\begin{proposition}\label{prop:ball-meas-upper-bound}
For any compact set $K\subset \Omega_f$ there exists $r_K>0$ such that
\begin{align*}
    \Ha^n(B_{\Omega_f}(z,r))\le \omega_nd^{n/2}K_IK_O r^n,\quad z\in K,\ r<r_K.
\end{align*}
\end{proposition}

\begin{theorem}\label{thm:higher-integrability}
If $f:\Omega\to \R^n$ is a non-constant quasiregular map of degree $d$, then there exists $p>n$ such that $\minv f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_d(\Omega))$.
\end{theorem}

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\R^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\minv{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\R^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\R^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

\begin{theorem}\label{thm:wug}
For $\Mod_n$-a.e. curve $\gamma$ in $f(\Omega)$, $\minv f\circ\gamma$ is an absolutely continuous curve in $\mathcal{A}_d(\Omega)$, and satisfies
\begin{align}\label{eq:ug-ineq}
H(\gamma_t)|\gamma_t'|\le |(\minv f\circ\gamma)'_t|\le (K_IK_O)^{1/n} H(\gamma_t)|\gamma_t'|
\end{align}
a.e. $t$, where
\begin{align}\label{eq:H}
    H(y):=\big(f_\ast(\|Df\|^{-2})(y)\big)^{1/2},\quad y\in f(\Omega).
\end{align}
\end{theorem}

\begin{lemma}\label{lem:homeo}
The generalized inverse $\minv f:f(X)\to \mathcal A_d(X)$ is a homeomorphism onto its image.
\end{lemma}

\begin{lemma}\label{lem:barycenter}
The barycenter map \eqref{eq:barycenter} is $1/\sqrt d$-Lipschitz.
\end{lemma}

\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \R^n,\quad g(y):=\sum_{x\in f\inv(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\R^n)$.
\end{corollary}

\begin{align}\label{eq:inf-n-PI}
|Dg|_p=\Lip g\quad \mu\textrm{-a.e. on }X
\end{align}

\begin{theorem}\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \R^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\R^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\R^n)^Q$, and either $k+1\le p$ of $\ud \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\R^m)$ and moreover
\begin{align*}
    \ud(f^*\omega)=f^*(\ud\omega)
\end{align*}
weakly.
\end{theorem}

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Quasiregular curves (QR curves) in turn were introduced by Pankka \cite{pankka20}, and they generalize quasiregular maps by allowing the dimension of the target to be higher than that of the domain; for example pseudoholomorhic curves \cite{gro85} are QR curves. Much of the classical theory of QR maps extends to QR curves \cite{pankka20,pan-onn21}, with one notable exception: Reshetnyak's theorem. Namely, QR curves need not be branched covers. Indeed, if $k>n$, a QR curve cannot be open, while the more non-trivial failure of discreteness follows from an example in \cite{IVV02}.

Compared to QR maps, one of the challenges in QR curves is understanding how the choice of $\omega$ reflects the behaviour of $f$, see for example \cite{hei-pan-pry23,iko-pan24} for some rigidity results and connections with calibrations.

\bigskip\noindent In this paper, we identify a fruther connection between QR maps and QR curves: we construct a multi-valued inverse $\minv f$ for a QR map $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$, and show that $\minv f$ is naturally a QR curve (see Theorem \ref{thm:QR-curve}). In particular, QR maps have a local (multi-valued) inverse across the singular branch set (and not only outside it) with suitable Sobolev regularity. Earlier, related constructions include a push-forward operator \cite[p. 263]{HKM06}, \cite[Section 4]{teripekka} and a generalized local inverse \cite[Chapter II.5]{ric93}, \cite[Section 8]{onn-raj09}, both of which are important for obtaining modulus inequalities.

Our approach is based on Almgren's space of unordered tuples \cite{del11} (also known as the symmetric product, cf. \cite[Chapter 4.K]{hat02}), and brings a new perspective to quasiregular theory; together with a pull-back theory by multi-valued maps developed in this paper (see Theorems \ref{thm:sob-pull-back} and \ref{thm:lip-pullback-flat-form}) we obtain e.g. higher integrability properties of the generalized inverse \cite[Chapter II.5]{ric93} (see Corollary \ref{cor:QR-curve} and Theorem \ref{thm:higher-integrability}). We describe the construction and briefly discuss some its advantages over alternative approaches below.

\subsection{Multi-valued inverse}

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\begin{definition}\label{def:multi-valued-inv}
Let $f:X\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\minv f:f(X)\to \mathcal A_d(X)$ of $f$ is defined by
\begin{align*}
\minv f(y)=\sum_{x\in f\inv(y)}\iota(f,x)\bb x
\end{align*}
\end{definition}

\subsection{Statement of results}

In particular, if $f:U\to \mathcal A_Q(\R^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

An element $\omega\in \Omega^k(\mathcal A_d(V))$ is a smooth map $\omega:V^d\to \bigwedge^k(\R^n)^d$ with
\begin{align}\label{eq:S_d-invariant}
\omega_x(v_1,\ldots,v_k)=\omega_{\sigma(x)}(\sigma v_1,\ldots,\sigma v_k),\quad x\in V^d,\ v_1,\ldots, v_k\in (\R^n)^d.
\end{align}
In particular, for each $\bb{x}\in \mathcal A_d(V)$, the value $\omega_{\bb{x}}:=\omega_{\sigma(x)}\circ \sigma\in \bigwedge^k(\R^n)^d$
is well-defined and independent of $\sigma\in S_d$. We define the comass $\|\omega\|:\mathcal A_d(V)\to \R$ of a $k$-form $\omega\in \Omega^k(\mathcal A_d(V))$ by
\begin{align*}
\|\omega\|_{\bb x}=\|\omega_{x}\|,\quad \bb x\in \mathcal A_d(V).
\end{align*}
This is well-defined by \eqref{eq:S_d-invariant}.

