[$2\pi^2$-Theorem, Marques-Neves~\cite{MN14}]\label{2pi2-theorem}
Let $\Sigma\subset\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$.
Then its canonical five-parameter family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\]
satisfies
\[
\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\mathrm{Area}\bigl(\Sigma_{(v,t)}\bigr)\;\geqslant\; 2\pi^2.
\]
Moreover, equality holds if and only if $\Sigma$ is the Clifford torus.

\begin{theorem}\label{Thm: main}
Let $V=\underline{v}(\Sigma,1)$ be an integral $2$-varifold in $\mathbb{S}^3$
with unit density and mean curvature $H\in L^2(d\mu)$, and assume its support
$\Sigma$ has genus at least $1$.
If the Willmore energy
\[
\mathcal{W}(V)=\int_\Sigma\Bigl(1+\tfrac14 |H|^2\Bigr)\,d\mu
\]
satisfies
\[
\mathcal{W}(V)\le 2\pi^2+\delta^2,
\qquad \delta>0 \text{ sufficiently small},
\]
then, after a suitable conformal transformation of $\mathbb{S}^3$, there exists
a homeomorphism
\[
f:\mathbb{S}^1\times\mathbb{S}^1 \longrightarrow \Sigma \subset\mathbb{S}^3
\]
such that in the standard coordinates $(\theta,\varphi)$ on
$\mathbb{S}^1\times\mathbb{S}^1$ the pull-back metric takes the form
\[
f^{*}g_{\mathbb{S}^3}
= e^{2u}\,(a\, d\theta^2 + 2 b\, d\theta\, d\varphi + c\, d\varphi^2).
\]

Let $f_0:\mathbb{S}^1\times\mathbb{S}^1\to\mathbb{S}^3$ denote the Clifford torus
embedding
\[
f_0(\theta,\varphi)
= \frac{1}{\sqrt{2}}(\cos\theta,\;\sin\theta,\;\cos\varphi,\;\sin\varphi),
\]
and write its image as $\mathbb{T}^2:=f_0(\mathbb{S}^1\times\mathbb{S}^1)$.
Then
\[
f_0^{*} g_{\mathbb{S}^3}
= \tfrac12 (d\theta^2+d\varphi^2).
\]

The following quantitative rigidity estimates hold:
\begin{enumerate}

\item[\rm(1)]
\[
\| f - f_0 \|_{W^{2,2}(\mathbb{S}^1\times\mathbb{S}^1)} \le C\,\delta.
\]

\item[\rm(2)]
\[
\|u\|_{L^\infty(\mathbb{S}^1\times\mathbb{S}^1)} \le C\,\delta.
\]

\item[\rm(3)]
\[
|a-\tfrac12| + |b| + |c-\tfrac12| \le C\,\delta.
\]

\end{enumerate}
Here $C$ is a universal constant independent of $\Sigma$.
\end{theorem}

For an immersed surface $f:\Sigma\to \varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\varmathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\varmathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\geqslant 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\varmathbb{T}^2=\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\varmathbb{S}^3.
\]

The purpose of this paper is to study the quantitative rigidity of this theorem.
Given $\delta<\delta_0\ll1$ and a surface $\Sigma$ with
\[
\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2,
\]
we ask how close $\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\infty$-bound of the conformal factor, and the conformal structure.
This parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\"uller~\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\"atzle~\cite{LS-2014} and to varifolds by Bi-Zhou~\cite{BZ25}.

A first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value.
Using existing existence and energy asymptotic results, one obtains that if a surface $\Sigma$ satisfies $g(\Sigma)\ge1$ and $\mathcal{W}(\Sigma)$ is sufficiently close to $2\pi^2$, then in fact $g(\Sigma)=1$.
This follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\cite{LS93} and Bauer-Kuwert for genus $\ge2$~\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\"atzle~\cite{KLS10} and the Willmore conjecture~\cite{MN14}.

Once the genus has been identified, Leon Simon's results~\cite{LS93} imply that any sequence of tori
\(\Sigma_k \subset \varmathbb{S}^3\) with \(\mathcal{W}(\Sigma_k)\to 2\pi^2\) admits, after suitable conformal reparametrization,
a subsequence converging as varifolds to a limit surface whose genus is at least one.
By lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \(2\pi^2\).
In view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \(2\pi^2\).
It is well known to experts (see for instance Kuwert-Li~\cite{KL-2012},  Sch\"atzle~\cite{Sch13} and Rivi\'ere\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model
surface may be any fixed genus~$\ge1$ minimal surface (not only the Clifford
torus), provided that no energy concentration occurs---in the precise sense
that the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.
Our approach proceeds by first establishing convergence of the surfaces
\emph{as subsets of $\varmathbb{S}^{3}$} under these assumptions, and only
afterwards upgrading the convergence to that of the associated conformal
parametrizations.  This is somewhat different in spirit from earlier
treatments, which work directly with compactness of $W^{2,2}$ conformal
immersions into Euclidean space.

The main difficulty is to promote this qualitative convergence to a quantitative, in particular linear, stability estimate.

A natural idea would be to expand the Willmore functional around the Clifford torus: one considers the second variation together with the higher-order remainder, analyzes the spectral properties of the resulting symmetric quadratic form, and then derives quantitative stability from the coercivity of this linearized operator.  However, in order for this approach to be effective, the remainder must be genuinely dominated by the quadratic term, which requires the surface to be $C^1$ or at least Lipschitz close to the Clifford torus.  Establishing such a priori closeness is delicate in our weak $W^{2,2}$-conformal setting.

For this reason, following the strategy of Marques-Neves, we avoid a direct perturbative analysis of the Willmore functional.  Instead, using the Heintze-Karcher-type inequality to relate the Willmore energy to the area of the canonical family $\Sigma_{(v,t)}$, we reformulate the problem in Section~3 as a minimal-surface stability problem.

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\varmathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\varmathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\leqslant \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

For an immersed surface $f:\Sigma\to \varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\varmathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\varmathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\geqslant 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\varmathbb{T}^2=\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\varmathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\varmathbb{S}^3.
\]

The purpose of this paper is to study the quantitative rigidity of this theorem.
Given $\delta<\delta_0\ll1$ and a surface $\Sigma$ with
\[
\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2,
\]
we ask how close $\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\infty$-bound of the conformal factor, and the conformal structure.
This parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\"uller~\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\"atzle~\cite{LS-2014} and to varifolds by Bi-Zhou~\cite{BZ25}.

A first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value.
Using existing existence and energy asymptotic results, one obtains that if a surface $\Sigma$ satisfies $g(\Sigma)\ge1$ and $\mathcal{W}(\Sigma)$ is sufficiently close to $2\pi^2$, then in fact $g(\Sigma)=1$.
This follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\cite{LS93} and Bauer-Kuwert for genus $\ge2$~\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\"atzle~\cite{KLS10} and the Willmore conjecture~\cite{MN14}.

Once the genus has been identified, Leon Simon's results~\cite{LS93} imply that any sequence of tori
\(\Sigma_k \subset \varmathbb{S}^3\) with \(\mathcal{W}(\Sigma_k)\to 2\pi^2\) admits, after suitable conformal reparametrization,
a subsequence converging as varifolds to a limit surface whose genus is at least one.
By lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \(2\pi^2\).
In view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \(2\pi^2\).
It is well known to experts (see for instance Kuwert-Li~\cite{KL-2012}, Sch\"atzle~\cite{Sch13} and Rivi\'ere\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model
surface may be any fixed genus~$\ge1$ minimal surface (not only the Clifford
torus), provided that no energy concentration occurs---in the precise sense
that the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.
Our approach proceeds by first establishing convergence of the surfaces
\emph{as subsets of $\varmathbb{S}^{3}$} under these assumptions, and only
afterwards upgrading the convergence to that of the associated conformal
parametrizations. This is somewhat different in spirit from earlier
treatments, which work directly with compactness of $W^{2,2}$ conformal
immersions into Euclidean space.

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\varmathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\varmathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\leqslant \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

For an immersed surface $f:\Sigma\to \mathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by
\[
\mathcal{W}(\Sigma)
=\int_{\Sigma}\!\left(1+\tfrac{1}{4}|H|^2\right)d\mu_f,
\]
where $H$ is the mean curvature and $d\mu_f$ is the area element of $f^{*}g_{\mathbb{S}^3}$.
A fundamental property of $\mathcal{W}$ is its conformal invariance.
By the theorem of Marques and Neves resolving the Willmore conjecture~\cite{MN14}, any closed oriented surface $\Sigma\subset\mathbb{S}^3$ with genus at least one satisfies
\[
\mathcal{W}(\Sigma)\ge 2\pi^2,
\]
with equality if and only if $\Sigma$ is, up to a conformal transformation, the Clifford torus
\[
\mathbb{T}^2=\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times
\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset\mathbb{S}^3.
\]

To describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\cite{MN14}.
For $v\in\mathring{B}^4$, let
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v
\]
be the corresponding conformal transformation.
If $\mathbb{S}^3\setminus\Sigma=A\cup A^*$ is the decomposition into connected components, we denote $\Sigma_v:=F_v(\Sigma)$ and $\mathbb{S}^3\setminus\Sigma_v=A_v\cup A_v^*$ with $A_v=F_v(A)$.
Let $d_v$ be the signed distance to $\Sigma_v$, and define
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in\mathring{B}^4\times[-\pi,\pi].
\]
The Heintze-Karcher-type inequality~\cite{MN14,HK78,R99} gives
\[
\mathrm{Area}(\Sigma_{(v,t)})\le \mathcal{W}(\Sigma),
\]
with equality only at $t=0$ when $\Sigma_v$ is minimal.

