Assume GRH for the non-principal character modulo $4$. Then, the natural density of the set $ \mathcal{P}_{1/2}(4;3,1)=\{ x\ge 2\, :\, \pi_{1/2}(x;4,3)>\pi_{1/2}(x;4,1)\}$
    exists and equals $1$.

\begin{thm}\label{main}
    Let $q\ge 3$ and assume \text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\, a,$ and $b$, such that the natural density of the set \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\M(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{\frac{1}{2}-\varepsilon}}\right\}\]
    exists and equals $1$, and the logarithmic density of the set
    \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\M(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{1-\varepsilon}}\right\}\]
    exists and equals $1$.
\end{thm}

\begin{equation}\label{DRH-estimate}
    \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b)=-\frac{\M(q;a,b)}{2\varphi(q)} \log \log x+C+o(1) \, ,
\end{equation}

In a letter~\cite{Chebyshev} sent to Fuss in 1853, Chebyshev observed that primes congruent to $3\bmod\,  4$ seem to appear more often than those congruent to $1\bmod\,  4$. Let $q\ge3$ and let $a\in \mathbb Z$ be coprime to $q$, and define $\pi(x;q,a)=\#\{ p\le x\, :\,  p\equiv a\bmod\,  q\}$. Although the prime number theorem in arithmetic progressions asserts that primes are equidistributed among invertible classes modulo $q$, it gives no control over the difference $\pi(x;q,a)-\pi(x;q,b)$ when $b\in\mathbb Z$ is coprime to $q$. In 1914, Littlewood~\cite{Littlewood} proved that the function $x\mapsto\pi(x;4,1)-\pi(x;4,3)$ changes sign infinitely often. This naturally raised the question of the existence of the natural density of the set $\mathcal{P}(q;a,b):=\{x\ge 2\, :\, \pi(x;q,a)>\pi(x;q,b)\}$, namely the limit \[ \lim_{X\to \infty}\frac{\boldsymbol{m}\{2\le x\le X\, :\, x\in \mathcal{P}(q;a,b) \}}{X},\] where $\boldsymbol{m}$ is the Lebesgue measure on $\mathbb R$. This question was first addressed by Kaczorowski~\cite{Kac} who proved under $\text{GRH}$ that the natural density of $\mathcal{P}(4;3,1)$ does not exist.

In 1994, Rubinstein and Sarnak~\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \[ \lim_{X\to \infty} \frac{1}{\log X}\int_{2}^X \mathds{1}_{\mathcal{P}(q;a,b)}(u)\frac{\mathrm{d}u}{u}\]
of the set $\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\mathcal{P}(4;3,1)$ is approximately $0.9959\dots$

More recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \begin{equation}\label{pi_1/2}
    \pi_{1/2}(x;q,a)=\sum_{\substack{p\le x\\ p\equiv a\bmod\, {q}}}\frac{1}{\sqrt{p}}
\end{equation}
and ask whether the natural density of the set $\mathcal{P}_{1/2}(q;a,b):=\{x\ge 2\, :\, \pi_{1/2}(x;q,a)>\pi_{1/2}(x;q,b)\}$ exists. In their framework, and that of subsequent authors~\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \begin{equation}\label{DRH} (\log x)^{m_\chi} \prod_{p\le x}\left(1-\frac{\chi(p)}{p^{1/2}}\right)^{-1}=\ell_\chi+o(1) \quad(x\to \infty) ,\end{equation}
where $\ell_\chi\ne 0$ and $m_\chi:=\text{ord}_{s=1/2}L(s,\chi)$ is the order of vanishing of $L(s,\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\sum_{p\le x} \chi(p^2)/p$, leads to an estimate of $\sum_{p\le x} \chi(p)/\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \begin{equation}\label{DRH-estimate}
    \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b)=-\frac{\text{M}(q;a,b)}{2\varphi(q)} \log \log x+C+o(1) \, ,
\end{equation}
    where $\text{M}(q;a,b)=r(a)-r(b)+2\sum_{\chi \ne \chi _0}(\chi(a)-\chi(b))m_\chi$, $r:(\mathbb Z/q\mathbb Z)^\times\to \Z_{\ge 0}$ defined by $r(a)=\#\{ x\in (\mathbb Z/q\mathbb Z)^\times\, :\, x^2=a\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\cite{Su} studied the weighted Chebyshev-type function $\psi_{1/2}(x):=\sum_{n\le x}\Lambda(n)/\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if
\[\sqrt{x}\left(\sum_{\substack{p\le x\\ p\equiv 1\bmod\, 4}}\frac{\log p}{\sqrt{p}}\log\left(\frac{x}{p}\right)-\sum_{ \substack{p\le x\\ p\equiv 3 \bmod\, 4}}\frac{\log p}{\sqrt{p}}\log \left(\frac{x}{p}\right) \right)\sim-(\log x)^2 \frac{\sqrt{x}}{4}\quad (x\to \infty)\, .\]

This suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\log p \log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.