\begin{lemma}\label{lem:pullback-est}
Suppose $U\subset \R^m$ and $V\subset \R^n$ are open sets, $\omega$ is a smooth $k$-form on $\mathcal A_d(V)$ and $h:U\to \mathcal A_d(V)$ a.e. approximately differentiable. Then $h^*\omega$ is a measurable $k$-form on $U$ and
\begin{align*}
\|h^*\omega\|_x\le \|D_xh\|_x^k\|\omega_{h(x)}\|\quad\textrm{a.e. }x\in U.
\end{align*}
\end{lemma}
\begin{remark}
In particular $h^*\omega\in L^{p/k}_{loc}(U)$ if $h\in W^{1,p}_{loc}(U,\mathcal A_d(V))$.
\end{remark}
\begin{proof}
The approximate differentials $Dh_1,\ldots,Dh_d$ are measurable, cf. ***. For all $x\in U$ where $h$ is approximately differentiable, we have
\begin{align*}
\|h^*\omega\|_x&=\max\{\omega_{h(x)}(Dh(v_1),\ldots,Dh (v_d)):\ |v_1|,\ldots,|v_d|\le 1\}\\
&\le \|D_xh\|^k\|\omega_{h(x)}\|,
\end{align*}
completing the proof.
\end{proof}

\begin{lemma}\label{lem:pull-back-of-decomp}
Suppose $f_i:U\to \mathcal A_{d_i}(V)$ is Lipschitz ($i=0,1$), $d=d_0+d_1$, and $\omega$ is a $S_{d_0}\times S_{d_1}$-invariant form on $V^d$. Then the pull-back $\bb{f_0,f_1}^*\omega$ is a well-defined $k$-form on $U$. Moreover,
\begin{align*}
\bb{f_0,f_1}^*(\omega_0\otimes\omega_1)=f_0^*\omega_0\wedge f_1^*\omega_1
\end{align*}
whenever $\omega_i\in \Omega^{k_i}(\mathcal A_{d_i}(V))$ ($i=0,1$) and $k=k_0+k_1$.
\end{lemma}
\begin{proof}
Let $x\in U$ be such that $f_0,f_1$ are differentiable at $x$, and let $f_i^1(x),\ldots, f_i^{d_i}(x)$ and $D_xf_i^1,\ldots,D_xf_i^{d_i}$ be as in Theorem \ref{thm:multi-valued-rademacher} (i) and (ii). (In particular, note that $f=\bb{f_0,f_1}$ is differentiable at $x$ and $Df=\bb{D_xf_0^1,\ldots,D_xf_0^{d_0},D_xf_1^1,\ldots,D_xf_1^{d_1}}$.) Given $(\sigma_0,\sigma_1)\in S_{d_0}\times S_{d_1}$, denote
\[
D_xf^{(\sigma_0,\sigma_1)}=(D_xf_0^{\sigma_0\inv(1)},\ldots D_xf_0^{\sigma_0\inv(d_0)},D_xf_1^{\sigma_1\inv(1)},\ldots,D_xf_1^{\sigma_1\inv(d_1)})
\]
and similarly
\[
(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))=(f_0^{\sigma_0\inv(1)}(x),\ldots,f_0^{\sigma_0\inv(d_0)},f_1^{\sigma_1\inv(1)},\ldots,f_1^{\sigma_1\inv(d_1)}).
\]
We define
\begin{align*}
 (f^*\omega)_x(v_1,\ldots,v_k)=\omega_{(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))}(D_xf^{(\sigma_0,\sigma_1)}v_1,\ldots,D_xf^{(\sigma_0,\sigma_1)}v_k)
\end{align*}
for $v_1,\ldots,v_k\in T_xU=\R^m$. By the $S_{d_0}\times S_{d_1}$-invariance of $\omega$, this expression is independent of $(\sigma_0,\sigma_1)$ and gives a well-defined element in $\bigwedge^k\R^m$, which we denote by $(f^*\omega)_x=\omega_{(f_0(x),f_1(x))}\circ(D_xf_0,D_xf_1)$.

\begin{proposition}\label{prop:flat-form}
Let $d_0,d_1\ge 1$ be natural numbers, $d:=d_0+d_1$, and let $U\subset \R^m$ be an open set. If $f_i:U\to \mathcal A_{d_i}(\R^n)$ are Lipschitz functions such that $\ud(f_i^*\omega)=f_i^*(\ud\omega)$ weakly for all $l$-forms $\omega\in \Omega^l(\mathcal A_{d_i}(\R^n))$, $l\le k$ ($i=0,1$), then $\ud(\bb{f_0,f_1}^*\omega)=\bb{f_0,f_1}^*(\ud\omega)$ weakly for all $k$-forms $\omega\in \Omega^k(\mathcal A_d(\R^n))$.
\end{proposition}
\begin{proof}
Denote $f:=\bb{f_0,f_1}$, and let $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Let
\[
(\omega_j)\subset \bigcup_{r+l=k}\Omega^l((\R^n)^{d_0})\otimes \Omega^r((\R^n)^{d_1})
\]
be a sequence such that $(\omega_j)_x\to \omega_x$ and $(\ud\omega_j)_x\to (\ud\omega)_x$ for all $x\in (\R^n)^d$ and $\sup_j\max\{\|\omega_j\|_\infty,\|\ud\omega_j\|_\infty\}<\infty$. Then
\begin{align*}
\tilde\omega_j&:=P_{S_{d_0}\times S_{d_1}}\omega_j\to P_{S_{d_0}\times S_{d_1}}\omega=\omega,\\
\ud\tilde\omega_j&=P_{S_{d_0}\times S_{d_1}}(\ud\omega_j)\to P_{S_{d_0}\times S_{d_1}}(\ud\omega)=\ud\omega
\end{align*}
and $\sup_j\max\{\|\tilde \omega_j\|_\infty,\|\ud\tilde \omega_j\|_\infty\}<\infty$, cf. Lemma \ref{lem:proj-onto-invariant-forms}. By Lemma \ref{lem:pull-back-of-decomp} each $\tilde\omega_j$ is a sum of tensor products $\omega_0\otimes\omega_1$ where $\omega_i$ is $S_{d_i}$-invariant and $f^*(\omega_0\otimes\omega_1)=(f_0^*\omega_0)\wedge(f^*_1\omega_1)$. Together with the assumption that $f_i^*(\ud\omega_i)=\ud(f^*_i\omega)$ weakly for $S_{d_i}$-invariant forms $\omega$, this yields
\begin{align*}
f^*(\ud(\omega_0\otimes\omega_1))&=f^*(\ud\omega_0\otimes \omega_1+(-1)^{\deg\omega_0}\omega_0\otimes\ud\omega_1)\\
&=f^*_0(\ud\omega_0)\wedge(f^*_1\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge (f_1^*(\ud\omega_1))\\
&=\ud(f_0^*\omega_0)\wedge(f_1^*\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge\ud(f_1^*\omega_1)\\
&=\ud(f_0^*\omega_0\wedge f_1^*\omega_1)=\ud f^*(\omega_0\otimes\omega_1)
\end{align*}
weakly. Consequently $f^*(\ud\tilde\omega_j)=\ud(f^*\tilde\omega_j)$ weakly. Now the pointwise convergence $f^*\tilde\omega_j\to f^*\omega$, $f^*(\ud\tilde\omega_j)\to f^*(\ud\omega)$ together with the dominated convergence theorem yield
\begin{align*}
\int_U \alpha\wedge f^*(\ud\omega)&=\lim_{j\to\infty}\int_U \alpha\wedge f^*(\ud\tilde\omega_j)=(-1)^{k+1}\lim_{j\to\infty}\int_U\ud\alpha\wedge f^*\tilde\omega_j\\
&=(-1)^{k+1}\int_U\ud\alpha\wedge f^*\omega
\end{align*}
for all $\alpha\in \Omega_c^{m-k-1}(U)$. This proves the claim.
\end{proof}

