[$2\pi^2$-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.


\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.


\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\left\lvert\lambda\right\rvert$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\left\lvert\lambda\right\rvert .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\left\lvert\lambda\right\rvert>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.

\begin{theorem}\label{theorem: radius of essential spectrum}
    On the half-plane $\Re(\beta) >(1-\alpha)/2$, the radius $\rho_e(L_\beta):= \sup_{\lambda\in \sigma_\varepsilon(L_\beta)}\abs{\lambda}$ of the essential spectrum satisfies
    \[\rho_e(L_\beta)\leq \rho(\mathcal{L}_{\Re(\beta) +\alpha}):=\sup_{\lambda\in \sigma(\mathcal{L}_{\beta+\alpha})}\abs{\lambda} .\]
    Furthermore, the generalised eigenspaces belonging to eigenvalues $\lambda$ with $\abs{\lambda}>\rho_e(L_\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\mathcal{L}_\beta$.
\end{theorem}

\begin{remark}\label{remark: abstract}
    The zeroes of $Z(s)$ on the line $s=1/2+it$ are those for which there are eigenvalues of $\Delta$ called \enquote{Maass cusp forms} with eigenvalue $s(1-s)$. Furthermore the multiplicity of the zeroes and the eigenspaces of $\Delta$ match. The other zeroes of $Z(s)$ are a simple zero at $s=1$ and at the values of $s$ for which $2s$ is a nontrivial zero of the Riemann zeta function $\zeta$. Since $\rho(\mathcal{L}_a)<1$ when $a>1$, we see that the eigenspaces of $\mathcal{L}_\beta$ and $L_\beta$ for the eigenvalues $\lambda=\pm 1$ are identical when $1>\alpha>1-\Re(\beta)$. In particular, the eigenspaces associated to the Maass cusp forms are preserved when $\alpha >1/2$, and the eigenspaces associated to the nontrivial zeroes of the Riemann zeta function\footnote{In fact, the zeroes of $Z(s)$ associated with the nontrivial zeroes of $\zeta$ always occur due to $1$ being an eigenvalue of $\mathcal{L}_\beta$.} to the right of the critical line are preserved when $\alpha=3/4$.
\end{remark}

Let $G:(0,1)\to [0,1)$ be the Gauss map
\begin{equation}\label{equation: gauss map}
 G(x) = \frac{1}{z}-\left\lfloor\frac{1}{z}\right\rfloor.
\end{equation}
We associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\to \mathbb{C}$:
\begin{equation}\label{equation: Mayer transfer operator}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}f\left(\frac{1}{n+z}\right),
\end{equation}
where $\beta \in \{z\in \mathbb{C}: \Re(\beta)>1/2\}$. When $\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\mapsto f\circ G$. The operators in the case $\beta \neq 1$ were first introduced by Mayer \cite{Mayer90}, who considered them acting on the Banach space of functions $A_\infty(D)$ which are holomorphic on $D:=\{z\in\mathbb{C}: \left\lvertz-1\right\rvert< 3/2\}$ with continuous extension to $\overline{D}$.

For this choice of Banach space, Mayer also showed that the operator-valued function $\beta\mapsto \mathcal{L}_{\beta}$ admits a meromorphic continuation on $\mathbb{C}$, with simple poles at $\beta=k/2$ for $k\in\{1,0,-1,-2,\ldots\}$. The continuation for $\Re(\beta) >0$ can be obtained by rewriting $f=f(0)+f^\star$ with $f^\star:=f-f(0)$ and noting that
\begin{equation}\label{equation: Mayer transfer extended}
 \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $\zeta_H(s,z)=\sum_{n=0}^\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\beta \geq -k/2$ for any $k\in\mathbb{N}$ by noticing that
\begin{equation}\label{equation: Mayer transfer extended2}
 \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}
where $f^\star(z)=f(z)- \sum_{n=0}^k\frac{d^nf}{d^nz}(0)z^n/n!$.
Furthermore $\mathcal{L}_{\beta}$ is a nuclear operator on $A_\infty(D)$ of order $0$ for all $\beta$.

The Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\mathbf{M}$ obtained from considering the Poincaré upper half plane $\mathbb{H}:\{x+iy:x,y\in \mathbb{R}, y>0\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\mathbb{H}\ni z\sim (az+b)/(cz+d)$ for all $a,b,c,d\in\mathbb{Z}$ with $ad-bc=1$. Since $\mathcal{L}_{\beta}$ is a nuclear operator, the Fredholm determinants $\det(1-\mathcal{L}_{\beta})$ and $\det(1+\mathcal{L}_{\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula
\begin{equation}\label{equation: zeta functions}
 Z(s)= \det(1-\mathcal{L}_s)(1+\mathcal{L}_s),
\end{equation}
see \cite{Mayer91}.
The Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \cite{Iwaniec}. Equation \eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\Delta$ with that of $\mathcal{L}_\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.

From a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\alpha\in (0,1)$, let $C^\alpha([0,1])$ denote the space of $\alpha$-Hölder functions on $[0,1]$. If $\alpha = k+\eta$ with $k\in\mathbb{N}$ and $\eta\in [0,1)$, let $C^\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\eta$-Hölder if $\eta=0$ or $\eta\in (0,1)$ respectively. Denote by $L_\beta$ the operator \eqref{equation: Mayer transfer extended2} acting on $C^\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\alpha$. These operators are not compact, but we can decompose the spectrum $\sigma(L_\beta)$ into the essential spectrum $\sigma_e(L_\beta)$ and the discrete spectrum $\sigma_{\mathrm{disc}}(L_\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.

\begin{equation}\label{equation: Mayer transfer extended}
    \mathcal{L}_{\beta}(f)(z)=f(0)\zeta_H(2\beta,z+1)+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

\begin{equation}\label{equation: Mayer transfer extended2}
    \mathcal{L}_{\beta}(f)(z)=\sum_{n=0}^k\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta,z+1)}{n!}+\sum_{n=1}^\infty \frac{1}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right),
\end{equation}

Many of the usual theorems in complex analysis continue to hold for Banach or topological vector space-valued functions. In particular, holomorphic implies infinitely differentiable with a locally converging Taylor expansion. The unique extension theorem also continues to hold.
Finally, note that it is straightforward to show that $\beta\to L_\beta$ is a meromorphic function on $\Re(\beta)>(1-\alpha)/2$ for any $\alpha\in\mathbb{R}_{>0}$ with derivative
\[ \left(\frac{d L_\beta}{d\beta}\right)(f)(z)= -\sum_{n=0}^k(n+2\beta)\frac{d^nf}{dz^n}(0)\frac{\zeta_H(n+2\beta+1,z+1)}{n!} -2 \sum_{n=1}^\infty \frac{\ln(n+z)}{(n+z)^{2\beta}}(f^\star)\left(\frac{1}{n+z}\right).\]
\section{Uniformly Bounded Essential Spectrum}\label{section: uniformly bounded essential spectrum}
By \eqref{equation: Nussbaum}, we can estimate the essential spectral radius by finding for each $l\in \mathbb{N}$ a compact operator $T'$ for which $\|L_\beta^l - T' \|$ is small. A first naive approach would be to consider the operator $P$ which sends $f$ to a piecwise linear approximation that interpolates $f$ on the points $0,\frac{1}{N},\frac{2}{N},\cdots,1$ and considering $\|L_\beta^l-L_\beta^l\circ P\|$ as $N\to \infty$. For $\Re(\beta)>0$, we indeed have by Lemma \ref{lemma: mayer operator inequalities} that for any $\varepsilon>0$ there exists some $N$ for which $\lvert (L_\beta-L_\beta\circ P) (f)\rvert_\alpha(x)\leq \mathfrak{L}_{\Re(\beta)+\alpha}((1+\varepsilon)|f-P(f)|_\alpha)(x)+\varepsilon$. However, we do not have sufficient control over the $\|\cdot\|_\infty$-norm. We remedy this by considering a \textit{countable} interpolation which ensures $(L_\beta^l\circ P)f$ remains a (nonlinear) interpolation of
$L_\beta^l f$.