In this paper, we show that GRH is sufficient to prove that the equality~\eqref{DRH-estimate} holds for a set of $x\ge 2$ of natural density $1$. More precisely:
\begin{thm}\label{main}
    Let $q\ge 3$ and assume \text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\, a,$ and $b$, such that the natural density of the set \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\text{M}(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{\frac{1}{2}-\varepsilon}}\right\}\]
    exists and equals $1$, and the logarithmic density of the set
    \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\text{M}(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{1-\varepsilon}}\right\}\]
    exists and equals $1$.
\end{thm}
Assuming that $m_\chi=0$ for all $\chi \bmod\,  q$, we have $\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:

In 1994, Rubinstein and Sarnak~\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \[ \lim_{X\to \infty} \frac{1}{\log X}\int_{2}^X \mathds{1}_{\mathcal{P}(q;a,b)}(u)\frac{\mathrm{d}u}{u}\]
of the set $\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\mathcal{P}(4;3,1)$ is approximately $0.9959\dots$

More recently, a new framework for studying Chebyshev's bias was introduced by Aoki and Koyama~\cite{Ao-Ko}. They observed that since one of the reasons for the existence of the logarithmic density is that the factor $1/u$ in the integral dampens the contribution of large primes, it would be natural to consider the prime counting function \begin{equation}\label{pi_1/2}
 \pi_{1/2}(x;q,a)=\sum_{\substack{p\le x\\ p\equiv a\bmod\, {q}}}\frac{1}{\sqrt{p}}
\end{equation}
and ask whether the natural density of the set $\mathcal{P}_{1/2}(q;a,b):=\{x\ge 2\, :\, \pi_{1/2}(x;q,a)>\pi_{1/2}(x;q,b)\}$ exists. In their framework, and that of subsequent authors~\cite{Ko-Ku, Sheth}, the idea was to avoid using a hypothesis as strong as LI. Therefore, they worked in the context of the Deep Riemann Hypothesis (DRH), a hypothesis stronger than GRH introduced by K. Conrad~\cite{Conrad}, which states that for all non-principal Dirichlet characters modulo $q$ we have \begin{equation}\label{DRH} (\log x)^{m_\chi} \prod_{p\le x}\left(1-\frac{\chi(p)}{p^{1/2}}\right)^{-1}=\ell_\chi+o(1) \quad(x\to \infty) ,\end{equation}
where $\ell_\chi\ne 0$ and $m_\chi:=\text{ord}_{s=1/2}L(s,\chi)$ is the order of vanishing of $L(s,\chi)$ at the central point $s=1/2$. Aoki and Koyama observed that taking the logarithm in~\eqref{DRH}, and using its Taylor expansion, then using Mertens' theorem to estimate $\sum_{p\le x} \chi(p^2)/p$, leads to an estimate of $\sum_{p\le x} \chi(p)/\sqrt{p}$, which allowed them to prove that DRH holds if and only if for all invertible residue classes $a,b$ modulo $q$ one has \begin{equation}\label{DRH-estimate}
 \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b)=-\frac{\text{M}(q;a,b)}{2\varphi(q)} \log \log x+C+o(1) \, ,
\end{equation}
 where $\text{M}(q;a,b)=r(a)-r(b)+2\sum_{\chi \ne \chi _0}(\chi(a)-\chi(b))m_\chi$, $r:(\mathbb Z/q\mathbb Z)^\times\to \Z_{\ge 0}$ defined by $r(a)=\#\{ x\in (\mathbb Z/q\mathbb Z)^\times\, :\, x^2=a\}$ and $C$ is a constant depending on $q,a,b$. In a very recent paper, Suzuki~\cite{Su} studied the weighted Chebyshev-type function $\psi_{1/2}(x):=\sum_{n\le x}\Lambda(n)/\sqrt{n}$ and its variants in arithmetic progressions. The author proved that GRH holds for the non-principal Dirichlet character modulo $4$ if and only if
\[\sqrt{x}\left(\sum_{\substack{p\le x\\ p\equiv 1\bmod\, 4}}\frac{\log p}{\sqrt{p}}\log\left(\frac{x}{p}\right)-\sum_{ \substack{p\le x\\ p\equiv 3 \bmod\, 4}}\frac{\log p}{\sqrt{p}}\log \left(\frac{x}{p}\right) \right)\sim-(\log x)^2 \frac{\sqrt{x}}{4}\quad (x\to \infty)\, .\]

This suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\log p \log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.