\begin{proof}
Denote $H(y):=(f_\ast(|Df|^{-2}))^{1/2}$ and observe that
\[
\lim_{h\to 0}\frac{d_\mathcal A(\minv f(y+hv),\minv f(y))}{|h|}=\md_y\minv f(v) \quad\textrm{ a.e. }y\in f(\Omega)
\]
for each $v\in \mathbb S^{n-1}$. By considering the family $\Gamma_v=\{\gamma\subset f(\Omega): \gamma'=v\}$, it follows from a Fubini type argument and Theorem \ref{thm:wug} that
\begin{align}\label{eq:metric-diff-est}
H(y)\le \md_y\minv f(v)\le (K_IK_O)^{1/n}H(y)\quad \textrm{ for all } v\in \mathbb S^{n-1}.
\end{align}
This implies \eqref{eq:jacob-vs-H}. The remaining claim follows from \eqref{eq:metric-diff-est}. Let $\Gamma$ be a path family in $f(\Omega)$ and let $\rho$ be admissible for $\minv f(\Gamma)$. If $\Gamma_0$ is the exceptional family in Theorem \ref{thm:wug} then $\Mod_v\Gamma_0=0$ and, for any $\gamma\in \Gamma\setminus \Gamma_1$ we have
\begin{align*}
1\le \int_{\minv f\circ\gamma}\rho\ud s&=\int_0^1\rho(\minv f(\gamma_t))|(\minv f\circ\gamma)_t'|\ud t\le (K_IK_O)^{1/n}\int_0^1\rho\circ \minv f(\gamma_t)H(\gamma_t)|\gamma_t'|\ud t\\
&=(K_IK_O)^{1/n}\int_\gamma(\rho\circ \minv f)\cdot H\ud s.
\end{align*}
Thus $(K_IK_O)^{1/n}(\rho\circ \minv f)H$ is admissible for $\Gamma\setminus \Gamma_0$ implying
\begin{align*}
\Mod_n\Gamma&=\Mod_n\Gamma\setminus\Gamma_0\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f H^n\ud y\\
&\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f\J \minv f\ud y=K_IK_O\int_{\Omega_f}\rho^n\ud\Ha^n.
\end{align*}
Taking infimum over $\rho$ yields $\Mod_n\Gamma\le K_IK_O\Mod_n\minv f(\Gamma)$ and completes the proof.
\end{proof}

In particular, if $f:U\to \mathcal A_Q(\mathbb{R}^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

Applying the definition of pull-back in the QR context we can show that, if $f:\Omega\to \mathbb{R}^n$ is a quasiregular map, the multi-valued inverse is a quasiregular curve.

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\accentset{\leftarrow}{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\mathbb{R}^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

Theorems \ref{thm:sob-pull-back} and \ref{thm:QR-curve} together with standard arguments found e.g. in \cite{pan-onn21} yield in particular the higher integrability of the generalized inverse of \cite[Chapter II.5]{ric93}.
\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \mathbb{R}^n,\quad g(y):=\sum_{x\in f^{-1}(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\mathbb{R}^n)$.
\end{corollary}

The multi-valued inverse can also be seen as a homeomorphism $\minv f:f(\Omega)\to \Omega_f$ onto its image $\Omega_f:=\minv f(f(\Omega))\subset \mathcal A_d(\mathbb{R}^n)$ (Lemma \ref{lem:homeo}). Thus, although $\mathcal A_d(\mathbb{R}^n)$ is not a manifold, $\Omega_f$ is a metric $n$-manifold when equipped with the metric from $\mathcal A_d(\mathbb{R}^n)$. In the following theorem we obtain that $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal.

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\mathbb{R}^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

We refer to \eqref{eq:inf-n-PI} in Section \ref{sec:Omega_f-prop} for the definition of infinitesimal Poincar\'e inequality and further commentary. We mention here that the infinitesimal Poincar\'e inequality is not a quantitative condition.

\subsection{Main ideas and further discussion}
Theorem \ref{thm:sob-pull-back} follows by an approximation argument from it Lipschitz counterpart, Theorem \ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. We outline the main idea.

In order to apply induction in the proof of Theorem \ref{thm:lip-pullback-flat-form}, we will need to approximate a given multi-valued Lipschitz map by maps which are locally decomposable. A given map $f:U\to \mathcal A_d(\mathbb{R}^n)$ fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$. The restriction of $f$ to $F$ agrees with the map $d\llbracket b(f)\rrbracket$, where $b:\mathcal A_d(\mathbb{R}^n)\to \mathbb{R}^n$ is the barycenter map
\begin{align}\label{eq:barycenter}
b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d},
\end{align}
which is $\frac 1{\sqrt d}$-Lipschitz, see Lemma \ref{lem:barycenter} below. The equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies that the differentials of the two functions agree almost everywhere. However, since $F$ is closed and not open, we need an approximation argument for the induction step to work. Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$.

The main step in the proof of Theorem \ref{thm:QR-curve} are the Sobolev estimates, which we obtain in Theorem \ref{thm:wug}. Our argument here uses Vitali's covering theorem, which is valid in high generality, and thus Theorem \ref{thm:wug} probably holds e.g. in the setting of \cite{onn-raj09}.

The proof of Theorem \ref{thm:geom-QC} employs a similar induction argument as in Theorem \ref{thm:lip-pullback-flat-form} together with some metric quasiconformal theory. We remark here that a large part of metric quasiconformal theory is based on the Ahlfors regularity of the spaces under consideration. Indeed, in our setting this property would imply the Loewner property of the image set $\Omega_f:=\minv f(f(\Omega)$ by a (deep) result of Semmes \cite{sem96}, which in turn implies the equivalence of the various notions of quasiconformality in the metric setting \cite{hei98}. In this paper, we establish the upper Ahlfors $n$-regularity of $\Omega_f$ (Proposition \ref{prop:ball-meas-upper-bound}), and an infinitesimal Poincar\'e inequality. It would be interesting to analyze the geometry of $\Omega_f$ further, in particular whether $\Omega_f$ admits local bi-Lipschitz parametrizations. By Almgren's bi-Lipschitz embedding $\mathcal A_d(\mathbb{R}^n)\hookrightarrow \mathbb{R}^N$, $\Omega_f$ can be regarded as a subset of Euclidean space, and the existence of bi-Lipschitz parametrizations is linked with the existence of Cartan--Whitney presentations in a suitable Sobolev class  \cite[[Theorem 1.2]{hei11}.

