\label{marker}
\end{figure}
The categories at the right are pushouts, which are determined by the rest of diagram and therefore equivalent.
\end{proof}

\bibliographystyle{alpha}
\bibliography{refere

Mirror symmetry was first observed as an unexpected phenomenon in string theory in the late $1980$s. This phenomenon bridges two seemingly unrelated areas of mathematics: algebraic geometry and symplectic geometry. The resulting conjectures have since revolutionized both fields. In $1994$, at the ICM, Kontsevich \cite{Kon94} proposed that mirror symmetry reflects an equivalence of two categories
\[
D^\pi \mathrm{Fuk} (X) \cong D^b \mathrm{Coh}(\check{X}),
\]
between the derived Fukaya category of a Calabi--Yau manifold $X$ and the derived category of coherent sheaves on the mirror Calabi-Yau $\check{X}$. This formulation, known as homological mirror symmetry, has been generalized beyond Calabi-Yau manifolds. While verifying homological mirror symmetry for various examples is an active field, it also serves as a tool for transferring ideas and insights between symplectic and algebraic geometry.

A particularly timely direction in the study of homological mirror symmetry is to understand how Fukaya categories and the equivalence conjectured by Kontsevich behave under geometric decompositions of spaces into simpler building blocks. The setting in which this has been most successful so far are Liouville manifolds, which are non-compact symplectic manifolds with exact symplectic structure and cylindrical ends.
However, when Liouville manifolds have boundaries, defining Floer homology becomes challenging because the pseudoholomorphic curves (whose counts are used to define operations in Floer theory) might escape through the boundary. This leads to the notion of \emph{Liouville sector}, first introduced in a recent work of Ganatra-Pardon-Shende \cite{MR4106794}.

This paper explores the application of this technology to the study of symmetric products of open Riemann surfaces, which are smooth complex manifolds. It is a particularly important class of Liouville manifolds in light of applications to low-dimensional topology, such as Heegaard Floer homology by Ozsv\'ath and Szab\'o \cite{MR2113020}. Tim Perutz \cite{Per05} gave a careful description of a symplectic structure on the symmetric product. (The naive symplectic structure on the symmetric product induced by the symplectic form on the surface is not smooth on the diagonal.) Auroux \cite{MR2755992, MR2827825} provided a symplectic interpretation of bordered Heegaard-Floer homology through Fukaya categories of symmetric products. This application provides further motivation to study the symplectic geometry and the Fukaya categories of symmetric products.

We study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.

We study the geometry of the second symmetric products of surfaces by constructing a decomposition of these spaces into Liouville sectors. This provides a symplectic insight into the structure of symmetric products and further exploration of its applications of the machinery of sectors in homological mirror symmetry. The main result of this paper is the following geometric decomposition theorem, which is restated and proved in section $5$.

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.

\begin{theoremB}
For any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\widehat X)$ of the completion of $X$.
\end{theoremB}
Furthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\Sym^2(\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.

The rest of this paper is organized as follows.
Section $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.
Section $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.
Section $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.
Section $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.
Finally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.

\begin{defn}
 Let $\Sigma $ be a Riemann surface and $\varphi$ be a proper plurisubharmonic function on $\Sigma$. Let $\{s_i\}_{i\in I}$ be the set of saddles of $\varphi$ and $\mathcal{N}(\gamma_i)$ be the tubular neighborhood of the stable manifold $\gamma_i$ of the saddle $s_i$.\\
 We say $(\Sigma, \varphi)$ is a Riemann surface with a quadratic Stein structure if $\varphi |_{\mathcal{N}(\gamma_i)}$ is quadratic in local coordinates.
\end{defn}

\begin{proposition}
\label{global}
 For any non compact topological surface $\Sigma$ with disjoint proper embedded arcs $\{\gamma_i \}, i \in I$,
 we can build a quadratic Stein structure with one saddle $s_i$ (and possibly other saddles as well)
 and on each $\gamma_i$ and one minimum $m_j$
 on each component of $\Sigma - \bigcup \gamma_i$.
\end{proposition}

\section{Construction of sectorial decomposition}
\subsection{Statement and notations}
\begin{theorem} \label{mainthm}
Let $(\Sigma,\varphi)$ be a Riemann surface with a quadratic Stein structure and let $\{s_i\}$ the set of saddles and $\{m_j\}$ the set of minimum. This determines a sectorial collection of hypersurfaces $H_{s_i}$ in $Sym^2(\Sigma)$, which decompose $Sym^2(\Sigma)$ into a union of sectors with corners $U_{m_i,m_j}$.
\end{theorem}

\begin{corollary}
 Let $m$ be the number of arcs and $n$ be the number of the connected components of $\Sigma - \cup_{i\in I} \gamma_i$.
 The decomposition constructed consists of $\binom{n+1}{2}$ Liouville sectors with corners separated by $mn$ pieces of smooth hypersurfaces that meet transversely at $\binom{m}{2}$ corners.
 \end{corollary}

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$. However, it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.

In general, this theorem states that given a surface's sectorial decomposition, we can construct a sectorial decomposition of its second symmetric product.

