\begin{lemma}[{\cite[Lemma 6.1]{MR1962933}}]\label{techlem}
Let $Z:[r_{0},r_{1}] \rightarrow \mathbb{R}$ be a nonnegative and bounded function; let $\vartheta \in (0,1)$ and $C_{1},C_{2},\chi > 0$ be numbers. Assume that
\begin{equation*}
Z(\varrho_{0}) \le \vartheta Z(\varrho_{1}) + \frac{C_{1}}{(\varrho_{1}-\varrho_{0})^{\chi}} + C_{2}
\end{equation*}
holds for every choice of $\varrho_{0}$ and $\varrho_{1}$ such that $r_{0} \le \varrho_{0} < \varrho_{1} \le r_{1}$. Then the following inequality holds with $c\equiv c(\vartheta,\chi)>0$:
\begin{equation*}
Z(r_{0}) \le c\left[\frac{C_{1}}{(r_{1}-r_{0})^{\chi}} + C_{2}\right].
\end{equation*}
\end{lemma}

\section{Proof of Theorem~\ref{thm.bdd}}\label{Caccio.bounded}
The following Caccioppoli type estimate can be proved in a standard way, see for instance \cite[Lemma~4.2]{BOS}.

\begin{lemma}\label{caccioppoli2}
Let $u\in\mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{a.bound}.
There exists a constant $c\equiv c(n,s,t,p,\nu,L)>0$ such that
\begin{equation}\label{caccio.2}
\begin{aligned}
& \int_{B_{\rho}}\int_{B_{\rho}}\left(a(x,y)\frac{|w_{\pm}(x)-w_{\pm}(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\frac{|w_{\pm}(x)-w_{\pm}(y)|^{p}}{|x-y|^{n+tp}}\right)\,dx\,dy  \\
& \le \frac{c}{(r-\rho)^{p}}\int_{B_{r}}\int_{B_{r}}\left(a(x,y)\frac{w_{\pm}^{p}(x)}{|x-y|^{n+(s-1)p}}+ b(x,y)\frac{w_{\pm}^{p}(x)}{|x-y|^{n+(t-1)p}}\right)\,dx\,dy \\
& \quad\; + c\left(\frac{r}{r-\rho}\right)^{n+tp}T(w_{\pm};r,r)\int_{B_{r}}w_{\pm}\,dx
\end{aligned}
\end{equation}
holds whenever $B_{\rho} \subset  B_{r} \Subset \Omega$ are concentric balls, where $w_{\pm} \coloneqq (u-k)_{\pm}$ for any $k\in\mathbb{R}$.
Moreover, estimate \eqref{caccio.2} continues to hold for $w_{+}$ (resp. $w_{-}$) when $u$ is merely a weak subsolution (resp. supersolution) to \eqref{main.eq}.
\end{lemma}

\begin{aligned}
\mathrm{P.V.}\int_{\mathbb{R}^{n}} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\right)\,dy = 0
\end{aligned}
\end{equation}
with $s,t \in (0,1)$ and $1<p\le q < \infty$, were first considered by De Filippis and Palatucci \cite{DP19}. More precisely, they proved H\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \cite{BKK,fang2021weak,SM} for various results for \eqref{model.one}. The papers \cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\mathbb{R}^{n}$ and are under the assumption that $t\le s$, which implies that the second term in \eqref{model.one} plays a role as a lower order term. Under these settings, $b(\cdot,\cdot)$ is imposed to be bounded and discontinuous.

On the contrary, in the paper \cite{BOS}, joint with Byun, the second and third authors of this paper proved the local boundedness and H\"older continuity of weak solutions to \eqref{model.one} in the case $s\le t$, under natural assumptions on the powers and the modulating coefficients. Specifically, it was proved that if
\[  b \in C^{0,\alpha}(\mathbb{R}^{n}\times\mathbb{R}^{n}) \;\; \text{for some}\;\; \alpha\in (0,1] \quad \text{and} \quad tq \le sp + \alpha, \]
then every locally bounded weak solution to \eqref{model.one} is locally H\"older continuous.
We emphasize that when $s \le t$, the second term in \eqref{model} or \eqref{model.one} has a higher order and, as in the local case, the interplay between the H\"older regularity of $b(\cdot,\cdot)$ and the growth/differentiability condition of the problem plays a key role in the analysis.
Moreover, in light of the Lavrentiev type phenomena considered in \cite{Balci},
the above assumption is essentially sharp.
We also refer to the paper \cite{BLS} concerned with regularity results and Harnack inequality for mixed local and nonlocal double phase problems.

In this paper, we prove the local boundedness and H\"older continuity of weak solutions to the general equation \eqref{main.eq}. Moreover, in the case of one modulating coefficient (i.e. when $a(\cdot,\cdot)\equiv 1$), we further prove a Harnack inequality for weak solutions to \eqref{main.eq}.
To our knowledge, each of our regularity results is the first one for \eqref{main.eq};
moreover, our Harnack inequality given in Theorem~\ref{thm.harnack} below is the first one for nonlocal nonautonomous problems.

\subsection{Assumptions and main results}\label{setting}

We say that a function $F:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ is symmetric if $F(x,y)=F(y,x)$ for every $x,y \in \mathbb{R}^{n}$.

The kernels $K_{sp},K_{tp}:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ are measurable, symmetric functions that satisfy
\begin{equation}\label{kernel.growth}
\frac{\nu}{|x-y|^{n+sp}} \leq K_{sp}(x,y) \leq \frac{L}{|x-y|^{n+sp}}, \qquad
\frac{\nu}{|x-y|^{n+tp}} \leq K_{tp}(x,y) \leq \frac{L}{|x-y|^{n+tp}}
\end{equation}
for a.e. $x,y \in\mathbb{R}^{n}$ with $x\neq y$, where $0 < \nu \le 1 \le L < \infty$ and
\begin{equation}\label{power}
 0 < s < t < 1 < p < \infty.
\end{equation}
The modulating coefficients $a,b:\mathbb{R}^{n}\times\mathbb{R}^{n} \rightarrow \mathbb{R}$ are nonnegative, measurable and symmetric functions that satisfy the bound
\begin{equation}\label{a.bound}
\nu \le a(x,y) + b(x,y) \le L
\end{equation}
for a.e. $x,y \in \ern$.

In addition, for H\"older regularity and Harnack inequality,
we further assume the H\"older continuity of $b(\cdot,\cdot)$ as follows: There exist two numbers $[b]_{\alpha}\geq 0$ and $\alpha \in (0,1]$ such that
\begin{equation}\label{a.holder}
|b(x_{1},y_{1}) - b(x_{2},y_{2})| \leq [b]_{\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\alpha}
\end{equation}
for every $(x_{1},y_{1}),(x_{2},y_{2})\in\ern\times\ern$, and
\begin{equation}\label{assumption.hol}
\quad (t-s)p \le \alpha.
\end{equation}

\begin{remark}
For any ball $B_{r}\subset \Omega$, we set
\begin{equation*}
b^{+}_{B_{r}} \coloneqq \sup_{x,y \in B_{r}}b(x,y), \qquad b^{-}_{B_{r}} \coloneqq \inf_{x,y \in B_{r}}b(x,y).
\end{equation*}
Moreover, when \eqref{a.holder} is in force, we choose a small radius $R_{0} \equiv R_{0}(\nu,\alpha,[b]_{\alpha}) \in (0,1)$ satisfying
\begin{equation}\label{R0.choice}
    [b]_{\alpha}(2R_{0})^{\alpha} \le \frac{\nu}{8}.
\end{equation}
We observe that, when $r \le R_{0}$,
\begin{align}
b^{+}_{B_{r}} \leq \frac{\nu}{4} & \;\; \Longrightarrow \;\; \frac{3\nu}{4} \leq a(\cdot,\cdot)\leq L \ \text{ in } \ B_r\times B_r, \label{alt.dp} \\
b^{+}_{B_{r}} > \frac{\nu}{4} & \;\; \Longrightarrow \;\;
b^{+}_{B_{r}} \overset{\eqref{a.holder}}{\le} b^{-}_{B_{r}} + [b]_{\alpha}(2r)^{\alpha} \overset{\eqref{R0.choice}}{\le} 2b^{-}_{B_{r}}. \label{alt.p}
\end{align}
Accordingly, if we set
\begin{equation}\label{G.def}
G_{B_{r}}(\tau) \coloneqq \sup_{x,y\in B_{r}}\left(a(x,y)\frac{\tau^{p}}{r^{sp}} + b(x,y)\frac{\tau^{p}}{r^{tp}}\right), \qquad g_{B_{r}}(\tau) \coloneqq \frac{G_{B_{r}}(\tau)}{\tau}, \qquad \tau \ge 0.
\end{equation}
then we have
\begin{align}\label{alt.dG}
b^{+}_{B_r} \le  \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{sp}} + b^+_{B_r}\frac{\tau^p}{r^{tp}}\approx \frac{\tau^p}{r^{sp}} + b^-_{B_r}\frac{\tau^p}{r^{tp}},
\end{align}
\begin{align}\label{alt.G}
b^{+}_{B_r} > \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{tp}}.
\end{align}
\end{remark}

We next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations.
For $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, $x_{0} \in \mathbb{R}^{n}$ and $r,\rho>0$, we denote
\begin{equation}\label{def.T}
T(f;x_{0},r,\rho) \coloneqq \sup_{x \in B_{\rho}(x_{0})}\int_{\mathbb{R}^{n}\setminus B_{r}(x_{0})}\left(a(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\right)\,dy.
\end{equation}
We will omit the point $x_{0}$ if it is clear from context.

With the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.

\begin{theorem}[Local boundedness]\label{thm.bdd}
Let  $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak subsolution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{a.bound}, and let $B_{r}\Subset \Omega$ be a ball with $r\le1$.
\begin{enumerate}
\item We have
\begin{equation}\label{est:bdd1}
\esssup_{B_{r/2}} u \le  c\,r^{-(t-s)/\beta} \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ r^{sp} T(u_+;r/2,r)\right]^{1/(p-1)}
\end{equation}
for a constant $c \equiv c(n,s,t,p,\nu,L)>0$, where
\begin{equation}\label{def.beta}
\beta \coloneqq \min\{sp/n,p-1\}.
\end{equation}
\item Assume further that $b(\cdot,\cdot)$ satisfies \eqref{a.holder} and \eqref{assumption.hol}. Then we have
\begin{equation}\label{est:bdd2}
\begin{aligned}
\esssup_{B_{r/2}} u & \le c \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ \frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\right]^{1/(p-1)} \\
& = c\left(\mean{B_{r}}u_{+}^{p}\,dx\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))
\end{aligned}

\begin{aligned}
\mean{B_{\rho}}G((u-k)_{+})\,dx
& \leq c \left(\mean{B_{\sigma}} \left[\frac{(u-h)_{+}}{k-h}\right]^p\,dx\right)^{sp/n}  \mean{B_{\sigma}}G((u-h)_{+})\,dx \\
& \quad \;\cdot \left[\frac{r^{p}}{(\sigma-\rho)^{p}}+ \frac{r^{n+tp}}{(\sigma-\rho)^{n+tp}}\frac{1}{(k-h)^{p-1}G(1)}T((u-k)_{+};r/2,r)\right] \\
& \quad + c \left(\mean{B_{\sigma}} \left[\frac{(u-h)_{+}}{k-h}\right]^p\,dx\right)^{p-1}  \mean{B_{\sigma}}G((u-h)_{+})\,dx.
\end{aligned}
\end{equation*}
Dividing both sides of the above inequality by $r^{-sp}+b^{+}_{B_r}r^{-tp}$, we obtain the estimate \eqref{cacciokh} with $r^{p+(s-t)p}$ and $r^{sp}$ replaced by $r^{p}$ and $1/G(1)$, respectively. Hence we can conclude with the desired estimate \eqref{est:bdd2}, and the proof is complete.
\end{proof}

\section{Expansion of positivity}\label{Holder}

Throughout this section, we assume that the modulating coefficient $b(\cdot,\cdot)$ satisfies \eqref{a.holder} with \eqref{assumption.hol}.
We start this section with the following logarithmic type estimate, whose proof is analogous to those of \cite[Corollary~3.2]{DKP16} and \cite[Corollary~5.2]{BOS}.