Theorem~\ref{thm:sob-pull-back} "follows by an approximation argument from its Lipschitz counterpart, Theorem~\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme." The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, to obtain the Sobolev pull-back statement of Theorem~\ref{thm:sob-pull-back}.

To prove the main theorem, we proceed by an approximation argument from its Lipschitz counterpart. We first prove the following theorem.

\begin{theorem}\label{thm:lip-pullback-flat-form}
Suppose $U\subset\R^m$ is an open set, $f:U\to\mathcal A_d(\R^n)$ is Lipschitz and $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Then $\ud(f^*\omega)=f^*(\ud\omega)$ weakly. In particular, $f^*\omega$ is a flat form on $U$ in the sense of Whitney.
\end{theorem}

Its proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, in establishing the main theorem.

\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \mathbb{R}^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\mathbb{R}^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\mathbb{R}^n)^Q$, and either $k+1\le p$ of $\mathrm {d} \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\mathbb{R}^m)$ and moreover
\begin{align*}
    \mathrm {d}(f^*\omega)=f^*(\mathrm {d}\omega)
\end{align*}
weakly.,

Question: Let $n,m,Q\ge 1$ be natural numbers, let $U\subset \mathbb{R}^m$ be open, and let $f\in W^{1,p}_{\mathrm{loc}}(U,\mathcal A_Q(\mathbb{R}^n))$, where $\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q$ is the Almgren space of unordered $Q$-tuples and $S_Q$ acts on $(\mathbb{R}^n)^Q$ by permuting the $Q$ factors. Let $\omega$ be a smooth $k$-form on $(\mathbb{R}^n)^Q$ that is $S_Q$-invariant, meaning $\sigma^*\omega=\omega$ for every permutation $\sigma\in S_Q$. Assume either (i) $k+1\le p$, or (ii) $\mathrm d\omega=0$ and $k\le p$. Under these hypotheses, which conclusion about the pull-back of $\omega$ by $f$ is valid?

The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover its weak exterior derivative satisfies
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$.

Choices:
B.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ whenever $k\le p$; moreover one always has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$, without any additional assumption such as $k+1\le p$ or $\mathrm d\omega=0$.
C. The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ under the stated hypotheses.
D.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover, if $\mathrm d\omega=0$ and $k\le p$, then
\[
\mathrm d(f^*\omega)=0
\]
weakly on $U$, whereas the identity \(\mathrm d(f^*\omega)=f^*(\mathrm d\omega)\) is guaranteed only in the case $k+1\le p$.
E.
For every smooth $k$-form $\omega$ on $(\mathbb{R}^n)^Q$, the pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; if in addition $\omega$ is $S_Q$-invariant, then one has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$ under the stated hypotheses.

integrability threshold k+1<=p versus merely k<=p for non-closed forms

differential identity conclusion dropped

closed-form case still yields full chain-rule identity, not just vanishing derivative

S_Q-invariance is required for the pull-back itself to be well defined on Almgren space

The stem does not explicitly state the conclusion. It gives hypotheses and asks for the valid consequence, so the correct answer is not leaked directly.

The item is quite close to a theorem-recall question: the correct option essentially states the theorem under the given hypotheses. However, it is not purely tautological because the distractors alter thresholds, invariance requirements, and the derivative conclusion.

Some reasoning is needed to distinguish the exact chain-rule conclusion from weaker or overgeneralized variants, especially around the conditions k+1<=p versus dω=0 and k<=p. Still, the question mainly tests precise recall of a theorem rather than substantial derivation.

The distractors are mathematically plausible and target common mistakes: overextending regularity assumptions, omitting the differential identity, confusing the closed-form case, and ignoring the need for S_Q-invariance for well-definedness on Almgren space.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than deep generative reasoning.


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question:
Let \(B_r\) 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:toric-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair over $\mathbb{D}$ for which $(\mathcal{X},\mathcal{X}_0)$ is plt.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is a toric pair\footnote{Meaning that $\mathcal{X}_t$ is a projective toric variety and $\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \in \mathbb{D}^*$,
then $(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a toric pair.
\end{theorem}

\begin{proposition}\label{prop:tor-deg}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\pp(1,1,n)$ with $n\geq 3$, then $\mathcal{X}_0$ is toric if and only if $\mathcal{X}_0\cong \pp(x^2,y^2,n)$ such that $x^2+y^2+n=(n+2)xy$.
\end{proposition}

\begin{definition}\label{def:pt}
{\em
A pair $(X,B)$ is said to be {\em purely terminal} if the following conditions are satisfied:
\begin{enumerate}
\item the pair $(X,B)$ is plt; and
\item for every prime component $S\subset \lfloor B\rfloor$ the pair $(S,B_S)$, obtained from adjunction of $(X,B)$ to $S$, is terminal.
\end{enumerate}
}
\end{definition}

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\pp(a^2,b^2,c^2)$ has no non-trivial $\qq$-Gorenstein klt degenerations.
\end{corollary}

\begin{example}\label{ex:P^2-toric-model-1}
{\em
We consider the pair
$(\pp^2,B)$ where $B$ is the sum of a quadric and a transversal line.
This example is of cluster type, as described by Gross, Hacking, and Keel~\cite{GHK15}.
The choice of a torus embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ is equivalent to the choice of a {\em toric model}
$\phi \colon (T,B_T)\dashrightarrow (\pp^2,B)$.
More precisely, $\phi$ is a crepant birational map that only extracts log canonical places and $(T,B_T)$ is a toric pair. In this example, we demonstrate
that the choice of such a toric model, or equivalently the choice of an embedding $\mathbb{G}_m^2\hookrightarrow \pp^2\setminus B$ naturally induces a degeneration of $\pp^2$ to a toric pair.
We will consider a toric model
$(\mathbb{F}_2,B_T)$ of $(\pp^2,B)$
where $\mathbb{F}_2$ is the second Hirzebruch surface. This toric model induces an embedding $j_0\colon \mathbb{F}_2\setminus B_T \simeq \mathbb{G}_m^2 \hookrightarrow \pp^2\setminus B$.