The proof of theorem A relies on two main ingredients. One is the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface. The other ingredient is the
calculation on a local model described in section $3$. The methods used to prove Theorem A also imply other results for sectors and symmetric products of Riemann surfaces. For example,

\begin{theoremB}
For any $2$-dimensional Liouville sector $X$, the second symmetric product of $X$ is deformation equivalent to a Liouville sector $Y$. Moreover, the completion $\widehat{Y}$ is deformation equivalent to the second symmetric product $Sym^2(\widehat X)$ of the completion of $X$.
\end{theoremB}
Furthermore, we can apply the decomposition theorem (Theorem A) to homological mirror symmetry. Namely, a sectorial decomposition of $\Sym^2(\Sigma)$ opens the perspective of constructing a mirror space of the symmetric product by finding mirrors to the various Liouville sectors into which it can be decomposed and gluing them to each other in a suitable manner.

Ganatra, Pardon, and Shende \cite{MR4106794, MR4695507} have introduced and studied  wrapped Fukaya categories of Liouville sectors. The upshot of their work is that the wrapped Fukaya category of a Liouville manifold can be determined from a decomposition into Liouville sectors by a local-to-global principle.

In section $7$, we discuss the example of the symmetric product of the $4$-punctured sphere $\Sym^2(\mathbb{P}^1 - 4 \mbox{pts})$, which is also known as the \emph{complex $2$-dimensional pair of pants}. Lekili and Polishchuk \cite{MR4120165} have established homological mirror symmetry for this example using a computation of Auslander algebras. A geometric interpretation of mirror symmetry can be deduced from our sectorial decomposition results. In this case, the local-to-global principle in  \cite{MR4695507} yields the top diamond of the commutative diagram shown in Figure \ref{marker},
which describes the wrapped Fukaya category of $\text{Sym}^2(\mathbb{P}^1 - 4\text{pts})$ as the pushout of a diagram of wrapped Fukaya
categories of simpler Liouville sectors. Homological Mirror Symmetry for the $2$-dimensional pair of pants follows by identifying the mirror of each Liouville sector and the gluing maps between them.

The rest of this paper is organized as follows.
Section $2$ reviews the basic theory of Liouville sectors and related notions such as sectorial domains, completions and truncations.
Section $3$ develops a local model on the symmetric square of the complex plane while section $4$ establishes the existence of quadratic Stein structures on surfaces with embedded arcs, providing the analytic framework for our main construction.
Section $5$ uses this structure to construct a global sectorial decomposition of the symmetric product of a surface.
Section $6$ studies the fibers of the sectorial boundaries and their completions, showing their Liouville equivalences.
Finally, Section $7$ applies these results to the symmetric product of the four-punctured sphere and proves homological mirror symmetry for this example.

\paragraph{Acknowledgement.}
I would like to thank my PhD advisor Denis Auroux for his guidance, support, and generosity, as well as for the many ideas he contributed to this work. I am also grateful to John Pardon for helpful conversations and suggestions. This work was partially supported by NSF grant DMS-2202984, by the Simons Foundation (grant \#$385573$, Simons Collaboration on Homological Mirror Symmetry), and by the Simons Collaboration on New Structures in Low-Dimensional Topology.

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”

“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described in section $3$.”

Roughly, $H_i$ corresponds to the set of configurations of points on $\Sigma$ where one of the points is on the arc $\gamma_i$, but “it is difficult to construct a well-behaved geometric decomposition of the symmetric product directly from a geometric decomposition of $\Sigma$, because symmetric products of manifolds with boundaries have badly behaved corners and singularities.”

“The proof of theorem A relies on two main ingredients.” First, “the observation that the ascending manifold of an index one critical point of a Stein potential defines a sectorial hypersurface.” Second, “the calculation on a local model described later.”,

Let $\Sigma$ be a topological surface with disjoint proper embedded arcs $\{\gamma_i\}_{i\in I}$. For suitably defined Stein structures on $\Sigma$ and $\Sym^2(\Sigma)$, the arcs $\{\gamma_i\}_{i\in I}$ determines a sectorial collection of smooth hypersurfaces $H_{_i}$ in $\Sym^2(\Sigma)$ which separates the symmetric product into Liouville sectors.,

Question: Let $\Sigma$ be a topological surface equipped with a collection of pairwise disjoint proper embedded arcs $\{\gamma_i\}_{i\in I}$. Assume one has suitably defined Stein structures on $\Sigma$ and on its second symmetric product $\operatorname{Sym}^2(\Sigma)=(\Sigma\times\Sigma)/\mathfrak{S}_2$. Under these hypotheses, which conclusion about $\operatorname{Sym}^2(\Sigma)$ holds?

The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$, and these hypersurfaces separate $\operatorname{Sym}^2(\Sigma)$ into Liouville sectors.