In this paper, we show that GRH is sufficient to prove that the equality~\eqref{DRH-estimate} holds for a set of $x\ge 2$ of natural density $1$. More precisely:
\begin{thm}\label{main}
 Let $q\ge 3$ and assume \text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $\varepsilon>0$ and let $a,b$ be distinct invertible residue classes modulo $q$. Then there exists a (unique) constant $C$ depending on $q,\, a,$ and $b$, such that the natural density of the set \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\text{M}(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{\frac{1}{2}-\varepsilon}}\right\}\]
 exists and equals $1$, and the logarithmic density of the set
 \[\left\{x\ge 2\, :\, \left| \pi_{1/2}(x;q,a)-\pi_{1/2}(x;q,b) +\frac{\text{M}(q;a,b)}{2\varphi(q)}\log \log x-C\right|\le \frac{1}{(\log x)^{1-\varepsilon}}\right\}\]
 exists and equals $1$.
\end{thm}
Assuming that $m_\chi=0$ for all $\chi \bmod\, q$, we have $\text{M}(q;a,b)=r(a)-r(b)$. Therefore, if $r(a)<r(b)$, then assuming GRH for all non-principal Dirichlet characters modulo $q$, we have the natural density of the set $\mathcal{P}_{1/2}(q;a,b)$ exists and equals $1$. Thus, Theorem~\ref{main} explains Chebyshev's bias, thanks to these weighted prime counting functions, without the need for any strong additional hypotheses (such as LI or DRH). For historical reasons, we state the particular case $q=4$:

In 1994, Rubinstein and Sarnak~\cite{RS94} established a theoretical framework that enabled them to address the question of Chebyshev's bias for general moduli. They proved, under GRH and a linear independence hypothesis on non-negative imaginary parts of the zeros of Dirichlet $L$-functions (LI), that the logarithmic density \[ \lim_{X\to \infty} \frac{1}{\log X}\int_{2}^X \mathds{1}_{\mathcal{P}(q;a,b)}(u)\frac{\mathrm{d}u}{u}\]
of the set $\mathcal{P}(q;a,b)$ exists. Furthermore, they quantified the bias for $q=4$, proving that the logarithmic density of the set $\mathcal{P}(4;3,1)$ is approximately $0.9959\dots$

This suggests that studying the asymptotic behavior of the difference of prime counting functions of the same type as~\eqref{pi_1/2} requires either hypotheses stronger than GRH (such as DRH in Aoki--Koyama's work), or additional smoothing weights ($\log p \log (x/p)$ in Suzuki's work) which make the contribution of large primes smaller.

Our last main result concerns the mean $\frac{1}{x}\int_2^x(\pi_{1/2}(u;q,a)-\pi_{1/2}(u;q,b))\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\eqref{DRH-estimate}:

\end{proof}
\section{Proofs of the main theorems}
As explained at the beginning of section~\ref{sec:Prel}, we will prove our theorems for a general function $t:(\Z/q\Z)^\times\to \C$ satisfying $\langle t,\chi_0\rangle=0$. Our main theorems are obtained by replacing $t$ with $\mathds{1}_{\{a\}}-\mathds{1}_{\{b\}}$. In what follows, we assume GRH for all Dirichlet characters $\chi$ modulo $q$ satisfying $\langle t,\chi\rangle \ne 0$.
\begin{proof}[Proof of Theorem~\ref{main}]
 We begin with a summation by parts, yielding, for all $x\ge 2$,
\[\pi_{1/2}(x;t)=\frac{\pi(x;t)}{\sqrt{x}}+\frac{1}{2}\int_2^x\frac{\pi(u;t)}{u^{3/2}}\mathrm{d}u\, .\]
Thus, for all $y\ge 1$
\begin{align*}\pi_{1/2}(e^y;t)&=\frac{\pi(e^y;t)}{\exp(y/2)}+\frac{1}{2}\int_{\log 2}^y\frac{\pi(e^u;t)}{\exp(u/2)}\mathrm{d}u\\
&=\frac{1}{y}\left(\Delta(y;t)-\frac{\M(t)}{\varphi(q)} \right)+\frac{1}{2}\int_{\log 2}^y\frac{1}{u}\left(\Delta(u;t)-\frac{\M(t)}{\varphi(q)}\right)\mathrm{d}u\, .
\end{align*}
Hence, by the second estimate of Lemma~\ref{ing}, denoting $C=(\varphi(q)L+\M(t)\log \log 2)/(2\varphi(q))$, we have
\begin{equation}\label{main-form}
 \pi_{1/2}(e^y;t)+\frac{\M(t)}{2\varphi(q)}\log y-C=\frac{1}{y}\left( \Delta(y;t)-\frac{\M(t)}{\varphi(q)}+R(y)\right)
\end{equation}
where $|R(y)|\ll \log y$. Let $Y_0=Y_0(t)$ be a large positive constant such that for all $y\ge Y_0$ we have $|R(y)-\M(t)/\varphi(q)|<\sqrt{y}/2$. Denote by $\m$ the Lebesgue measure on $\R$, fix $\varepsilon>0$, and define \[E:=\left\{ x\ge 2\, :\, \left|\pi_{1/2}(x;t)+\frac{\M(t)}{2\varphi(q)}\log \log x-C\right|>\frac{1}{(\log x)^{\frac{1}{2}-\varepsilon}}\right\}\, .\]
Let $Y>Y_0$, by the triangle inequality and Markov's inequality
\begin{align*}
 \m \left\{ \log 2\le y\le Y\, :\, e^y\in E\right\} &\le Y_0+ \m\left\{ Y_0\le y\le Y\, :\, \left| \Delta(y;t)-\frac{\M(t)}{\varphi(q)}+R(y)\right|>y^{\frac{1}{2}+\varepsilon} \right\}\\
 &\le Y_0+ \m\left\{ Y_0\le y\le Y\, :\, \bigl| \Delta(y;t) \bigr|>\frac{y^{\frac{1}{2}+\varepsilon}}{2}\right\}\\
 &\le Y_0+ 4 \int_{\log 2}^Y \frac{|\Delta(y;t)|^2}{y^{1+2\varepsilon}}\mathrm{d}y\ll1\, .
\end{align*}
We just proved that \[ \int_{\log 2}^\infty \mathds{1}_E(e^y)\mathrm{d}y <\infty \, . \]
Thus, by~\eqref{nat-density}, the natural density of the set $E$ exists and equals $0$. Hence, the natural density of its complement exists and equals $1$. We proceed similarly for the statement regarding the logarithmic density. Define
\[E':=\left\{ x\ge 2\, :\, \left|\pi_{1/2}(x;t)+\frac{\M(t)}{2\varphi(q)}\log \log x-C\right|>\frac{1}{(\log x)^{1-\varepsilon}}\right\}\, .\]
Let $Y_1>\log 2$ be large enough so that for all $y\ge Y_1$ we have $|R(y)-\M(t)/\varphi(q)|<y^\varepsilon/2$. We deduce that
\[\m \left\{ \log 2\le y\le Y\, :\, e^y\in E'\right\}\le Y_1+4\int_{\log 2}^Y \frac{|\Delta(y;t)|^2}{y^{2\varepsilon}}\mathrm{d}y\, .\]