We start with the trivial family
\begin{equation}\label{fam1}
(\mathcal{T},B_{\mathcal{T}})\rightarrow \mathbb{D},
\end{equation}
where all the fibers are isomorphic
to the second Hirzebruch surface plus its toric boundary.
In other words, we have
\[
(\mathcal{T},B_{\mathcal{T}}) \simeq
(\mathbb{F}_2,B_T)\times \mathbb{D}.
\]
Let $B_{\mathcal{T}}=S_{\mathcal{T}}+\Delta_{0,\mathcal{T}}+F_{0,\mathcal{T}}+F_{\infty,\mathcal{T}}$ where $S_{\mathcal{T}}\sim \Delta_0+2F\subset \mathbb{F}_2$ and the $f$'s are the fibers of the projection $\mathbb{F}_2\rightarrow \pp^1$. We blow-up the image of a section
$s(\mathbb{D})\subset \mathcal{T}$ whose image is contained in
$F_{0,\mathcal{T}}$ and $s(0)$
is the intersection of
$F_{0,\mathcal{T},0}$ and $\Delta_{0,\mathcal{T}}$.
This leads to a family
\begin{equation}\label{fam2}
(\mathcal{T}',B_{\mathcal{T}'})\rightarrow \mathbb{D}
\end{equation}
where the general fiber is a blow-up of $\mathbb{F}_2$ along a general point of a fiber of $\mathbb{F}_2\rightarrow \pp^1$
and the central fiber is a toric surface.
We may blow-down the strict transform of
$F_{0,\mathcal{T}}$ in $\mathcal{T}'$ to obtain a family
\begin{equation}\label{fam3}
(\mathcal{Y}^+,B_{\mathcal{Y}^+}) \rightarrow \mathbb{D}.
\end{equation}
The general fiber of this family is
isomorphic to $\mathbb{F}_1$ and
$\mathcal{B}_{Y^+,t}$ has three components;
a fiber, a section with negative self-intersection, and a section with positive self-intersection.
The central fiber of~\eqref{fam3} is $\mathbb{F}_3$ with its toric boundary.
By \cite[Corollary 3.23]{HTU17}, the family admits an anti-flip which leads to a degeneration
\begin{equation}\label{fam4}
(\mathcal{Y},B_{\mathcal{Y}})\rightarrow \mathbb{D}.
\end{equation}
The general fiber of~\eqref{fam4}
is isomorphic to the general fiber of~\eqref{fam3}.
The central fiber of~\eqref{fam4} is
isomorphic to the blow-up of
$\mathbb{P}(1,1,4)$ at a torus invariant point (see for example \cite[Section 7]{UZ24}).
The family $\mathcal{Y}\rightarrow \mathbb{D}$ admits a blow-down
$\mathcal{X}\rightarrow \mathbb{D}$.
We obtain a family of Calabi--Yau pairs
\begin{equation}\label{fam5}
(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}.
\end{equation}

Below, we give an explicit description, modulo change of coordinates, of the family~\eqref{fam5} and the blow-up
$\mathcal{Y}\rightarrow \mathcal{X}$.
The family is given by
\[
\mathcal{X}_t=\{ [x:y:z:w] \mid xy=z^2+tw = 0\} \subset \pp(1,1,1,2)
\]
for $t\in \mathbb{D}$.
It is straightforward to check that $\mathcal{X}_t\cong \pp^2$ for $t\neq 0$ and $\mathcal{X}_0\cong \pp(1,1,4)$.
Indeed, $\mathcal{X}_t$, for $t\neq 0$, is a smooth del Pezzo surface of Picard rank one. The family of curves $\mathcal{B}\to \mathbb{D}$, where
\begin{equation}
\mathcal{B}_t=\{[x:y:z:w] \mid xyw^2+tz^2w^2=0\} \subset \mathcal{X}_t \subset \pp(1,1,1,2).
\end{equation}
When $t=0$, $\mathcal{B}_0$ corresponds to the toric boundary of $\pp(1,1,4)$, while for $t\neq 0$ it follows that
\begin{equation*}
B_t=\{[x:y:z:w] \mid w^2((1+t)z^2+tw)=0\}
\end{equation*}
 which corresponds to the union of the line $L_t=\{w=0\}$ and the irreducible conic
\begin{equation*}
Q_t=\{[x:y:z:w]\mid (1+t)z^2+tw=0\}.
\end{equation*}
The intersection points of $L_t$ and $Q_t$ are $[1:0:0:0]$ and $[0:1:0:0]$ which specialize to the smooth toric invariant points of $\mathcal{X}_0$.
The family~\eqref{fam4} described above is obtained by blowing up the section $s:\mathbb{D}\to \mathcal{X}$ such that $s(t)=[1:0:0:0]$, we obtain a $\qq$-Gorenstein degeneration $\mathbb{F}_1\rightsquigarrow Bl_p\pp(1,1,4)$.

By construction, the family~\eqref{fam5} admits the embedding of a product
\[
\xymatrix{
\mathbb{D} \times \mathbb{G}_m^2 \ar@{^{(}->}[rr]\ar[rd]_-{\pi_1} & &  \mathcal{X}\setminus \mathcal{B} \ar[ld] \\
& \mathbb{D}.&
}
\]
The restriction of the previous embedding to the general fiber recovers $j_0$
while the restriction to the central fiber
is just the embedding of the open torus into the toric variety $\pp(1,1,4)$.
}
\end{example}

In this article, we study degenerations of varieties and pairs.
We focus on degenerations with central fiber having klt singularities.
These are known as {\em klt degenerations}.
The problem is particularly interesting when the general fiber $\mathcal{X}_t$ of the degeneration $\mathcal{X}\rightarrow \mathbb{D}$ is a Fano variety.
In this case, there are many possible degenerations $\mathcal{X}_0$
and these degenerations are interesting even from the combinatorial perspective.
For instance, it is known that the toric degenerations of $\mathbb{P}^2$ are given by
\begin{equation}\label{eq:markov-triple}
\mathbb{P}(a^2,b^2,c^2) \text{ with $a^2+b^2+c^2=3abc$.}
\end{equation}
These triples $(a,b,c)\in \zz_{\geq 1}^3$ are known as {\em Markov triples}.
The topic of degenerations of projective surfaces is classic in algebraic geometry.
In~\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\eqref{eq:markov-triple}.
These surfaces were studied by Manetti and are nowadays known as
{\em Manetti surfaces} (see~\cite{Man91}).
In~\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\mathbb{Q}$-Gorenstein smoothings.
In~\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\mathbb{Q}$-Gorenstein families.
In~\cite{UZ24}, Urzua and the second author explain how to relate
the degenerations as in~\eqref{eq:markov-triple} via birational transformations.
Given two degenerations $\mathcal{X} \rightarrow \mathbb{D}$
and $\mathcal{Y}\rightarrow \mathbb{D}$ of $\mathbb{P}^2$ into
$\mathcal{X}_0 \simeq \mathbb{P}(a,b,c)$
and $\mathcal{Y}_0\simeq \mathbb{P}(d,e,f)$,
the authors explain how to perform small birational modifications to go from
$\mathcal{X}\rightarrow \mathbb{D}$ to $\mathcal{Y}\rightarrow \mathbb{D}$
and how these small modifications reflect on the combinatorics of the Markov triples.
In higher dimensions, H\"oring and Peternell proved that a klt degeneration $X$ of $\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\cite{HP24}).
They also classify normal degenerations of $\mathbb{P}^3$ with canonical singularities.
In~\cite{HKW25}, Hausen, Kir\'aly, and Wrobel explicitly determine all log terminal,
rational, degenerations of $\mathbb{P}^2$ that admit non-trivial torus actions.
In~\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear
in the degenerations of del Pezzo surfaces of degree $d$,
extending the work of Manetti and Hacking-Prokhorov in degree $9$.
Recently in~\cite{Pen25}, Peng studied special $\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$
induced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.
Peng proved that the space of special valuations of $(X,C)$ is connected and admits a
partition, which is locally finite, and each interval corresponds to a different $\mathbb{G}_m$-degeneration of $X$.