Choices:
B. The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$, and these hypersurfaces separate $\operatorname{Sym}^2(\Sigma)$ into Weinstein domains with boundary.
C. The arcs $\{\gamma_i\}_{i\in I}$ determine a collection of smooth hypersurfaces $H_i$ in $\operatorname{Sym}^2(\Sigma)$ which decompose $\operatorname{Sym}^2(\Sigma)$ into pieces compatible with the given Stein structures.
D. The arcs $\{\gamma_i\}_{i\in I}$ determine a sectorial collection of smooth hypersurfaces $H_i$ directly by taking configurations in which one of the points lies on $\gamma_i$, and this immediately separates $\operatorname{Sym}^2(\Sigma)$ into Liouville sectors.
E. For any suitably defined Stein structures on $\Sigma$ and $\operatorname{Sym}^2(\Sigma)$, there is a single Liouville sector $Y$ deformation equivalent to $\operatorname{Sym}^2(\Sigma)$ whose boundary is the union of the hypersurfaces $H_i$ determined by the arcs $\{\gamma_i\}_{i\in I}$.

target pieces are Liouville sectors, not Weinstein domains

drops the sectorial and Liouville-sector conclusions

ignores the singular-corner obstruction to the naive configuration-space construction

replaces a decomposition into multiple sectors by a single deformation-equivalent Liouville sector

The stem does not explicitly reveal the correct choice or uniquely signal option A. It states hypotheses and asks for the resulting conclusion, without giving away the key phrases 'sectorial collection' or 'Liouville sectors' as part of the prompt itself.

This is essentially a theorem-recall item: the stem gives the hypotheses and asks for the theorem's conclusion. The correct option is a direct statement of that conclusion rather than a nontrivial application or comparison-based inference.

There is some pressure to distinguish the strongest valid conclusion from weaker or tampered alternatives, especially against option C and the Liouville/Weinstein confusion in B. However, the item mainly tests recognition of the theorem rather than genuinely generating or deriving the answer from the setup.

The distractors are mathematically plausible and target realistic failure modes: weakening the conclusion (C), confusing related structures (B), using a naive geometric construction that ignores a subtle obstruction (D), and overstrengthening the conclusion (E). They are distinct and well aligned with expert-level misunderstandings.

A solid theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests reasoning rather than generative understanding.


\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 $M$ be an integer and $N\ge 1$, and for a primitive Dirichlet character $\chi \pmod q$ 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
\[
a(r)=2\log 2\,\bigl(3.0758r+1.38402\log(4r)-1.5379\bigr).
\]
Assume that $\chi$ is primitive modulo $q$, that
\[
q\ge \max\{10^{1145},e^{e^{a(r)}}\},
\]
and that either $r=2$ or $q$ is cubefree. Let $C(r)$ and $D(r)$ be given by
\[
\begin{array}{c|ccccccccc}
r&2&3&4&5&6&7&8&9&\ge 10\\ \hline
C(r)&15.219&5.359&3.671&2.953&2.549&2.290&2.108&1.973&1.869\\
D(r)&8.362&4.581&3.396&2.811&2.462&2.229&2.063&1.938&1.841
\end{array}
\]
Which of the following conclusions about $S_\chi(M,N)$ is valid under these hypotheses?

For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]

Choices:
B.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
C.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
D.
For every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r},
\]
provided $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and $q$ is cubefree.
Moreover, as $q\to\infty$, one has the asymptotic refinement
\[
|S_{\chi}(M,N)|\le (D(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]
E.
There exists a constant $C(r)$ depending only on $r$ such that for every primitive Dirichlet character $\chi\pmod q$ with $q\ge \max\{10^{1145},e^{e^{a(r)}}\}$ and either $r=2$ or $q$ is cubefree, one has for every integers $M$ and $N\ge 1$,
\[
|S_{\chi}(M,N)|\le C(r)
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r},
\]
and moreover the same asymptotic statement holds with $D(r)$ replaced by $C(r)$:
\[
|S_{\chi}(M,N)|\le (C(r)+o(1))
N^{1-\frac1r}q^{\frac{r+1}{4r^2}}(\log q)^{\frac1{2r}}
\left((4r)^{\omega(q)}m_r(q)\right)^{\frac1{2r}-\frac1{2r^2}}
\left(\frac{q}{\phi(q)}\right)^{\frac1r}.
\]

exponent of ((4r)^{\omega(q)}m_r(q)) in main theorem vs weaker theorem

dropped the asymptotic refinement involving D(r)+o(1)

special exception allowing r=2 without cubefree q

replacement of the sharper asymptotic constant D(r) by C(r)

The stem does not state the conclusion itself. It supplies hypotheses and notation, but the exact bound, exponents, and asymptotic refinement must still be identified from the options.

This is essentially a theorem-statement recognition item: the correct choice is a near-verbatim restatement of the intended result, with distractors formed by small perturbations.

There is some reasoning in checking logical strength and boundary cases (for example, the r=2 cubefree exception and the sharper D(r)+o(1) refinement), but the item mostly tests precise recall/comparison rather than generating a conclusion from mathematical argument.

The distractors are strong: they are close to the target theorem, differ in meaningful ways, and reflect realistic failure modes such as a wrong exponent, omission of the asymptotic refinement, mishandling the cubefree exception, or confusing C(r) with D(r).