This proves that the logarithmic density of the set $E'$ exists and equals $0$, hence the logarithmic density of its complement exists and equals $1$.
\end{proof}
We now move to Theorem~\ref{GRHalmimpliesDRH}
\begin{proof}[Proof of Theorem~\ref{GRHalmimpliesDRH}]
 Define, for $x\ge 2$, $F(x)=(\log x)^{m_\chi} \prod_{p\le x} \bigl(\,1-\chi(p)/\sqrt{p}\,\bigr)$. Following the proof of~\cite{Ao-Ko}*{Proposition 2.1}, and using the following form of Mertens' theorem \[ \sum_{p\le x} \frac{\chi(p^2)}{p}=\langle \chi,r\rangle \log \log x+c+O\left(\frac{1}{\log x}\right) \, ,\]
 where $c\in \C$ is a constant, we deduce that there exists a constant $C'\in \C$ such that
 \[ \log F(x)= \pi_{1/2}(x,\chi)+\frac{\M(\chi)}{2\varphi(q)}\log \log x+C'+O\left(\frac{1}{\log x}\right)\, .\]
 Assume that $t=\chi$, and let $\varepsilon>0$ and $C$ be the constant given by~\eqref{main-form}. Define $\ell_\chi =\exp(C+C')\ne 0$, and fix $x\ge 2$ sufficiently large satisfying the inequality
 \[ \left|\pi_{1/2}(x,\chi)+\frac{\M(\chi)}{2\varphi(q)}\log \log x-C\right|<\frac{1}{(\log x)^{(1-\varepsilon)/2}}\, . \]
 Using the inequality $|\exp(z)-1|\le 2|z|$ for $|z|<1/2$, we deduce that \begin{align*} |F(x)-\ell_\chi|&=|\ell_\chi|\,\bigl|\exp(\log F(x)-C-C')-1\bigr|\\
 &\le 2|\ell_\chi|\, \left|\pi_{1/2}(x,\chi)+\frac{\M(\chi)}{2\varphi(q)}\log \log x-C+O\left(\frac{1}{\log x}\right)\right|\\
 &< \frac{1}{(\log x)^{\frac{1}{2}-\varepsilon}}\, ,
 \end{align*}
 since $x$ is sufficiently large. Since $\mathcal{E}_\chi$ contains all sufficiently large values of a set of natural density $1$. It follows that the natural density of $\mathcal{E}_\chi$ exists and equals $1$.
\end{proof}
We now prove Theorem~\ref{mean}.
\begin{proof}[Proof of Theorem~\ref{mean}]
 Let $x\ge 2$ and set $Y=\log x$.
 From~\eqref{main-form} we deduce that
 \[\int_{\log 2}^Y e^y\pi_{1/2}(e^y;t)\mathrm{d}y=\int_{\log 2}^Y\left(-\frac{\M(t)}{2\varphi(q)}e^y \log y+Ce^y\right)\mathrm{d}y+\int_{\log 2}^Y\frac{e^y}{y}\left(\Delta(y;t)-\frac{\M(t)}{\varphi(q)}+R(y)\right)\mathrm{d}y\, .\]
 We now use Lemma~\ref{ing} together with the bound $ \int_{\log 2}^Y\tfrac{e^y}{y}\mathrm{d}y\ll e^Y/Y$ to deduce that \[ \left| \int_{\log 2}^Y \frac{e^y \Delta(y;t)}{y}\mathrm{d}y\right|=\left| \frac{e^Y}{Y}G(Y;t)-\int_{\log 2}^Y G(y;t)\frac{ye^y-e^y}{y^2}\mathrm{d}y\right|\ll \log Y \frac{e^Y}{Y}\, .\]
 Since $|R(y)|\ll \log y$, we have
 \[ \left|\int_{\log 2}^Y \frac{e^y}{y}\left(\Delta(y;t)-\frac{\M(t)}{\varphi(q)}+R(y)\right)\mathrm{d}y\right|\ll \log Y\frac{e^Y}{Y}\, .\]
 Using the classical estimate
 \[ \int_{\log 2}^Y e^y \log y\, \mathrm{d}y=e^Y \log Y+O\left(\frac{e^Y}{Y}\right)\, , \]
 we conclude that
 \[\int_2^x\pi_{1/2}(u;t)\mathrm{d}u =\int_{\log 2}^Ye^y\pi_{1/2}(e^y;t) \mathrm{d}y=-\frac{\M(t)}{2\varphi(q)}e^Y\log Y+Ce^Y+O\left(\frac{e^Y \log Y}{Y}\right)\, .\]
 The result follows by substituting $Y=\log x$.
\end{proof}