\begin{lemma}\label{log.lemma}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak supersolution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol}, which is nonnegative in a ball $B_{R} \equiv B_{R}(x_{0})$ with $R \le R_{0}$. For any $d,\zeta > 0$ and $\xi>1$, define
\begin{equation*}
v \coloneqq \min\{(\log(\zeta+d)-\log(u+d))_{+}, \log\xi\}.
\end{equation*}
Then for any $r \in (0, R/2]$ and $d>0$, we have
\begin{equation}\label{log.est}
\mean{B_{r}}|v-(v)_{B_{r}}|\,dx
\leq c + cd^{1-p}\frac{1}{G_{B_{2r}}(1)}T(u_{-};R,2r)
\end{equation}
for a constant $c \equiv c(\data)>0$.
\end{lemma}
\begin{proof}
In the case $b^{+}_{B_{R}} \le \nu/4$, since $\nu/4 \le a(x,y) \le L$, the estimate is proved in \cite[Lemma~5.1]{BOS} with $q=p$. Note that, in the setting of the present paper, \cite[(5.10)]{BOS} can be replaced by
\begin{equation*}
\begin{aligned}
& \int_{B_{3r/2}}\int_{\mathbb{R}^{n}\setminus B_{R}}\frac{1}{g_{B_{2r}}(u(x)+d)}\left(a(x,y)\frac{(u_{-}(y))^{p-1}}{|x-y|^{n+sp}}+b(x,y)\frac{(u_{-}(y))^{p-1}}{|x-y|^{n+tp}}\right)\,dy\,dx \\
& \le \frac{cr^{n}}{g_{B_{2r}}(d)}\sup_{x\in B_{2r}}\int_{\mathbb{R}^{n}\setminus B_{R}}\left(a(x,y)\frac{(u_{-}(y))^{p-1}}{|y-x_{0}|^{n+sp}}+b(x,y)\frac{(u_{-}(y))^{p-1}}{|y-x_{0}|^{n+tp}}\right)\,dy \\
& = \frac{cr^{n}d^{1-p}}{G_{B_{2r}}(1)}T(u_{-};R,2r).
\end{aligned}

\begin{theorem}[Local boundedness]\label{thm.bdd}
Let  $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak subsolution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{a.bound}, and let $B_{r}\Subset \Omega$ be a ball with $r\le1$.
\begin{enumerate}
\item We have
\begin{equation}\label{est:bdd1}
\esssup_{B_{r/2}} u \le  c\,r^{-(t-s)/\beta} \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ r^{sp} T(u_+;r/2,r)\right]^{1/(p-1)}
\end{equation}
for a constant $c \equiv c(n,s,t,p,\nu,L)>0$, where
\begin{equation}\label{def.beta}
\beta \coloneqq \min\{sp/n,p-1\}.
\end{equation}
\item Assume further that $b(\cdot,\cdot)$ satisfies \eqref{a.holder} and \eqref{assumption.hol}. Then we have
\begin{equation}\label{est:bdd2}
\begin{aligned}
\esssup_{B_{r/2}} u & \le c \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ \frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\right]^{1/(p-1)} \\
& = c\left(\mean{B_{r}}u_{+}^{p}\,dx\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))
\end{aligned}
\end{equation}
whenever $r \le R_{0}$, where $R_{0}$ is given in \eqref{R0.choice} and $c\equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{enumerate}
Consequently, if $u \in \mathcal{A}(\Omega)\cap \mathcal{T}(\mathbb{R}^{n})$ is a weak solution to \eqref{main.eq}, then $u \in L^{\infty}_{\loc}(\Omega)$ and estimate \eqref{est:bdd1} or \eqref{est:bdd2} holds with $|u|$ in each case.
\end{theorem}

This paper is concerned with a class of nonlinear nonlocal equations whose differentiability order shows a drastic change depending on the point. More precisely, with $\Omega\subset \mathbb{R}^{n}$ being a bounded domain, we consider the following equation
\begin{equation}\label{main.eq}
\mathcal{L}u = 0 \quad \text{in } \Omega,
\end{equation}
whose leading operator $\mathcal{L}(\cdot)$ is given by
\begin{equation*}
\begin{aligned}
\mathcal{L}u(x) &\coloneqq \mathrm{P.V.}\int_{\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\left[a(x,y)K_{sp}(x,y)+b(x,y)K_{tp}(x,y)\right]\,dy
\end{aligned}
\end{equation*}
for $x\in\mathbb{R}^n$. Here, $K_{sp},K_{tp}: \mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ are measurable kernels of orders $(s,p)$ and $(t,p)$, respectively, for some $0<s < t <1< p <\infty$, and $a,b:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ are nonnegative functions. We refer to Section~\ref{setting} below for the detailed assumptions.

There have been several researches on various kinds of anisotropic nonlocal equations, see for instance
 \cite{CK20,FKV15,KW24} for the linear case and \cite{BOS,CK22} for the nonlinear case.
 The equation \eqref{main.eq} is modeled on the following example, which is the case where $K_{sp}(x,y) \equiv |x-y|^{-n-sp}$ and $K_{tp}(x,y) \equiv |x-y|^{-n-tp}$:
\begin{equation}\label{model}
\begin{aligned}
&\mathrm{P.V.}\int_{\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\left(\frac{a(x,y)}{|x-y|^{n+sp}}+\frac{b(x,y)}{|x-y|^{n+tp}}\right)\,dy = 0 \quad \text{in }\Omega.
\end{aligned}
\end{equation}
It is straightforward to see that \eqref{model} is the Euler-Lagrange equation of the functional
\begin{equation}\label{functional}
v \mapsto \iint_{\mathcal{C}_{\Omega}}\left(a(x,y)\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+tp}}\right)\,dx\,dy,
\end{equation}
where
\begin{equation}\label{COmega}
\mathcal{C}_{\Omega} \coloneqq (\mathbb{R}^{n}\times\mathbb{R}^{n})\setminus((\mathbb{R}^{n}\setminus\Omega)\times(\mathbb{R}^{n}\setminus\Omega)).
\end{equation}
A main point in \eqref{main.eq} is that the (possibly degenerate) coefficients $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ make the associated integro-differential operator $\mathcal{L}(\cdot)$ switch between two different fractional elliptic phases.
In this respect, our problem is closely related to the following local equation
\[ \mathrm{div}\left(a(x)|Du|^{p-2}Du + b(x)|Du|^{q-2}Du \right) = 0, \]
where $1 < p < q < \infty$ and $a(\cdot), b(\cdot) \ge 0$ satisfy $\nu \le a(x) + b(x) \le L$ for some constants $0 < \nu \le L < \infty$. This is called a double phase problem with two modulating coefficients. Since the pioneering works of Colombo and Mingione \cite{CM15a, CM15b}, the regularity theory for local double phase problems with a single modulating coefficient (i.e., the case $a(\cdot)\equiv 1$ in the equation above) has been extensively developed; see, for instance, \cite{BCM1, BCM2, CM16, DM20} and the references therein. More general classes of problems, including the double phase type, have subsequently been investigated in \cite{HO22, KO24}. In particular, \cite{KO24} studies various regularity results for equations of the form above, including the case in which $a(\cdot)$ may be degenerate, meaning that $a(\cdot)$ can vanish. A key idea in \cite{KO24} is the following observation: when $a(x)$ is close to zero, the equation can be regarded as a $q$‑Laplace equation with a lower‑order perturbation of $p$‑Laplace type, whereas when $a(x)$ stays away from zero, the equation behaves genuinely as a double phase problem with a single modulating coefficient.

We now turn to regularity results for nonlinear nonlocal equations. The De Giorgi-Nash-Moser theory for the $(s,p)$-Laplacian was first investigated by Di Castro, Kuusi and Palatucci \cite{DKP14,DKP16}; they proved local regularity and Harnack inequality for weak solutions to fractional $p$-Laplacian type equations involving measurable kernels. Cozzi \cite{coz} extended such results to minimizers of non-differentiable functionals with lower-order dependencies, via a slightly different approach using fractional De Giorgi classes. We  also refer to \cite{BDNS,Lind} and references therein for similar results concerned with more general equations.

Very recently, the results and techniques in \cite{coz,DKP14,DKP16} were further developed and extended to nonlocal problems with nonstandard growth and/or differentiability conditions. Similarly to the case of local problems, one can consider several typical examples, such as Orlicz growth \cite{BKO,CKW2}, variable growth \cite{CK23,Ok23}, and double phase \cite{BOS,DP19,fang2021weak}. These papers are concerned with local boundedness and H\"older continuity of weak solutions.
Moreover, in the case of Orlicz growth, Harnack inequalities were also proved in \cite{BKS,CKW1}.
We also mention the paper \cite{OS} concerned with regularity results and Harnack inequalities for nonlinear nonlocal equations with kernels of general differentiability order.

Nonlocal double phase problems with one modulating coefficient, whose prototype is
\begin{equation}\label{model.one}
\begin{aligned}
\mathrm{P.V.}\int_{\mathbb{R}^{n}} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\right)\,dy = 0
\end{aligned}
\end{equation}
with $s,t \in (0,1)$ and $1<p\le q < \infty$, were first considered by De Filippis and Palatucci \cite{DP19}. More precisely, they proved H\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \cite{BKK,fang2021weak,SM} for various results for \eqref{model.one}. The papers \cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\mathbb{R}^{n}$ and are under the assumption that $t\le s$, which implies that the second term in \eqref{model.one} plays a role as a lower order term. Under these settings, $b(\cdot,\cdot)$ is imposed to be bounded and discontinuous.

On the contrary, in the paper \cite{BOS}, joint with Byun, the second and third authors of this paper proved the local boundedness and H\"older continuity of weak solutions to \eqref{model.one} in the case $s\le t$, under natural assumptions on the powers and the modulating coefficients. Specifically, it was proved that if
\[  b \in C^{0,\alpha}(\mathbb{R}^{n}\times\mathbb{R}^{n}) \;\; \text{for some}\;\; \alpha\in (0,1] \quad \text{and} \quad tq \le sp + \alpha, \]
then every locally bounded weak solution to \eqref{model.one} is locally H\"older continuous.
We emphasize that when $s \le t$, the second term in \eqref{model} or \eqref{model.one} has a higher order and, as in the local case, the interplay between the H\"older regularity of $b(\cdot,\cdot)$ and the growth/differentiability condition of the problem plays a key role in the analysis.
Moreover, in light of the Lavrentiev type phenomena considered in \cite{Balci},
the above assumption is essentially sharp.
We also refer to the paper \cite{BLS} concerned with regularity results and Harnack inequality for mixed local and nonlocal double phase problems.

In this paper, we prove the local boundedness and H\"older continuity of weak solutions to the general equation \eqref{main.eq}. Moreover, in the case of one modulating coefficient (i.e. when $a(\cdot,\cdot)\equiv 1$), we further prove a Harnack inequality for weak solutions to \eqref{main.eq}.
To our knowledge, each of our regularity results is the first one for \eqref{main.eq};
moreover, our Harnack inequality given in Theorem~\ref{thm.harnack} below is the first one for nonlocal nonautonomous problems.

\subsection{Assumptions and main results}\label{setting}

We say that a function $F:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ is symmetric if $F(x,y)=F(y,x)$ for every $x,y \in \mathbb{R}^{n}$.

The kernels $K_{sp},K_{tp}:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow \mathbb{R}$ are measurable, symmetric functions that satisfy
\begin{equation}\label{kernel.growth}
\frac{\nu}{|x-y|^{n+sp}} \leq K_{sp}(x,y) \leq \frac{L}{|x-y|^{n+sp}}, \qquad
\frac{\nu}{|x-y|^{n+tp}} \leq K_{tp}(x,y) \leq \frac{L}{|x-y|^{n+tp}}
\end{equation}
for a.e. $x,y \in\mathbb{R}^{n}$ with $x\neq y$, where $0 < \nu \le 1 \le L < \infty$ and
\begin{equation}\label{power}
 0 < s < t < 1 < p < \infty.
\end{equation}
The modulating coefficients $a,b:\mathbb{R}^{n}\times\mathbb{R}^{n} \rightarrow \mathbb{R}$ are nonnegative, measurable and symmetric functions that satisfy the bound
\begin{equation}\label{a.bound}
\nu \le a(x,y) + b(x,y) \le L
\end{equation}
for a.e. $x,y \in \mathbb{R}^n$.