Marques and Neves further established the following fundamental result, often referred to as the \emph{$2\pi^2$-Theorem}:

With the five-parameter family $\{\Sigma_{(v,t)}\}$ of
Marques-Neves recalled above, we now indicate how it enters our proof.

In this section we study the almost minimizing property for surfaces whose
Willmore energy is sufficiently close to $2\pi^{2}$.
A central object in our analysis is the canonical five-parameter family
$\{\Sigma_{(v,t)}\}$ introduced by Marques-Neves.
Rather than repeating the full construction and discussion from the
Introduction, we recall only the variational problem that plays a decisive role:
\begin{equation}\label{eq:max-canonical-area}
\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]}
\mathrm{Area}\bigl(\Sigma_{(v,t)}\bigr)\,.
\end{equation}

The fundamental result of Marques-Neves, stated as Theorem~\ref{2pi2-theorem} in
the Introduction, asserts that for any smooth embedded surface of genus at least
$1$, the canonical family satisfies
\[
\max_{(v,t)} \mathrm{Area}(\Sigma_{(v,t)}) \ge 2\pi^{2},
\]
with equality if and only if the surface is the
Clifford torus.

Let $\Sigma\subset\mathbb{S}^3$ be a smooth closed torus
and assume that $\Sigma$ is $(\gamma,r_0)$-regular and $\mathcal{W}(\Sigma)\leq 8\pi$
Consider the canonical family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]},
\]
and let $(v_0,t_0)$ be any maximizer of the area , i.e.
\[
\mathrm{Area}(\Sigma_{(v_0,t_0)})
 = \max_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]}
 \mathrm{Area}(\Sigma_{(v,t)}).
\]
Then
\[
|v_0|\le 1-\eta_1.
\]
\end{proposition}

Now let $(v_{0},t_{0})$ be a maximizer of the canonical area functional as in
Proposition~\ref{away from boundary of conformal group}.
By the $2\pi^{2}$-Theorem~\ref{2pi2-theorem} and \eqref{Area formula},
\begin{align*}
2\pi^{2}
\;\le\; \mathrm{Area}(\Sigma_{(v_{0},t_{0})})
\;\le\; \int_{\Sigma_{v_{0}}^{*}(t_{0})} J_{\varphi_{t_{0}}}\, d\mathcal{H}^{2}.
\end{align*}
Using again \cite[Lemma~3.5]{MN14} (see also \cite{R99}), we have
\[
J_{\varphi_{t_{0}}}(p)
=\Bigl(1+\frac{|H|^{2}}{4}\Bigr)
 -\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2}
 -\sin^{2}t_{0}\, |\mathring{A}|^{2},
\]
where $H$ is the mean curvature and $ \mathring{A}=A-\frac{H}{2}g$. Hence
\begin{align*}
2\pi^{2}
&\le \int_{\Sigma_{v_{0}}^{*}(t_{0})}
\Bigl(1+\frac{|H|^{2}}{4}\Bigr)
 -\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2}
 -\sin^{2}t_{0}\, |\mathring{A}|^{2}\, d\mathcal{H}^{2} \\
&\le \int_{\Sigma_{v_{0}}^{*}(t_{0})} \Bigl(1+\frac{|H|^{2}}{4}\Bigr)\, d\mathcal{H}^{2} \\
&\le \mathcal{W}(\Sigma_{v_{0}}) \\
&=\mathcal{W}(\Sigma)\;\le\; 2\pi^{2}+\delta^{2}.
\end{align*}
This yields the three estimates
\begin{align}
\int_{E_{v_{0}}(t_{0})}\Bigl(1+\frac{|H|^{2}}{4}\Bigr)\, d\mathcal{H}^{2}
&\le \delta^{2}, \label{negative small} \\
\int_{\Sigma_{v_{0}}^{*}(t_{0})}
\Bigl(\frac{H}{2}\cos t_{0}-\sin t_{0}\Bigr)^{2} d\mathcal{H}^{2}
&\le \delta^{2}, \label{mean curvature small} \\
\sin^{2}t_{0} \int_{\Sigma_{v_{0}}^{*}(t_{0})} |\mathring{A}|^{2}\, d\mathcal{H}^{2}
&\le \delta^{2}. \label{small time}
\end{align}

\begin{proposition}\label{away from boundary of conformal group}
There exist constants $\eta_1>0$, $\delta_0>0$, and a function
$\gamma=\gamma(\delta_0)>0$ with $\gamma(\delta_0)\to 0$ as $\delta_0\to 0$,
and a universal radius $r_0>0$, such that the following holds.

Let $\Sigma\subset\mathbb{S}^3$ be a smooth closed torus
and assume that $\Sigma$ is $(\gamma,r_0)$-regular and $\mathcal{W}(\Sigma)\leq 8\pi$
Consider the canonical family
\[
\{\Sigma_{(v,t)}\}_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]},
\]
and let $(v_0,t_0)$ be any maximizer of the area , i.e.
\[
\mathrm{Area}(\Sigma_{(v_0,t_0)})
 = \max_{(v,t)\in\mathring{B}^4\times[-\pi,\pi]}
   \mathrm{Area}(\Sigma_{(v,t)}).
\]
Then
\[
|v_0|\le 1-\eta_1.
\]
\end{proposition}

With the five-parameter family $\{\Sigma_{(v,t)}\}$ of
Marques-Neves recalled above, we now indicate how it enters our proof.

The key point is to show that if a genus--one surface $\Sigma$ satisfies
$\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then a \emph{conformal}
transformation of $\varmathbb{S}^3$ yields a surface $\Sigma'$ with
\[
\mathrm{Area}(\Sigma')\;\geqslant\;2\pi^2 - C\delta^2.
\]

The $2\pi^2$ theorem guarantees that the full $(v,t)$-family contains a slice
whose area is at least $2\pi^2$.  Since the variables $v$ encode conformal
transformations and $t$ is the signed distance parameter, our aim is to
realize such a high-area slice \emph{within the conformal}
four-parameter subfamily.
For this, it is necessary to control the distance parameter $t$ in terms of
the energy gap~$\delta$, ensuring that the maximizing slice must lie close to
the $t=0$ slice.

A closer examination of the Heintze-Karcher inequality used by
Marques-Neves shows that, \emph{once the five-parameter family is built
from the quantitatively normalized surface obtained in Section~2}, the
problem reduces to proving that the maximizing parameter $(v_0,t_0)$ stays a
definite distance away from the boundary of the conformal group.
Equivalently, there exists a universal constant $\eta_1>0$ such that
\[
|v_0|\;\leqslant\;1-\eta_1 .
\]
This reduction is carried out in Section~3, where the technical
proof of Proposition~\ref{away from boundary of conformal group} is deferred to the Appendix.

To clarify why such a uniform bound is true, we now outline the geometric
intuition behind it.  When the conformal parameter $v$ approaches the boundary of
$\mathring{B}^4$, the transformed surface $\Sigma_{(v,t)}$ becomes
quantitatively close to a geodesic sphere, which forces its
area to stay strictly below $2\pi^2$.
Thus such $v$ cannot maximize the canonical family.

More precisely, when $t=0$, the qualitative stability implies that $\Sigma_v= \Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, and the region enclosed by $\Sigma_v$ is similarly close to that enclosed by~$S$.
A direct computation then shows that
\[
\mathrm{Area}(\Sigma_v)
\]
is close to $\mathrm{Area}(S)$, which is at most $4\pi$. Since $\Sigma_{(v,t)}$ is obtained from $\Sigma_v$ by shifting along its normal directions, the area difference satisfies an estimate of the form
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|
    \;\lesssim\; |t|\,\mathcal{W}(\Sigma_v).
\]
Hence, for sufficiently small $|t|$, the area remains below~$5\pi$.

For larger $|t|$, we distinguish two cases.
If $\Sigma_v$ stays a definite distance from the poles, we may apply the
Euclidean model: a region trapped between two nearly parallel half–spaces of
separation $\varepsilon\ll1$ has the property that, for any fixed
$t_0\sim1$, the boundaries of its $t$-parallel sets ($t\geqslant t_0$) become Lipschitz graphs over one
bounding plane with Lipschitz constant $\to0$ as $\varepsilon/t_0\to0$.
Transferring this to the spherical setting yields that $\Sigma_{(v,t)}$ is a
small Lipschitz graph over a geodesic sphere and hence has area $<5\pi$.

If instead $\Sigma_v$ lies very close to a pole, its diameter is already
tiny, and a simple comparison–geometry argument shows that
$\Sigma_{(v,t)}$ remains of area $<5\pi$.