Throughout this work, we focus on understanding the degenerations of cluster type
varieties and cluster type pairs.
A {\em cluster type pair} is a pair $(X,B)$ with mild singularities
for which $K_X+B\sim 0$ and there is an embedding in codimension one
$\mathbb{G}_m^n \dashrightarrow X\setminus B$ (see Definition~\ref{def:ct}).
A {\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure.
In this case, we call $B$ a cluster type boundary.
Cluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs.
Indeed, any toric pair is a cluster type pair.
However, the realm of cluster type pairs is much broader.
A del Pezzo surface of degree $d\geq 2$ as well as a general del Pezzo
surface of degree $d=1$ is cluster type (see, e.g.,~\cite[Theorem 2.1 and Remark 2.2]{ALP23}).
Thus, all the examples discussed in the introduction are cluster type varieties.
Our first aim is to understand degenerations of cluster type pairs
for which the limit of the cluster type boundary still has reasonable singularities.
In the next subsection, we prove that there are not many such degenerations
for toric pairs, but quite a lot for cluster type pairs.

\subsection{Degenerations of cluster type pairs}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

\begin{definition}\label{def:ct}
{\em
Let $(X,B)$ be log Calabi--Yau pair of dimension $n$.
We say that $(X,B)$ is of {\em cluster type}
or that $(X,B)$ is a {\em cluster type pair} if there exists an embedding
in codimension one
\[
\mathbb{G}_m^n \dashrightarrow X\setminus B,
\]
i.e., there exists a closed subset of codimension at least two $Z\subset \mathbb{G}_m^n$
and an embedding $\mathbb{G}_m^n\setminus Z \hookrightarrow X\setminus B$.
}
\end{definition}

Our first theorem states that degenerations of toric pairs
are finite quotients of toric pairs. We impose some conditions
on the singularities of the central fiber.

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

\begin{proof}[Proof of Theorem~\ref{thm:toric-deg}]
Let $f\colon \mathcal{X'}\rightarrow \mathcal{X}$ be the finite cover given by Lemma~\ref{lem:finite-cover}.
By Lemma~\ref{lem:finite-cover}.(2), we have an induced
finite cover of log Calabi--Yau pairs $(\mathcal{X}',\mathcal{B}')\rightarrow (\mathcal{X},\mathcal{B})$.
Then, up to shrinking $\mathbb{D}$ near $\{0\}$ we have a commutative diagram
\[
\xymatrix{
(\mathcal{X},\mathcal{B})\ar[d]_-{\pi} & (\mathcal{X}',\mathcal{B}') \ar[l]_-{f}\ar[d]^-{\pi'} \\
\mathbb{D} & \mathbb{D}\ar[l]_-{f_\mathbb{D}}
}
\]
where $f_\mathbb{D}$ is simply given by $t\mapsto t^k$ for some suitable positive integer $k$.
The morphism $f'\colon \mathcal{X}'\rightarrow \mathbb{D}$ is a fibration by Lemma~\ref{lem:finite-cover}.(3).
By assumption $(\mathcal{X},\mathcal{X}_0)$ is plt so by Riemann-Hurwitz we conclude that $(\mathcal{X}',\mathcal{X}_0')$ is plt as well. In particular, the variety $\mathcal{X}_0'$ is irreducible.
By Lemma~\ref{lem:finite-cover}.(4), every component of $\mathcal{B}'_t$
is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}'$.
Note that $\pi$ and $\pi'$ have the same general log fibers; indeed $f$ is induced by a finite cover of $\mathbb{D}$ ramified over $\{0\}$.
Therefore, the general fiber $(\mathcal{X'},\mathcal{B'})$ is a projective toric variety of dimension $n$
and Picard rank $\rho$.
Shrinking $\mathbb{D}$ around $\{0\}$, we may assume that all the fibers over $\pi'$ are irreducible.
Therefore, we conclude that $\rho(\mathcal{X}'/\mathbb{D})\leq \rho$.
On the other hand, as every component of $\mathcal{B}'_t$ is the restriction to $\mathcal{X}'_t$ of a component of $\mathcal{B}$,
we conclude that $\mathcal{B}$ has at least $n+\rho$ components.
Thus, we can compute the relative complexity of
the log Calabi--Yau pair $(\mathcal{X}',\mathcal{B}'+\mathcal{X}_0')$ over $\{0\} \in \mathbb{D}$.
We obtain
\[
c_{\{0\}}(\mathcal{X}'/\mathbb{D},\mathcal{B}'+\mathcal{X}_0') =
\dim \mathcal{X}' + \rho(\mathcal{X}'/\mathbb{D}) - |\mathcal{B}'+\mathcal{X}_0'| \leq
n+1+\rho - (n+\rho+1).
\]
Therefore, by~\cite[Theorem 1]{MS21}, we conclude that
$(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ is a formally toric morphism near $\{0\}$.
Thus, the pair $(\mathcal{X}'_0,\mathcal{B}'_0)$ obtained from adjunction
of $(\mathcal{X}',\mathcal{B}'+\mathcal{X}'_0)$ must be a projective
toric log Calabi--Yau pair.
Henceforth, we have a finite crepant morpshim
of log Calabi--Yau pairs
\[
f_0\colon (\mathcal{X}'_0,\mathcal{B}'_0)
\rightarrow (\mathcal{X}_0,\mathcal{B}).
\]
We conclude that the pair $(\mathcal{X}_0,\mathcal{B}_0)$
is a finite quotient of a toric pair.
\end{proof}

\begin{theorem}\label{thm:1-comp}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a Fano projective morphism where $\mathcal{X}_0$ is reduced in $\mathcal{X}$ and
$(\mathcal{X},\mathcal{X}_0)$ plt.
Assume that the following conditions hold for the general fiber $\mathcal{X}_t$ with $t\neq 0$:
\begin{enumerate}
\item $\mathcal{X}_t$ is a toric surface of Picard rank one;
\item ${\rm mld}(\mathcal{X}_t)<\frac{1}{6}$;
\item $\mathcal{X}_t$ does not have singularities in the baskets
$\mathcal{F}_1,\dots,\mathcal{F}_4,\mathcal{D}$.
\end{enumerate}
Then, the pair $(\mathcal{X},\mathcal{X}_0)$ admits a $1$-complement over $\mathbb{D}$.
\end{theorem}
\begin{proof}
We break the proof into five steps.\\