In addition, for H\"older regularity and Harnack inequality,
we further assume the H\"older continuity of $b(\cdot,\cdot)$ as follows: There exist two numbers $[b]_{\alpha}\geq 0$ and $\alpha \in (0,1]$ such that
\begin{equation}\label{a.holder}
|b(x_{1},y_{1}) - b(x_{2},y_{2})| \leq [b]_{\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\alpha}
\end{equation}
for every $(x_{1},y_{1}),(x_{2},y_{2})\in\mathbb{R}^n\times\mathbb{R}^n$, and
\begin{equation}\label{assumption.hol}
\quad (t-s)p \le \alpha.
\end{equation}

\begin{remark}
For any ball $B_{r}\subset \Omega$, we set
\begin{equation*}
b^{+}_{B_{r}} \coloneqq \sup_{x,y \in B_{r}}b(x,y), \qquad b^{-}_{B_{r}} \coloneqq \inf_{x,y \in B_{r}}b(x,y).
\end{equation*}
Moreover, when \eqref{a.holder} is in force, we choose a small radius $R_{0} \equiv R_{0}(\nu,\alpha,[b]_{\alpha}) \in (0,1)$ satisfying
\begin{equation}\label{R0.choice}
    [b]_{\alpha}(2R_{0})^{\alpha} \le \frac{\nu}{8}.
\end{equation}
We observe that, when $r \le R_{0}$,
\begin{align}
b^{+}_{B_{r}} \leq \frac{\nu}{4} & \;\; \Longrightarrow \;\; \frac{3\nu}{4} \leq a(\cdot,\cdot)\leq L \ \text{ in } \ B_r\times B_r, \label{alt.dp} \\
b^{+}_{B_{r}} > \frac{\nu}{4} & \;\; \Longrightarrow \;\;
b^{+}_{B_{r}} \overset{\eqref{a.holder}}{\le} b^{-}_{B_{r}} + [b]_{\alpha}(2r)^{\alpha} \overset{\eqref{R0.choice}}{\le} 2b^{-}_{B_{r}}. \label{alt.p}
\end{align}
Accordingly, if we set
\begin{equation}\label{G.def}
G_{B_{r}}(\tau) \coloneqq \sup_{x,y\in B_{r}}\left(a(x,y)\frac{\tau^{p}}{r^{sp}} + b(x,y)\frac{\tau^{p}}{r^{tp}}\right), \qquad g_{B_{r}}(\tau) \coloneqq \frac{G_{B_{r}}(\tau)}{\tau}, \qquad \tau \ge 0.
\end{equation}
then we have
\begin{align}\label{alt.dG}
b^{+}_{B_r} \le  \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{sp}} + b^+_{B_r}\frac{\tau^p}{r^{tp}}\approx \frac{\tau^p}{r^{sp}} + b^-_{B_r}\frac{\tau^p}{r^{tp}},
\end{align}
\begin{align}\label{alt.G}
b^{+}_{B_r} > \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{tp}}.
\end{align}
\end{remark}

We next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations.
For $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, $x_{0} \in \mathbb{R}^{n}$ and $r,\rho>0$, we denote
\begin{equation}\label{def.T}
T(f;x_{0},r,\rho) \coloneqq \sup_{x \in B_{\rho}(x_{0})}\int_{\mathbb{R}^{n}\setminus B_{r}(x_{0})}\left(a(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\right)\,dy.
\end{equation}
We will omit the point $x_{0}$ if it is clear from context.

With the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.

There have been several researches on various kinds of anisotropic nonlocal equations, see for instance
 \cite{CK20,FKV15,KW24} for the linear case and \cite{BOS,CK22} for the nonlinear case.
 The equation \eqref{main.eq} is modeled on the following example, which is the case where $K_{sp}(x,y) \equiv |x-y|^{-n-sp}$ and $K_{tp}(x,y) \equiv |x-y|^{-n-tp}$:
\begin{equation}\label{model}
\begin{aligned}
&\mathrm{P.V.}\int_{\mathbb{R}^{n}} |u(x)-u(y)|^{p-2}(u(x)-u(y))\left(\frac{a(x,y)}{|x-y|^{n+sp}}+\frac{b(x,y)}{|x-y|^{n+tp}}\right)\,dy = 0 \quad \text{in }\Omega.
\end{aligned}
\end{equation}
It is straightforward to see that \eqref{model} is the Euler-Lagrange equation of the functional
\begin{equation}\label{functional}
v \mapsto \iint_{\mathcal{C}_{\Omega}}\left(a(x,y)\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+tp}}\right)\,dx\,dy,
\end{equation}
where
\begin{equation}\label{COmega}
\mathcal{C}_{\Omega} \coloneqq (\mathbb{R}^{n}\times\mathbb{R}^{n})\setminus((\mathbb{R}^{n}\setminus\Omega)\times(\mathbb{R}^{n}\setminus\Omega)).
\end{equation}
A main point in \eqref{main.eq} is that the (possibly degenerate) coefficients $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ make the associated integro-differential operator $\mathcal{L}(\cdot)$ switch between two different fractional elliptic phases.
In this respect, our problem is closely related to the following local equation
\[ \mathrm{div}\left(a(x)|Du|^{p-2}Du + b(x)|Du|^{q-2}Du \right) = 0, \]
where $1 < p < q < \infty$ and $a(\cdot), b(\cdot) \ge 0$ satisfy $\nu \le a(x) + b(x) \le L$ for some constants $0 < \nu \le L < \infty$. This is called a double phase problem with two modulating coefficients. Since the pioneering works of Colombo and Mingione \cite{CM15a, CM15b}, the regularity theory for local double phase problems with a single modulating coefficient (i.e., the case $a(\cdot)\equiv 1$ in the equation above) has been extensively developed; see, for instance, \cite{BCM1, BCM2, CM16, DM20} and the references therein. More general classes of problems, including the double phase type, have subsequently been investigated in \cite{HO22, KO24}. In particular, \cite{KO24} studies various regularity results for equations of the form above, including the case in which $a(\cdot)$ may be degenerate, meaning that $a(\cdot)$ can vanish. A key idea in \cite{KO24} is the following observation: when $a(x)$ is close to zero, the equation can be regarded as a $q$‑Laplace equation with a lower‑order perturbation of $p$‑Laplace type, whereas when $a(x)$ stays away from zero, the equation behaves genuinely as a double phase problem with a single modulating coefficient.

Nonlocal double phase problems with one modulating coefficient, whose prototype is
\begin{equation}\label{model.one}
\begin{aligned}
\mathrm{P.V.}\int_{\mathbb{R}^{n}} \left(\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}+ b(x,y)\frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))}{|x-y|^{n+tq}}\right)\,dy = 0
\end{aligned}
\end{equation}
with $s,t \in (0,1)$ and $1<p\le q < \infty$, were first considered by De Filippis and Palatucci \cite{DP19}. More precisely, they proved H\"older continuity of viscosity solutions for nonhomogeneous equations with bounded source terms. We also refer to \cite{BKK,fang2021weak,SM} for various results for \eqref{model.one}. The papers \cite{BKK,DP19,fang2021weak,SM} consider solutions bounded in $\mathbb{R}^{n}$ and are under the assumption that $t\le s$, which implies that the second term in \eqref{model.one} plays a role as a lower order term. Under these settings, $b(\cdot,\cdot)$ is imposed to be bounded and discontinuous.

\begin{remark}
For any ball $B_{r}\subset \Omega$, we set
\begin{equation*}
b^{+}_{B_{r}} \coloneqq \sup_{x,y \in B_{r}}b(x,y), \qquad b^{-}_{B_{r}} \coloneqq \inf_{x,y \in B_{r}}b(x,y).
\end{equation*}
Moreover, when \eqref{a.holder} is in force, we choose a small radius $R_{0} \equiv R_{0}(\nu,\alpha,[b]_{\alpha}) \in (0,1)$ satisfying
\begin{equation}\label{R0.choice}
 [b]_{\alpha}(2R_{0})^{\alpha} \le \frac{\nu}{8}.
\end{equation}
We observe that, when $r \le R_{0}$,
\begin{align}
b^{+}_{B_{r}} \leq \frac{\nu}{4} & \;\; \Longrightarrow \;\; \frac{3\nu}{4} \leq a(\cdot,\cdot)\leq L \ \text{ in } \ B_r\times B_r, \label{alt.dp} \\
b^{+}_{B_{r}} > \frac{\nu}{4} & \;\; \Longrightarrow \;\;
b^{+}_{B_{r}} \overset{\eqref{a.holder}}{\le} b^{-}_{B_{r}} + [b]_{\alpha}(2r)^{\alpha} \overset{\eqref{R0.choice}}{\le} 2b^{-}_{B_{r}}. \label{alt.p}
\end{align}
Accordingly, if we set
\begin{equation}\label{G.def}
G_{B_{r}}(\tau) \coloneqq \sup_{x,y\in B_{r}}\left(a(x,y)\frac{\tau^{p}}{r^{sp}} + b(x,y)\frac{\tau^{p}}{r^{tp}}\right), \qquad g_{B_{r}}(\tau) \coloneqq \frac{G_{B_{r}}(\tau)}{\tau}, \qquad \tau \ge 0.
\end{equation}
then we have
\begin{align}\label{alt.dG}
b^{+}_{B_r} \le \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{sp}} + b^+_{B_r}\frac{\tau^p}{r^{tp}}\approx \frac{\tau^p}{r^{sp}} + b^-_{B_r}\frac{\tau^p}{r^{tp}},
\end{align}
\begin{align}\label{alt.G}
b^{+}_{B_r} > \frac{\nu}{4} & \;\; \Longrightarrow \;\; G_{B_r}(\tau) \approx \frac{\tau^p}{r^{tp}}.
\end{align}
\end{remark}

We next introduce the nonlocal tail, which is one of the essential tools in analyzing local regularity for fractional equations.
For $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$, $x_{0} \in \mathbb{R}^{n}$ and $r,\rho>0$, we denote
\begin{equation}\label{def.T}
T(f;x_{0},r,\rho) \coloneqq \sup_{x \in B_{\rho}(x_{0})}\int_{\mathbb{R}^{n}\setminus B_{r}(x_{0})}\left(a(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+sp}} + b(x,y)\frac{|f(y)|^{p-1}}{|y-x_{0}|^{n+tp}}\right)\,dy.
\end{equation}
We will omit the point $x_{0}$ if it is clear from context.

With the definitions of weak solutions and relevant function spaces to be introduced in the next section, here we state our main results.

\begin{theorem}[Harnack inequality]\label{thm.harnack}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol} with $a(\cdot,\cdot)\equiv 1$. If $u$ is nonnegative in a ball $B_{10r} \Subset \Omega$ with $10r \leq R_{0}$, where $R_{0}$ is given in \eqref{R0.choice}, then we have
\begin{equation}\label{Harnack}
\sup_{B_{r}}u \le c\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r))
\end{equation}
for a constant $c \equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{theorem}
\begin{remark}
In Theorem~\ref{thm.harnack}, the condition $a(\cdot,\cdot)\equiv 1$ can be weakened as $\inf_{\mathbb{R}^n \times \mathbb{R}^n}a(\cdot,\cdot)>0$. Such a restriction is concerned with the tail estimate given in Lemma~\ref{tail.pm} below.
Indeed, even if $a(\cdot,\cdot)\not\equiv1$, we can prove an estimate similar to \eqref{Harnack} which involves nonlocal tails of $u$ instead of $u_{-}$.
However, in view of the natural Harnack inequalities obtained in \cite{coz,DKP14,Kas11}, we confine ourselves to the case $a(\cdot,\cdot)\equiv1$ in Theorem~\ref{thm.harnack}.
\end{remark}

\begin{lemma}\label{caccioppoli2}
Let $u\in\mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{a.bound}.
There exists a constant $c\equiv c(n,s,t,p,\nu,L)>0$ such that
\begin{equation}\label{caccio.2}
\begin{aligned}
& \int_{B_{\rho}}\int_{B_{\rho}}\left(a(x,y)\frac{|w_{\pm}(x)-w_{\pm}(y)|^{p}}{|x-y|^{n+sp}} + b(x,y)\frac{|w_{\pm}(x)-w_{\pm}(y)|^{p}}{|x-y|^{n+tp}}\right)\,dx\,dy \\
& \le \frac{c}{(r-\rho)^{p}}\int_{B_{r}}\int_{B_{r}}\left(a(x,y)\frac{w_{\pm}^{p}(x)}{|x-y|^{n+(s-1)p}}+ b(x,y)\frac{w_{\pm}^{p}(x)}{|x-y|^{n+(t-1)p}}\right)\,dx\,dy \\
& \quad\; + c\left(\frac{r}{r-\rho}\right)^{n+tp}T(w_{\pm};r,r)\int_{B_{r}}w_{\pm}\,dx
\end{aligned}
\end{equation}
holds whenever $B_{\rho} \subset B_{r} \Subset \Omega$ are concentric balls, where $w_{\pm} \coloneqq (u-k)_{\pm}$ for any $k\in\mathbb{R}$.
Moreover, estimate \eqref{caccio.2} continues to hold for $w_{+}$ (resp. $w_{-}$) when $u$ is merely a weak subsolution (resp. supersolution) to \eqref{main.eq}.
\end{lemma}

\begin{lemma}\label{density.improve}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak supersolution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol}, which is nonnegative in a ball $B_{R} \equiv B_{R}(x_{0}) \Subset \Omega$ with $R \le R_{0}$. Let $k > 0$ and assume that there exists $\sigma \in (0,1]$ satisfying
\begin{equation}\label{density.assumption}
|B_{2r} \cap \{ u \ge k \}| \ge \sigma|B_{2r}| \qquad \text{for some} \quad r \in (0,R/4].
\end{equation}
Then there exists a constant $\delta \equiv \delta(\data,\sigma) \in (0,1/4)$ such that if
\begin{equation}\label{T.assumption}
d \coloneqq \left[\frac{1}{G_{B_{2r}}(1)}T(u_{-};R,2r) \right]^{1/(p-1)} = g_{B_{2r}}^{-1}(T(u_{-};R,2r)) \le \delta k,
\end{equation}
then
\begin{equation}\label{lower.bound}
\essinf_{B_{r}}u \ge \delta k.
\end{equation}
\end{lemma}
\begin{proof}
We divide the proof into two steps.