A solid recognition-style theorem MCQ with high-quality distractors and little answer leakage, but it is fundamentally a restatement/recall question rather than one that strongly tests generative mathematical 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 \(B_r\) denote the ball of radius \(r\) centered at a fixed point of \(\partial\Omega\), and let the fractional Laplacian be defined by \(( -\Delta)^s u(x)=C_{n,s}\,\mathrm{P.V.}\!\int_{\mathbb R^n}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy\). 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
\[
\begin{cases}
(-\Delta)^s u=g,&\text{in }\Omega\cap B_4,\\
u=0,&\text{in }B_4\setminus\Omega.
\end{cases}
\]
Which of the following conclusions about \(u\) on \(\Omega\cap B_{1/2}\) is valid under these hypotheses?

The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).

Choices:
B.
The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{\mathcal L_{2s}}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
C.
The solution satisfies \(u\in C^{\alpha_0}(\Omega\cap B_{1/2})\), where \(\alpha_0=\min\{s,1-s\}\), and
\[
\|u\|_{C^{\alpha_0}(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
D.
The solution satisfies \(u\in C^s(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^s(\Omega\cap B_{1/2})}\le C\|g\|_{L^\infty(\Omega\cap B_4)},
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).
E.
The solution satisfies \(u\in C^{2s}(\Omega\cap B_{1/2})\) and
\[
\|u\|_{C^{2s}(\Omega\cap B_{1/2})}\le C\Big(\|g\|_{L^\infty(\Omega\cap B_4)}+\|u\|_{L^\infty(\Omega\cap B_4)}\Big),
\]
where \(C\) depends only on \(n\), \(s\), and the local \(C^{1,1}\) norm of \(\partial\Omega\).

replacement of local \(L^\infty\) control by tail norm \(\mathcal L_{2s}\) in the final estimate

dropped the iteration step upgrading boundary exponent from \(\alpha_0=\min\{s,1-s\}\) to \(s\)

omission of the local \(L^\infty\) dependence on \(u\) needed for the harmonic part

upgrading boundary regularity from \(C^s\) to \(C^{2s}\) by misreading interior estimates as boundary estimates

The stem states only the hypotheses and asks for the valid conclusion; it does not explicitly reveal the exact regularity class or estimate. There is no direct wording that singles out choice A.

This is close to a theorem-recall item: the assumptions are presented and the student is asked for the corresponding conclusion. However, it is not purely tautological because the options include weaker, stronger, and subtly altered estimates, so one must distinguish among competing statements.

The item creates real pressure to identify the strongest valid boundary regularity statement, not just any true one. In particular, the weaker true option C, the overstrong option E, and the estimate-structure mistakes in B and D force careful reasoning about boundary regularity and norm dependence.

The distractors are mathematically plausible and target common failure modes: confusing local L^\infty control with tail norms, settling for a weaker exponent, omitting necessary dependence on u, or overextending interior regularity to the boundary. They are distinct and well aligned with the topic.

A strong MCQ overall: no meaningful answer leakage, high-quality distractors, and good reasoning demand. Its main limitation is that it still functions largely as a theorem-conclusion recognition question rather than a deeply generative problem.


\begin{theorem}  \label{thm: factor hazard}
Let the hazard function, $h_0:(0,\infty)\to(0,\infty)$, admit the factorization:
\[
h_0(x)=C\,x^{p-1}\,g(x),\qquad C>0,\; p\ge 0,
\]
where $g:(0,\infty)\to(0,\infty)$ is non-decreasing. For $\tau>0$ define:
\[
r(x,\tau)\;=\;\tau\,\frac{h_0(\tau x)}{h_0(x)}.
\]
Then, for every $x>0$,
\[
0\le \tau\le 1
\quad\Longleftrightarrow\quad
0\le r(x,\tau)\le 1.
\]
\end{theorem}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0(\tau x)}{h_0(x)}, \qquad x>0.
\]
Then:
\[
0 \le r(x,\tau) \le 1 \quad \forall x>0
\quad \Longleftrightarrow \quad
0 \le \tau \le 1.
\]
\end{theorem}

\label{app: thms}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0

\begin{theorem} \label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. Then:
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]
\end{theorem}

