinger equation
\[
\begin{cases}
 i\partial_t u+\Delta_g^2u-\beta\Delta_g u+|u|^{2k}u=1_{\omega}h(t,x), &(t,x)\in[0,T]\times M,\\
 u(0,x)=u_0(x),
\end{cases}
\]
which uniform exact controllability statement holds for \(H^2\)-data bounded by a prescribed radius \(R_0>0\), and what is the corresponding statement in the special five-dimensional cubic case?

If \(d\le 4\), then for every open set \(\omega\subset M\) satisfying GCC and every \(R_0>0\), there exist \(T=T(R_0)>0\) and \(C>0\) such that for every \(u_0,u_1\in H^2(M)\) with \(\|u_0\|_{H^2(M)}\le R_0\) and \(\|u_1\|_{H^2(M)}\le R_0\), there exists a control \(h\in L^2([0,T];H^2(M))\) with \(\|h\|_{L^2([0,T];H^2(M))}\le C\) and \(\operatorname{supp} h\subset [0,T]\times\overline{\omega}\) such that the uniquely defined solution of
\[
\begin{cases}
 i\partial_t u+\Delta_g^2u-\beta\Delta_g u+|u|^{2k}u=1_{\omega}h(t,x),\\
 u(0,x)=u_0,
\end{cases}
\]
satisfies \(u(T)=u_1\). Moreover, when \(d=5\), the same controllability conclusion holds for the cubic equation
\[
\begin{cases}
 i\partial_t u+\Delta_g^2u-\beta\Delta_g u+|u|^2u=1_{\omega}h(t,x),\\
 u(0,x)=u_0,
\end{cases}
\]
in the case \(M=\mathbb S^5\): for every \(R_0>0\) and every \(u_0,u_1\in H^2(\mathbb S^5)\) with \(\|u_0\|_{H^2(\mathbb S^5)}\le R_0\) and \(\|u_1\|_{H^2(\mathbb S^5)}\le R_0\), there exist such \(T\), \(C\), and a control \(h\in L^2([0,T];H^2(\mathbb S^5))\), supported in \([0,T]\times\overline{\omega}\), for which the unique solution reaches \(u(T)=u_1\).

Choices:
B. If \(d\le 4\), then for every open set \(\omega\subset M\) satisfying GCC and every \(R_0>0\), there exist \(T=T(R_0)>0\) and \(C>0\) such that for every \(u_0,u_1\in H^2(M)\) with \(\|u_0\|_{H^2(M)}\le R_0\) and \(\|u_1\|_{H^2(M)}\le R_0\), there exists a control \(h\in L^2([0,T];H^2(M))\) with \(\|h\|_{L^2([0,T];H^2(M))}\le C\) and \(\operatorname{supp} h\subset [0,T]\times\overline{\omega}\) such that the uniquely defined solution reaches \(u(T)=u_1\). Moreover, when \(d=5\), the same controllability conclusion holds for the cubic equation on an arbitrary compact five-dimensional manifold \(M\) satisfying GCC, with every \(u_0,u_1\in H^2(M)\) of norm at most \(R_0\).
C. If \(d\le 4\), then for every open set \(\omega\subset M\) satisfying GCC and every \(R_0>0\), there exists \(T=T(R_0)>0\) such that for every \(u_0,u_1\in H^2(M)\) with \(\|u_0\|_{H^2(M)}\le R_0\) and \(\|u_1\|_{H^2(M)}\le R_0\), one can find a control \(h\in L^2([0,T];H^2(M))\) supported in \([0,T]\times\overline{\omega}\) for which the unique solution satisfies \(u(T)=u_1\). Moreover, in the special case \(d=5\), the analogous conclusion holds for the cubic equation on \(\mathbb S^5\) for \(H^2\)-data bounded by \(R_0\).
D. If \(d\le 4\), then for every open set \(\omega\subset M\) satisfying GCC there exists a time \(T_*>0\), depending only on \((M,g)\) and \(\omega\) but independent of \(R_0\), and a constant \(C>0\) such that for every \(R_0>0\) and every \(u_0,u_1\in H^2(M)\) with \(\|u_0\|_{H^2(M)}\le R_0\) and \(\|u_1\|_{H^2(M)}\le R_0\), there exists a control \(h\in L^2([0,T_*];H^2(M))\) with \(\|h\|_{L^2([0,T_*];H^2(M))}\le C\) and \(\operatorname{supp} h\subset [0,T_*]\times\overline{\omega}\) such that the uniquely defined solution satisfies \(u(T_*)=u_1\). Moreover, when \(d=5\), the same statement holds for the cubic equation on \(\mathbb S^5\).
E.
If \(d\le 5\), then for every open set \(\omega\subset M\) satisfying GCC and every \(R_0>0\), there exist \(T=T(R_0)>0\) and \(C>0\) such that for every \(u_0,u_1\in H^2(M)\) with \(\|u_0\|_{H^2(M)}\le R_0\) and \(\|u_1\|_{H^2(M)}\le R_0\), there exists a control \(h\in L^2([0,T];H^2(M))\) with \(\|h\|_{L^2([0,T];H^2(M))}\le C\) and \(\operatorname{supp} h\subset [0,T]\times\overline{\omega}\) such that the uniquely defined solution of
\[
\begin{cases}
 i\partial_t u+\Delta_g^2u-\beta\Delta_g u+|u|^{2k}u=1_{\omega}h(t,x),\\
 u(0,x)=u_0,
\end{cases}
\]
satisfies \(u(T)=u_1\); in particular, the five-dimensional case requires no restriction to the cubic nonlinearity and no assumption \(M=\mathbb S^5\).