Thus in all cases $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Since $5\pi < 2\pi^2$, no parameter $(v,t)$ with $|v|$ sufficiently close to $1$ can maximize the canonical area.
This yields the desired estimate
\[
|v_0| \leqslant 1 - \eta_1.
\]

With this uniform control of the conformal parameter, the remaining task is to
derive a quantitative stability estimate for minimal surfaces.
Assembling the arguments developed so far leads to our main quantitative
rigidity theorem:

\begin{theorem}\label{Thm: main}
Let $V=\underline{v}(\Sigma,1)$ be an integral $2$-varifold in $\varmathbb{S}^3$
with unit density and mean curvature $H\in L^2(d\mu)$, and assume its support
$\Sigma$ has genus at least $1$.
If the Willmore energy
\[
\mathcal{W}(V)=\int_\Sigma\Bigl(1+\tfrac14 |H|^2\Bigr)\,d\mu
\]
satisfies
\[
\mathcal{W}(V)\leqslant 2\pi^2+\delta^2,
\qquad \delta>0 \text{ sufficiently small},
\]
then, after a suitable conformal transformation of $\varmathbb{S}^3$, there exists
a homeomorphism
\[
f:\varmathbb{S}^1\times\varmathbb{S}^1 \longrightarrow \Sigma \subset\varmathbb{S}^3
\]
such that in the standard coordinates $(\theta,\varphi)$ on
$\varmathbb{S}^1\times\varmathbb{S}^1$ the pull-back metric takes the form
\[
f^{*}g_{\varmathbb{S}^3}
= e^{2u}\,(a\, d\theta^2 + 2 b\, d\theta\, d\varphi + c\, d\varphi^2).
\]

Let $f_0:\varmathbb{S}^1\times\varmathbb{S}^1\to\varmathbb{S}^3$ denote the Clifford torus
embedding
\[
f_0(\theta,\varphi)
= \frac{1}{\sqrt{2}}(\cos\theta,\;\sin\theta,\;\cos\varphi,\;\sin\varphi),
\]
and write its image as $\varmathbb{T}^2:=f_0(\varmathbb{S}^1\times\varmathbb{S}^1)$.
Then
\[
f_0^{*} g_{\varmathbb{S}^3}
= \tfrac12 (d\theta^2+d\varphi^2).
\]

The following quantitative rigidity estimates hold:
\begin{enumerate}

\item[\rm(1)]
\[
\| f - f_0 \|_{W^{2,2}(\varmathbb{S}^1\times\varmathbb{S}^1)} \leqslant C\,\delta.
\]

\item[\rm(2)]
\[
\|u\|_{L^\infty(\varmathbb{S}^1\times\varmathbb{S}^1)} \leqslant C\,\delta.
\]

\item[\rm(3)]
\[
|a-\tfrac12| + |b| + |c-\tfrac12| \leqslant C\,\delta.
\]

\end{enumerate}
Here $C$ is a universal constant independent of $\Sigma$.
\end{theorem}

Recent work of Rupp-Scharrer~\cite{RS25}, making use of the regularity theorem
developed in~\cite{BZ-2022b}, establishes density of smooth surfaces among
integral $2$-varifolds with square-integrable second fundamental form and
Willmore energy below $8\pi$.
Since unit density together with $H\in L^2$ already yields square-integrable
second fundamental form by~\cite{BZ-2022b}, their result is closely related to
the class of varifolds considered here.

In the appendix we prove a density statement suited to our setting:
compactly supported integral $2$-varifolds with unit density and
square-integrable mean curvature can be approximated in $W^{2,2}$ by smooth
embedded surfaces.
Our construction provides global bilipschitz control between the limiting
parametrization $F$ and its smoothings $F_\eta$, which ensures
embeddedness of the approximating surfaces without requiring any additional
smallness assumptions on the Willmore energy.

\medskip
\noindent\textbf{Outline of the paper.}
In Section~2, we establish the topological stability and qualitative geometric stability for surfaces with Willmore energy sufficiently close to $2\pi^2$.
In particular, we show that such a surface must have genus one and is, after suitable conformal normalization, qualitatively close to the Clifford torus.

In Section~3, we analyze the Marques-Neves canonical family and prove the key estimate that under the assumption $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, one can find a conformal transformation of $\varmathbb{S}^3$ such that the transformed surface $\Sigma'$ satisfies
\[
\mathrm{Area}(\Sigma')\geqslant 2\pi^2 - C\delta^2,
\]
for a universal constant $C>0$.
This reduces the Willmore stability problem to a quantitative stability statement for minimal surfaces.

Section~4 contains the linearization and stability analysis.
Using the estimate from Section~3, we derive the quantitative bounds on the conformal factor, the conformal structure, and the $W^{2,2}$-distance to the Clifford torus, thereby proving Theorem~\ref{Thm: main}.

Finally, in the Appendix we provide the proof of Proposition~\ref{away from boundary of conformal group} and we also establish the density and parametrization results for compactly
supported integral $2$-varifolds with unit density and $L^2$-mean curvature,
showing in particular that such varifolds admit global conformal
parametrizations and can be approximated by smooth embedded surfaces.

Using the Marques--Neves five-parameter family $\{\Sigma_{(v,t)}\}$, the key point is to show that if a genus--one surface $\Sigma$ satisfies $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then after a \emph{conformal} transformation one gets a surface $\Sigma'$ with $\mathrm{Area}(\Sigma')\geqslant 2\pi^2-C\delta^2$.

The $2\pi^2$-Theorem~\ref{2pi2-theorem} ensures the full $(v,t)$-family contains a slice of area at least $2\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\delta$, so the maximizing slice lies close to $t=0$.

A closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\eta_1>0$ such that $|v_0|\leqslant 1-\eta_1$ (reduced in Section~3; technical proof deferred to the Appendix).

Geometric intuition for this uniform bound: as $|v|\to1$, $\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\Sigma_v=\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\mathrm{Area}(\Sigma_v)$ is close to $\mathrm{Area}(S)\le 4\pi$, and for small $|t|$ one uses
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|\lesssim |t|\,\mathcal{W}(\Sigma_v)
\]
so the area stays $<5\pi$. For larger $|t|$, either (i) if $\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\pi$, or (ii) if $\Sigma_v$ lies close to a pole, comparison geometry gives $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Thus for $|v|$ sufficiently close to $1$, $\mathrm{Area}(\Sigma_{(v,t)})<5\pi<2\pi^2$, proving $|v_0|\le 1-\eta_1$.

With this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the Section~3 area lower bound to reduce Willmore stability to minimal-surface stability, and in Section~4 performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\ref{Thm: main}.

Using the Marques--Neves five-parameter family $\{\Sigma_{(v,t)}\}$, the key point is to show that if a genus--one surface $\Sigma$ satisfies $\mathcal{W}(\Sigma)\leqslant 2\pi^2+\delta^2$, then after a \emph{conformal} transformation one gets a surface $\Sigma'$ with $\mathrm{Area}(\Sigma')\geqslant 2\pi^2-C\delta^2$.

To prove the main theorem, we use that the full $(v,t)$-family contains a slice of area at least $2\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\delta$, so the maximizing slice lies close to $t=0$.

A closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\eta_1>0$ such that $|v_0|\leqslant 1-\eta_1$ (reduced later; technical proof deferred to an appendix).

Geometric intuition for this uniform bound: as $|v|\to1$, $\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\Sigma_v=\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\mathrm{Area}(\Sigma_v)$ is close to $\mathrm{Area}(S)\le 4\pi$, and for small $|t|$ one uses
\[
\bigl|\mathrm{Area}(\Sigma_{(v,t)})-\mathrm{Area}(\Sigma_v)\bigr|\lesssim |t|\,\mathcal{W}(\Sigma_v)
\]
so the area stays $<5\pi$. For larger $|t|$, either (i) if $\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\pi$, or (ii) if $\Sigma_v$ lies close to a pole, comparison geometry gives $\mathrm{Area}(\Sigma_{(v,t)})<5\pi$. Thus for $|v|$ sufficiently close to $1$, $\mathrm{Area}(\Sigma_{(v,t)})<5\pi<2\pi^2$, proving $|v_0|\le 1-\eta_1$.

With this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the earlier area lower bound to reduce Willmore stability to minimal-surface stability, and later performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\ref{Thm: main}.

Inequality or Bound
Biconditional or Equivalence

Question:
Let \(\Sigma\subset \mathbb{S}^3\) be a smooth embedded surface of genus at least \(1\). For \(v\in \mathring{B}^4\), define the conformal transformation
\[
F_v(x)=(1-|v|^2)\frac{x-v}{|x-v|^2}-v.
\]
If \(\mathbb{S}^3\setminus \Sigma=A\cup A^*\) is the decomposition into connected components, set \(\Sigma_v:=F_v(\Sigma)\), \(A_v:=F_v(A)\), let \(d_v\) be the signed distance to \(\Sigma_v\), and define the canonical five-parameter family
\[
\Sigma_{(v,t)}=\partial\{x\mid d_v(x)<t\}, \qquad (v,t)\in \mathring{B}^4\times[-\pi,\pi].
\]
Which quantitative area estimate holds for this family, including the equality case?

\[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2.\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus \(\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times \mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset \mathbb{S}^3\).