As per \citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form:
\begin{align}
 \bar{F}_X(x) = P(X>x) = \bar{F}_0(x),\quad x>0, \label{eq: general marginal}
\end{align}
for some survival function $\bar{F}_0(x) \in [0, 1] \ \forall x> 0$, and for each $x>0$:
\begin{equation}
 P(Y>y\mid X>x) = \bar{F}_1\left(\beta(x)y\right), \quad x, y >0,\label{eq: general cond}
\end{equation}
for some survival function $\bar{F}_1(\beta(x)y) \in [0, 1] \ \forall x, y >0$ and a suitable acceleration function $\beta(x)$. As per \citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\{X > x\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:
\begin{align}
 P(X>x, Y>y) = \bar{F}_0(x) \bar{F}_1\left(\beta(x)y\right), \quad x, y>0. \label{eq: general joint}
\end{align}
Assuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:
\[
f_X(x) = -\frac{d}{dx}P(X>x) = -\frac{d}{dx}\bar{F}_0(x) = f_0(x),
\]
and since $\lim_{x\to 0^+}\bar{F}_0(x)=1$, where $\beta(0):=\lim_{x\to 0^+}\beta(x)$:
\[
P(Y>y) = \lim_{x\to 0^+}P(Y>y,X>x) = \bar{F}_1\bigl(\beta(0)y\bigr).
\]
Now, the density of $Y$ is:
\[
f_Y(y) = -\frac{d}{dy}P(Y>y)
= -\frac{d}{dy}\bar{F}_1(\beta(0)y)
= \beta(0)\,f_1(\beta(0)y),
\]
where $f_1(t)=-\bar{F}_1'(t)$ denotes the density associated with $\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\bar{F}_1$, but with its argument scaled by $\beta(0)$. For \eqref{eq: general joint} to be a valid survival function, it must have a non-negative
mixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\bar{F}_0$ and $\bar{F}_1$, and differentiating, we obtain:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) &=
f_0(x) f_1(\beta(x) y)\beta(x)
- \bar{F}_0(x) f_1'(\beta(x) y) \beta(x) \beta'(x) y
- \bar{F}_0(x) f_1(\beta(x) y) \beta'(x) \nonumber \\
&= f_0(x) f_1(\beta(x) y)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(\beta(x) y) + f_1'(\beta(x) y) \beta(x) y\right]. \nonumber
\end{align}
Denoting $t = \beta(x)y$, we have:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y)
&= f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(t) + f_1'(t)t\right] \nonumber \\
& = f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ t f_1(t) \right]' .\nonumber
\end{align}
Thus a necessary and sufficient condition for $ \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) \geq 0$ is:
\[
\beta'(x)\;\begin{cases}
\le\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'>0,\\[10pt]
\ge\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'<0, \\
\text{unconstrained}, & \text{if } [t f_1(t)]'=0.
\end{cases}
\]
Denoting the hazard function $h_0(x) = \frac{f_0(x)}{\bar{F}_0(x)}$ and $S(t) = \frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:
\begin{align}
\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\beta(x)
\;\leq\; \beta'(x)
\;\leq\;
\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\beta(x). \label{eq: B'(x) bounds}
\end{align}
A special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \frac{f_1(t)}{\left[tf_1(t) \right]'} = \frac{f_1(t)}{f_1(t) + tf_1'(t) } \geq1$ for $t \geq 0$, hence $\beta'(x)
\;\leq\;
 h_0(x)\beta(x)$.\\

It should be noted that multiple functional forms for $\beta(x)$ are possible. The question of selecting a form of $\beta(x)$ that provides the best model fit has already been investigated by \citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\beta(x)$ that arise from the extreme case in which $\beta'(x)$ attains its theoretical upper bound, that is, given $\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:
\begin{align}
 \beta'(x) &= c^*h_0(x) \beta(x), \nonumber \\
 \therefore \int \frac{1}{\beta(x)} d\beta(x) &= \int c^* h_0(x) dx, \nonumber\\
 \therefore \log\left( \beta(x) \right) &= -c^* \log\left( \bar{F}_0 (x)\right) + K. \nonumber \\
 \therefore \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(x)\right]^{c^*}}, \label{eq: beta(x) general}
 \end{align}
for $\gamma = e^K>0$. This suggests $\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\beta(0): = \lim_{x\to 0^+} \beta(x) = \gamma$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$. We also note that since $\bar{F}_0(x)$ approaches zero only as $x\rightarrow \infty$, the expression in \eqref{eq: beta(x) general} remains finite for all finite $x$.\\

The role of the acceleration function $\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.

\subsection{Moments}
Noting $X \sim Weibull(\alpha, \lambda)$ and $Y \sim Weibull(\gamma, \nu)$, we have:
\begin{align}
 E(X) &= \frac{1}{\alpha} \Gamma\left(1+ \frac{1}{\lambda}\right), \label{eq: Weibull ex} \\
 E(Y) &= \frac{1}{\gamma} \Gamma\left(1+ \frac{1}{\nu}\right), \label{eq: Weibull ey} \\
 {Var}(X) &= \frac{1}{\alpha^2}\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right),\label{eq: Weibull varx} \\
 {Var}(Y) &= \frac{1}{\gamma^2}\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right).\label{eq: Weibull vary}
\end{align}
Accordingly:
\begin{align}
\operatorname{Cov}(X, Y)
&= \int_0^\infty \int_0^\infty
 \big( P(Y>y , X> x) - P(X> x)P(Y> y) \big) \,dy\,dx \nonumber \\
&= \int_0^\infty \int_0^\infty
 \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr) \right) \,dy\,dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \int_0^\infty \left( e^{-(\beta(x)y )^\nu}- e^{-(\beta(0)y )^\nu} \right) dy \right) dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(x)} -\frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(0)} \right) dx \nonumber \\
&= \frac{1}{\gamma } \Gamma\!\left( 1+ \frac{1}{\nu}\right)
 \left[\int_0^\infty e^{-(\alpha x)^\lambda - \frac{1}{\nu}(\alpha \tau x)^\lambda} \, dx
 - \int_0^\infty e^{-(\alpha x)^\lambda} \, dx \right] \nonumber \\
&= \frac{1}{\alpha \gamma} \Gamma\!\left(1+ \frac{1}{\nu} \right)\Gamma\!\left(1+ \frac{1}{\lambda} \right)
 \left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right].
\label{eq: Weibull cov}
\end{align}
Now, using \eqref{eq: Weibull varx}, \eqref{eq: Weibull vary} and \eqref{eq: Weibull cov}, the correlation function is:
\begin{align}
 \rho(X,Y)
 &= \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}} \nonumber\\
 &= \frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}}.
 \label{eq: Weibull rho}
\end{align}
Clearly, \eqref{eq: Weibull rho} is strictly decreasing in $\tau \in [0, 1]$. Hence:
\begin{align}
 \rho_{\text{min}}(\lambda, \nu):=\frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + 1} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}} \leq \rho(X, Y) \leq 0.
\end{align}
Now, $\inf_{\lambda, \nu > 0} \rho_{\text{min}}(\lambda, \nu) = \lim_{\nu \to \infty}\rho_{\text{min}}(\lambda = 1, \nu) = -\frac{\sqrt{6}}{\pi}$. Hence, this Weibull model with acceleration function $\beta(x) = \gamma e^{\frac{1}{\nu}(\alpha \tau x)^\lambda}$ will be able to accommodate correlations in the range $\rho(X,Y)\in\big[-\tfrac{\sqrt{6}}{\pi},\,0\big]$.