special d=5 sphere/Bourgain-space restriction

uniform control-norm bound \(\|h\|_{L^2H^2}\le C\)

dependence of controllability time on radius \(R_0\)

dimension/nonlinearity range in five dimensions

The stem does not state the exact theorem, but it strongly cues the shape of the correct answer by mentioning a 'uniform exact controllability statement' and a 'special five-dimensional cubic case,' which narrows the field substantially.

This is largely a theorem-recall item: the correct option is essentially the full formal statement of the result. However, it is not a pure tautology because the alternatives vary in scope, quantifiers, and the 5D exception.

Some reasoning is needed to distinguish the precise strongest valid conclusion from overstatements and a weaker true variant, but the task is mainly careful recognition/recall rather than genuine mathematical generation.

The distractors are plausible and mathematically targeted: they test overgeneralization to all 5D manifolds, omission of uniform control bounds, incorrect dependence of control time on the radius, and an invalid extension of the dimension/nonlinearity range.

A solid but theorem-centric MCQ: good distractors and some precision-testing, but it leans more toward exact statement recall than deeper generative reasoning.


[Müller\&Vallarino \cite{MuVa}]\label{thm:MullerVallarino}
Let $S$ be a Damek–Ricci space of dimension $n$, and let $\mathcal{L}$ denote the left-invariant distinguished Laplacian and $\mathrm{d}\rho$ the right Haar measure on $S$. Then, for all $1<p<\infty$, there exists $C_p>0$ such that the solution to the Cauchy problem
\begin{align}\label{eq: wave}
\begin{cases}
\partial_t^2 u(t,x)-\mathcal{L} u(t,x)=0,
\qquad t\in\mathbb{R},\,x\in S,  \\
u(0,t)=f(x),\,\partial_t u(t,x)|_{t=0}=g(x),
\end{cases}
\end{align}
satisfies the estimate
\begin{align}\label{est:mainthm}
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\leq C_p\,
\Big( (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}}\!f\|_{L^p(S,\mathrm{d}\rho)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_1}{2}}\!g\|_{L^p(S,\mathrm{d}\rho)}\Big),
\end{align}
for all $t\in\mathbb{R}^*$, provided that
\begin{align*}
\alpha_0>(n-1)\left|{1\over p}-{1\over 2}\right|
\quad\text{and}\quad \alpha_1>(n-1)\left|{1\over p}-{1\over 2}\right|-1.
\end{align*}

\begin{theorem}[Müller\&Vallarino \cite{MuVa}]\label{thm:MullerVallarino}
Let $S$ be a Damek–Ricci space of dimension $n$, and let $\L$ denote the left-invariant distinguished Laplacian and $\d\rho$ the right Haar measure on $S$. Then, for all $1<p<\infty$, there exists $C_p>0$ such that the solution to the Cauchy problem
\begin{align}\label{eq: wave}
\begin{cases}
\partial_t^2 u(t,x)-\L u(t,x)=0,
\qquad t\in\mathbb{R},\,x\in S,  \\
u(0,t)=f(x),\,\partial_t u(t,x)|_{t=0}=g(x),
\end{cases}
\end{align}
satisfies the estimate
\begin{align}\label{est:mainthm}
\|u(t,\cdot)\|_{L^p(S,\d\rho)}
\leq C_p\,
\Big( (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\Id+\L)^{\frac{\alpha_0}{2}}\!f\|_{L^p(S,\d\rho)}+(1+|t|)\,\|(\Id+\L)^{\frac{\alpha_1}{2}}\!g\|_{L^p(S,\d\rho)}\Big),
\end{align}
for all $t\in\mathbb{R}^*$, provided that
\begin{align*}
\alpha_0>(n-1)\left|{1\over p}-{1\over 2}\right|
\quad\text{and}\quad \alpha_1>(n-1)\left|{1\over p}-{1\over 2}\right|-1.
\end{align*}
\end{theorem}

\begin{align}\label{e3.4}
k_\psi(x)=\delta(x)^{\frac12}\kappa_\psi(x)={2^{m_\fz-2}\Gamma({n\over 2})\over \uppi^{{n\over 2}+1}}\delta(x)^{\frac12}\int_0^\infty \psi(\lambda)\,\varphi_\lambda(R(x))\,|\bc(\lambda)|^{-2}\,\d \lambda.
\end{align}

\smallskip
\subsection{The local Hardy space}\label{h1}
We recall in the section the definition of the local atomic Hardy space $\mathfrak{h}^1(S)$, which can be viewed as the analogue, in the setting of Damek--Ricci spaces, of the local Hardy space introduced by Goldberg in the Euclidean case \cite{G}. The local Hardy space was further developed and studied by Meda and Volpi \cite{MV} and by Taylor \cite{T} in more general contexts. It is straightforward to verify that Damek--Ricci spaces satisfy the geometric assumptions of \cite{MV} and \cite{T}, so that the corresponding theory applies to our setting.