Choices:
B. \[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\le 2\pi^2.\] Moreover, equality holds if and only if for some \((v_0,t_0)\) the slice \(\Sigma_{(v_0,t_0)}\) is a conformal image of the Clifford torus.
C. \[\max_{(v,t)\in \mathring{B}^4\times[-\pi,\pi]} \operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2.\]
D. \[\operatorname{Area}(\Sigma_{(v,t)})\ge 2\pi^2\qquad\text{for every }(v,t)\in \mathring{B}^4\times[-\pi,\pi].\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus.
E. \[\max_{v\in \mathring{B}^4} \operatorname{Area}(\Sigma_v)\ge 2\pi^2.\] Moreover, equality holds if and only if \(\Sigma\) is the Clifford torus \(\mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\times \mathbb{S}^1\!\left(\tfrac{1}{\sqrt2}\right)\subset \mathbb{S}^3\).

direction of the min-max area bound and misplaced equality on a slice

max over the five-parameter family replaced by pointwise lower bound for every slice

removed the signed-distance parameter t and restricted to the conformal four-parameter subfamily

The stem only defines the geometric family and asks for the correct area estimate; it does not state or strongly hint at the bound 2π^2 or the Clifford torus equality case.

This is close to recalling a known theorem statement about the canonical family, so it is not deeply non-tautological. However, the options do force discrimination among nearby formulations (max vs pointwise, full family vs subfamily, exact vs weaker statement).

Some reasoning is needed to reject plausible variants, especially the weaker true statement and the parameter-confusion distractor. Still, for a well-prepared student the item is mainly theorem recognition rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: reversing the inequality, omitting the equality characterization, replacing a max statement by a pointwise one, and confusing the 5-parameter family with the 4-parameter conformal subfamily.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it tests recall more than genuinely generative reasoning.


\begin{theorem}\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \ref{Table 1}. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \ref{Table 1} such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

Let $q$ be a positive integer.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}

Burgess, in a series of papers~\cite{burgess:1962,Bur1962,Bur19632,Bur1963,burgess86}, proved$$\left|S_{\chi}(M,N)\right| \ll N^{1-\frac{1}{r}}q^{\frac{r+1}{4r^2}+\eps},$$
for $r = 2,3$ and for any $r\ge 4$ when $q$ is cubefree. In recent years, there has been progress in obtaining explicit estimates for $S_{\chi}(M,N)$. In the case when $q$ is prime, Booker \cite{Boo2006} proved a result for $\chi$ quadratic, and McGown \cite{mcgown2012} and Trevi\~no \cite{ET2} made improvements for general $\chi$. The current best result for $q$ prime and $2\le r\le 10$ is due to Francis \cite{francis2021}. An example of an explicit Burgess inequality for $q = p$ prime is the following uniform bound (from \cite{ET2})
$$|S_{\chi}(M,N)| \le 2.74 N^{1-\frac{1}{r}}p^{\frac{r+1}{4r^2}}(\log{p})^{\frac{1}{r}},$$
for all integers $r\ge 2$ and for all primes $p\ge 10^7$.

These explicit results have been used to determine the class number of a quadratic field of large discriminant \cite{Boo2006}, to bound the least quadratic non-residue modulo a prime \cite{francis2021,ET2}, to classify Norm-Euclidean cyclic fields \cite{mcgown2012}, to bound the least primitive root modulo a prime \cite{COT-2016}, to give an explicit bound for $L(1,\chi)$ in the case of quadratic characters \cite{JRT-2023}, to compute the rank of an elliptic curve with large rank \cite{Elkies}, among other applications.

In the case of composite moduli $q$, the only explicit result is the following due to Jain-Sharma, Khale, and Liu \cite{JKL}.
\begin{theoremA*}
 Let $\chi$ be a primitive character mod $q$ with $q \ge e^{e^{9.594}}$. Then, for $N\le q^{\frac{5}{8}}$,
 $$\left|S_{\chi}(M,N)\right|\le 9.07 \sqrt{N} q^{\frac{3}{16}} \log^{\frac{1}{4}}(q)\left(2^{\omega(q)}\tau(q)\right)^{\frac{3}{4}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{2}},$$
 where $\tau(q)$ is the number of divisors of $q$ and $\omega(q)$ is the number of distinct prime factors of~$q$.
\end{theoremA*}

In other words, the only explicit result for composite moduli $q$ is for the case $r =2$. We improve this result for $r =2$ and $q$ large enough, and we also generalize the result to all $r\ge 3$ when $q$ is cubefree.

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{

To state our theorem, we use the following standard labels of arithmetic functions:
let $\omega(q)$ denote the number of distinct prime factors of~$q$, let $\phi(q)$ be Euler's totient function, and let $\tau_{k}(q)$ denote the number of ways to write $q$ as an ordered product of~$k$ integers, so that $\tau(n) = \tau_2(n)$ is the ordinary number-of-divisors function. Also define
\begin{equation} \label{meet Mr. Q}
m_r(q) = \min \biggl\{ \tau_{2r}(q), \biggl( \frac{\tau(q)}2 \biggr)^{2r-1} , \frac{q}{2r} \biggr\}.
\end{equation}

\begin{table}[h!]
\begin{center}
 \begin{tabular}{ | c | c | c|}
 \hline
 $r$ & $C(r)$ & $D(r)$ \\ \hline
 2 & 15.219 & 8.362 \\ \hline
 3 & 5.359 & 4.581\\ \hline
 4 & 3.671 & 3.396\\ \hline
 5 & 2.953 & 2.811\\ \hline
 6 & 2.549 & 2.462\\ \hline
 7 & 2.290 & 2.229\\ \hline
 8 & 2.108 & 2.063\\ \hline
 9 & 1.973 & 1.938\\ \hline
 $\ge$10 & 1.869 & 1.841 \\ \hline
 \end{tabular}
 \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

\begin{theorem}\label{main theorem 2}
 Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
 $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
 S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

\begin{lemma}\label{lem: alpha bound}
Let $r,q\ge 2$ and $N \ge q^{\frac{1}{4}+\frac{1}{4r}} $ be integers and let $i \in \{1,2\}$. Let $A = A_i(r,q)$, $B = B_i(r,q)$, $\kappa$ be defined as in \eqref{choice of kappa}, and $t_i(r)$ be defined as in Lemma \ref{A lower bound lemma}.
Let
\begin{equation*}
 \alpha
 =
 \left[ \frac{\left(\kappa\left(\frac{\phi(q)}{q}\right)+\frac{B2^{\omega(q)-1}}{N} \right)^2}{B}
 +
 2\kappa\left(\frac{6}{\pi^2}\log{(A)}+\delta+\frac{2\log{(A)}+2}{A-1}\right)\right],
\end{equation*}
where $\delta = \frac6{\pi^2} - \frac{36\zeta'(2)}{\pi^4} \approx 0.954422$.
Then, for $q\ge \max\{10^{1145}, (10r)^{18}, t_i(q)\}$,
$$\alpha \le \frac{4}{3}\kappa \log{q}.$$
\end{lemma}

\begin{lemma}\label{lemma to set up the main theorems}
For $i\in\{1, 2\}$ and $T_i$ be defined as in \eqref{eq T1} and \eqref{eq T2}. Let $s, \alpha, u, $ and $w$ be defined as in \eqref{eq s}, \eqref{eq alpha}, \eqref{eq u}, and \eqref{w}, respectively. Then
 \begin{equation*}
\begin{split}
 \left| S_\chi(M, N) \right|
 \le
 \left(\frac{B}{B-1}\right)N^{1-1/r}q^{(r+1)/{4r^2}}(\log q)^{\frac{1}{2r}}
 \left(
 u(\alpha w)^{\frac{1}{2r}}
 +sT_i(q)
 \right).
\end{split}
\end{equation*}
\end{lemma}
\begin{proof}
 Fix $i\in \{1,2\}$ and let $q\ge \max\{10^{1145},t_i(r)\}$. From Lemma \ref{eq after Holder}, once we choose $A$ and $B$ as described in Definition \ref{def A B v}, replacing $T(q)$ with $T_i(q)$ and $C(r)$ with $C$, we have
\begin{align*}
 \left| S_\chi\left(M, N\right)\right|
 &\le
 \frac{1}{\#\cA\cdot \#\cB}\left(\sum_{x=1}^q v_\cA(x)\right)^{1-\frac{1}{r}}
 \left(\sum_{x=1}^q v^2_{\cA}(x)\right)^{\frac{1}{2r}}\left(\sum_{x=1}^q\left|\sum_{b \in \cB}\chi\left(x+b\right)\right|^{2r}\right)^\frac{1}{2r}
 \\
 &\qquad{}+
\frac{2C}{2-1/r} A^{1-1/r} \frac{(\lfloor B\rfloor+1)^{2-1/r}}{\#\cB} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T_i(q). \end{align*}