Using Lemmas~\ref{density.improve} and \ref{KrySaf}, we obtain a weak Harnack type estimate.
\begin{lemma}\label{weak.harnack}
Let $u\in\mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak supersolution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol} with $a(\cdot,\cdot)\equiv 1$, which is nonnegative in a ball $B_{10r} \equiv B_{10r}(x_{0}) \Subset \Omega$ with $10r \le R_{0}$. Then there exist constants $p_{0} \in (0,1)$ and $c \ge 1$, both depending only on $\data$, such that
\begin{equation*}
\left(\mean{B_{2r}}u^{p_{0}}\,dx\right)^{1/p_{0}} \le c \essinf_{B_{r}}u + cg_{B_{6r}}^{-1}(T(u_{-};10r,10r)).
\end{equation*}
\end{lemma}
\begin{proof}
We set $\bar{\delta} \coloneqq 2^{-n-1}$ and $T_{0}\coloneqq c_{2}g_{B_{6r}}^{-1}(T(u_{-};10r,10r))$, with $c_{2}\equiv c_{2}(\data)$ being a constant whose precise value will be determined in \eqref{dT} below.
For each $i \in \mathbb{N}\cup\{0\}$ and $\tau>0$, we define
\begin{equation*}
A^{i}_{\tau} \coloneqq \left\{ x\in B_{2r}:u(x)>\tau\delta^{i}-\frac{T_{0}}{1-\delta}\right\},
\end{equation*}
where $\delta \equiv \delta(\data) \in (0,1/4)$ is the constant determined in Lemma~\ref{density.improve} in the case $\sigma=\bar{\delta}/6^{n}$.
It is obvious that $A^{i-1}_{\tau} \subset A^{i}_{\tau}$.

The following lemma allows us to capture the precise tail contribution in the Harnack inequality.
\begin{lemma}\label{tail.pm}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol} with $a(\cdot,\cdot) \equiv 1$, which is nonnegative in a ball $B_{R}\equiv B_{R}(x_{0}) \Subset \Omega$ with $R\le R_{0}$. Then
\[ T(u_{+};r,2r) \le cg_{B_{r}}\left(\sup_{B_{r}}u\right) + cT(u_{-};R,r) \]
holds for any $r\in(0,R/2]$, where $c\equiv c(\data)>0$.
\end{lemma}
\begin{proof}
We set $w \coloneqq u-2k$ for $k \coloneqq \sup_{B_{r}}u$, and take a cut-off function $\phi \in C^{\infty}_{0}(B_{r})$ such that $\supp\,\phi \subseteq B_{3r/4}$, $0 \le \phi \le 1$, $\phi \equiv 1$ in $B_{r/2}$ and $|D\phi| \le 8/r$. Recalling the notation \eqref{phip}, we test \eqref{main.eq} with $w\phi^{p}$ in order to have
\begin{equation}\label{test}
\begin{aligned}
0 & = \int_{B_{r}}\int_{B_{r}}\Phi_{p}(u(x)-u(y))(w(x)\phi^{p}(x)-w(y)\phi^{p}(y))\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\right] \,dx\,dy \\
& \quad\; + 2\int_{\mathbb{R}^{n}\setminus B_{r}}\int_{B_{r}}\Phi_{p}(u(x)-u(y))w(x)\phi^{p}(x)\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\right]\,dx\,dy \\
& \eqqcolon I_{1} + I_{2}.
\end{aligned}
\end{equation}
We first estimate $I_{1}$. By the calculations done in the proof of \cite[Lemma 4.2]{BOS}, we have
\begin{equation*}
\begin{aligned}
& \Phi_{p}(w(x)-w(y))(w(x)\phi^{p}(x)-w(y)\phi^{p}(y)) \\
& \ge \frac{1}{4}|w(x)-w(y)|^{p}(\phi^{p}(x)+\phi^{p}(y)) - c|\phi(x)-\phi(y)|^{p}(w(x)+w(y))^{p} \\
& \ge -ck^{p}\frac{|x-y|^{p}}{r^{p}}
\end{aligned}
\end{equation*}
and therefore
\begin{equation}\label{I1}
\begin{aligned}
I_{1} & \ge -c\int_{B_{r}}\int_{B_{r}}\left(\frac{k^{p}}{r^{p}}\frac{1}{|x-y|^{n+(s-1)p}} + b(x,y)\frac{k^{p}}{r^{p}}\frac{1}{|x-y|^{n+(t-1)p}}\right)\,dx\,dy \\
& \ge -c\left(k^{p}r^{n-sp} + b^{+}_{B_{r}}k^{p}r^{n-tp}\right) \\
& \ge -cr^{n}G_{B_{r}}(k).
\end{aligned}
\end{equation}
We next split $I_{2}$ as
\begin{equation}\label{I2.split}
\begin{aligned}
I_{2} & \ge 2 \int_{\mathbb{R}^{n}\setminus B_{r}}\int_{B_{r}}k(u(y)-k)_{+}^{p-1}\phi^{p}(x)\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\right]\,dx\,dy \\
& \quad\; -2\int_{\mathbb{R}^{n}\setminus B_{r}}\int_{B_{r}}2k\chi_{\{u(y)<k\}}(u(x)-u(y))_{+}^{p-1}\phi^{p}(x)\left[K_{sp}(x,y) + b(x,y)K_{tp}(x,y)\right]\,dx\,dy \\
& \eqqcolon I_{2,1} - I_{2,2}.
\end{aligned}
\end{equation}
We note that
\[ \frac{1}{4}|y-x_{0}| \le |y-x_{0}| - |x-x_{0}| \le |x-y| \le |x-x_{0}| + |y-x_{0}| \le \frac{7}{4}|y-x_{0}|\]
for any $x\in B_{3r/4}$ and $y\in\mathbb{R}^{n}\setminus B_{r}$,
in order to estimate $I_{2,1}$ as
\begin{equation}\label{I21}
\begin{aligned}
I_{2,1} & \ge \frac{k}{c}\int_{\mathbb{R}^{n}\setminus B_{r}}\int_{B_{r}}\left(\frac{(u_{+}(y))^{p-1}}{|x-y|^{n+sp}}+b(x,y)\frac{(u_{+}(y))^{p-1}}{|x-y|^{n+tp}}\right)\phi^{p}(x)\,dx\,dy \\
& \quad\; - c\int_{\mathbb{R}^{n}\setminus B_{r}}\int_{B_{r}}\left(\frac{k^{p}}{|x-y|^{n+sp}} + b(x,y)\frac{k^{p}}{|x-y|^{n+tp}}\right)\phi^{p}(x)\,dx\,dy \\
& \ge \frac{kr^{n}}{c}\int_{\mathbb{R}^{n}\setminus B_{r}}\left(\frac{(u_{+}(y))^{p-1}}{|y-x_{0}|^{n+sp}} + \inf_{B_{r/2}}b(\cdot,y)\frac{(u_{+}(y))^{p-1}}{|y-x_{0}|^{n+tp}}\right)\,dy - cr^{n}G_{B_{r}}(k).
\end{aligned}
\end{equation}
On the other hand, we observe that
\begin{itemize}
\item[(i)] If $x\in B_{r}$ and $y \in B_{R}$, then $(u(x)-u(y))_{+}^{p-1} \le (u(x))^{p-1}$;

\begin{theorem}[H\"older regularity]\label{thm.hol}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol}. Then there exists an exponent $\gamma \equiv \gamma(n,s,t,p,\nu,L,\alpha,[b]_{\alpha}) \in (0,1)$ such that $u \in C^{0,\gamma}_{\loc}(\Omega)$. Moreover, for any ball $B_{r} \Subset \Omega$ with $r\le R_{0}$, where $R_{0}$ is given in \eqref{R0.choice}, we have
\begin{equation}\label{est:hol}
[u]_{C^{0,\gamma}(B_{r/2})} \le cr^{-\gamma}\left[\left(\mean{B_{r}}|u|^{p}\,dx\right)^{1/p} + g_{B_{r}}^{-1}(T(u;r/2,r))\right]
\end{equation}
for a constant $c\equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{theorem}

\begin{theorem}[Harnack inequality]\label{thm.harnack}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \eqref{main.eq} under assumptions \eqref{kernel.growth}--\eqref{assumption.hol} with $a(\cdot,\cdot)\equiv 1$. If $u$ is nonnegative in a ball $B_{10r} \Subset \Omega$ with $10r \leq R_{0}$, where $R_{0}$ is given in \eqref{R0.choice}, then we have
\begin{equation}\label{Harnack}
\sup_{B_{r}}u \le c\inf_{B_{r}}u + cg_{B_{r}}^{-1}(T(u_{-};10r,10r))
\end{equation}
for a constant $c \equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{theorem}
\begin{remark}
In Theorem~\ref{thm.harnack}, the condition $a(\cdot,\cdot)\equiv 1$ can be weakened as $\inf_{\mathbb{R}^n \times \mathbb{R}^n}a(\cdot,\cdot)>0$. Such a restriction is concerned with the tail estimate given in Lemma~\ref{tail.pm} below.
Indeed, even if $a(\cdot,\cdot)\not\equiv1$, we can prove an estimate similar to \eqref{Harnack} which involves nonlocal tails of $u$ instead of $u_{-}$.
However, in view of the natural Harnack inequalities obtained in \cite{coz,DKP14,Kas11}, we confine ourselves to the case $a(\cdot,\cdot)\equiv1$ in Theorem~\ref{thm.harnack}.
\end{remark}

Our proof of regularity results in Theorems~\ref{thm.bdd} and \ref{thm.hol} is based on the Moser type approach in \cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \cite{BOS}.
Moreover, for the Harnack inequality in Theorem~\ref{thm.harnack}, we additionally modify and develop the techniques used in \cite{DKP14} adapted to our problem.
We stress that, in obtaining both regularity results and Harnack inequalities for \eqref{main.eq}, these processes exhibit several substantial difficulties which did not appear in \cite{BOS}.
First, while we only had to consider the case $sp \le n$ in \cite{BOS}, here we also treat the case $sp > n$ as well; note that in the latter case we automatically obtain the local boundedness of weak solutions via the fractional Sobolev-Morrey embedding. However, in both cases, we have to obtain explicit and precise local sup-estimates which extend the one in \cite[Theorem~1.1]{DKP16}.
Second, since we are dealing with weak solutions which are not necessarily bounded in $\mathbb{R}^{n}$, we need a delicate analysis to handle the nonlocal tails in proceeding towards the proof of our Harnack inequality.
More precisely, with the same spirit as in the case of fractional $p$-Laplacian \cite{coz,DKP14}, we aim to prove a Harnack inequality involving the tail of $u_{-} \coloneqq \max\{-u,0\}$ only, which reduces to the classical Harnack inequality when dealing with solutions nonnegative in the whole $\mathbb{R}^{n}$. This is indeed the most problematic issue in the case of nonlocal nonautonomous problems. In order to address this issue, we establish a weak Harnack type estimate (Lemma~\ref{weak.harnack}) and a tail estimate (Lemma~\ref{tail.pm}), and then combine them with the local sup-estimate \eqref{est:bdd2}. All of these estimates encode the long-range interactions of solutions in an optimal way.