A function $a$ in $L^1(S)$ is called a \emph{standard $\fh^1$-atom} if $a$ is  supported in a ball $B$ of radius less than $1$, and satisfies
\begin{enumerate}[(i)]
\item size condition: $\|a\|_{L^2(S)}\leq \rho(B)^{-1/2}$;
\item cancellation condition: $\int a \,\d\rho=0$.
\end{enumerate}
A \emph{global $\fh^1$-atom} is a function $a$ in $L^1(S)$ supported in a ball $B$ of radius $1$ such that $\|a\|_{L^2(S)}\leq \rho(B)^{-1/2}$. Standard and global $\fh^1$-atoms will be referred to as \emph{admissible atoms}.
The Hardy space $\fh^1(S)$ is the space of functions $f$ in $L^1(S)$ such that $f=\sum_jc_ja_j$, where $\sum_j|c_j|<\infty$ and $a_j$ are admissible atoms. The norm $\|f\|_{\fh^1(S)}$ is defined as the infimum of $\sum_j|c_j|<\infty$ over all atomic decompositions of $f$. We now introduce several technical lemmas that will be useful in the subsequent analysis.

\begin{lemma}\label{lem2.7.0}
Let $T$ be a bounded linear operator on $L^2(S)$ with Schwartz kernel $K_T$, which satisfies the following two conditions:
\begin{asparaenum}[(i)]
\item The local H\"{o}rmonder condition:
\begin{align*}
\sup_B\sup_{y,y'\in B}\int_{(2B)^c}|K_T(x,y)-K_T(x,y')|\,\d \rho(x)<\infty,
\end{align*}
where the first supremum is taken over all balls $B$ of radii at most $1$;
\item The size condition:
\begin{align*}
\sup_{y\in S}\int_{(B(y,2))^c}|K_T(x,y)|\,\d\rho(x)<\infty.
\end{align*}
\end{asparaenum}
Then, the operator $T$ is bounded from $\fh^1(S)$ to $L^1(S)$.
\end{lemma}
\begin{proof}
The proof is a combination of \cite[Theorem 8.2]{CaMaMe09} and \cite[Proposition 4.5 (ii)]{CaMaMe10}.
\end{proof}
Recall that the symbol class is defined by
\begin{align*}
\cS^\sigma=\left\{m\in C^\infty(\R):\|m\|_{\cS^\sigma,k}:=\sup_{\lambda\in\R}\,(1+\lambda^2)^{-{\sigma-k\over 2}}|m^{(k)}(\lambda)|\leq C_{\alpha,k}\text{ for all }k\in \mathbb{N}\right\}.
\end{align*}
\begin{lemma}\label{lem2.6}
Suppose that $m\in\cS^\sigma$ is an even symbol. Then the following statements hold.
\begin{asparaenum}[(i)]
\item If $\sigma=0$, then the multiplier $m(\L)$ is bounded from $\fh^1(S)$ to $L^1(S)$, and is bounded on $L^p(S)$ for all $1<p<\infty$.
\item If $\sigma=-1$, then the operator $|\nabla m(\L)\cdot|_\g$, where $\nabla$ denotes the Riemannian gradient on $S$, is bounded from $\fh^1(S)$ to $L^1(S)$, and from $L^p(S)$ to $L^p(S)$ for all $1<p<\infty$.
\end{asparaenum}
\end{lemma}

\begin{proof}
We briefly outline the proof of this lemma. Following the argument of \cite[Lemma A.1]{WaYa}, we obtain that the Schwartz kernel $K$ of any operator in (i) or (ii) satisfies
\begin{align*}
|K(x,y)|+|(\nabla K(x,\cdot))(y)|_\g
\lesssim \delta(xy)^{1\over 2}(d(x,y))^{-N}
\quad\text{if }d(x,y)\geq 1,
\end{align*}
for any $N\in\mathbb{N}^*$, and
\begin{align*}
|(\nabla K(x,\cdot))(y)|_\g
\lesssim \delta(xy)^{1\over 2}(d(x,y))^{-n-1}
\quad\text{if } 0<d(x,y)\leq 1.
\end{align*}
Using Lemma \ref{lem2.2} and reasoning as in the proof of \cite[Lemma 2.11]{WaYa}, we verify that both the local Hörmander condition and the size condition are satisfied. Finally, a standard interpolation argument yields the $L^p$-boundedness.
\end{proof}

We conclude this section by recalling the following analytic interpolation property.