Similar to the proof of Theorem \ref{main thm 1}, $10^{1145}\ge (10r)^{18}$ for $r\le 10^{50}$ while $t_2(r) \ge (10r)^{18}$ for $r\ge 10^{50}$, thus from \eqref{roughly u} and \eqref{eq alpha upper bound},
\begin{equation*}
 u \le \frac{2^{1/r}}{q^{\frac{1}{2r^2}}}\left(\frac{N}{A\phi^*}\right)^{1/r},
\end{equation*}
and
\[
\alpha \le \frac{4}{3}\kappa \log{q}.
\]
Furthermore,
\begin{equation}\label{f2}
f_2 = (4r)^{\omega(q)}m_r(q)(r-1)+1 \le r(4r)^{\omega(q)}m_r(q),
\end{equation}
and \begin{equation}\label{beta B def 2}\beta = \left(\frac{r-1}{(2r)r!}\right)^{\frac{1}{r}},
\end{equation}
therefore we bound $w$ in \eqref{w} as
\begin{equation}\label{bounding w 2}
 w \le \frac{2^{1+1/r}(r\cdot r!)^{1/r}}{(r-1)^{1+1/r}\log q} \left(4r\right)^{\omega(q)}m_r(q).
\end{equation}
Using \eqref{beta B def 2}, $B = r^2q^{\frac{1}{2r}} \beta$, and $AB = \kappa N$, we get
\begin{equation}\label{bounding N over A 2}
\frac{N}{A q^{\frac{1}{2r}}} = \frac{r^2}{\kappa}\left(\frac{r-1}{r!(2r)}\right)^{\frac{1}{r}}.
\end{equation}
Therefore, combining \eqref{eq s}, \eqref{roughly u}, \eqref{eq alpha upper bound}, \eqref{bounding w 2}, and \eqref{bounding N over A 2}, we get
\begin{align*}
 &u(\alpha w)^{\frac{1}{2r}}+sT_2(q)\\
 &\le 2^{\frac{1}{r}}\left(\frac{N}{Aq^{\frac{1}{2r}}\phi^*}\right)^{\frac{1}{r}}\left(\frac{4}{3}\kappa \log{q}\right)^{\frac{1}{2r}}\left(\frac{2^{1+\frac{1}{r}}(r\cdot r!)^{\frac{1}{r}}r^{\frac{1}{r}}}{(r-1)^{1+\frac{1}{r}}\log{q}}\right)^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\\ &+\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}T_2(q)\\
 &= \left(2^{\frac{5}{2r}-\frac{1}{2r^2}}3^{-\frac{1}{2r}}(r!)^{-\frac{1}{2r^2}}(r-1)^{\frac{1}{2r^2}-\frac{1}{2r}}r^{\frac{2}{r}-\frac{1}{2r^2}}\kappa^{-\frac{1}{2r}} +\frac{2C\kappa^{1-\frac{1}{r}}}{2-\frac{1}{r}}\left(\frac{B+1}{B}\right)^{2-\frac{1}{r}}\right)T_2(q).
\end{align*}
This produces the same conditions on $C$ and $D$ as in the proof of Theorem \ref{main thm 1}, thus we obtain the constants in Table \ref{Table 1}.
\end{proof}

\begin{table}[h!]
\begin{center}
  \begin{tabular}{ | c | c | c|}
    \hline
    $r$ & $C(r)$ & $D(r)$ \\ \hline
    2 & 15.219 & 8.362 \\ \hline
    3 & 5.359 & 4.581\\ \hline
    4 & 3.671 & 3.396\\ \hline
    5 & 2.953 & 2.811\\ \hline
    6 & 2.549 & 2.462\\ \hline
    7 & 2.290 & 2.229\\ \hline
    8 & 2.108 & 2.063\\ \hline
    9 & 1.973 & 1.938\\ \hline
    $\ge$10 & 1.869 & 1.841 \\ \hline
  \end{tabular}
  \caption{Constants in the Burgess inequality for values of $r$.}\label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{theorem}\label{main theorem 2}
    Let $\chi$ be a primitive Dirichlet character modulo $q$ . Let $C(r)$ be defined as in Table \ref{Table 1}. Then, for $q \ge \max\{10^{1145}, 2^{4r-2}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$
\end{theorem}

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} we need the following explicit Weil-type inequality, which we consider of independent interest.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

The paper is organized as follows. We prove Theorem \ref{weil} in Section~\ref{sec: weil}. In Section~\ref{General setup} we describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}. We prove other technical lemmas in Section~\ref{sec: technical}, some important bounds in Section~\ref{section asymptotics}, and we prove our main theorems in Section~\ref{sec: main theorems}.

\subsection*{Notation}
Most of the notation in the paper is standard, for example, $\phi(n)$ is Euler's totient function, for a finite set $S$, $\#S$ is the number of elements of $S$, $\lfloor x\rfloor$ is the floor of $x$, for a pair of integers $a,b$, $(a,b) = \gcd(a,b)$. We also use Landau's $O$ notation, i.e., $f(x) = O(g(x))$ if there exists a positive constant $C$ such that $|f(x)| \le C|g(x)|$ for large enough $x$. A notation we use that is less well-known is $O^*$. We say that $f(x) = O^*(g(x))$ if for all $x\ge x_0$, $|f(x)| \le |g(x)|$.

To prove Theorems \ref{main thm 1} and \ref{main theorem 2} the authors say they "need the following explicit Weil-type inequality" (Theorem \ref{weil}). They then describe the structure of the proof: they prove Theorem \ref{weil} in Section~\ref{sec: weil}, in Section~\ref{General setup} they "describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in \cite[Theorem 12.6]{IK2004}," they prove "other technical lemmas" in Section~\ref{sec: technical} and obtain "some important bounds" in Section~\ref{section asymptotics}, and finally they "prove our main theorems" in Section~\ref{sec: main theorems}.

To prove the main theorem and Theorem \ref{main theorem 2} the authors say they “need the following explicit Weil-type inequality”. We first record this Weil-type inequality.

\begin{theorem}[Weil-type inequality]\label{weil}
Let $r\ge 2$ and $q$ be positive integers such that $r=2$ or $q$ is cubefree. Let $\chi$ be a primitive Dirichlet modulo $q$. Let $B \ge 2$ be a real number.
Then
\begin{equation*}
\sum_{x=1}^q\left|\sum_{1\le b\le B}\chi(x+b)\right|^{2r} \le 2r(4r)^{\omega(q)} B^{2r}m_r(q)\sqrt{q} + \frac{r^{2r}}{r!} B^r q.
\end{equation*}
\end{theorem}

They then describe the structure of the proof as follows: they prove Theorem \ref{weil} earlier in the paper; next, in

\begin{align}\label{eq bad tuples}
    \sum_{(b_1,b_2,\ldots,b_{2r})\in \cB_2}\sum_{x\bmod{q}}\chi(f_1(x)f_2^{(\phi(q)-1)}(x)) \le r^{2r} \binom{\lfloor B\rfloor}{r} q \le \frac{r^{2r} \lfloor B\rfloor^r}{r!}q.
\end{align}
Combining the bounds in \eqref{eq good tuples} and \eqref{eq bad tuples}, we get the desired upper bound in the Weil-type inequality.
\end{proof}

\section{General setup}\label{General setup}

Let $r\ge 2, N$ be positive integers, and let $q$ be a positive integer that is cubefree when $r\ge 3$ for the remainder of the paper.
Given a primitive Dirichlet character $\chi\pmod q$ and integers~$M$ and ~$N\ge1$, we consider the character sum
\begin{equation*}
    S_\chi\left(M, N\right)=\sum_{M<n \le M+N}\chi(n).
\end{equation*}
Given an integer $r\ge2$, we use induction to prove a bound of the form
\begin{equation} \label{eq desired bound}
| S_\chi(M,N) | \le E_{q,r}(N), \quad\text{where }
E_{q,r}(N) = C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q),
\end{equation}
where $C\ge 1$ is a constant to be determined later, $m_r(q)$ is defined in \eqref{meet Mr. Q}, and
\begin{equation}\label{meet T(q)}
T(q) = \begin{cases}\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}-\frac{1}{2r^2}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main thm 1},\\
\bigl((4r)^{\omega(q)} m_r(q)\bigr)^{\left({\frac{1}{2r}}\right)} \bigl( \frac q{\phi(q)} \bigr)^{1/r}& \text{ under the conditions of Theorem } \ref{main theorem 2}
.\end{cases}
\end{equation}

We take $1\le N\le q^{\frac{1}{4}+\frac{1}{4r}}$ as the base case. In this case, the inequality~\eqref{eq desired bound} follows from the trivial bound $|S_\chi(M,N)| \le N$, since
\begin{align*}
&N \le C N^{1-1/r} q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N^{1/r} \le C q^{\frac{r+1}{4r^2}} (\log q)^{\frac{1}{2r}} T(q) \\
\implies &N \le q^{\frac{r+1}{4r}} \bigl( C (\log q)^{\frac{1}{2r}} T(q) \bigr)^r,
\end{align*}
when $N \le q^{\frac{1}{4}+\frac{1}{4r}}$ (because $C(\log q)^{\frac{1}{2r}} T(q) \ge 1$).
\\
Therefore, in the induction step, we assume
\begin{align}\label{inductive hypothesis}
\text{for $N > q^{\frac{1}{4}+\frac{1}{4r}}$, the bound in \eqref{eq desired bound} holds for all smaller positive integers.}
\end{align}

they “describe the overall plan for the proofs of our main theorems following the approach for the explicit Burgess inequality detailed in E. Kowalski and H. Iwaniec, \cite[Theorem 12.6]{IK2004}.” They then prove other technical lemmas later, obtain some important bounds later, and finally establish the main theorem together with Theorem \ref{main theorem 2}.