\appendix
\section*{Appendices}
\section{Joint Densities and Log-likelihoods} \label{app: joint lik}
\subsection{Exponential}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: exp joint survival}:
\[
f_{X,Y}(x,y)
= \alpha \gamma \,\bigl(\gamma \tau y e^{\alpha \tau x} - \tau + 1 \bigr)
 e^{\!\left(\alpha x(\tau - 1) - \gamma y e^{\alpha \tau x}\right)},
 \qquad x,y>0.
\]
where $\alpha, \gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\gamma,\tau)
&= n \log \alpha + n \log \gamma
+ \sum_{i=1}^n \log\!\Big( \gamma \tau y_i e^{\alpha \tau x_i} - \tau + 1 \Big) \nonumber \\
&\quad + \alpha (\tau - 1)\sum_{i=1}^n x_i
- \gamma \sum_{i=1}^n y_i e^{\alpha \tau x_i}. \nonumber
\end{align}
\subsection{Lomax}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Lomax joint survival}:
\begin{align*}
f_{X,Y}(x,y)
&= \nu \,(1+\alpha x)^{-3\lambda - 1}\,
 \bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{-3\nu - 4} \\[6pt]
&\quad \times \Biggl[
 \frac{\alpha \gamma^{2} \lambda \tau y (\nu+1)\,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{2\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+2}}
 {1+\alpha \tau x} \\[6pt]
&\qquad - \frac{\alpha \gamma \lambda \tau \,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}}
 {1+\alpha \tau x} \\[6pt]
&\qquad + \alpha \gamma \lambda \,\bigl((1+\alpha x)^{2}(1+\alpha \tau x)\bigr)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}
 \Biggr], \quad x,y >0,
\end{align*}
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \log \nu
 - (3\lambda+1)\sum_{i=1}^n \log(1+\alpha x_i) \nonumber \\
&\quad - (3\nu+4)\sum_{i=1}^n \log\!\bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr) \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Biggl\{
 \frac{\alpha \gamma^{2} \lambda \tau y_i (\nu+1)\,(1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{2\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+2} \nonumber \\
&\qquad\qquad
 - \frac{\alpha \gamma \lambda \tau (1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3} \nonumber \\
&\qquad\qquad
 + \alpha \gamma \lambda \,\bigl((1+\alpha x_i)^{2}(1+\alpha \tau x_i)\bigr)^{\lambda}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3}
 \Biggr\}. \nonumber
\end{align}
\subsection{Weibull}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Weibull joint survival}:
\[
f_{X, Y}(x,y) = \alpha^{\lambda} \gamma^{\nu} \lambda \nu x^{\lambda - 1} y^{\nu - 1}
\left( \tau^{\lambda} (\gamma y)^{\nu} e^{(\alpha \tau x)^\lambda}
- \tau^{\lambda} + 1 \right)
e^{(\alpha x)^\lambda(\tau^\lambda - 1)
- (\gamma y)^{\nu} e^{(\alpha\tau x)^{\lambda}}}, \quad x, y>0,
\]
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \lambda \log \alpha + n \nu \log \gamma
 + n \log \lambda + n \log \nu \nonumber\\
&\quad + (\lambda - 1) \sum_{i=1}^n \log x_i
 + (\nu - 1) \sum_{i=1}^n \log y_i \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Bigl(\tau^{\lambda} (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}
 - \tau^{\lambda} + 1 \Bigr) \nonumber \\
&\quad + (\tau^{\lambda} - 1)\sum_{i=1}^n (\alpha x_i)^\lambda
 - \sum_{i=1}^n (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}. \nonumber
\end{align}
\subsection{Log-logistic}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Log joint survival}:
\[
\begin{aligned}
f_{X,Y}(x,y)
&= \frac{\lambda \nu\,(\gamma y)^{\nu}}
{x\,y\,\bigl((\alpha x)^{\lambda}+1\bigr)^{2}
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)^{3}}
\\[4pt]
&\quad\times
\Biggl\{
(\alpha x)^{\lambda}\bigl((\alpha \tau x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
+ (\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
- 2\,(\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Biggr\},
\qquad x,y>0,
\end{aligned}
\]