\begin{lemma}\label{lem2.7}
Denote by $\Sigma$ the closed strip $\{\zeta\in\mathbb{C}:\Re \zeta\in[0,1]\}$. Suppose that $\{T_\zeta\}_{\zeta\in\Sigma}$ is a family of uniformly bounded operators on $L^2(S)$ such that the map $\zeta \mapsto \int_G T_\zeta(f)\, g\,\d \rho$ is continuous on $\Sigma$ and analytic in the interior of $\Sigma$, whenever $f,g\in L^2(S)$. Moreover, assume that
\begin{align*}
\|T_\zeta f\|_{L^1(S)}\leq A_1\|f\|_{\fh^1(S)}\quad\text{for all }f\in L^2(S)\cap \fh^1(S) \text{ and }\Re\zeta=0,
\end{align*}
and
\begin{align*}
\|T_\zeta f\|_{L^2(S)}\leq A_2\|f\|_{L^2(S)}\quad\text{for all }f\in L^2(S)\text{ and }\Re\zeta=1,
\end{align*}
where $A_1$ and $A_2$ are positive constants independent of $\Im \zeta\in\R$. Then for every $\vartheta\in(0,1)$ the operator $T_\vartheta$ is bounded on $L^{q_\vartheta}$ with $1/q_\vartheta=1-\vartheta/2$ and
\begin{align*}
\|T_\vartheta f\|_{L^{q_\vartheta}(S)}\leq C\,A_1^{1-\vartheta} A_2^\vartheta\|f\|_{L^{q_\vartheta}(S)}\quad\text{for all }f\in L^2(S)\cap L^{q_\vartheta}(S).
\end{align*}
\end{lemma}
\begin{proof}
This follows from \cite[Theorem 5 (ii)]{MV} and \cite[Theorem 1]{CwJa}.
\end{proof}

\section{Proof of the main theorems}\label{sec3}
We prove our main theorems in this section, focusing on Theorem \ref{thm:main2}, since Theorem \ref{thm:main} follows from an interpolation argument. The proof of the necessity part of Theorem \ref{thm:main2} is a straightforward adaptation of the argument for the $ax+b$ group given in \cite{WaYa}, and will therefore be omitted. The sufficiency part of Theorem \ref{thm:main2} follows from the results below concerning the corresponding multipliers.

\begin{proposition}\label{prop: main}
Let $m$ be an even symbol. Then, for all $t\in\mathbb{R}^*$, the following results hold.
\begin{asparaenum}[\rm (a)]
\item If $m\in\cS^{-(n-1)/2}$, then
\begin{align*}
\|m(\sqrt{\L})\,\cos(t\sqrt{\L})\|_{\fh^1(S)\to L^1(S)}\lesssim 1+|t|.
\end{align*}

\item If $m\in\cS^{-(n-1)/2+1}$, then
\begin{align*}
\left\|m(\sqrt{\L})\,{\sin(t\sqrt{\L})\over\sqrt{\L}}\right\|_{\fh^1(S)\to L^1(S)}\lesssim 1+|t|.
\end{align*}
\end{asparaenum}
\end{proposition}

The proof of Proposition \ref{prop: main} relies on the idea, originating from the critical case on Euclidean space, of relating the wave propagator to the spherical measure. See, for instance, \cite{Pe,Ta}. In our setting, the spherical measure is closely related to the spherical function given by \eqref{e2.10}, which therefore requires a detailed analysis. The following expression will be useful later on.

\begin{lemma}
Let $f$ be a suitable function on $S$. Then, for all $t>0$ and $z\in S$, we have
\begin{align}
\varphi_{\!\sqrt{\L}}(t)(f)(z)
={1\over\nu_n}\int_{\SS^{n-1}}\delta(x(t,\omega))^{-{1\over 2}} f(z\cdot x(t,\omega))\,\d \omega,\label{e4.1}
\end{align}

\label{sec1}

The wave equation has long played a central role in analysis and partial differential equations on manifolds, where the non-flat geometry strongly influences wave propagation. In particular, on negatively curved manifolds, the long-time dispersive estimates for the wave equation differ significantly from the classical results in the Euclidean setting. This phenomenon has been observed in various contexts, for instance, in \cite{AP14,MT11,Tat01} on real hyperbolic spaces, in \cite{AZ24,Zha21} on their higher-rank generalizations, that is, on noncompact symmetric spaces, in \cite{SSWZ20} on non-trapping asymptotically hyperbolic manifolds, and in \cite{APV15} on the so-called Damek--Ricci spaces.

Damek–Ricci spaces are solvable extensions $S = N \rtimes \mathbb{R}^+$ of Heisenberg-type groups $N$, endowed with a left-invariant Riemannian structure. As Riemannian manifolds, these solvable Lie groups include important examples such as real hyperbolic spaces and all other noncompact rank-one symmetric spaces. Most of them, however, are not symmetric, thereby providing counterexamples to the Lichnerowicz conjecture \cite{DaRi1}. We refer to Section \ref{sec2} for more details on the structure of $S$ and the analysis carried out on it.

The dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \cite{MuVa}.

\begin{abstract}
Let $\mathcal{L}$ be the left-invariant distinguished Laplacian, and let $\mathrm{d}\rho$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\partial_t^2 u-\mathcal{L} u=0$ with initial data $(u,\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$ bounds
\begin{align*}
\|u(t,\cdot)\|_{L^p(S ,\mathrm{d}\rho)}
\lesssim_p(1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}}\!f\|_{L^p(S ,\mathrm{d}\rho)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_1}{2}}\!g\|_{L^p(S,\mathrm{d}\rho)}
\end{align*}
for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $\alpha_0 = n\left|1/p-1/2\right|$ and $\alpha_1 = n\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino.
\end{abstract}

The dispersive effect of the wave equation, manifested through the $L^{p'}$-$L^p$ estimates of the propagator, describes how the solution spreads out over time. In contrast, the $L^p$ ($p\neq2$) norm estimates for solutions to the wave equation may not exhibit decay but instead show growth in time. The following Sobolev estimate was first established by Müller and Thiele \cite{MuTh} on the group model of the real hyperbolic space, namely the $ax+b$ group, which corresponds to the case where the nilpotent part $N$ is abelian and thus represents the simplest instance of a Damek–Ricci space. It was later extended to the full class of Damek–Ricci spaces by Müller and Vallarino \cite{MuVa}.

Müller and Thiele on the $ax+b$ group \cite[Remark 8.1]{MuTh}, and Müller and Vallarino on Damek–Ricci spaces \cite[Remark 2]{MuVa}, conjectured that estimate \eqref{est:mainthm} should remain valid at the endpoint cases $\alpha_0=(n-1)\left|1/p-1/2\right|$ and $\alpha_1=(n-1)\left|1/p-1/2\right|-1$. This conjecture was recently confirmed on the $ax+b$ group by Wang and Yan \cite{WaYa}. The following result extends this conclusion to the entire class of Damek–Ricci spaces.

\begin{theorem}\label{thm:main2}
Following the notation of Theorem \ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \eqref{eq: wave} satisfies, for all $t\in\mathbb{R}^*$,
\begin{align*}
\|u(t,\cdot)\|_{L^1(S,\d\rho)}\leq C\,(1+|t|)\, \big(\|(\Id+\L)^{\frac{\alpha_0}{2}}f\|_{\fh^1(S)}+\|(\Id+\L)^{{\frac{\alpha_1}{2}} }g\|_{\fh^1(S)}\big).
\end{align*}
if and only if
\begin{align*}
\alpha_0\geq {n-1\over 2}\quad \text{and}\quad \alpha_1\geq {n-1\over 2}-1.
\end{align*}
\end{theorem}

\paragraph{\bf Damek--Ricci spaces.}
Let $H$ be the unit vector generating the one-dimensional subalgebra $\fa$, and define its adjoint action on $\fn = \fv \oplus \fz$ by
\begin{align*}
\operatorname{ad}(H)v = \frac{1}{2}v \quad \forall\,v \in \fv
\qquad \text{and} \qquad
\operatorname{ad}(H)z = z \quad \forall\,z \in \fz.
\end{align*}
We extend the inner product on $\fn$ to the direct sum $\fs = \fn\oplus\fa$ by requiring $\fa$ to be orthogonal to $\fn$. Then $\fs$ is a solvable Lie algebra, and the corresponding simply connected Lie group $S=N \rtimes A$ is called the \textit{Damek–Ricci space}, where $A= \mathbb{R}_+$ acts on $N$ by the dilation
\begin{align*}
\delta_a(v,z)&=(a^{1\over2}v,az)
\quad\forall\,(v,z)\in N,\,\,\,\forall\,a\in\R_+.
\end{align*}
The product in $S$ is then defined by the rule
\begin{align*}
(v,z,a)(v',z',a')
=
\left(v+a^{1\over2}v',z+az'+\frac12 a^{1\over2}[v,v'],aa'\right)
\qquad\forall\,(v,z,a),\,(v',z',a')\in S.
\end{align*}
We denote by $n=m_{\fv}+m_{\fz}+1$ the dimension of $S$. Recall that the group $S$ is nonunimodular, the right and left Haar measures on $S$ are given, respectively, by
\begin{align*}
\d\rho(v,z,a)=a^{-1}\d v\d z\d a
\qquad\text{and}\qquad
\d\lambda(v,z,a)=a^{-(Q+1)}\d v\d z\d a
\end{align*}
The modular function is thus given by $\delta(v,z,a)=a^{-Q}$. Hereafter, all function spaces on $S$ are defined with respect to the right Haar measure $\d\rho$, unless specified otherwise.