\label{main thm 1}
Let $r\ge 2$ be an integer and $\chi$ be a primitive Dirichlet character modulo~$q$. Let $C(r)$ be defined as in Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{. Let $a(r) = 2\log{2}\left(3.0758r+1.38402\log(4r)-1.5379\right)$. Then, for $q \ge \max\{10^{1145},e^{e^{a(r)}}\}$, if $r=2$ or $q$ is cubefree, we have
    $$|S_{\chi}(M,N)| \le C(r)N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$

    Furthermore, as $q\rightarrow\infty$, we have a constant $D(r)$ from Table \label{Table 1}
\end{center}
\end{table}

The theorem above requires $q$ to be very large, so we have the following theorem which is weaker asymptotically but works for smaller values of $q$.

\begin{ such that
    $$|S_{\chi}(M,N)| \le (D(r)+o(1))N^{1-\frac{1}{r}} q^{\frac{r+1}{4r^2}}(\log{q})^{\frac{1}{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac{1}{2r}-\frac{1}{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac{1}{r}}.$$,

Question:
Let $q$ be a positive integer, let $\chi$ be a primitive Dirichlet character modulo $q$, and for integers $M$ and $N\ge 1$ define
$$S_{\chi}(M,N)=\sum_{M<n\le M+N}\chi(n).$$
For arithmetic functions, let $\omega(q)$ be the number of distinct prime factors of $q$, let $\phi(q)$ be Euler’s totient function, let $\tau_k(q)$ be the number of ordered factorizations of $q$ into $k$ positive integers (so $\tau(q)=\tau_2(q)$), and define
$$m_r(q)=\min\left\{\tau_{2r}(q),\left(\frac{\tau(q)}{2}\right)^{2r-1},\frac{q}{2r}\right\}.$$
Fix an integer $r\ge 2$, and set
$$a(r)=2\log 2\,\bigl(3.0758r+1.38402\log(4r)-1.5379\bigr).$$
Let $C(r)$ and $D(r)$ be the constants given by
$$(C(r),D(r))=\begin{cases}
(15.219,8.362),&r=2,\\
(5.359,4.581),&r=3,\\
(3.671,3.396),&r=4,\\
(2.953,2.811),&r=5,\\
(2.549,2.462),&r=6,\\
(2.290,2.229),&r=7,\\
(2.108,2.063),&r=8,\\
(1.973,1.938),&r=9,\\
(1.869,1.841),&r\ge 10.
\end{cases}$$
Assume that either $r=2$ or $q$ is cubefree. Which quantitative estimate holds for $S_\chi(M,N)$?

For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$

Choices:
B.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
C.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
D.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, if $q$ is cubefree then
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
E.
For all integers $M$ and $N\ge 1$, if $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, then
$$|S_{\chi}(M,N)|\le C(r)N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1r}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$
Furthermore, as $q\to\infty$, under the same hypothesis $r=2$ or $q$ cubefree,
$$|S_{\chi}(M,N)|\le (D(r)+o(1))N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1r}\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}\left(\frac{q}{\phi(q)}\right)^{\frac1r}.$$

The stem does not explicitly reveal the correct option. It supplies the hypotheses and notation, but the exact bound must still be identified from the choices.

This is essentially a theorem-recall item: the question asks which estimate holds after presenting all the setup for a specific result. The correct choice is basically the theorem statement itself.

Some comparison is needed among close variants of the bound, especially exponents, hypotheses, and the asymptotic clause. However, the task is mainly recognition of the exact stated theorem rather than generating a conclusion from mathematical reasoning.

The distractors are plausible and mathematically targeted: they alter a power, remove the asymptotic clause, restrict the hypothesis incorrectly, or change the logarithmic exponent. These reflect realistic failure modes.

A technically well-constructed recall/recognition MCQ with strong distractors, but it is largely tautological and only moderately tests reasoning rather than theorem memorization.


\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \lVertu\rVert_{C^s(\Omega\cap B_{1/2})}\le C(\lVertg\rVert _{L^\infty(\Omega\cap B_4)} + \lVertu\rVert _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.

\begin{theorem}\label{1bdry C^s}
    Suppose $\Omega$ is a locally $C^{1,1}$ domain, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
    \begin{equation}
        \begin{cases}
            \flap u = g,&\text{in } \Omega\cap B_4,\\
            u=0, &\text{in } B_4\backslash \Omega.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
    \[
    \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
    \]
    where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}

\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
    \end{equation}
\end{theorem}

In this paper, we consider the following fractional problem in  unbounded domain $\Omega$
 \begin{equation}\label{main1}
        \begin{cases}
            \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}
First, we derive a boundary regularity estimate for  solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain  a priori estimates for solutions to nonlinear equations on unbounded domains with  boundaries \begin{equation}\label{main2}
\begin{cases}
    \mbox{$(-\triangle)^s$} u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$  denotes the fractional Laplacian,  defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

\[\lVertu\rVert_{{\mathcal{L}}_{2s}}:=\int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx.\]

The fractional Laplacian is a nonlocal integro-differential operator that effectively models memory effects and long-range diffusion phenomena (Chaves \cite{Chave1998}; Metzler and Klafter \cite{MK2000}). Specific applications include random walks with memory, Lévy flights \cite{MS1984, SZK1993, N1986}, and kinetic theories of systems exhibiting chaotic \cite{SZ1997, Z2002} and pseudo-chaotic \cite{ZE2001} dynamics. Beyond these, the operator finds significant applications across diverse domains including probability theory and finance, image processing and computer vision, mathematical biology and ecology, as well as geophysics and climate modeling.

 Nonlocal operators, particularly the fractional Laplacian, have attracted considerable attention in the mathematical community owing to their broad applications. For foundational developments in this field, we refer to the seminal work of Caffarelli and Silvestre \cite{CS07}, along with subsequent contributions by  Stinga \cite{Stinga2019}, Dipierro et al. \cite{Dipierro2019}, and the comprehensive monograph by Bucur and Valdinoci \cite{BV2016}. Regarding qualitative properties of solutions, extensive results can be found in \cite{CDQ,CDG,CG,CGLO,CHL17,DQ, FL,FLS}, with specific advances in a priori estimates documented in \cite{CGL,Chen2016directblowup,ouyang2023}.

    The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}.  Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
    For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
    The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
    For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an  unbounded domain {\em without boundary}.

Then for general unbounded domains  with  boundaries, can one derive a priori estimates for the solutions?

This requires boundary H\"older estimates in terms of only the local $L^{\infty}$ norm of the solutions.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors  require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which  an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for  a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution  to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident   from their proof that  in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary  regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions,  in which  we use {\em only the local $L^{\infty}$  norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

In this paper, we consider the following fractional problem in unbounded domain $\Omega$
 \begin{equation}\label{main1}
 \begin{cases}
 \mbox{$(-\triangle)^s$} u(x) = g(x),&\text{in } \Omega,\\
 u(x)=0, &\text{in } \Omega^c.\\
 \end{cases}
 \end{equation}
First, we derive a boundary regularity estimate for solutions
in terms of only the local $L^{\infty}$ norm of $u$ and $g$.
Then as an important application, we obtain a priori estimates for solutions to nonlinear equations on unbounded domains with boundaries \begin{equation}\label{main2}
\begin{cases}
 \mbox{$(-\triangle)^s$} u (x) = f(x,u) , & x\in \Omega,\\
 u (x) = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}
Here, $(-\Delta)^s$ denotes the fractional Laplacian, defined as
\begin{eqnarray}\label{eq1-1}
\begin{aligned}
(-\Delta)^s u(x)&=C_{n, s}P.V. \int_{\mathbb{R}^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy\\
&=C_{n, s}\lim_{\varepsilon \to 0}\int_{\mathbb{R}^n\backslash B_\varepsilon(0)}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,
\end{aligned}
\end{eqnarray}
where $P.V.$
stands for the Cauchy principal value, and $C_{n, s}$ is a
dimensional constant.
In order that the integral on the right hand side of \eqref{eq1-1} is well defined in the classical sense, we require that $u\in C_{loc}^{1, 1}(\Omega)\cap \mathcal{L}_{2s}$, where
$$
\mathcal{L}_{2s}=\left\{u\in L_{loc}^1(\mathbb{R}^n) \,\Big| \int_{\mathbb{R}^n}\frac{|u(x)|}{1+|x|^{n+2s}}dx<\infty \right\}
$$
endowed with norm

The regularity properties of solutions to fractional order equations have also been extensively studied. The foundation was laid by Silvestre \cite{Silvestre2007Regularity}, who established Schauder and H\"older estimates for entire solutions. Then the interior type estimates are studied in \cite{Chen2010frac}. Later, Li and Wu \cite{Li2021Pointwise} established interior pointwise regularity estimates for fractional equations, a result analogous in spirit to Caffarelli's seminal work \cite{Caffarelli1989Ann} on pointwise estimates for viscosity solutions of fully nonlinear elliptic equations.
 For boundary regularity, Ros-Oton and Serra \cite{Rosoton2014Dirichlet} proved optimal regularity for the Dirichlet problem in domains satisfying an exterior ball condition (with zero exterior data), while their follow-up work \cite{Rosoton2014extremal} treated general exterior data in $C^{1,1}$ domains.
 The regularity theory has been further developed from different perspectives: Servadei and Valdinoci \cite{Servadei2014weak} investigated both weak and viscosity solutions, while Caffarelli-Stinga \cite{CS2016} and Stinga-Torrea \cite{ST2010} provided additional important regularity criteria.
 For more results concerning the nonlocal elliptic equations, please see \cite{Fall2019regulariy,Fall2020reg,Dyda2020reg}, while for fractional parabolic equations, we refer the readers to \cite{CGL,Chen2010Heat,Bogdan2010heat,Fernandezreal2016boundary,Stinga2017reg,Kassmann2024parabolic}.