\begin{proof}
Since $P(Y>y , X> x) = \bar{F}_0(x)\bar{F}_1\bigl(\beta(x)y\bigr)$ then $P(Y>y) = \lim_{x\to 0^+}P(Y>y , X> x) = \bar{F}_0(0)\bar{F}_1\bigl(\beta(0)y\bigr) = \bar{F}_1\bigl(\beta(0)y\bigr)$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$ and $\beta(0): = \lim_{x\to 0^+} \beta(x)$. We note from Hoeffding's covariance identity:
\begin{align}
    Cov(X, Y) & = \int_0^\infty \int_0^\infty\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \right) \ dx dy \nonumber \\
    & = \int_0^\infty \int_0^\infty  \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr)  \right) \ dx dy.\nonumber
\end{align}
Now since $\bar{F}_1'(\cdot)\le 0$ (as $\bar{F}_1$ is a survival function), if $\beta'(x) \geq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \leq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \leq 0$ for $x, y \geq 0$. Conversely, if $\beta'(x) \leq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \geq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \geq 0$ for $x, y \geq 0$. Finally, if $\beta(x)$ is constant, say $\beta(x)\equiv c$, then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.
\end{proof}
Furthermore, as done in \citet{arnold2020bivariate}, we introduce a dependence parameter $\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \eqref{eq: beta(x) general}, we define the acceleration function as such:
\begin{align}
    \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(\tau x)\right]^{c^*}}. \label{eq: beta(x) general 2}
\end{align}
Given the bounds established in \eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\beta(x)$ as defined in \eqref{eq: beta(x) general 2}, and again assuming $\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\tau$ as such:
\begin{align}
    0 &\leq \beta'(x) \leq c^* h_0(x)\, \beta(x), \nonumber\\
    \therefore
    0 &\leq
    \frac{c^* \tau \gamma f_0(\tau x)}{\big[\bar{F}_0(\tau x)\big]^{c^* + 1}}
    \leq
    c^* \gamma \,
    \frac{f_0(x)}{\big[\bar{F}_0(x)\big]^{c^*}}
    \frac{1}{\big[\bar{F}_0(\tau x)\big]^{c^*}}, \nonumber\\
    \therefore
    0 &\leq \tau \leq \frac{h_0(x)}{h_0(\tau x)}. \nonumber\end{align}
We re-write this inequality as such:
\begin{align}
    0 \le r(x, \tau) \le 1,
    \label{eq: tau bounds}
\end{align}
where $r(x, \tau) = \tau \frac{h_0(\tau x)}{h_0(x)}$, where the endpoint $\tau=0$ is included by continuity, with $r(x,0):=\lim_{\tau\to 0^+}r(x,\tau)=0$. Theorem \ref{thm: ABC} and Theorems \ref{thm: ifr hazard} and \ref{thm: factor hazard} in Appendix \ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\tau$ that satisfy \eqref{eq: tau bounds}.

\begin{theorem}
\label{thm: ABC}
Let $h_0:(0,\infty)\to(0,\infty)$ be a hazard function. For $x>0$ and $\tau>0$ define:
\[
r(x,\tau):=\tau\,\frac{h_0(\tau x)}{h_0(x)},\qquad
\phi(x):=x\,h_0(x).
\]
Let
\[
A:\ \ 0\le r(x,\tau)\le 1\ \ \text{for all }x>0,\qquad
B:\ \ \tau\in(0,1],\qquad
C:\ \ \phi \text{ is non-decreasing on }(0,\infty).
\]
Then:
\begin{enumerate}
\item[(1)] $(A \wedge B)\ \Rightarrow\ C$.
\item[(2)] $(C \wedge B)\ \Rightarrow\ A$.
\item[(3)] $(C \wedge A)\ \Rightarrow\ B$, \emph{unless} $\phi$ is identically constant on $(0,\infty)$, in which case $A$ holds for all $\tau>0$ and $B$ cannot be inferred.
\end{enumerate}

\end{theorem}

\begin{proof}
\textbf{(1) $(A \wedge B)\Rightarrow C$.}
Fix $0<y<x$ and set $\tau:=y/x\in(0,1]$ so that $B$ holds. By $A$,
\[
1 \ \ge\ r(x,\tau)=\tau\,\frac{h_0(\tau x)}{h_0(x)}=\frac{y\,h_0(y)}{x\,h_0(x)}.
\]
Hence $y\,h_0(y)\le x\,h_0(x)$ for all $0<y<x$, i.e.\ $\phi$ is non-decreasing. Thus $C$ holds.

\medskip
\textbf{(2) $(C \wedge B)\Rightarrow A$.}
Assume $C$ and $\tau\in(0,1]$. Then $\tau x\le x$ and, by the non-decreasing nature of $\phi$,
\[
\phi(\tau x)\le \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x)\le x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau)\le 1.
\]
Since $h_0>0$, we also have $r(x,\tau)\ge 0$. Hence $A$ holds.