\begin{lemma}\label{lem2.7.0}
Let $T$ be a bounded linear operator on $L^2(S)$ with Schwartz kernel $K_T$, which satisfies the following two conditions:
\begin{asparaenum}[(i)]
\item The local H\"{o}rmonder condition:
\begin{align*}
\sup_B\sup_{y,y'\in B}\int_{(2B)^c}|K_T(x,y)-K_T(x,y')|\,\d \rho(x)<\infty,
\end{align*}
where the first supremum is taken over all balls $B$ of radii at most $1$;
\item The size condition:
\begin{align*}
\sup_{y\in S}\int_{(B(y,2))^c}|K_T(x,y)|\,\d\rho(x)<\infty.
\end{align*}
\end{asparaenum}
Then, the operator $T$ is bounded from $\fh^1(S)$ to $L^1(S)$.
\end{lemma}
\begin{proof}
The proof is a combination of \cite[Theorem 8.2]{CaMaMe09} and \cite[Proposition 4.5 (ii)]{CaMaMe10}.
\end{proof}
Recall that the symbol class is defined by
\begin{align*}
\cS^\sigma=\left\{m\in C^\infty(\R):\|m\|_{\cS^\sigma,k}:=\sup_{\lambda\in\R}\,(1+\lambda^2)^{-{\sigma-k\over 2}}|m^{(k)}(\lambda)|\leq C_{\alpha,k}\text{ for all }k\in \mathbb{N}\right\}.
\end{align*}
\begin{lemma}\label{lem2.6}
Suppose that $m\in\cS^\sigma$ is an even symbol. Then the following statements hold.
\begin{asparaenum}[(i)]
\item If $\sigma=0$, then the multiplier $m(\L)$ is bounded from $\fh^1(S)$ to $L^1(S)$, and is bounded on $L^p(S)$ for all $1<p<\infty$.
\item If $\sigma=-1$, then the operator $|\nabla m(\L)\cdot|_\g$, where $\nabla$ denotes the Riemannian gradient on $S$, is bounded from $\fh^1(S)$ to $L^1(S)$, and from $L^p(S)$ to $L^p(S)$ for all $1<p<\infty$.
\end{asparaenum}
\end{lemma}

\begin{proof}
It follows from \cite[Theorem 3.1]{As1} that, for all $0<t\le1$,
\begin{align*}
\varphi_\lambda(t)
=\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n-1}{2})}\,
t^{\frac{n-1}{2}} A(t)^{-\frac12}
\sum_{\ell=0}^\infty 2^{\frac{n}{2}-1+\ell}\Gamma\!\left({n-1\over 2}+\ell\right)
c_\ell(t)\,J_{\frac{n}{2}-1+\ell}(\lambda t)\,
(\lambda t)^{-\frac{n}{2}+1-\ell}t^{2\ell},
\end{align*}
where $J_{\mu}$ denotes the classical Bessel function of order $\mu$, and the coefficients $a_\ell$ satisfy
\begin{align*}
c_0 \equiv 1
\qquad\textnormal{and}\qquad
|c_\ell(t)|\leq C\,4^{-\ell}\quad\forall\,\ell\in\mathbb{N}^*.
\end{align*}
It is well known that $J_{\mu}$ has the asymptotic expansion
\begin{align}\label{e3.1.2}
J_\mu(r)=\left({\uppi\,r\over 2}\right)^{-\frac12}
\cos\!\left(r-\frac{\uppi\mu}{2}-\frac{\uppi}{4}\right)
+\e^{\i r}s_\mu(r)+\e^{-\i r}s_\mu(-r)\quad \text{as }r \to \infty,
\end{align}
where $s_\mu$ is a symbol of order $-3/2$, see, for instance, \cite[p. 356]{St}. Hence, there exists a symbol $a_t \in\cS^{-(n+1)/2}$, uniformly in $0<t\leq 1$, such that
\begin{align*}
\varphi_\lambda(t)
={2^{\frac{n-1}{2}}\Gamma(\frac{n}{2})\over \sqrt{\uppi}}\,
A(t)^{-\frac12}\lambda^{-\frac{n-1}{2}}
\cos\!\left(\lambda t-\frac{n-1}{4}\uppi\right)
+\e^{\i \lambda t}a_t(\lambda t)+\e^{-\i \lambda t}a_t(-\lambda t).
\end{align*}
Thus, we have proved (a).

\begin{proof}[Proof of Theorem \ref{thm:main2} (sufficient part)]
According to the spectral theorem, the solution to the Cauchy problem \eqref{eq: wave} is given by
\begin{align*}
u(t,x)=\cos (t\sqrt{\L})(f)(x)+{\sin(t\sqrt{\L})\over \sqrt{\L}}(g)(x).
\end{align*}
We deduce from Proposition \ref{prop: main} that the estimate
\begin{align*}
\|u(t,\cdot)\|_{L^1(S)}\lesssim (1+|t|) \big(\|(\Id+\L)^{\frac{\alpha_0}{2}}f\|_{\fh^1(S)}+\|(\Id+\L)^{{\frac{\alpha_1}{2}} }g\|_{\fh^1(S)}\big)
\end{align*}
holds in the critical case where
\begin{align*}
\alpha_0=(n-1)\left|{1\over p}-{1\over 2}\right|
\quad\text{and}\quad \alpha_1=(n-1)\left|{1\over p}-{1\over 2}\right|-1,
\end{align*}
and therefore also in the supercritical cases.
\end{proof}

Müller and Thiele on the $ax+b$ group \cite[Remark 8.1]{MuTh}, and Müller and Vallarino on Damek–Ricci spaces \cite[Remark 2]{MuVa}, conjectured that estimate \eqref{est:mainthm} should remain valid at the endpoint cases $\alpha_0=(n-1)\left|1/p-1/2\right|$ and $\alpha_1=(n-1)\left|1/p-1/2\right|-1$. This conjecture was recently confirmed on the $ax+b$ group by Wang and Yan \cite{WaYa}. The following result extends this conclusion to the entire class of Damek–Ricci spaces.

\begin{theorem}\label{thm:main}
Theorem \ref{thm:MullerVallarino} remains valid at the endpoint cases
\begin{align*}
\alpha_0=(n-1)\left|{1\over p}-{1\over 2}\right|
\quad\text{and}\quad \alpha_1=(n-1)\left|{1\over p}-{1\over 2}\right|-1,
\end{align*}
and these regularity conditions are sharp.
\end{theorem}

Let us briefly comment on Theorem \ref{thm:main}. This endpoint result represents a strengthening of Theorem \ref{thm:MullerVallarino}. In fact, we now know that estimate \eqref{est:mainthm} holds if and only if $\alpha_0 \ge (n-1)\left|1/p-1/2\right|$ and $\alpha_1 \ge (n-1)\left|1/p-1/2\right|-1$, which corresponds to the analogue of the related results in the Euclidean setting, see \cite{Mi,Pe}.