In particular, in \cite{chen2025refinedregularitynonlocalelliptic}, the authors establish interior H\"older and Schauder regularity estimates for fractional equation \eqref{main1}. This makes it possible to derive a priori estimate for nonnegative solutions of \eqref{main2} when $\Omega=\mathbb{R}^n$, an unbounded domain {\em without boundary}.

However, in the existing results, such as the ones in \cite{BDGQ2018} and \cite{Barrios2019note}, in order to obtain the $C^s$ boundary H\"older continuity of the solutions for \eqref{main1} when $\Omega=\mathbb{R}^n_+$, the authors require that solution $u$ is globally bounded. Their proof mainly based on the proposition 1.1 in \cite{Rosoton2014Dirichlet}, in which an optimal boundary H\"older estimate in terms of $\lVertg\rVert_{L^\infty(\Omega)}$ is established for a {\em bounded domian} $\Omega$ with zero exterior Dirichlet condition. It is well known that in this situation, one can construst a supersolution to show that the global $L^{\infty}$ norm of $u$ is bounded by $\lVertg\rVert_{L^\infty(\Omega)}$. When $\Omega$ is an {\em unbounded domain}, it is evident from their proof that in order to obtain local boundary H\"older continuity, the solution $u$ is required to be globally bounded, as indicated by \cite{BDGQ2018} and \cite{Barrios2019note}. For general unbounded domains, Dipierro, Soave and Valdinoci in \cite{DSV17} obtain $C^{\alpha}$ boundary regularity for some $0<\alpha<s$ under the condition that the solution $u$ of \eqref{main1} be globally bounded.

With these existing boundary regularity estimates, it is inadequate to carry out blow up and rescaling arguments on unbounded domains aimed at obtaining a priori estimates, because the rescaled functions may be globally unbounded.

One of our main result is to establish the local version of the boundary regularity for nonnegative solutions, in which we use {\em only the local $L^{\infty}$ norm of $u$ instead of the global ones} to control its local H\"older norm up to the boundary.

\begin{equation}\label{main1}
        \begin{cases}
            \flap u(x) = g(x),&\text{in } \Omega,\\
            u(x)=0, &\text{in }  \Omega^c.\\
        \end{cases}
    \end{equation}

\begin{equation}\label{main2}
\begin{cases}
    \flap u (x) = f(x,u)  , & x\in \Omega,\\
    u (x)  = 0 , &x\in \Omega ^c.
\end{cases}
\end{equation}

As an important application, we establish a priori estimate for the solutions to \eqref{main2}. We assume that $\Omega \subset \R ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
 \item $f(x,t):\Omega \times [0,\infty)\to\R$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
 f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
 \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
 where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

To begin with, let us recall a classical result in
\cite{Rosoton2014Dirichlet}, in which the authors establish an optimal boundary regularity for the fractional equation with zero exterior Dirichlet condition.
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
 Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
 \begin{equation}\label{eq1}
 \begin{cases}
 \flap u = g,&\text{in } \Omega,\\
 u=0, &\text{in } \Omega ^c.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
 \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
Notice that here $\Omega$ is a {\em bounded domian}. In this situation, a supersolution and a subsolution can be constructed to show that the global $L^{\infty}$ norm of $u$ is bounded by $\norm{g}_{L^\infty(\Omega)}$ as the following. Let
\[\psi(x):=(1-|x|^2)_+^s.\]
It is well known that
\[\flap \psi (x)=a>0.\]
Choose $R$ sufficiently large, so that $\Omega\subset B_R(0).$ Denote
\[\bar{u}(x):=\norm{g}_{L^{\infty}(\Omega)}\frac{R^{2s}}{a}\psi_R(x), ~\mbox{with}~ \psi_R(x):={\psi(\frac{x}{R})}.\]
Then it can be easily verified that $\bar{u}$ and $-\bar{u}$ are super-solution and sub-solution of \eqref{eq1}, respectively. Therefore,
\begin{equation*}
 \norm{u}_{L^{\infty}(\Omega)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation*}
From the above construction, one can see that in order to obtain \eqref{hd}, it is necessary that $\Omega$ be bounded.
When $\Omega$ is an {\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\em unbounded domains with boundaries}.

This motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\infty}}$ norm of the solution is involved.
\begin{theorem}\label{bdry C^s}
 Suppose $\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $0<s<1$, $g\in L^\infty(\Omega\cap B_4)$ and $u$ is a nonnegative classical solution of
 \begin{equation}
 \begin{cases}
 \flap u = g,&\text{in } \Omega\cap B_4,\\
 u=0, &\text{in } B_4\backslash \Omega.\\
 \end{cases}
 \end{equation}
 Then $u\in C^{s}(\Omega\cap B_{1/2})$ and
 \[
 \norm{u}_{C^s(\Omega\cap B_{1/2})}\le C(\norm{g} _{L^\infty(\Omega\cap B_4)} + \norm{u} _{L^\infty(\Omega\cap B_4)} ),
 \]
 where $C$ depends on $n,s$ and $C^{1,1}_{loc}$ norm of $\partial\Omega$, and $B_r$ denotes a ball centered at any fixed point on the boundary $\partial\Omega$ with radius $r$.
\end{theorem}
The idea of the proof is that we divide the solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Theorem \ref{otonserra}. For the harmonic part, we firstly derived an $\alpha$ power order decay near $\partial\Omega$, here $\alpha$ is a certain positive number which may be less than $s$. Combining the decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Finally we refine the boundary regularity by an iteration progress.

To continue estimate $\norm{u}_{C^{\alpha_0}(\Omega\cap B_{1/2})}$, we need the interior H\"older estimate from \cite{chen2025refinedregularitynonlocalelliptic}.
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
 Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
 \begin{equation}
 \flap u(x) = g(x),\quad x\in B_1
 \end{equation}
 where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
 \begin{equation}
 \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
 \end{equation}
\end{proposition}
Based on this, we arrive at
\begin{lemma}\label{global holder lemma}
 The decay estimate \eqref{decay estimate} and Proposition \ref{Holder thm} implies that
 \begin{equation}
 \norm{u} _{C^{\alpha _0}(\Omega\cap B_{1/2})}\le C(\norm{u}_{L^\infty(\Omega\cap B_4)}+\norm{g}_{L^\infty(\Omega\cap B_4)})
 \end{equation}
 where $C$ depends on $s,n$, and $\norm{\partial \Omega}_{C^{1,1}_{loc}}$, $\alpha_0:=\min\{s,1-s\}\in (0,1/2)$.
\end{lemma}

\section{Appendix}
Here is a generalization of \cite[Theorem 1.1]{Du2023blowup}Du-Jin-Xiong-Yang's convergence result.
\begin{theorem}
 Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
 converges to a function $u\in \mathcal{L}_{2s}$ in $C^{2s+\beta}_{loc}(\Omega)$, and $\{\flap u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
 \begin{equation}
 \lim _{i\to\infty}\flap u_i(x) = \flap u(x) -b.
 \end{equation}
\end{theorem}
\begin{proof}
 Fix any $x\in\Omega$, consider $R\gg|x|+1$.
 \begin{align*}
 \flap u(x)-\flap u_i(x) & = c_{n,s}\int _{B_R}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\ &+c_{n,s}\int _{B_R^c}\frac{(u-u_i)(x)-u(y)}{|x-y|^{n+2s}}dy\\
 & + c_{n,s}\int _{B_R^c}\frac{u_i(y)}{|x-y|^{n+2s}}dy\\
 & := A_i(x,R) + E_i(x,R) + F_i(x,R).
 \end{align*}
 For $E_i$, since $u\in \mathcal{L}_{2s}$,
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}E_i(x,R) = -c_{n,s}\lim _{R\to\infty}\int _{B_R^c}\frac{u(y)}{|x-y|^{n+2s}}dy =0.
 \end{equation}
 The consideration of $A_i$ is a little bit different with that in \cite{Du2023blowup}. In our cases, $B_R$ is not contained in $\Omega$ for sufficiently large $R$. Fix $\varepsilon >0$ such that $B_\varepsilon(x)\subset \Omega$. Then
 \begin{align*}
 A_i(x,R) & = c_{n,s}\int _{B_R\backslash B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\
 &+c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy
 \end{align*}
 By dominated convergence theorem, the first term converges to 0 as $i\to\infty$. For the second term,
 \[
 \lim _{i\to\infty} c_{n,s}\int _{B_\varepsilon(x)}\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\le c_{n,s}\lim _{i\to\infty}[u-u_i]_{C^{2s+\beta}(B_\varepsilon(x))}\varepsilon ^\beta = 0.
 \]
 Hence $\lim _{i\to\infty}A_i(x,R)=0$. Therefore
 \begin{equation}
 \lim _{R\to\infty}\lim _{i\to\infty}A_i(x,R)=0.
 \end{equation}
 Then the same argument in \cite{Du2023blowup} implies $\lim _{i\to\infty}F_i(x,R)$ exists and
 \[
 \lim _{R\to\infty}\lim _{i\to\infty}F_i(x,R) =\lim _{R\to\infty}\lim _{i\to\infty}F_i(0,R) = :b\ge 0.
 \]
\end{proof}

The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we  first carry out a detailed analysis to   derive an $\alpha$ power order decay near $\partial\Omega$,
\begin{equation*}
        |h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\operatorname{dist}(x,\partial \Omega)^{\alpha },
    \end{equation*}
    where $\alpha = \min\{s,1-s\}$. Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}), we derive the $C^\alpha$ boundary regularity. Then by an iteration process, we increase the power $\alpha$ successively until it reaches the desired power $s$.

