The fourth stable homotopy group of spheres is trivial, meaning that $\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\in\{2,3,4,5\}$, as we will briefly explain.

Curtis proved that $\pi_n(S^4)\neq 0$ for all $n\geq 4$ \cite{Cur69}. Curtis also proved that $\pi_n(S^2)\neq 0$ for all $n\not\equiv 1\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \cite{MMO75}. The proof that $\pi_n(S^5)\neq 0$ for all $n\geq 5$ was given by Mori \cite{Mor75} and Mahowald \cite{Mah75,Mah82}.

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
    Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

In fact, it appears that every positive integer has at least two Toda primes.

\begin{conj}\label{conj:at least two}
    If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
 Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{conj}\label{conj:at least two}
 If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{figure}[p]
    \includegraphics[width=.75\linewidth]{tn.png}
    \caption{$t(n)$ for $n\leq 100000$}\label{fig:tn}
\end{figure}

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\begin{lem}\label{lem:not 5 or not 3 and 5}
 Let $p\in\{3,5\}$. If $p\nmid n$, then $p\in T(n)$. In particular:
 \begin{itemize}
 \item If $5\nmid n$, then $t(n)\geq 1$.
 \item If $3,5\nmid n$, then $t(n)\geq 2$.
 \end{itemize}
\end{lem}
\begin{proof}
 Let $p\in\{3,5\}$. Then $p-1\mid 4n$, and $\gcd(4n,p)=1$ by assumption. Thus $\gcd(\frac{4n}{p-1},p)=1$, so $p\in T(n)$.
\end{proof}

\begin{lem}\label{lem:t(n)=3}
 Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}
\begin{proof}
 We will induct on $\omega(n)$. Our base cases will consist of $\omega(n)\leq 4$. Note that if $p\nmid n$ for each $p\in\{5,7,13\}$, then $T(n)\supseteq\{5,7,13\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\geq t(a)$ for any prime $p\not\in T(a)$ by \cref{cor:multiply by prime}, so we may assume that every prime factor $p\mid n$ is a Toda prime of some divisor of $n$.
 \begin{itemize}
 \item The case of $\omega(n)=1$ is just the calculation $T(3)=\{5,7,13\}$.
 \item For $\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.
 \item For $\omega(n)=3$, we first compute $t(3\cdot 5\cdot 7)=9$ and $t(3\cdot 5\cdot 13)=t(3\cdot 7\cdot 13)=8$. It remains to compute, for each $p\in\{5,7,13\}$, the Toda primes of $3pq$ for each $q\in T(3p)$. Using the code provided in \cref{sec:code}, we find that $t(3pq)\geq 4$ for all such $p,q$.
 \item For $\omega(n)=4$, we first compute $t(3\cdot 5\cdot 7\cdot 13)=16$. For the remaining computations in this case, we use the code in \cref{sec:code}.
 \begin{itemize}
 \item If $\{p,q\}\subseteq\{5,7,13\}$ and $r\in T(3pq)$, then $t(3pqr)\geq 9$.
 \item If $p\in\{5,7,13\}$ and $\{q,r\}\subseteq T(3p)$, then $t(3pqr)\geq 7$.
 \item If $p\in\{5,7,13\}$, $q\in T(3p)$, and $r\in T(3pq)$, then $t(3pqr)\geq 5$.
 \end{itemize}
 \end{itemize}

\begin{cor}\label{prop:divisible by 3}
 Assume \cref{conj:main}. If $3\mid n$, then $t(n)\geq 3$.
\end{cor}
\begin{proof}
 By \cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \cref{lem:t(n)=3}.
\end{proof}

Firstly, if there exists $q'\in T(3p)-\{5,7,13,q\}$ such that $q'\nmid n$, then $\{5,7,13,q,q'\}-\{p\}\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\{2d+1:d\mid 2n\}$ that are not factors of $n$. Thus if
\begin{equation}\label{eq:heuristic}
\{2d+1\text{ prime}:d\mid 2n\}-(\Omega(n)\cup\{5,7,13,q\})
\end{equation}
is non-empty, then $t(n)\geq 4$. Our heuristic for \cref{conj:main} is that the set $\{2d+1:d\mid 2n\}$ consists of $2^{\omega(n)+1}$ elements, while $\Omega(n)\cup\{5,7,13,q\}$ consists of $\omega(n)+3$ elements.

\begin{prop}\label{prop:t(p)}
Assume $p\geq 7$ is a prime. Let $\vphi$ denote the totient function. If $\vphi(x)=4p$ for some integer $x$, then $T(p)=\{3,5,2p+1\}$ or $\{3,5,4p+1\}$. Otherwise, $T(p)=\{3,5\}$.
\end{prop}
\begin{proof}
 One can directly check that $3,5\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\vphi(x)=p_1^{e_1-1}(p_1-1)\cdots p_m^{e_m-1}(p_m-1)$, where $x=\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\vphi(x)=4p$ if and only if one of the following cases holds:
 \begin{enumerate}[(i)]
 \item $x=2^2\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot 3\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=1$.
 \item $x=2^r\cdot 5^2$, where $r\in\{0,1\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\geq 7$.
 \end{enumerate}
 It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).

\begin{lem}\label{lem:strategy for denoms}
 Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\{2pi+1:i\mid 2a\}$ contains a prime number for each $p\in T(a)$, then \cref{conj:general bernoulli} \eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.
\end{lem}
\begin{proof}
 We know that $p-1\mid 4a$ with $\gcd(\frac{4a}{p-1},p)=1$ for all $p\in T(a)$. Thus $p-1\mid 4am$, and we have $\gcd(\frac{4am}{p-1},p)=1$ if and only if $p\nmid m$. It therefore suffices to show that if $p\mid m$ for some $p\in T(a)$, then $D_{4am}>D_{4a}$.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{lem}\label{lem:t(n)=3}
    Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}

\begin{thm}\label{thm:main}
    Assume \cref{conj:main}. If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.
\end{thm}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\subsection*{Acknowledgements}
We thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.

\label{thm:main}
Assume
\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}
If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.,

Question: For a positive integer m, let T(m) be the set of odd primes p such that p-1 divides 4m and \(\gcd\!\left(p,\frac{4m}{p-1}\right)=1\), and let \(t(m)=|T(m)|\). Assume the following statement is true: whenever m is an odd, square-free multiple of 3, there exists some \(p\in\{5,7,13\}\) with \(p\mid m\), no prime \(r\in\{5,7,13\}\setminus\{p\}\) divides m, and there exists \(q\in T(3p)\setminus\{5,7,13\}\) such that \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which statement about a positive integer n holds?

For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

Choices:
B. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\nmid n\), then \(t(n)\ge 3\).
C. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer n, \(t(n)\ge 2\). In addition, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer n, \(t(n)\ge 1\). In addition, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

divisibility condition on 3 in the final clause

dropped the strongest conclusion \(3\mid n\Rightarrow t(n)\ge 3\)

replaced the specific hypothesis \(5\nmid n\) by the unrelated \(3\nmid n\)


\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


\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