However, as pointed out by Müller and Thiele \cite[p. 147]{MuTh}, the exponent $2|1/p - 1/2|$ of $(1 + |t|)$ in estimate \eqref{est:mainthm} is independent of the topological dimension $n$. This contrasts with the corresponding estimates of Miyachi \cite{Mi} and Peral \cite{Pe} in the Euclidean case, where the exponent is $(n - 1)|1/p - 1/2|$. We observe here that the corresponding exponent could be $(\nu - 1)|1/p - 1/2|$, where $\nu$ denotes the pseudo-dimension (or dimension at infinity), and $\nu=3$ when $S$ is a Damek–Ricci space. This natural conjecture is motivated by previously known results on dispersive estimates in similar settings, such as real hyperbolic spaces, Damek–Ricci spaces, and general noncompact symmetric spaces, where the long-time dispersive decay rate depends on the dimension at infinity rather than on the topological dimension, see, for instance, \cite{AP14,APV15,AZ24}. Establishing estimate \eqref{est:mainthm} under sharp regularity conditions for higher rank groups would therefore be an interesting direction for further investigation on this point.

It is noteworthy that Ionescu proved in \cite{Ion00} a sharp result analogous to estimate \eqref{est:mainthm} for the wave equation associated with the Laplace--Beltrami operator on noncompact symmetric spaces of rank one. In that case, the time growth on the right-hand side of the estimate is exponential rather than polynomial as in our setting. See also \cite[Remark 3]{MuVa}. Although Damek--Ricci spaces also have exponential volume growth, we still obtain polynomial growth with respect to $|t|$ in estimate \eqref{est:mainthm}. This reflects the difference between the Laplace--Beltrami operator and the distinguished Laplacian.

Theorem \ref{thm:main} follows from an interpolation argument based on the following improved $L^1$-norm estimate. We denote by $\mathfrak{h}^1(S)$ the local Hardy space on $S$, and $\|\cdot\|_{\mathfrak{h}^1(S)}$ the corresponding norm (see Subsection \ref{h1} for more details).

\begin{theorem}\label{thm:main2}
Following the notation of Theorem \ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \eqref{eq: wave} satisfies, for all $t\in\mathbb{R}^*$,
\begin{align*}
\|u(t,\cdot)\|_{L^1(S,\mathrm{d}\rho)}\leq C\,(1+|t|)\, \big(\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}}f\|_{\mathfrak{h}^1(S)}+\|(\mathrm{Id}+\mathcal{L})^{{\frac{\alpha_1}{2}} }g\|_{\mathfrak{h}^1(S)}\big).
\end{align*}
if and only if
\begin{align*}
\alpha_0\geq {n-1\over 2}\quad \text{and}\quad \alpha_1\geq {n-1\over 2}-1.
\end{align*}
\end{theorem}

The proof of Theorem \ref{thm:main2} is based on a careful study of the corresponding multipliers of the wave propagators. It combines refined spherical analysis with techniques inspired by Peral \cite{Pe} and Tao \cite{Ta}, and the argument relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives. It is well known that extending results from the $ax+b$ group to the full class of Damek–Ricci spaces is far from automatic. The nilpotent part of the group is no longer abelian but an $H$-type group endowed with a more intricate algebraic structure. Moreover, such a generalization provides the first insights into the study of the higher rank situations mentioned above.

Before presenting the detailed proofs of the main theorems in Section \ref{sec3}, we review in Section \ref{sec2} the structure of the Damek–Ricci space $S$ and the analysis on it, where we also recall the definition of the local Hardy space on $S$. Throughout this paper, the notation $A \lesssim B$ between two positive quantities means that there exists a constant $C>0$, independent of all possible variables, such that $A \le CB$.

Theorem~\ref{thm:main} “follows from an interpolation argument based on the following improved $L^1$-norm estimate” (Theorem~\ref{thm:main2}). Theorem~\ref{thm:main2} is proved by “a careful study of the corresponding multipliers of the wave propagators”: it “combines refined spherical analysis with techniques inspired by Peral \cite{Pe} and Tao \cite{Ta},” and “relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives.”