Let us now prove most of Theorem \ref{theorem: radius of essential spectrum}.
\begin{theorem}\label{theorem: spectral theorem}
Let $0<\alpha<1 $. Then for all $\beta$ with $\Re(\beta)>(1-\alpha)/2$ the transfer operator $L_\beta$ satisfies
\[\rho_e(L_\beta) \leq \rho(\mathfrak{L}_{\Re(\beta)+\alpha}).\]
\end{theorem}
\begin{proof}
 By Nussbaum's formula and Proposition \ref{proposition: compact operator}, it suffices to prove that there is some constant $C''>0$ for which \[\limsup_{N\to\infty} \|L_{\beta}^l -L_{\beta}^l \circ P_{l,N} \| \leq C''\left\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\right\| \text{ for all } l.\]

\end{proof}
\begin{remark}\label{remark: spectral radius}
By using the spectral radius formula $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})=\limsup_{l\to\infty}\|\mathfrak{L}_{\Re(\beta)+\alpha}^l\|^{1/l}$ and bounding every function $f\in L^\infty([0,1])$ by a constant, we obtain that $\rho(\mathfrak{L}_{\Re(\beta)+\alpha})\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$ and hence that
\[\rho(\mathfrak{L}_{\Re(\beta)+\alpha}) = \lambda_1(\Re(\beta)+\alpha),\]
where $\lambda_1(t)$ for $t>1/2$ is the eigenvalue of maximum modulus of $\mathcal{L}_{t}$. The eigenvalue is guaranteed to be simple and positive by the Perron-Frobenius theorem. In fact $\lambda_1(t)$ is a real-analytic and decreasing function of $t$ with $\lambda(1)=1$, see \cite{Mayer90,Vallee98}.
\end{remark}
Theorem \ref{theorem: radius of essential spectrum} for $\alpha \in (0,1)$ then follows from Remark \ref{remark: spectral radius} and from the following corollary of Theorem \ref{theorem: spectral theorem}.
\begin{corollary}\label{Corollary: Ruelle}
 Let $\lambda \in \mathbb{C}$ satisfy $\abs{\lambda}>\rho(\mathfrak{L}_{\Re(\beta)+\alpha})$. Then $\lambda$ is an eigenvalue of $L_\beta$ if and only if it is an eigenvalue of $\mathcal{L}_\beta$. In this case the respective eigenspaces $E_\lambda^\alpha$ and $E_\lambda^\omega$ are equal.
\end{corollary}
This is an example of a more general principle for operators on Banach spaces, see e.g. \cite{Grabiner}. For our purposes, we may directly adapt a proof due to Ruelle (Corollary 3.3 in \cite{Ruelle89}), which we provide in the appendix.

\end{proof}
We see by Proposition $4.1$ that a generic solution of \eqref{equation: three term functional} in $C^\alpha((-1,\infty))$ satisfies $f= O(z^{1-2\Re(\beta)})$ if $\Re(\beta)>(1-\alpha)/2$. On the other hand, we see that the condition $Q=0$ in \eqref{equation: Qinfinity} or in \eqref{equation: Q0} is equivalent to the condition $L_\beta f= \lambda f$, which by Theorem \ref{theorem: radius of essential spectrum} implies $f$ extends to a holomorphic function in $\mathbb{C}\setminus (-\infty,-1]$ if $\abs{\lambda}\leq \rho(\mathcal{L}_{\Re(\beta)+\alpha})$. For example, we have the following corollary of Proposition \ref{equation: proposition: }, which we can view as a stronger version of the Bootstrapping procedure by Lewis and Zagier in \cite{ZagierLewis}.
\begin{corollary}
 Let $\Re(\beta)=1/2$. Suppose $f:(-1,\infty)\to \mathbb{C}$ is some solution of \eqref{equation: three term functional} for $\abs{\lambda} = 1$. Assume furthermore it satisfies the mild growth condition $f(z)= C_0x^{1-2\beta}+o(1)$ as $z\to \infty$.
 Then if $f$ is $1/2+\varepsilon$-Hölder for some $\varepsilon>0$, it is automatically holomorphically extendable to the cut plane $\mathbb{C}\setminus (-\infty,-1]$.
\end{corollary}
This corollary immediately follows from Theorem \ref{theorem: radius of essential spectrum} and the fact that $\rho(\mathcal{L}_a)<1$ if $a>1$. We remark that when $\Re(\beta)=1/2$ and $\lambda =1$ or $-1$, the space of eigenvalues of $\mathcal{L}_{\beta}$ is in bijection with the space of even, respectively uneven Maass cusp forms on the modular surface with eigenvalue $\beta(1-\beta)$ and we have that $C_0=0$, i.e. $f(z)= o(1)$ as $z\to \infty$, see \cite{Lewis97, ZagierLewis}.