\begin{example}\label{ex:non-toric-central}
{\em
We give an example of a degeneration of a toric pair where the central fiber is not a toric variety.
Fix $n\geq 3$ and consider the variety $\mathcal{Y}:=(\pp^1)^n \times \mathbb{D}$
and let $\pi_2\colon \mathcal{Y}\rightarrow \mathbb{D}$ be the projection onto the second component.
Let $B_T$ be the torus-invariant boundary of $(\pp^1)^n$
and $\mathcal{B}_\mathcal{Y}:=B_T\times \mathbb{D}$.
Therefore, the pair $(\mathcal{Y},\mathbb{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is a log Calabi--Yau pair for which every fiber of $\pi_2$ is isomorphic to
$((\pp^1)^n,B_T)$.
Consider the group $\zz_2$ acting on $\mathcal{Y}$ via
\begin{align*}
\mu & \colon (\pp^1)^n\times \mathbb{D} \rightarrow
(\pp^1)^n \times \mathbb{D} \\
\mu & \cdot ([x_1:y_1],[x_2:y_2],\dots,[x_n:y_n],t) :=
([y_1:x_1],[y_2:x_2],\dots,[y_n:x_n],-t).
\end{align*}
The log Calabi--Yau pair $(\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0)$
is invariant under the $\zz_2$-action.
Let $(\mathcal{X},\mathcal{B}+\mathcal{X}_0)$ be the induced quotient, so
we obtain a commutative diagram:
\[
\xymatrix{
(\mathcal{X},\mathcal{B}+\mathcal{X}_0) \ar[d]^-{\pi} & (\mathcal{Y},\mathcal{B}_{\mathcal{Y}}+\mathcal{Y}_0) \ar[d]^-{\pi_2} \ar[l]^-{/\zz_2} \\
\mathbb{D} & \mathbb{D} \ar[l]^-{/\zz_2}
}
\]
The morphism $\pi$ is a degeneration of the toric log Calabi--Yau pair
$((\pp^1)^n,B_T)$ into $(\mathcal{X}_0,\mathcal{B}_0)=((\pp^1)^n,B_T)/\zz_2$.
We have that $\mathcal{D}(\mathcal{X}_0,\mathcal{B}_0)\simeq_{\rm PL} \mathbb{P}_\rr^{n-1}$.
Thus, the central fiber $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair.
}
\end{example}

In the previous theorem,
the boundary $\mathcal{B}_0$ on the central fiber $\mathcal{X}_0$
is defined via adjunction.
This means that $\mathcal{B}_0$ is defined via the formula.
\[
K_{\mathcal{X}_0}+\mathcal{B}_0 = (K_{\mathcal{X}}+\mathcal{B}+\mathcal{X}_0)|_{\mathcal{X}_0}.
\]
In Example~\ref{ex:non-toric-central},
we show that in general $(\mathcal{X}_0,\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\ref{thm:toric-deg}.
Our next theorem gives a similar statement for degenerations of cluster type pairs.
We need to impose stronger conditions on the singularities in this case.
In the proof of Theorem~\ref{thm:toric-deg}, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.
Now, we turn to discuss the case of cluster type pairs.

\begin{theorem}\label{thm:ct-deg}
Let $\pi\colon \mathcal{X}\rightarrow \mathbb{D}$ be a projective Fano fibration.
Let $(\mathcal{X},\mathcal{X}_0+\mathcal{B})$ be a log Calabi--Yau pair of index one over $\mathbb{D}$.
Assume that $(\mathcal{X},\mathcal{X}_0)$ is plt
and purely terminal on the complement of $\mathcal{B}$.
If $(\mathcal{X}_t,\mathcal{B}_t)$ is of cluster type for $t\in \mathbb{D}^*$, then
$(\mathcal{X}_0,\mathcal{B}_0)$ is a finite quotient of a cluster type pair.
\end{theorem}

We refer the reader to Definition~\ref{def:pt} for the concept of purely terminal pairs.
In contrast to the toric case, a cluster type pair may have several interesting degenerations for which the degeneration of the cluster type boundary still has lc singularities. In Example~\ref{ex:P^2-toric-model-1}, we show that some toric degenerations of $\mathbb{P}^2$ can be regarded as cluster type degenerations for different embeddings of algebraic tori
$\mathbb{G}_m^2 \hookrightarrow \mathbb{P}^2\setminus C$ where $C$ is a nodal cubic.

\subsection{Degenerations of singular surfaces}

Now, we restrict ourselves to the study of degenerations of singular toric surfaces.
One of our aims is to understand how the singularities affect the possible degenerations.
Our third theorem states that almost every weighted projective plane has no interesting degenerations.

\begin{theorem}\label{no-deg-almost}
For almost all well-formed triples $(a,b,c)\in \zz_{\geq 1}^3$
the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial
$\mathbb{Q}$-Gorenstein klt degenerations.
\label{weighteddeg}
\end{theorem}

In the previous theorem, when we write {\em almost all}, we mean that the statement holds up to a subset $S\subsetneq \zz_{\geq 1}^3$ that has density zero with the natural density endowed from $\mathbb{Z}^3$.
We say that a degeneration is {\em trivial} if the central fiber is isomorphic to the general fiber.
However, in the setting of Theorem~\ref{no-deg-almost}, we will prove something stronger; the family is a product near the origin of the disk.
In order to prove Theorem~\ref{weighteddeg}, we will prove Theorem~\ref{thm:1-comp}, which is a general statement about
the existence of complements for degenerations of singular toric surfaces.
This means that, under some mild conditions, we prove that given a degeneration
$\mathcal{X}\rightarrow \mathbb{D}$ of a singular toric surface,
there exists some boundary $\mathcal{B}\in |-K_{\mathcal{X}}|$ for which $(\mathcal{X}_t,\mathcal{B}_t)$ has log canonical singularities for $t$ near $\{0\}\in \mathbb{D}$.
In the setting of Theorem~\ref{weighteddeg}, in most cases, we can argue that $(\mathcal{X}_t,\mathcal{B}_t)$ is toric for every $t$ and so
the statement is similar to that of Theorem~\ref{thm:toric-deg}, which states that there are no interesting such toric degenerations.
Theorem~\ref{thm:1-comp} is rather technical and depends on some meticulous analysis of basket of singularities.
The idea of using the theory of complements to understand degenerations of del Pezzo surfaces goes back to Hacking and Prokhorov.
The situation becomes a bit more delicate when we allow the general fiber of the degeneration to have singularities.
We will argue that for a Markov triple $(a,b,c)\in \zz_{\geq 2}^3$ the triple
$(a^2,b^2,c^2)$ belongs to the complement of the subset $S\subsetneq \zz_{\geq 0}^3$ of density zero mentioned above.
Thus, we conclude the following corollary.