In our proofs, we utilize assumptions \eqref{kernel.growth}--\eqref{assumption.hol} to deal with the two modulating coefficients $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$ in \eqref{main.eq}. Specifically, with $B_r \subset \Omega$ being a fixed ball, we divide cases as follows:
\begin{itemize}
\item[(i)] If $b^{+}_{B_r}\leq \nu/4$, then \eqref{alt.dp} implies that $a(x,y)$ satisfies uniform ellipticity in the sense that the first term $a(x,y)K_{sp}(x,y) \approx K_{sp}(x,y)$  in $B_r \times B_r$, while the second term $b(x,y)K_{tp}(x,y)$ has a higher order with $b(x,y)$ being possibly degenerate in $B_r \times B_r$. As a consequence, the equation \eqref{main.eq} features a nonlocal double phase structure analogous to \eqref{model.one} with $p=q$.
\item[(ii)] If $b^{+}_{B_r}>\nu/4$, then \eqref{alt.p} implies that $b(x,y)$ satisfies uniform ellipticity in $B_r \times B_r$ with ellipticity ratio $2b^-_{B_{r}}/b^-_{B_{r}}=2$. In other words, the term $b(x,y)K_{tp}(x,y) \approx K_{tp}(x,y)$ becomes a dominating term and $a(x,y)K_{sp}(x,y)$ is regarded as a lower order term in $B_{r} \times B_r$. In this case, the equation \eqref{main.eq} becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.
\end{itemize}

The remaining part of this paper is organized as follows. In the next section, we introduce basic notation and function spaces which will be used throughout this paper. In Section~\ref{Caccio.bounded}, we employ a Caccioppoli type estimate to prove Theorem~\ref{thm.bdd}. After obtaining a logarithmic lemma and an expansion of positivity lemma in Section~\ref{Holder}, we finally prove Theorems~\ref{thm.hol} and \ref{thm.harnack} in Section~\ref{sec.mainthm}.

Our proof of regularity results in Theorems~\ref{thm.bdd} and \ref{thm.hol} is based on the Moser type approach in \cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \cite{BOS}. The authors note additional difficulties compared to \cite{BOS}: they treat both cases $sp\le n$ and $sp>n$ (in the latter, local boundedness follows from the fractional Sobolev--Morrey embedding), but “in both cases” they “have to obtain explicit and precise local sup-estimates which extend the one in \cite[Theorem~1.1]{DKP16}.”

They also explain how the assumptions \eqref{kernel.growth}--\eqref{assumption.hol} are used “to deal with the two modulating coefficients $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$”: for a fixed ball $B_r\subset\Omega$ they split into two regimes:
\begin{itemize}
\item[(i)] If $b^{+}_{B_r}\le \nu/4$, then $a(x,y)$ is uniformly elliptic in $B_r\times B_r$ so that $a(x,y)K_{sp}(x,y)\approx K_{sp}(x,y)$, while $b(x,y)K_{tp}(x,y)$ is “higher order” and may be degenerate; thus \eqref{main.eq} has a nonlocal double phase structure analogous to the model with $p=q$.
\item[(ii)] If $b^{+}_{B_r}>\nu/4$, then $b(x,y)$ is uniformly elliptic in $B_r\times B_r$ (ellipticity ratio $2$), so $b(x,y)K_{tp}(x,y)\approx K_{tp}(x,y)$ is dominating and $a(x,y)K_{sp}(x,y)$ is “lower order”; hence \eqref{main.eq} becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.
\end{itemize}

Finally, the paper roadmap gives the proof structure for Theorem~\ref{thm.bdd}: “In Section~\ref{Caccio.bounded}, we employ a Caccioppoli type estimate to prove Theorem~\ref{thm.bdd}.”

Our proof of regularity results in To prove the main theorem, and in the following theorem.

We first state the following theorem.
\begin{theorem}[H\"older regularity]\label{thm.hol}
Let $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak solution to \begin{equation}\label{main.eq}
\mathcal{L}u = 0 \quad \text{in } \Omega,
\end{equation}
 under assumptions \begin{equation}\label{kernel.growth}
\frac{\nu}{|x-y|^{n+sp}} \leq K_{sp}(x,y) \leq \frac{L}{|x-y|^{n+sp}}, \qquad
\frac{\nu}{|x-y|^{n+tp}} \leq K_{tp}(x,y) \leq \frac{L}{|x-y|^{n+tp}}
\end{equation}
--\begin{equation}\label{assumption.hol}
\quad (t-s)p \le \alpha.
\end{equation}
 Then there exists an exponent $\gamma \equiv \gamma(n,s,t,p,\nu,L,\alpha,[b]_{\alpha}) \in (0,1)$ such that $u \in C^{0,\gamma}_{\loc}(\Omega)$. Moreover, for any ball $B_{r} \Subset \Omega$ with $r\le R_{0}$, where $R_{0}$ is given in \eqref{R0.choice}, we have
\begin{equation}\label{est:hol}
[u]_{C^{0,\gamma}(B_{r/2})} \le cr^{-\gamma}\left[\left(\mean{B_{r}}|u|^{p}\,dx\right)^{1/p} + g_{B_{r}}^{-1}(T(u;r/2,r))\right]
\end{equation}
for a constant $c\equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{theorem}

Our proof is based on the Moser type approach in \cite{DKP16} which employs a logarithmic estimate; the extension of this approach to the double phase setting was first presented in \cite{BOS}. The authors note additional difficulties compared to \cite{BOS}: they treat both cases $sp\le n$ and $sp>n$ (in the latter, local boundedness follows from the fractional Sobolev--Morrey embedding), but “in both cases” they “have to obtain explicit and precise local sup-estimates which extend the one in \cite[Theorem~1.1]{DKP16}.”

They also explain how the assumptions \begin{equation}\label{kernel.growth}
\frac{\nu}{|x-y|^{n+sp}} \leq K_{sp}(x,y) \leq \frac{L}{|x-y|^{n+sp}}, \qquad
\frac{\nu}{|x-y|^{n+tp}} \leq K_{tp}(x,y) \leq \frac{L}{|x-y|^{n+tp}}
\end{equation}
--\begin{equation}\label{assumption.hol}
\quad (t-s)p \le \alpha.
\end{equation}
 are used “to deal with the two modulating coefficients $a(\cdot,\cdot)$ and $b(\cdot,\cdot)$”: for a fixed ball $B_r\subset\Omega$ they split into two regimes:
\begin{itemize}
\item[(i)] If $b^{+}_{B_r}\le \nu/4$, then $a(x,y)$ is uniformly elliptic in $B_r\times B_r$ so that $a(x,y)K_{sp}(x,y)\approx K_{sp}(x,y)$, while $b(x,y)K_{tp}(x,y)$ is “higher order” and may be degenerate; thus \begin{equation}\label{main.eq}
\mathcal{L}u = 0 \quad \text{in } \Omega,
\end{equation}
 has a nonlocal double phase structure analogous to the model with $p=q$.
\item[(ii)] If $b^{+}_{B_r}>\nu/4$, then $b(x,y)$ is uniformly elliptic in $B_r\times B_r$ (ellipticity ratio $2$), so $b(x,y)K_{tp}(x,y)\approx K_{tp}(x,y)$ is dominating and $a(x,y)K_{sp}(x,y)$ is “lower order”; hence \begin{equation}\label{main.eq}
\mathcal{L}u = 0 \quad \text{in } \Omega,
\end{equation}
 becomes a $(t,p)$-Laplacian type equation with an $(s,p)$-Laplacian type lower-order term.
\end{itemize}

Finally, the paper roadmap gives the proof structure for Theorem~\ref{thm.bdd}: Next, we employ a Caccioppoli type estimate to prove the main theorem.

[Local boundedness]\label{thm.bdd}
Let  $u \in \mathcal{A}(\Omega) \cap \mathcal{T}(\mathbb{R}^{n})$ be a weak subsolution to
\begin{equation}\label{main.eq}
\mathcal{L}u = 0 \quad \text{in } \Omega,
\end{equation}
under assumptions
\begin{equation}\label{kernel.growth}
\frac{\nu}{|x-y|^{n+sp}} \leq K_{sp}(x,y) \leq \frac{L}{|x-y|^{n+sp}}, \qquad
\frac{\nu}{|x-y|^{n+tp}} \leq K_{tp}(x,y) \leq \frac{L}{|x-y|^{n+tp}}
\end{equation}
and
\begin{equation}\label{a.bound}
\nu \le a(x,y) + b(x,y) \le L
\end{equation}
and let $B_{r}\Subset \Omega$ be a ball with $r\le1$.
\begin{enumerate}
\item We have
\begin{equation}\label{est:bdd1}
\esssup_{B_{r/2}} u \le  c\,r^{-(t-s)/\beta} \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ r^{sp} T(u_+;r/2,r)\right]^{1/(p-1)}
\end{equation}
for a constant $c \equiv c(n,s,t,p,\nu,L)>0$, where
\begin{equation}\label{def.beta}
\beta \coloneqq \min\{sp/n,p-1\}.
\end{equation}
\item Assume further that $b(\cdot,\cdot)$ satisfies
\begin{equation}\label{a.holder}
|b(x_{1},y_{1}) - b(x_{2},y_{2})| \leq [b]_{\alpha}(|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\alpha}
\end{equation}
and
\begin{equation}\label{assumption.hol}
\quad (t-s)p \le \alpha.
\end{equation}
Then we have
\begin{equation}\label{est:bdd2}
\begin{aligned}
\esssup_{B_{r/2}} u & \le c \left(\mean{B_{r}}u_{+}^p\,dx\right)^{1/p} + \left[ \frac{1}{G_{B_r}(1)} T(u_+;r/2,r)\right]^{1/(p-1)} \\
& = c\left(\mean{B_{r}}u_{+}^{p}\,dx\right)^{1/p} + g_{B_{r}}^{-1}(T(u_{+};r/2,r))
\end{aligned}
\end{equation}
whenever $r \le R_{0}$, where $R_{0}$ is given in
\begin{equation}\label{R0.choice}
    [b]_{\alpha}(2R_{0})^{\alpha} \le \frac{\nu}{8}.
\end{equation}
and $c\equiv c(n,s,t,p,\nu,L,\alpha,[b]_{\alpha})>0$.
\end{enumerate}
Consequently, if $u \in \mathcal{A}(\Omega)\cap \mathcal{T}(\mathbb{R}^{n})$ is a weak solution to the equation above, then $u \in L^{\infty}_{\loc}(\Omega)$ and the estimate \eqref{est:bdd1} or \eqref{est:bdd2} holds with $|u|$ in each case.

Question:
Let
\[
\mathcal L u(x):=\operatorname{P.V.}\int_{\mathbb R^n}|u(x)-u(y)|^{p-2}(u(x)-u(y))\big(a(x,y)K_{sp}(x,y)+b(x,y)K_{tp}(x,y)\big)\,dy,
\]
and let \(u\) belong to the admissible class \(\mathcal A(\Omega)\cap \mathcal T(\mathbb R^n)\) used for weak solutions of this nonlocal equation. Assume that \(u\) is a weak subsolution of \(\mathcal Lu=0\) in \(\Omega\), and that for some \(\nu,L>0\),
\[
\frac{\nu}{|x-y|^{n+sp}}\le K_{sp}(x,y)\le \frac{L}{|x-y|^{n+sp}},\qquad
\frac{\nu}{|x-y|^{n+tp}}\le K_{tp}(x,y)\le \frac{L}{|x-y|^{n+tp}},
\]
with
\[
\nu\le a(x,y)+b(x,y)\le L.
\]
Let \(B_r\Subset \Omega\) be a ball with \(r\le 1\), write \(u_+=\max\{u,0\}\),
\(\fint_{B_r}f\,dx=|B_r|^{-1}\int_{B_r}f\,dx\),
\[
\beta:=\min\{sp/n,\,p-1\},
\]
and
\[
G_{B_r}(\tau):=\sup_{x,y\in B_r}\left(a(x,y)\frac{\tau^p}{r^{sp}}+b(x,y)\frac{\tau^p}{r^{tp}}\right),
\qquad g_{B_r}(\tau):=\frac{G_{B_r}(\tau)}{\tau}.
\]
If, in addition, \(b\) is Hölder continuous in the sense that
\[
|b(x_1,y_1)-b(x_2,y_2)|\le [b]_\alpha\,(|x_1-x_2|+|y_1-y_2|)^\alpha,
\]
and \((t-s)p\le \alpha\), with \(R_0\in(0,1)\) chosen so that
\[
[b]_\alpha(2R_0)^\alpha\le \nu/8,
\]
which statement holds for every such \(u\)?