\begin{proposition}\label{Proposition: Constructing Q}
 Let $\alpha>0$, $\lambda\in\mathbb{C}\setminus\{0\}$ and let $Q\in C^\alpha(\mathbb{R}/\mathbb{Z})$.
 For all $\beta$ for which $\Re(\beta)> (1-\alpha)/2$ and \[r(\mathcal{L}_{\Re(\beta)+\alpha})< \abs{\lambda},\]
 the function
 \[f=(1 - \lambda^{-1}L_\beta)^{-1}Q,\]
 is a meromorphic expression over all $\beta$ satisfying the conditions in this proposition. Furthermore $f$ satisfies \eqref{equation: three term functional}, \eqref{equation: Qinfinity} and \eqref{equation: Q0}.
\end{proposition}
\begin{proof}
 We interpret $(\lambda - L_\beta)^{-1}$ in the sense of Remark \ref{remark: extending L_beta}. More specifically, for $L_\beta$ acting on $ C^r([a,(1-a)^{-1}])$, the expression $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ is meromorphic and satisfies
\begin{equation}\label{equation: expansion for resolvent}
 \begin{split}
 &(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z)=\sum_{n=0}^\infty \lambda^{-n}L_\beta^n\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)(z).\\
 &= Q(z)+\sum_{n=1}^\infty \lambda^{-n}\sum_{a_1,\ldots,a_n\in\mathbb{N}} Q([a_1,\ldots,a_n+z])\Pi_{k=1}^n[a_k,\ldots,a_n+z]^{2\beta}
 \end{split}
\end{equation}
if $\Re(\beta)$ is large enough. By Theorem \ref{theorem: radius of essential spectrum}, we see that $(1 - \lambda^{-1}L_\beta)^{-1}$ is a meromorphic operator-valued function for all $a\leq1$. Hence the functions $(1 - \lambda^{-1}L_\beta)^{-1}\left(\rstr{Q}{[a,(1-a)^{-1}]}\right)$ are
meromorphic functions over the half-plane consisting of all $\beta$ satisfying the conditions in the proposition. We see by the second line of \eqref{equation: expansion for resolvent} and uniqueness of analytic continuation that these functions agree for different $a$ on the intersection of their domains. Hence we may define $(1 - \lambda^{-1}L_\beta)^{-1}Q(z)$ for any $z\in (-1,\infty)$ by choosing $a$ in \eqref{equation: expansion for resolvent} small enough.

The statement in the abstract concerning Maass cusp forms and nontrivial zeroes of the Riemann zeta function, which we make more precise in Remark \ref{remark: abstract}, turns out to follow immediately from this theorem.

Comparing the behaviour of transfer operators for different levels of regularity is a somewhat common theme in dynamics. One typically expects for a transfer operator associated to a sufficiently \enquote{nice} expanding map that the essential spectral radius vanishes for the operator acting on spaces of increasingly smooth functions, see e.g. \cite{Butterley}. In a sense the above theorem is therefore not too suprising. Indeed, we shall show that if $\alpha$ is a natural number, the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.

 However, the proof of the case $1/2<\alpha < 1$ is perhaps more interesting. In that case, we still obtain uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma < 1/2$, where the function $\beta\mapsto L_\beta$ is unbounded. However, by \eqref{equation: Mayer transfer operator}, the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. Nevertheless, we do not obtain this result using a Doeblin-Fortet-Lasota-Yorke inequality, so our proof is substantially different from the standard approaches to these type of problems.

In Section \ref{section: three-term functional equation}, we interpret this result in terms of Lewis' three-term functional equation. In particular, we can interpret this as an extension of the \enquote{bootstrapping} result of Lewis and Zagier, who proved in \cite{ZagierLewis} that real-analytic solutions of this equation on $(1,\infty)$ satisfying certain growth conditions automatically extend to holomorphic functions on the cut plane $\mathbb{C}\backslash(-\infty,-1]$. We show that for $\Re(\beta)>0$, we may even replace real-analyticity with a \textit{Hölder condition} and obtain that it is still the restriction of a holomorphic function on $\mathbb{C}\backslash(-\infty,-1]$.

If $\alpha$ is a natural number, “the proof of Theorem \ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.”

For $1/2<\alpha<1$, the proof is “substantially different from the standard approaches”: one “still obtain[s] uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded,” and this is because “the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius.” The result is “not obtain[ed]… using a Doeblin-Fortet-Lasota-Yorke inequality.”