As an important application, we establish a priori estimate for the solutions  to \eqref{main2}.  We assume that $\Omega \subset \mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:
\begin{itemize}
    \item $f(x,t):\Omega \times [0,\infty)\to\mathbb{R}$ is uniformly H\"older continuous with respect to $x$ and continuous with respect to $t$.
\end{itemize}
\begin{theorem}\label{1A1}
Assume $1<p<\frac{n+2s}{n-2s}$, $f$ satisfies
\begin{equation}\label{growth f}
    f(x,t)\le C_0(1+t^p)
\end{equation}
uniformly for all $x$ in $\Omega$, and
\begin{equation}\label{lim f}
    \lim _{t\to\infty}\frac{f(x,t)}{t^p} = K(x),
\end{equation}
    where $K(x)\in (0,\infty)$ is uniformly continuous and $K(\infty):=\lim _{|x|\to\infty }K(x) \in (0,\infty)$.
Then there exists a constant $C$, such that
\begin{equation}\label{bd1}
u(x)\le C,\quad\forall \, x\in \Omega
\end{equation}
holds for all nonnegative solutions $u$ of \eqref{main2}.
\end{theorem}

\begin{remark}
    The assumptions on $f$ are essential for the a priori estimate. When $f=0$ and $\Omega = B_1$, Abatangelo, Jarohs and Salda\~na \cite{Abatangelo2018Green} constructed nontrivial nonnegative solutions to \eqref{main2} that are unbounded, see also \cite{Li2023unique}.
\end{remark}

During the process of blow-up and rescaling, it can be shown that a sequence of the rescaled functions (still denoted by $u_i$) converges to a limiting function $u$ in the sense of $C^{2s+\beta}_{loc}$. In the case of a local operator, say $-\Delta$, we already have
\[   \lim _{i\to\infty}(-\Delta) u_i(x) = (-\Delta) u(x).\]
However, this is not the case for nonlocal operators, as shown in \cite[Theorem 1.1]{Du2023blowup} by Du-Jin-Xiong-Yang, in which they work on the whole space $\mathbb{R}^n$. In order to establish a priori estimate on general  unbounded domains with boundaries, we generalize  their convergence result as follows.
\begin{theorem}\label{conv}
    Assume $n\ge 1$, $s\in (0,1)$, $\beta\in (0,1)$. $\Omega$ is an unbounded domain with boundary. Suppose nonnegative functions $\{u_i\}\subset \mathcal{L}_{2s}\cap C^{2s+\beta}_{loc}(\Omega)$ vanishes outside $\Omega$, and $u\in \mathcal{L}_{2s}$ vanishes outside $\Omega$. If $\{u_i\}$
    converges in $C^{2s+\beta}_{loc}(\Omega)$ to a function $u\in \mathcal{L}_{2s}$ , and $\{\mbox{$(-\triangle)^s$} u_i \}$ converges pointwisely in $\Omega$, then there exists a constant $b\ge 0 $ such that
    \begin{equation}
        \lim _{i\to\infty}\mbox{$(-\triangle)^s$} u_i(x) = \mbox{$(-\triangle)^s$} u(x) -b.
    \end{equation}
\end{theorem}

 In Section 2, we establish local boundary H\"older regularity  for \eqref{main1} and prove Theorem \ref{1bdry C^s}. In Section 3, we derive  a priori estimates for \eqref{main2} and validate Theorem \ref{1A1}. Finally, the proof of Theorem \ref{conv} is included in the Appendix.

To prove Theorem~\ref{1bdry C^s}, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by Proposition \ref{otonserra}.” For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. “Combining this decay estimate and the interior regularity result (Theorem \ref{Holder thm}),” one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

To prove the main theorem, the proof “divide[s] a given solution into two parts: the potential part and the harmonic part.” The “regularity for the potential part is obtained by
\begin{proposition}[\cite{Rosoton2014Dirichlet}]\label{otonserra}
    Suppose $\Omega$ is a bounded Lipschitz domain with exterior ball condition, $g\in L^\infty(\Omega)$, $s\in (0,1)$, $u$ is a classical solution of the following Dirichlet problem:
    \begin{equation}\label{eq1}
        \begin{cases}
            \flap u = g,&\text{in } \Omega,\\
            u=0, &\text{in } \Omega ^c.\\
        \end{cases}
    \end{equation}
    Then $u\in C^{s}(\R^n)$ and
\begin{equation}\label{hd}
    \norm{u}_{C^s(\R^n)}\le C\norm{g}_{L^\infty(\Omega)},
\end{equation}
where $C=C(n,s)$ is a positive constant.
\end{proposition}
For the harmonic part $h$, one “rewrite[s] it in terms of the Poisson representation formula in balls,” and from this explicit expression one derives an $\alpha$-power decay near $\partial\Omega$:
\[
|h(x)|\le C\{\lVertu\rVert_{L^\infty(\Omega \cap B_4)}+\lVertu\rVert_{\cL_{2s}}\}\,\operatorname{dist}(x,\partial \Omega)^{\alpha},
\]
with $\alpha=\min\{s,1-s\}$. Combining this decay estimate and the interior regularity result,
\begin{proposition}[Interior H\"older regularity]\label{Holder thm}
    Assume $0<s<1$, and $u\in C^{1,1}_{loc}(\R^n)\cap \mathcal{L}_{2s}$ is a nonnegative solution of
    \begin{equation}
        \flap u(x) = g(x),\quad x\in B_1
    \end{equation}
    where $g\in L^\infty(B_1)$. Then for any $0<\varepsilon <2s$,
    \begin{equation}
        \norm {u}_{C^{[2s-\varepsilon],\{2s-\varepsilon\}}(B_{1/2})}\le C(\norm{g}_{L^\infty(B_1)}+\norm{u}_{L^\infty(B_1)}).
    \end{equation}
\end{proposition}
one gets “the $C^\alpha$ boundary regularity.” Then “by an iteration process,” one “increase[s] the power $\alpha$ successively until it reaches the desired power $s$,” yielding the stated $C^s$ boundary regularity and estimate.

Question:
Let \(\Omega\subset \mathbb{R}^n\) be a domain with locally \(C^{1,1}\) boundary, let \(0<s<1\), and fix a boundary point \(x_0\in \partial\Omega\). Write \(B_r:=B_r(x_0)\). Assume \(g\in L^\infty(\Omega\cap B_4)\), and let \(u\ge 0\) be a classical solution of
\[
\begin{cases}
(-\Delta)^s u=g,& \text{in }\Omega\cap B_4,\\
u=0,& \text{in }B_4\setminus \Omega,
\end{cases}
\]
where the fractional Laplacian is
\[
(-\Delta)^s v(x)=C_{n,s}\,\mathrm{P.V.}\int_{\mathbb{R}^n}\frac{v(x)-v(y)}{|x-y|^{n+2s}}\,dy.
\]
Which quantitative estimate holds for \(u\) up to the boundary in \(\Omega\cap B_{1/2}\)?

One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]

Choices:
B.
One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\,\|g\|_{L^\infty(\Omega\cap B_4)}.
\]
C.
One has \(u\in C^{\alpha_0}(\Omega\cap B_{1/2})\), where \(\alpha_0:=\min\{s,1-s\}\), and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{\alpha_0}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]
D.
One has \(u\in C^{s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{\mathcal L_{2s}}\bigr).
\]
E.
One has \(u\in C^{2s}(\Omega\cap B_{1/2})\) and there exists a constant \(C\), depending only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\), such that
\[
\|u\|_{C^{2s}(\Omega\cap B_{1/2})}
\le C\bigl(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\bigr).
\]

The stem gives hypotheses and asks for the boundary estimate, but it does not explicitly state or strongly telegraph the correct conclusion. The correct choice must be inferred from regularity theory.

This is close to a theorem-recall question: the hypotheses are presented and the user is asked to identify the matching quantitative conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement.

Some reasoning is required to distinguish the optimal Hölder exponent, the necessity of the local L-infinity term, and the incorrect stronger/weaker variants. Still, the task mainly tests recognition of the standard estimate rather than deeper derivation.

The distractors are strong: one is a weaker true statement, one omits a necessary term, one swaps in a tail norm, and one overstates regularity to C^{2s}. These are plausible and reflect common mathematical mistakes.

A solid theorem-discrimination MCQ with high-quality distractors and little answer leakage, though it leans more toward recall of a known result than genuinely generative reasoning.