One always has
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\,r^{-(t-s)/\beta}\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\big[r^{sp}T(u_+;r/2,r)\big]^{1/(p-1)},
\]
where \(c=c(n,s,t,p,\nu,L)>0\). If, moreover, \(b\) satisfies the Hölder condition above and \((t-s)p\le \alpha\), then for every \(r\le R_0\),
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\left[\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\right]^{1/(p-1)}
= c\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}+g_{B_r}^{-1}(T(u_+;r/2,r)),
\]
with \(c=c(n,s,t,p,\nu,L,\alpha,[b]_\alpha)>0\). Consequently, every weak solution \(u\in \mathcal A(\Omega)\cap\mathcal T(\mathbb R^n)\) of \(\mathcal Lu=0\) belongs to \(L^\infty_{\mathrm{loc}}(\Omega)\), and the corresponding estimate above holds with \(|u|\) in place of \(u\).

Choices:
B.
One always has
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\,r^{-(t-s)/(p-1)}\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\big[r^{sp}T(u_+;r/2,r)\big]^{1/(p-1)},
\]
where \(c=c(n,s,t,p,\nu,L)>0\). If, moreover, \(b\) satisfies the Hölder condition above and \((t-s)p\le \alpha\), then for every \(r\le R_0\),
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\left[\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\right]^{1/(p-1)},
\]
with \(c=c(n,s,t,p,\nu,L,\alpha,[b]_\alpha)>0\). Consequently, every weak solution \(u\in \mathcal A(\Omega)\cap\mathcal T(\mathbb R^n)\) of \(\mathcal Lu=0\) belongs to \(L^\infty_{\mathrm{loc}}(\Omega)\), and the corresponding estimate above holds with \(|u|\) in place of \(u\).
C.
One always has
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\,r^{-(t-s)/\beta}\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\big[r^{sp}T(u_+;r/2,r)\big]^{1/(p-1)},
\]
where \(c=c(n,s,t,p,\nu,L)>0\). Consequently, every weak solution \(u\in \mathcal A(\Omega)\cap\mathcal T(\mathbb R^n)\) of \(\mathcal Lu=0\) belongs to \(L^\infty_{\mathrm{loc}}(\Omega)\), and this estimate holds with \(|u|\) in place of \(u\).
D.
One always has
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\,r^{-(t-s)/\beta}\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\big[r^{sp}T(u_+;r/2,r)\big]^{1/(p-1)},
\]
where \(c=c(n,s,t,p,\nu,L)>0\). If, moreover, \(b\) satisfies the Hölder condition above and \((t-s)p\le \alpha\), then there exists a radius \(R_0\in(0,1)\), depending only on \(\nu,\alpha,[b]_\alpha\), such that for every ball \(B_r\Subset\Omega\) with \(r\le 1\),
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\left[\frac{1}{G_{B_r}(1)}T(u_+;r/2,r)\right]^{1/(p-1)},
\]
with \(c=c(n,s,t,p,\nu,L,\alpha,[b]_\alpha)>0\). Consequently, every weak solution \(u\in \mathcal A(\Omega)\cap\mathcal T(\mathbb R^n)\) of \(\mathcal Lu=0\) belongs to \(L^\infty_{\mathrm{loc}}(\Omega)\), and the corresponding estimate above holds with \(|u|\) in place of \(u\).
E.
One always has
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\,r^{-(t-s)/\beta}\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+\big[r^{tp}T(u_+;r/2,r)\big]^{1/(p-1)},
\]
where \(c=c(n,s,t,p,\nu,L)>0\). If, moreover, \(b\) satisfies the Hölder condition above and \((t-s)p\le \alpha\), then for every \(r\le R_0\),
\[
\operatorname*{ess\,sup}_{B_{r/2}} u
\le c\left(\fint_{B_r}u_+^p\,dx\right)^{1/p}
+g_{B_r}^{-1}(T(u_+;r/2,r)),
\]
with \(c=c(n,s,t,p,\nu,L,\alpha,[b]_\alpha)>0\). Consequently, every weak solution \(u\in \mathcal A(\Omega)\cap\mathcal T(\mathbb R^n)\) of \(\mathcal Lu=0\) belongs to \(L^\infty_{\mathrm{loc}}(\Omega)\), and the corresponding estimate above holds with \(|u|\) in place of \(u\).


\begin{theorem}\label{Thm:IGP}
    For any prime $p$ for which there exists an $\ell \in \cL$ with the Kronecker symbol $\legendre{\ell}{p}$ equal to $-1$, the group $\PSL_2(\F_{p^2})$ is a \emph{regular} Galois group over $\Q$.\footnote{Concretely, this implies that $\PSL_2(\F_{p^2})$ is realisable as a regular Galois group over $\Q$ for all $p<3,267,289$ (compared to the previous implied bound $p<1,009$).}
\end{theorem}

\begin{remark}\label{Rem:shih}
Theorem~\ref{Thm:cover} should be compared to the analogous cover $X_0(p)\to X(1)$ of modular curves corresponding to the level structure $\Gamma_0(p)$.

In the one-dimensional setting, $X_0(p)$ has field of definition $\Q(\zeta_p)$ and one only obtains a $\PGL_2(\F_p)$-cover. Under certain congruence conditions, Atkin-Lehner involutions can be used to turn this into a $\PSL_2(\F_p)$-cover by what has become known as Shih's trick \cite{Shih}. In our situation such a trick is unnecessary.
\end{remark}

The following popular question, going back to at least Noether, has remained open for over a century:
\begin{question}[Inverse Galois Problem]
 Is every finite group $G$ the Galois group of a finite extension of $\Q$?
\end{question}

We also have the variant:
\begin{question}[Regular Inverse Galois Problem]
 Is every finite group $G$ the Galois group of a finite geometrically integral cover $C \to \P^1_{\Q}$?
\end{question}

By the Hilbert Irreducibility Theorem (a positive answer to) the regular Inverse Galois Problem implies (a positive answer to) the classical one. On the other hand, the regular version has various interesting implications: for instance, it gives not just one $G$-extension of $\Q$ but infinitely many linearly disjoint $G$-extensions over any number field by specialisation.

A fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\PSL_2(\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\ell\in\{2,3,5,7\}$ with $\legendre{\ell}{p}=-1$ \cite{Shih, Malle} (although non-regularly for all $p$ \cite{Zywina}) and for $k=2$ when $\legendre{5}{p}=-1$ \cite{Feit, Mestre, Mestre2}, when $\legendre{2}{p}=-1$ or $\legendre{3}{p}=-1$ \cite{Shiina}, and finally when $\legendre{-3}{p}=-1$ or $\legendre{-7}{p}=-1$ \cite{DW} (although non-regularly for more $p$ \cite{DV}).

In this paper we introduce new methods for the Inverse Galois Problem to prove the following. Let
\[
\cL:=\{\ell\ \text { prime}: \ell \leq 41, \ell \neq 31\}.
\]

A fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\PSL_2(\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\ell\in\{2,3,5,7\}$ with $\legendre{\ell}{p}=-1$ \cite{Shih, Malle} (although non-regularly for all $p$ \cite{Zywina}) and for $k=2$ when $\legendre{5}{p}=-1$ \cite{Feit, Mestre, Mestre2}, when $\legendre{2}{p}=-1$ or $\legendre{3}{p}=-1$ \cite{Shiina}, and finally when $\legendre{-3}{p}=-1$ or $\legendre{-7}{p}=-1$ \cite{DW} (although non-regularly for more $p$ \cite{DV}).

In particular, this realises the group $\PSL_2(\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\ell \leq 17, \ell \neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \cite{EK} have proved $\Q$-rationality. For the primes $\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \cite{CZ}, and further studied by the first-named author \cite{Demeio} and by the second-named author in collaboration with Mezzedimi \cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.

In fact, we prove an even stronger statement, namely that $X(\Q(t))$ is non-thin in $X_{\bar \Q(t)}(\bar \Q(t))$ (Theorem \ref{Thm:K3regular}), thus producing an abundance of {\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.
The application to the Inverse Galois Problem is found through the following
\begin{theorem}\label{Thm:cover}
 Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

The particular Shimura varieties under our consideration {\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \to X$ to {\em these} rational curves does not produce surjective representations $\Gal(\overline{\Q(t)}/\Q(t)) \to \PSL_2(\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \emph{not} sufficient to construct Galois representations of $\PSL_2(\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \ref{Thm:K3}.

\begin{proposition}\label{Prop331regular}
 Let $k$ be a field of characteristic $0$, and $k(V)$ be a regular finitely generated transcendental extension of $k$. Let $\phi:Y\to X$ be a Galois cover with group $G$ of geometrically integral $k(V)$-varieties. Let $S \subset X(k(V))$ be a subset whose natural image in $X(\bar k(V))=X_{\bar k(V)}(\bar k(V))$ is a non-thin set of points of the base-changed variety $X_{\bar k(V)}$. Then there are Zariski-dense $x \in S$ for which the fibre $\phi^{-1}(x)$ is irreducible with regular function field over $k$, and with $\Gal(\phi^{-1}(x)/k(V))\simeq G$.
\end{proposition}

We write $\H^+$ and $\H^-$ for the upper and lower half-plane. For each genus $g \in \Cl^+K/2$ of $\mathfrak o$ we may pick a representative fractional ideal $\mathfrak a$ (of norm $A$) and define, for each finite index subgroup $\Gamma \subseteq \PSL(\mathfrak o \oplus \mathfrak a)$, the \emph{Hilbert modular surface}
\[
X_g(\Gamma):=\Gamma \backslash (\H^+)^2,
\]
where the quotient is by the properly discontinuous action:
\[
\begin{pmatrix}
 a & b \\ c&d
\end{pmatrix}\cdot (z_1,z_2) = \left( \frac{az_1+b}{cz_1+d} , \frac{a'z_2+b'}{c'z_2+d'} \right).
\]
Let $\Lambda=\mathfrak o \oplus \mathfrak a$. We call the surface $X_g:= X_g(\PSL(\Lambda))$ of \emph{base level}.\footnote{One could take further quotients of $(\H^+)^2$, the maximum being by the symmetrised Hurwitz-Maass extension of $\PSL(\Lambda)$, but we shall not be interested in these in this paper.} The isomorphism class of $X_g$ is independent of the choice of representative $\mathfrak a$.

The classes in $\Cl(K)$ are in bijection with $\PSL(\Lambda)$-orbits of points in $\P^1(K)$ via the map sending $(x_0:x_1)\in\P^1(K)$ to the class $[x_0\cdot \mathfrak o + x_1\cdot \mathfrak a]\in\Cl(K)$. Fix a class $[\mathfrak c]\in\Cl(K)$ and write shorthand $s=s_{[\mathfrak c]}$. To $s$ we associate $M_s=\mathfrak c^{-2}\mathfrak a^{-1}$ viewed as an \emph{ordered} $\mathfrak o$-module, i.e.\ equipped with an order on $M_s\otimes_\sigma \R$ for each embedding $\sigma:K\hookrightarrow\R$. The isomorphism class of $M_s$ does not depend on the chosen representative $\mathfrak c$.
\begin{proposition}\label{Prop:CoverQ}\hfill
 \begin{enumerate}
 \item The action of $\Gamma_\Q$ on $X_g^\infty(\ov\Q)$ is trivial, in other words all cusps are defined over $\Q$.
 \item The exceptional divisor $E_s$ of the minimal resolution $\tilde X_g\to X_g$ over any cusp $s$ is a cycle of smooth rational curves intersecting transversally.
 \end{enumerate}
\end{proposition}

Combining the above, we proved the theorem for all $p$ for which there exists $D_p$ in
\[
\cD \coloneqq \{5,8,12,13,17,21,24,28,29,33,37,40,41,44,56,57,69,105\}
\]
with $\legendre{D_p}{p}=-1$. One easily checks by hand that this includes all primes $p \leq 41$. For $p > 41$, such a $D_p$ exists unless
\begin{center}
 $\legendre{D}{p}=1$ for all $D \in \cD$,
\end{center}
or, equivalently, unless $\legendre{D}{p}=1$ for all $D$ in the multiplicative subgroup $\langle \cD \rangle \subset \Q^*$ generated by $\cD$. Rewriting $\cD$ as
\[
\{5,2^3,2^2 \cdot 3,13,17,3 \cdot 7,2^3 \cdot 3,2^2 \cdot 7,29,3 \cdot 11,37,2^3 \cdot 5,41,2^2 \cdot 11,2^3 \cdot 7,3\cdot 19,3 \cdot 23,3 \cdot 5 \cdot 7\}
\]
one sees that $\langle \cD \rangle=\langle \cL \rangle$, where $\cL=\{\ell \text{ prime}: \ell \leq 41, \ell \neq 31\}$ is the list of prime factors appearing in $\cD$. So $D_p$ exists unless $\legendre{\ell}{p}=1$ for all $\ell$, concluding the proof.
\end{proof}

In particular, this realises the group $\PSL_2(\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\ell \leq 17, \ell \neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \cite{EK} have proved $\Q$-rationality. For the primes $\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \cite{CZ}, and further studied by the first-named author \cite{Demeio} and by the second-named author in collaboration with Mezzedimi \cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.

\begin{theorem}\label{Thm:K3}
 Let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\Q)$ is not thin.
\end{theorem}

In fact, we prove an even stronger statement, namely that $X(\Q(t))$ is non-thin in $X_{\bar \Q(t)}(\bar \Q(t))$ (Theorem \ref{Thm:K3regular}), thus producing an abundance of {\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.
The application to the Inverse Galois Problem is found through the following
\begin{theorem}\label{Thm:cover}
    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

Compared to the well-known modular curve setting, the existence of the Galois cover which is geometrically integral may be surprising at first, see Remark~\ref{Rem:shih}.