\begin{corollary}\label{cor:markov-no-deg}
Let $(a,b,c)\in \zz_{\geq 2}^3$ be a Markov triple.
Then, the weighted projective plane $\mathbb{P}(a^2,b^2,c^2)$ has no non-trivial $\mathbb{Q}$-Gorenstein klt degenerations.
\end{corollary}

We note that Corollary~\ref{cor:markov-no-deg} can also be concluded from the work of Hacking and Prokhorov~\cite{HP10}. Indeed, every iterated degeneration of $\mathbb{P}(a^2,b^2,c^2)$ is indeed a degeneration of $\mathbb{P}^2$.
In upcoming work~\cite{Zun25}, the second author will prove
some structural theorems about $\mathbb{Q}$-Gorenstein klt degenerations
of Hirzebruch surfaces.
This will aim to finish the classification of $\mathbb{Q}$-Gorenstein
klt degenerations of minimal smooth rational surfaces.
This motivates us to pay particular attention to the weighted projective plane
$\mathbb{P}(1,1,n)$.
In this direction, using the tools introduced above, we can give
a complete classification of klt degenrations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

\begin{theorem}\label{thm:1-1-n}
Let $\mathcal{X}\rightarrow \mathbb{D}$ be a klt Fano degeneration of $\mathbb{P}(1,1,n)$ with $n \geq 3$,
then  for $\mathcal{X}_0$ one of the following holds:
\begin{enumerate}
\item $\mathcal{X}_0$ is a weighted projective plane, or
\item $\mathcal{X}_0$ is a $\mathbb{G}_m$-surface which is not toric.
\end{enumerate}
Furthermore, in the second case $\mathcal{X}_0$ is a $\mathbb{Q}$-Gorenstein deformation of a weighted projective plane.
\end{theorem}

In Proposition~\ref{prop:tor-deg},
we give a explicit classification
of the weighted projective planes
which are $\mathbb{Q}$-Gorenstein klt degenerations of $\mathbb{P}(1,1,n)$ with $n\geq 3$.

The paper is organized as follows.
In Section~\ref{sec:prelims}, we write some preliminary results regarding cluster type pairs, theory of complements, dual complexes, T-singularities, and Wahl singularities.
In Section~\ref{sec:degen-cluster-type}, we prove Theorem~\ref{thm:toric-deg} and Theorem~\ref{thm:ct-deg} regarding degenerations of toric pairs as well as cluster type pairs.
In Section~\ref{sec:complements-degen-klt-surfaces}, we prove some general statements regarding the existence of complements for degenerations of singular toric surfaces of Picard rank one. In this section, we also prove Theorem~\ref{weighteddeg} regarding the degenerations of weighted projective planes $\mathbb{P}(a,b,c)$.
In Section~\ref{sec:degen-wps}, we classify the degenerations of weighted projective planes $\mathbb{P}(1,1,n)$ with $n\geq 3$.
Finally, in Section~\ref{sec:ex-and-quest}, we give some examples and propose some questions for further research.

\subsection*{Acknowledgements}

The authors would like to thank Audric Lebovitz, Tomoki Oda, Giancarlo Urzúa, and Jos\'e Ignacio Y\'a\~nez
for many discussions related to this article.

In the proof of the main theorem, we will see that $(\mathcal{X},\mathcal{B})\rightarrow \mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.,

Question: Let \(\pi\colon \mathcal{X}\to \mathbb{D}\) be a projective fibration, let \(\mathcal{X}_0=\pi^{-1}(0)\), and let \((\mathcal{X},\mathcal{X}_0+\mathcal{B})\) be a log Calabi--Yau pair over \(\mathbb{D}\) (so \(K_{\mathcal{X}}+\mathcal{X}_0+\mathcal{B}\) is trivial over \(\mathbb{D}\)). Assume that \((\mathcal{X},\mathcal{X}_0)\) is plt. For every \(t\in \mathbb{D}^*:=\mathbb{D}\setminus\{0\}\), suppose the fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) is a toric pair, meaning that \(\mathcal{X}_t\) is a projective toric variety and \(\mathcal{B}_t\) is its reduced torus-invariant boundary. Which conclusion about the central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) holds?

The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is a finite quotient of a toric pair; equivalently, there exists a toric pair \((Y,B_Y)\), with \(Y\) a projective toric variety and \(B_Y\) its reduced torus-invariant boundary, together with a finite morphism of pairs \((Y,B_Y)\to (\mathcal{X}_0,\mathcal{B}_0)\).

Choices:
B. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is itself a toric pair; equivalently, \(\mathcal{X}_0\) is a projective toric variety and \(\mathcal{B}_0\) is its reduced torus-invariant boundary.
C. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) is birational to a toric pair; that is, there exists a projective toric pair \((Y,B_Y)\) and a birational map of pairs \((Y,B_Y) \dashrightarrow (\mathcal{X}_0,\mathcal{B}_0)\).
D. There exists a single projective toric pair \((Y,B_Y)\) such that, after possibly shrinking \(\mathbb{D}\), every fiber pair \((\mathcal{X}_t,\mathcal{B}_t)\) for \(t\in\mathbb{D}\) is isomorphic to \((Y,B_Y)\); in particular, the family is isotrivial and the central fiber is a toric pair isomorphic to the general fiber.
E. The central fiber pair \((\mathcal{X}_0,\mathcal{B}_0)\) admits a finite crepant morphism to a toric pair; equivalently, there exists a projective toric pair \((Y,B_Y)\) and a finite morphism of pairs \((\mathcal{X}_0,\mathcal{B}_0)\to (Y,B_Y)\).

The stem does not explicitly state or strongly hint at the key conclusion that the central fiber is a finite quotient of a toric pair. The relevant distinction only appears in the answer choices.

Although the question is theorem-driven, it is not a bare restatement: the options force the test-taker to distinguish among nearby conclusions such as toricity, birational toricity, isotriviality, and the direction of a finite morphism.

Some reasoning is required to select the strongest valid conclusion and reject stronger, weaker, or directionally incorrect variants. However, for someone who knows the theorem, the answer is largely recall-based rather than deeply generative.

The distractors are mathematically plausible and well-targeted: one is too strong, one is weaker-but-true-looking, one confuses family-level isotriviality with fiberwise toricity, and one reverses the finite morphism direction.

A strong MCQ with little answer leakage and high-quality distractors; it tests careful theorem discrimination well, though it remains somewhat recall-oriented rather than fully generative.