To prove the main theorem, it “follows from an interpolation argument based on the following improved $L^1$-norm estimate”. We first prove the following theorem.

\begin{theorem}\label{thm:main2}
Following the notation of Theorem \ref{thm:MullerVallarino}, there exists $C>0$ such that the solution to the Cauchy problem \eqref{eq: wave} satisfies, for all $t\in\mathbb{R}^*$,
\begin{align*}
\|u(t,\cdot)\|_{L^1(S,\d\rho)}\leq C\,(1+|t|)\, \big(\|(\Id+\L)^{\frac{\alpha_0}{2}}f\|_{\fh^1(S)}+\|(\Id+\L)^{{\frac{\alpha_1}{2}} }g\|_{\fh^1(S)}\big).
\end{align*}
if and only if
\begin{align*}
\alpha_0\geq {n-1\over 2}\quad \text{and}\quad \alpha_1\geq {n-1\over 2}-1.
\end{align*}
\end{theorem}

This theorem is proved by “a careful study of the corresponding multipliers of the wave propagators”: it “combines refined spherical analysis with techniques inspired by Peral \cite{Pe} and Tao \cite{Ta},” and “relies crucially on the explicit asymptotic expansions of the spherical function and its derivatives.”

Existential–Universal
Inequality or Bound

Question:
Let $S$ be a Damek–Ricci space of dimension $n$, let $\mathcal{L}$ be the left-invariant distinguished Laplacian on $S$, and let $\mathrm{d}\rho$ be the right Haar measure. For $1<p<\infty$, consider the wave equation on $\mathbb{R}\times S$
\[
\partial_t^2 u(t,x)-\mathcal{L}u(t,x)=0,
\]
with initial data
\[
u(0,x)=f(x),\qquad \partial_t u(0,x)=g(x).
\]
Assume the Sobolev exponents satisfy
\[
\alpha_0>(n-1)\left|\frac1p-\frac12\right|,
\qquad
\alpha_1>(n-1)\left|\frac1p-\frac12\right|-1.
\]
Which statement holds?

For every $1<p<\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\in\mathbb{R}^*=\mathbb{R}\setminus\{0\}$,
\[
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\le C_p\Big((1+|t|)^{2\left|\frac1p-\frac12\right|}\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_0/2}f\|_{L^p(S,\mathrm{d}\rho)}
+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_1/2}g\|_{L^p(S,\mathrm{d}\rho)}\Big).
\]

Choices:
B.
For every $1<p<\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\in\mathbb{R}$,
\[
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\le C_p\Big((1+|t|)^{2\left|\frac1p-\frac12\right|}\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_0/2}f\|_{L^p(S,\mathrm{d}\rho)}
+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_1/2}g\|_{L^p(S,\mathrm{d}\rho)}\Big),
\]
and moreover the same conclusion remains valid at the endpoint values $\alpha_0=(n-1)\left|\frac1p-\frac12\right|$ and $\alpha_1=(n-1)\left|\frac1p-\frac12\right|-1$.
C.
For every $1<p<\infty$ and every $t\in\mathbb{R}^*$, the solution $u$ satisfies an estimate of the form
\[
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\le C_{p,t}\Big(\|(\mathrm{Id}+\mathcal{L})^{\alpha_0/2}f\|_{L^p(S,\mathrm{d}\rho)}+\|(\mathrm{Id}+\mathcal{L})^{\alpha_1/2}g\|_{L^p(S,\mathrm{d}\rho)}\Big)
\]
for some constant $C_{p,t}>0$ depending on $p$ and $t$.
D.
For every $1<p<\infty$, there exists a constant $C>0$, independent of $p$, such that for every solution $u$ of the above Cauchy problem and every $t\in\mathbb{R}^*$,
\[
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\le C\Big((1+|t|)^{2\left|\frac1p-\frac12\right|}\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_0/2}f\|_{L^p(S,\mathrm{d}\rho)}
+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_1/2}g\|_{L^p(S,\mathrm{d}\rho)}\Big).
\]
E.
For every $1<p<\infty$, there exists a constant $C_p>0$ such that for every solution $u$ of the above Cauchy problem and every $t\in\mathbb{R}^*$,
\[
\|u(t,\cdot)\|_{L^p(S,\mathrm{d}\rho)}
\le C_p\Big((1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_0/2}f\|_{L^p(S,\mathrm{d}\rho)}
+(1+|t|)^{2\left|\frac1p-\frac12\right|}\,\|(\mathrm{Id}+\mathcal{L})^{\alpha_1/2}g\|_{L^p(S,\mathrm{d}\rho)}\Big).
\]

The stem does not explicitly reveal the correct choice. It gives the PDE and Sobolev threshold assumptions, but the exact conclusion must still be identified from the options.

This is close to theorem recall: the correct option is essentially the precise statement of the relevant estimate, with distractors formed by small perturbations. It is not a pure restatement in the stem, but it is only a mild reformulation of a known result.

Some reasoning is needed to distinguish strict vs endpoint assumptions, dependence on p, the role of t=0, and the placement of growth factors. However, the task mainly tests exact theorem recall/comparison rather than substantial generative mathematical reasoning.

Most distractors are plausible and target realistic failure modes (endpoint strengthening, p-independence, swapped time factors). However, choice C is a weaker true statement implied by A, so it is not a clean false distractor and makes the single-correct format ambiguous.

Reasonably well-constructed theorem-recognition item with plausible distractors, but it is not strongly generative and is weakened by ambiguity because at least one distractor is also true in a weaker sense.