Theorem \ref{Thm:IGP} is proven by combining Theorems \ref{Thm:K3} and \ref{Thm:cover}.
The proofs of Theorems \ref{Thm:K3} and \ref{Thm:cover} rely on the theory of canonical models of compactified Shimura varieties as applied to Hilbert modular surfaces. More precisely, Theorem \ref{Thm:K3} relies on an analysis of the intersection graph of these modular surfaces, while Theorem \ref{Thm:cover} follows naturally by looking at the Shimura data involved. (Alternatively, Theorem \ref{Thm:cover} can be inferred by looking at the $p$-torsion representation of the abelian surfaces with real multiplication parametrized by Hilbert Modular Surfaces.)

The particular Shimura varieties under our consideration {\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \to X$ to {\em these} rational curves does not produce surjective representations $\Gal(\overline{\Q(t)}/\Q(t)) \to \PSL_2(\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \emph{not} sufficient to construct Galois representations of $\PSL_2(\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \ref{Thm:K3}.

The Shimura theory is particularly explicit in our case but it would be interesting to see if a similar approach could apply to other settings.

It would also be interesting to study the Hilbert Property for honestly elliptic Hilbert modular surfaces, where the multiple fibration method does not apply. If instead of proving non-thinness of rational points the goal is to only specialise a specific Galois cover like in Theorem~\ref{Thm:cover}, then one may use less than the full Hilbert Property, although knowledge of the intersection graph of the modular surface would still be necessary. We will pursue this approach in a separate forthcoming work.

The article is organised as follows: In \S\ref{Sec:hilbert}, we review and extend the multiple fibration method to nonconstant $\Q(t)$-points. We develop the necessary Shimura theory of Hilbert modular surfaces in \S\ref{Sec:shimura} and construct the cover from Theorem~\ref{Thm:cover} in \S\ref{Sec:cover}. We then study the algebraic cycles on our surfaces in \S\ref{Sec:cycles}. Finally, we prove Theorems~\ref{Thm:IGP} and \ref{Thm:K3} in \S\ref{Sec:main}.

\paragraph{Acknowledgements}
The authors thank Gerard Van der Geer and John Voight for helpful comments. They gratefully acknowledge the hospitality of ICMS Edinburgh, the Center for Mathematical Sciences at Technion, and the University of Bath, where part of this research was carried out, and thank Daniel Loughran for the invitation to the latter. The second author was supported by UKRI award UKRI094.

To prove Theorem~\ref{Thm:IGP}, the authors "combin[e] Theorems \ref{Thm:K3} and \ref{Thm:cover}." Theorem~\ref{Thm:cover} supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a "Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$." Theorem~\ref{Thm:K3} provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then "$X(\Q)$ is not thin" (indeed they prove the stronger non-thinness statement over $\Q(t)$ in Theorem~\ref{Thm:K3regular}, yielding "an abundance of rational curves").

For small $\ell$ with $\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they "apply the classical Hilbert Irreducibility Theorem." For the remaining listed primes $\ell$ where the surfaces are "not (uni)rational," they instead "prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces" by establishing that "the Hilbert Property holds" for the relevant low-discriminant surfaces via the "multiple fibration method." Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, "specialising the cover $Y_p \to X$ to these rational curves does not produce surjective representations" to $\PSL_2(\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature "allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject," with the propagation formalised by the multiple fibration method used in proving Theorem~\ref{Thm:K3}.

To prove the main theorem, the authors “combin[e] Theorems \ref{Thm:K3} and \ref{Thm:cover}.” We first record the two inputs.

\begin{theorem}\label{Thm:cover}
    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

\begin{theorem}\label{Thm:K3}
 Let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\Q)$ is not thin.
\end{theorem}

The first theorem supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$. The second theorem provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then $X(\Q)$ is not thin (indeed they prove the stronger non-thinness statement over $\Q(t)$ in
\begin{theorem}\label{Thm:K3regular}
     Let $X/\Q$ be the Hilbert modular surface of discriminant $D\in \cS_+$ (resp.\ $\cS_-$) and principal genus (resp.\ non-principal genus) at base level. Then the natural image of $X(\Q(t))$ in $X_{\bar \Q(t)}$ is non-thin.
\end{theorem}
, yielding “an abundance of rational curves”).

For small $\ell$ with $\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they “apply the classical Hilbert Irreducibility Theorem.” For the remaining listed primes $\ell$ where the surfaces are “not (uni)rational,” they instead “prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces” by establishing that “the Hilbert Property holds” for the relevant low-discriminant surfaces via the “multiple fibration method.” Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, “specialising the cover $Y_p \to X$ to these rational curves does not produce surjective representations” to $\PSL_2(\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature “allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject,” with the propagation formalised by the multiple fibration method used in proving the theorem above.

Question: Let \(\mathcal L=\{\ell\text{ prime}: \ell\le 41,\ \ell\ne 31\}\). Suppose \(p\) is a prime such that there exists some \(\ell\in \mathcal L\) with Kronecker symbol \(\left(\frac{\ell}{p}\right)=-1\). Under this hypothesis, which statement about \(\operatorname{PSL}_2(\mathbb F_{p^2})\) holds over \(\mathbb Q\)? Here, saying that a finite group \(G\) is a regular Galois group over \(\mathbb Q\) means that there exists a finite geometrically integral cover \(C\to \mathbb P^1_{\mathbb Q}\) whose induced function-field extension is Galois with group \(G\) (equivalently, there is a Galois extension \(E/\mathbb Q(t)\) with Galois group \(G\) and \(E\cap \overline{\mathbb Q}=\mathbb Q\)).

The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\); equivalently, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) such that \(E\cap \overline{\mathbb Q}=\mathbb Q\).

Choices:
B. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a Galois group over \(\mathbb Q\); equivalently, there exists a finite Galois extension \(L/\mathbb Q\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\).
C. There exists a finite geometrically integral cover \(C\to \mathbb P^1_{\mathbb Q}\) whose induced function-field extension has Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\).
D. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\) for every prime \(p\) such that \(\left(\frac{\ell}{p}\right)=-1\) for all \(\ell\in \mathcal L\); equivalently, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) and \(E\cap \overline{\mathbb Q}=\mathbb Q\) whenever this stronger condition holds.
E. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\); equivalently, for every real quadratic field \(K\) in which \(p\) is inert, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) such that \(E\cap \overline{\mathbb Q}=\mathbb Q\).


[Hirose--Sato~\cite{HiroseSato2019A}]\label{thm:HiroseSato1}
For positive integers $a, b$, and $c$, we have
\begin{align*}
&\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)\\
&=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\end{align*}

\begin{theorem}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]
The drop 1 operator $\cD$ satisfies the following properties$:$
\label{thm:drop1}
\begin{enumerate}[label=\textnormal{(\roman*)}]
\item\label{it:HMSW1} $\{w-\cD(w) \mid w\in \cH^0\}=\Ker(Z^{\diamondsuit})\subset\Ker(Z)$,
\item\label{it:HMSW2} $\cD(w)=w$ for $w\in\cH^{\geq 2}$,
\item\label{it:HMSW3} $\cD(w)=\cD(\tau(w))$ for $w\in \cH^0$.
\end{enumerate}
\end{theorem}

\begin{definition}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]\label{def:drop1operator}
We define the \emph{drop 1 operator} $\cD\colon\cH^0\to\cH^{\geq 2}$ to be a $\ZZ$-linear map given as follows.
It suffices to define $\cD(w)$ for each word $w\in\cH^0$.
For $\bc=(c_1,\dots, c_{2t})$, we write $\fD(\bc)=\cD(y^{c_1}x^{c_2}\cdots y^{c_{2t-1}}x^{c_{2t}})$.
The image $\fD(\bc)$ is defined inductively as follows:
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)x^{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{oe}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{oe}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}.
\end{align*}
Here the initial condition is $\fD(\varnothing)=\cD(1)=1$.
\end{definition}

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\cH^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}

For positive integers $k_1, \dots, k_r$ with $k_r\geq 2$, the \emph{multiple zeta value} (MZV) $\zeta(k_1,\dots, k_r)$ is defined by
\[
\zeta(k_1,\dots, k_r)\coloneqq\sum_{0<n_1<\cdots <n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
Linear relations among MZVs have been studied for many years.
For each positive integer $c$, the relation
\begin{equation}\label{eq:Hoffman-conj}
\zeta(3,\{2\}^{c-1},1,2)=\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})
\end{equation}
is known as \emph{Hoffman's conjectural identity} and had remained open for about seventeen years.
Here the notation $\{2\}^n$ denotes the string $2,\dots, 2$ in which the entry 2 is repeated $n$ times.
Charlton proved in \cite{Charlton2016} that there exists a rational number $q_c$ such that $\zeta(3,\{2\}^{c-1},1,2)=q_c\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\wt(w)$ denotes the total degree of $w$, that is, $\wt(w)=k_1+\cdots + k_r$.
A $\ZZ$-linear map $Z_{\cA}\colon \ZZ+y\cH=\ZZ\langle z_1,z_2,\dots\rangle\to\cA$ is defined by $Z_{\cA}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\cA}(k_1,\dots, k_r)$ and $Z_{\cA}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\cH^{2,3}$, we have
\[
(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\cA}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.
\section*{Acknowledgements}
The author would like to thank Professor Minoru Hirose for kindly answering his questions about the previous works by Hirose and Sato.
This research was inspired by a question that came to mind when the author got married, and the author would like to thank his wife, Yui, for marrying him.
\section{The drop 1 operator}
We introduce the following notation to define the main operator.
Let $t$ be a positive integer, let $\bc=(c_1,\dots, c_{2t})$ be a tuple of positive integers, and let $S\subset\ZZ$.
\begin{itemize}[leftmargin=2.5em]
\item $\#A$ denotes the cardinality of a finite set $A$.
\item $[n]\coloneqq\{1,2,\dots,n\}$ for $n\in\ZZ_{\geq 0}$. $[0]=\varnothing$.
\item $[2t]_{\bc}^1\coloneqq\{i\in[2t] \mid c_i=1\}.$
\item $[2t]_{\bc}^{>1}\coloneqq\{i\in[2t] \mid c_i>1\}.$
\item $\cP^{\text{eo}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longleftrightarrow i+1\in A\setminus 2\ZZ\}.$
\item $\cP^{\text{oe}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longleftrightarrow i+1\in A\cap 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{eo}}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longrightarrow i+1\not\in A\setminus 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{oe}}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longrightarrow i+1\not\in A\cap 2\ZZ\}.$
\item $\bc_{(-A)}\coloneqq (c_{i_1},\dots, c_{i_r})$, where $[2t]\setminus A=\{i_1,\dots, i_r\}$ with $i_1<\cdots<i_r$, for $A\subset[2t]$.
\item $\bc_{(-\varnothing)}=\bc$, $\bc_{(-[2t])}=\varnothing$.
\item $\delta_{i\in B}\coloneqq\begin{cases}1 & \text{if } i\in B, \\ 0 & \text{otherwise}\end{cases}$ for a set $B$.
\item $\bc_{(-A)}-\bdelta_B\coloneqq(c_{i_1}-\delta_{i_1\in B},\dots, c_{i_r}-\delta_{i_r\in B})$ for the above $\bc_{(-A)}$ and $B\subset [2t]$.
\end{itemize}