If $\alpha$ is a natural number, the proof of the main theorem is a mundane application of standard techniques.

For $1/2<\alpha<1$, the proof is substantially different from the standard approaches: one still obtains uniformly bounded estimates for $\rho_e(L_\beta)$ on vertical lines $\Re(\beta)=\sigma$ even when $\sigma<1/2$, where the function $\beta\mapsto L_\beta$ is unbounded, and this is because the unboundedness in $\beta$ is due to the contribution rank one operator $f\mapsto f(0)\zeta_H(2\beta,z+1)$, which does not contribute to the essential spectral radius. The result is not obtained using a Doeblin-Fortet-Lasota-Yorke inequality.

Question:
Let lpha>0, write lpha=k+b7 with k\in\mathbb N and \eta\in[0,1), and let C^\alpha([0,1]) be the usual Hf6lder space. For \beta\in\mathbb C, define the analytically continued Mayer transfer operator L_\beta on C^\alpha([0,1]) by
\[
L_{\beta}(f)(z)=\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}\,\zeta_H(n+2\beta,z+1)+\sum_{m=1}^\infty \frac{1}{(m+z)^{2\beta}}\,f^*\!\left(\frac{1}{m+z}\right),
\]
where
\[
f^*(z)=f(z)-\sum_{n=0}^k\frac{f^{(n)}(0)}{n!}z^n,
\]
and \zeta_H(s,z)=\sum_{m=0}^\infty (m+z)^{-s} is the Hurwitz zeta function. Let \mathcal L_\gamma denote the corresponding holomorphic Mayer transfer operator
\[
\mathcal L_\gamma(g)(z)=\sum_{m=1}^\infty \frac{1}{(m+z)^{2\gamma}}\,g\!\left(\frac{1}{m+z}\right).
\]
If \sigma_e(L_\beta) denotes the essential spectrum of L_\beta and
\[
\rho_e(L_\beta):=\sup_{\lambda\in\sigma_e(L_\beta)}|\lambda|,
\qquad
\rho(\mathcal L_{\Re(\beta)+\alpha}):=\sup_{\mu\in\sigma(\mathcal L_{\beta+\alpha})}|\mu|,
\]
which statement holds for all \beta with \Re(\beta)>(1-\alpha)/2?

For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.

Choices:
B.
For every \beta with \Re(\beta)\ge (1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\beta+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|\ge \rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_\beta.
C.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
D.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)< \rho(\mathcal L_{\Re(\beta)+\alpha}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of holomorphic functions on a neighborhood of [0,1], and every such generalized eigenfunction extends uniquely to an eigenfunction of \mathcal L_\beta with the same eigenvalue.
E.
For every \beta with \Re(\beta)>(1-\alpha)/2, the essential spectral radius satisfies
\[
\rho_e(L_\beta)\le \rho(\mathcal L_{\Re(\beta)}).
\]
Moreover, for any eigenvalue \lambda of L_\beta with |\lambda|>\rho_e(L_\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \mathcal L_{\Re(\beta)}.

strict half-plane condition and dependence on \Re(\beta) rather than full \beta

analyticity and extension of generalized eigenspaces above the essential spectral radius

generalized eigenfunctions vs eigenfunctions; non-strict radius inequality strengthened to strict and extension target sharpened

loss of the +\alpha shift coming from Hölder regularity in the comparison operator

The stem gives definitions and asks for the valid conclusion, but it does not explicitly state the theorem’s conclusion or otherwise reveal the correct option. The answer must be identified from the choices.

The item is close to a theorem-restatement question: the correct option essentially reproduces the full theorem, while distractors are perturbed variants. It is not fully tautological because the choices encode competing conclusions, but it still mainly tests exact theorem recall.

Some reasoning is needed to distinguish subtle mathematical differences such as strict vs. non-strict inequalities, dependence on Re(β) versus β, the +α shift, and generalized eigenspaces versus eigenfunctions. However, the task is still mostly recognition of the precise theorem statement rather than substantial derivation.

The distractors are strong: they are mathematically close to the correct statement, vary in realistic ways, and reflect common failure modes such as boundary-condition weakening, unjustified strengthening, dropping the regularity shift, or omitting part of the conclusion.

A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known statement than genuine generative mathematical reasoning.