\medskip
\textbf{(3) $(C \wedge A)\Rightarrow B$, unless $\phi$ is constant.}
Assume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\tau)\ge 0$ implies $\tau \ge 0$. Now suppose, toward a contradiction, that $\tau>1$, then $\tau x > x$ and, by assuming $\phi$ is now strictly increasing:
\[
\phi(\tau x) > \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x) > x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau) > 1.
\]
contradicting the upper bound. Hence $\tau \le 1$. If instead $\phi$ is identically constant,
say $\phi(x)\equiv K>0$, then $h_0(x)=K/x$ and
\[
r(x,\tau)=\tau\,\frac{K/(\tau x)}{K/x}=1 \quad (\forall x>0,\ \forall \tau>0),
\]
so $A$ holds for all $\tau>0$ and no restriction $B$ follows.
\end{proof}
\begin{remark} When convenient, extend $B$ to $\tau\in[0,1]$ by defining $r(x,0):=\lim_{\tau\to 0^+} r(x,\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.
\end{remark}
\begin{remark}
If $h_0$ is non-decreasing, then $\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\ref{thm: ABC} recovers Theorem \ref{thm: ifr hazard}.
If $h_0(x)=C\,x^{p-1}g(x)$ with $C>0$, $p\ge 0$, and $g(x)$ non-decreasing, then $\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \ref{thm: ABC} recovers Theorem \ref{thm: factor hazard}.
\end{remark}

The following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \sim Family (\alpha, \lambda)$ and $Y \sim Family (\beta(0), \nu)$ where $\alpha$ and $\beta(0)$ denote the rate parameters and $\lambda$ and $\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).

Using $P(Y>y,X>x)=\bar F_0(x)\,\bar F_1(\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\to 0^+$:
\[
P(Y>y)=\lim_{x\to 0^+}P(Y>y,X>x)=\bar F_1(\beta(0)y),\qquad \beta(0):=\lim_{x\to 0^+}\beta(x),
\]
(since $\lim_{x\to 0^+}\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:
\[
\operatorname{Cov}(X,Y)=\int_0^\infty\!\int_0^\infty \Big(P(Y>y,X>x)-P(X>x)P(Y>y)\Big)\,dx\,dy
=\int_0^\infty\!\int_0^\infty \bar F_0(x)\Big(\bar F_1(\beta(x)y)-\bar F_1(\beta(0)y)\Big)\,dx\,dy.
\]
Since $\bar F_1'(\cdot)\le 0$, if $\beta'(x)\ge 0$ then $\bar F_1(\beta(x)y)\le \bar F_1(\beta(0)y)$, making the integrand non-positive for $x,y\ge 0$, hence $\operatorname{Cov}(X,Y)\le 0$; conversely, if $\beta'(x)\le 0$ then $\bar F_1(\beta(x)y)\ge \bar F_1(\beta(0)y)$, so $\operatorname{Cov}(X,Y)\ge 0$. Finally, if $\beta(x)\equiv c$ is constant then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\operatorname{Cov}(X,Y)=0$.

\label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]

Question:
Let \(X\) and \(Y\) be random variables in an accelerated conditional model with survival structure
\[
P(X>x)=\bar F_0(x), \qquad P(Y>y\mid X>x)=\bar F_1\bigl(\beta(x)y\bigr),
\]
where \(\bar F_0\) and \(\bar F_1\) are survival functions and \(\beta(x)\) is the acceleration function. Which statement about the covariance \(\operatorname{Cov}(X,Y)\) holds under these assumptions?

If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)\le 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)\ge 0\).

Choices:
B. If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)\ge 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)\le 0\).
C. If \(\beta(x)\equiv c\) is constant for some \(c>0\), then \(X\) and \(Y\) are independent, and hence \(\operatorname{Cov}(X,Y)=0\).
D. If \(\beta'(x)\ge 0\), then \(\operatorname{Cov}(X,Y)< 0\); and if \(\beta'(x)\le 0\), then \(\operatorname{Cov}(X,Y)> 0\).
E. If \(\beta'(x)\ge 0\) for all sufficiently large \(x\), then \(\operatorname{Cov}(X,Y)\le 0\); and if \(\beta'(x)\le 0\) for all sufficiently large \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

monotonicity-sign transfer through decreasing survival function

dropped the full sign-of-\(\beta'\) conclusion and retained only the constant-\(\beta\) independence case

equality case \(\beta'(x)\equiv 0\) leading to independence and zero covariance

global monotonicity in \(x\) needed to control the integrand for all \(x,y\)

The stem does not state or strongly hint at the covariance sign conclusion; the correct answer must be inferred from the model and monotonicity of the acceleration function.

The item is very close to a direct theorem recall question: it asks exactly which covariance-sign statement holds under the stated accelerated conditional model assumptions.

Some reasoning is needed to track how monotonicity of β affects Y through the decreasing survival function and to reject stronger/weaker variants, but the task is still mostly theorem recognition rather than deep generation.

The distractors are plausible and diagnostically useful: sign reversal, a weaker true statement, an overly strong strict-inequality version, and an insufficient tail-condition variant all reflect common mathematical mistakes.

A solid MCQ with strong distractors and no answer leakage, but it is mainly a theorem-restatement/recognition item rather than a highly generative reasoning question.