We consider the case $c>1$.
Then $\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc})=\cP([2t]^{>1}_{\bc})$ holds.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, decompose it as $A=A_1\cup A_2 \cup A_3$, where
\[
A_1\coloneqq A \cap [2l_r-1],\quad A_2\coloneqq A\cap ([2l_r+2c-2]\setminus [2l_r]),\quad A_3\coloneqq A\setminus [2l_r+2c-1].
\]
We define a map $\iota_1\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_1(A)\coloneqq (A_1-1)\cup (A_2-1)\cup\{2l_r+2c-3,2l_r+2c-2\}\cup(A_3+1)
\]
and a map $\iota_2\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_2(A)\coloneqq (A_1-1)\cup (A_2-1)\cup(A_3+1).
\]
A bijection $\cP^{\text{eo}}([2t]^1_{\bc})\sqcup\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ is obtained by sending an element $A$ in the first copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_1(A)$, and an element $A$ in the second copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_2(A)$.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, we have $\bc_{(-A)}=\bc_{(-\iota_2(A))}$.
Therefore, if we compute the defining recurrence for the drop 1 operator using this bijection, every term in the second sum is canceled by a term in the third sum, and we are left with
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\biggl(\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2\biggr)x^{\#A+\#B}\\
&\quad+\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2.
\end{align*}
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$ or $\cP^{\text{oe}}([2t]^1_{\bc})$, there exist $a'_1$, $\dots$, $a'_r$, $b'_1$, $\dots$, $b'_s$, and $c'$ such that
\[
\bc_{(-A)}=[a'_1,\dots, a'_r;b'_1,\dots, b'_s;c'].
\]
Here $a'_1\in[a_1]$, $\dots$, $a'_r\in[a_r]$, $b'_1\in[b_1]$, $\dots$, $b'_s\in[b_s]$, $c'\in[c]$, and $a'_1+\cdots+a'_r+b'_1+\cdots+b'_s+c'=t+1-\#A/2$.
Moreover, upon adding $-\bdelta_B$, addition occurs between the corresponding components; for example, when
\begin{align*}
\bc&=[2,1,3;4,1,1,2;3]\\
&=(\stackrel{1}{1},\stackrel{2}{1},\stackrel{3}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{8}{1},\stackrel{9}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{14}{1},\stackrel{15}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{18}{1},\stackrel{19}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{22}{1},\stackrel{23}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{30}{1},\stackrel{31}{1},\stackrel{32}{1}),
\end{align*}
\[
A=\{2,3,8,9,14,15,18,19,22,23,30,31\}\in\cP^{\text{eo}}([32]^1_{[2,1,3;4,1,1,2;3]}),
\]
and
\[
B=\{4,17,27\}\subset[32]^{>1}_{[2,1,3;4,1,1,2;3]}=\{4,6,12,17,25,27,29\},
\]
we have
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}=[1,1,2;2,1,1,1;2]\\
&=(\stackrel{1}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1})
\end{align*}
and
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}-\bdelta_{\{4,17,27\}}\\
&=[1+1,2;1+1,1;2+2]=[2,2;2,1;4]\\
&=(\stackrel{1}{1},\stackrel{\color{blue}{4}}{1},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{blue}{17}}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{blue}{27}}{1},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1}).
\end{align*}
Therefore, for any $A, B$ with $\#A+\#B\geq 1$, the induction hypothesis implies that there exist words $v_1, v_2$ in $\cH^{2,3}$ such that $\bc_{(\iota_1(-A))}-\bdelta_B$ corresponds to $v_1\tau(v_2)$, and we have
\[
\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2=\cD(v_1z_2\tau(v_2))-\cD(v_1\tau(v_2))z_2=v_1z_2\star v_2-(v_1\star v_2)z_2=0.
\]
Since we are in the case $c>1$, we can write $w_1=wz_2$ with a certain word $w$ in $\cH^{2,3}$.
Then we obtain
\begin{align*}
\fD(\bc)&=\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2=\fD([a_1,\dots, a_r;b_1,\dots, b_s;c-1])z_2=\cD(w\tau(w_2))z_2\\
&=(w\star w_2)z_2=wz_2\star w_2=w_1\star w_2.
\end{align*}

Their proof can be viewed as showing that the above identity is a consequence of the \emph{confluence relation} later formulated in \cite{HiroseSato2019B}.

Hirose and Sato gave a further generalization in \cite{HiroseSato2022+}, and we set up notation in order to state their result.
Let $\mathcal{H}\coloneqq\mathbb{Z}\langle x,y\rangle$ be the non-commutative polynomial ring over $\mathbb{Z}$ in two variables $x$ and $y$.
Let $z_k\coloneqq yx^{k-1}$ for each positive integer $k$.
Subrings $\mathcal{H}^{2,3}\subset\mathcal{H}^0\subset\mathcal{H}$ are defined by $\mathcal{H}^{2,3}\coloneqq\mathbb{Z}\langle z_2,z_3\rangle$ and $\mathcal{H}^0\coloneqq\mathbb{Z}+y\mathcal{H} x$, respectively.
A $\mathbb{Z}$-linear map $Z\colon\mathcal{H}^0\to\mathbb{R}$ is defined by $Z(z_{k_1}\cdots z_{k_r})=\zeta(k_1,\dots, k_r)$ and $Z(1)=1$.
A product $\star$ on $\mathcal{H}^{2,3}$ is defined inductively by
\begin{itemize}[leftmargin=2.5em]
\item $w\star 1=1\star w=w$,
\item $wz_2\star w'=w\star w'z_2=(w\star w')z_2$,
\item $wz_3\star w'z_3=(w\star w'z_3)z_3+(wz_3\star w')z_3+(w\star w')z_2^3$
\end{itemize}
for any words $w$ and $w'$ in $\mathcal{H}^{2,3}$, and then extended by $\mathbb{Z}$-bilinearity.
Let $\tau$ be an anti-automorphism on $\mathcal{H}$ defined by $\tau(x)=y$ and $\tau(y)=x$.
Using this notation, we can now state the following theorem.
\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\mathcal{H}^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}
By specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$, we recover Theorem~\ref{thm:HiroseSato1}.
Hirose and Sato derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity} (\cite[Theorem~10]{HiroseSato2022+}).

Hoffman conjectured in \cite{Hoffman1997} that any multiple zeta value can be expressed as a $\mathbb{Q}$-linear combination of the elements of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\in\{2,3\}\}.
\]
This conjecture was proved by Brown in \cite{Brown2012}.
Although the linear independence of these elements remains open, they are called the \emph{Hoffman basis}.
Theorem~\ref{thm:HiroseSato2} provides a family for which the expansion in the Hoffman basis can be described explicitly, and, even more remarkably, it shows that the expansion can be taken with \emph{integer coefficients}.
In general, the coefficients are not integral; for instance,
\[
\zeta(3,1,4)=-\frac{2048}{4125}\zeta(2,2,2,2)-\frac{17}{275}\zeta(2,3,3)+\frac{2}{275}\zeta(3,2,3)+\frac{113}{275}\zeta(3,3,2).
\]

In contrast, Hirose, Maesaka, Seki, and Watanabe recently proved in \cite{HiroseMaesakaSekiWatanabe2025+}, together with an explicit algorithm, that any multiple zeta value can be expressed as a $\mathbb{Z}$-linear combination of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\geq 2\}.
\]
Their explicit expansions give rise to a family of relations among MZVs, which we refer to as the \emph{drop 1 relation}.
Since their expansions have integer coefficients, it is natural to ask whether Theorem~\ref{thm:HiroseSato2} can be viewed as an expansion arising from the drop 1 relation.
However, in general, expansions coming from the drop 1 relation involve MZVs with entries greater than $3$, so it is not obvious whether the expansion in Theorem~\ref{thm:HiroseSato2}, in which only 2 and 3 appear, can be obtained from the drop 1 relation.
The main result of this paper is that this is indeed the case.

A subring $\mathcal{H}^{\geq 2}\subset\mathcal{H}^0$ is defined by $\mathcal{H}^{\geq 2}\coloneqq\mathbb{Z}+z_2\mathbb{Z}\langle x,z_2\rangle=\mathbb{Z}\langle z_2, z_3, z_4,\dots\rangle$.
A linear map $\mathcal{D}\colon\mathcal{H}^0\to\mathcal{H}^{\geq 2}$, called the \emph{drop 1 operator}, is defined in Definition~\ref{def:drop1operator}, and its properties are summarized in Theorem~\ref{thm:drop1}.
\begin{theorem}\label{thm:main}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
\mathcal{D}(w_1\tau(w_2))=w_1\star w_2.
\]
\end{theorem}
Not only does this theorem yield a new proof of Theorem~\ref{thm:HiroseSato2}, but it also implies, by Theorem~\ref{thm:drop1}~\ref{it:HMSW1}, that the same relations hold for \emph{multiple zeta diamond values}.
We omit the definition of $\zeta^{\diamondsuit}(k_1,\dots, k_r)$ here; see \cite[Definition~2.3]{HiroseMaesakaSekiWatanabe2025+}.
A $\mathbb{Z}$-linear map $Z^{\diamondsuit}\colon\mathcal{H}^0\to\mathbb{Q}^{\mathbb{N}}$ is defined by $Z^{\diamondsuit}(z_{k_1}\cdots z_{k_r})=\zeta^{\diamondsuit}(k_1,\dots, k_r)$ and $Z^{\diamondsuit}(1)=1$.

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}
For example, the relation corresponding to \eqref{eq:Hoffman-conj},
\[
\zeta^{\diamondsuit}(3,\{2\}^{c-1},1,2)=\zeta^{\diamondsuit}(\{2\}^{c+2})+2\zeta^{\diamondsuit}(3,3,\{2\}^{c-1})
\]
means that, for any positive integer $N$,
\begin{align*}
&\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}<n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2n_{c+1}^{}n_{c+2}^2}\\
&+\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}\leq n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2(N-n_{c+1}^{})n_{c+2}^2}\\
&=\sum_{0<n_1<\cdots < n_{c+2}<N}\frac{1}{n_1^2\cdots n_{c+2}^2}+2\sum_{0<n_1<n_2<n_3<\cdots<n_{c+1}<N}\frac{1}{n_1^3n_2^3n_3^2\cdots n_{c+1}^2}
\end{align*}
holds.
Letting $N\to\infty$ yields \eqref{eq:Hoffman-conj}.
On the other hand, if we take $N=p$ for a prime $p$ and take modulo $p$, then we have the following relation among \emph{finite multiple zeta values} (FMZVs):
\[
\zeta^{}_{\mathcal{A}}(3,\{2\}^{c-1},3)+2\zeta^{}_{\mathcal{A}}(3,3,\{2\}^{c-1})=0.
\]
Here, for positive integers $k_1,\dots, k_r$, the FMZV $\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ is defined as an element of $\mathcal{A}\coloneqq\left.\prod_p\mathbb{Z}/p\mathbb{Z}\right/\bigoplus_p\mathbb{Z}/p\mathbb{Z}$ by
\[
\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)\coloneqq (\zeta^{}_p(k_1,\dots,k_r)\bmod{p})_p,
\]
where $p$ runs over all prime numbers, and
\[
\zeta^{}_p(k_1,\dots,k_r)\coloneqq\sum_{0<n_1<\cdots<n_r<p}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
This definition is due to Kaneko and Zagier (\cite{KanekoZagier2025+}).
We have used a well-known formula: $\zeta^{}_{\mathcal{A}}(\{2\}^{c+2})=0$.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\mathrm{wt}(w)$ denotes the total degree of $w$, that is, $\mathrm{wt}(w)=k_1+\cdots + k_r$.
A $\mathbb{Z}$-linear map $Z_{\mathcal{A}}\colon \mathbb{Z}+y\mathcal{H}=\mathbb{Z}\langle z_1,z_2,\dots\rangle\to\mathcal{A}$ is defined by $Z_{\mathcal{A}}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ and $Z_{\mathcal{A}}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\mathcal{A}}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.

Their proof (of Theorem~\ref{thm:HiroseSato1}) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that Theorem~\ref{thm:HiroseSato1} is recovered by “specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$,” and that Hirose--Sato “derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity}.”

Their proof (of the main theorem) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that the main theorem is recovered by specializing the following theorem to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$.

\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\cH^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}

It is also stated that Hirose--Sato “derived the theorem above from their \emph{block shuffle identity}.”

Question:
For positive integers $k_1,\dots,k_r$ with $k_r\ge 2$, the multiple zeta value is defined by
\[
\zeta(k_1,\dots,k_r)=\sum_{0<n_1<\cdots<n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
For an integer $n\ge 0$, let $\{2\}^n$ denote the string consisting of $n$ copies of $2$ (so $\{2\}^0$ is the empty string). Which identity holds for every choice of positive integers $a,b,c$?

\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

Choices:
B.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+2\,\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1}).
\]
C.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+R_{a,b,c},
\]
where
\[
R_{a,b,c}=\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]
D.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{c-1},3,\{2\}^{a-1}).
\]
E.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c+1})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

symmetrized two-term contribution collapsed to a doubled single term

explicit decomposition of the remainder into the two displayed zeta terms

precise placement of the blocks in the second 3-term on the right-hand side

total weight balance between the left-hand side and the pure even zeta term