Our second main result shows that if GRH holds, then the equality~\eqref{DRH} holds for a set of natural density $1$.
\begin{thm}\label{GRHalmimpliesDRH}
    Assume GRH for a non-principal Dirichlet character $\chi$ modulo $q$. Then, there exists $\ell_\chi\ne 0$ such that the natural density of the set
    \begin{equation}\label{Echi}
        \mathcal{E}_\chi:=\left\{x\ge 2\, :\, \left|  (\log x)^{m_\chi} \prod_{p\le x}\left(1-\frac{\chi(p)}{p^{1/2}}\right)^{-1}-\ell_\chi  \right|\le \frac{1}{(\log x)^{1/2 - \varepsilon}} \right\}
    \end{equation}
    exists and equals $1$.
\end{thm}

Our last main result concerns the mean $\frac{1}{x}\int_2^x(\pi_{1/2}(u;q,a)-\pi_{1/2}(u;q,b))\mathrm{d}u$, our method allows us to obtain, under GRH, the exact estimate that one would obtain assuming the DRH estimate~\eqref{DRH-estimate}:

\begin{thm}\label{mean}
    Let $q\ge 3$ and assume \text{GRH} for all non-principal Dirichlet characters modulo $q$. Let $a,b$ be distinct invertible residue classes modulo $q$, we have
    \[\frac{1}{x}\int_{ 2}^x \bigl(\pi_{1/2}(u;q,a)-\pi_{1/2}(u;q,b)\bigr)\mathrm{d}u=-\frac{\text{M}(q;a,b)}{2\varphi(q)}\log \log x+C+O\left(\frac{\log \log x}{\log x}\right)\, .\]
\end{thm}
Although our arguments extend straightforwardly to the context of global fields, we restrict our focus to the classical case of primes in arithmetic progressions for the sake of clarity and to highlight the underlying ideas.
To establish our results, we rely on the explicit formula for prime counting functions. However, a direct application of this formula to the natural auxiliary Chebyshev function $\psi_{1/2}$ associated with the weighted function $\pi_{1/2}$ would be inconclusive: the resulting error term is of order $O(\log x)$, while its main term is not expected to be that large; in fact that main term is expected to be of order $O((\log \log \log x)^2)$ by Montgomery's conjecture~\cite{Montg}. Instead, our strategy is to relate $\pi_{1/2}$ to $\pi$ via summation by parts. We show (Lemma~\ref{ing}) that, thanks to the explicit formula for $\pi$, the integral term arising from this summation yields the main bias term
\[-\frac{\text{M}(q;a,b)}{2\varphi(q)} \log \log x +C + O\left(\frac{\log \log x}{\log x}\right).\]
The remaining non-integral term is essentially $(\pi(x;q,a)-\pi(x;q,b))/\sqrt{x}$. Relying on the fact that the limiting distribution of $y\mapsto y(\pi(e^y;q,a)-\pi(e^y;q,b))/\exp(y/2)$ has a finite variance, we apply Markov's inequality to bound the measure of the set of $x\ge 2$ for which the deviation exceeds $1/(\log x)^{1/2-\varepsilon}$. Since our estimates are naturally formulated on a logarithmic scale, we deduce the existence of the natural density, by showing that the set of large deviations has finite logarithmic measure. Indeed, any Borel set $A\subset (1,\infty)$ satisfying $\int_0^\infty \mathds{1}_A(e^y)\mathrm{d}y<\infty$ has zero natural density. To see this, note that for any $L>1$,
 \begin{equation}\label{nat-density}
    0\le \limsup_{x\to \infty}\frac{1}{x} \int_1^x \mathds{1}_A(u)\mathrm{d}u\le \limsup_{x\to \infty}\int_L^x \mathds{1}_A(u) \frac{\mathrm{d}u}{u}=\int_{\log L}^\infty \mathds{1}_{A}\bigl(e^y\bigr)\mathrm{d}y\, .
\end{equation}
The result follows by letting $L\to \infty$.
\subsection*{Acknowledgments} I would like to thank Florent Jouve and Daniel Fiorilli for their constant encouragement and for their suggestions, which were essential to the development of this paper. I am also grateful to Alexandre Bailleul for his detailed reading and for several important remarks.

To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\log x)$ while the main term “is expected to be of order $O((\log\log\log x)^2)$.”

Instead, “our strategy is to relate $\pi_{1/2}$ to $\pi$ via summation by parts.” They show (Lemma~\ref{ing}) that, using the explicit formula for $\pi$, “the integral term arising from this summation yields the main bias term”
\[
-\frac{\text{M}(q;a,b)}{2\varphi(q)}\log\log x + C + O\left(\frac{\log\log x}{\log x}\right).
\]
The “remaining non-integral term is essentially $(\pi(x;q,a)-\pi(x;q,b))/\sqrt{x}$.” Using that “the limiting distribution of $y\mapsto y(\pi(e^y;q,a)-\pi(e^y;q,b))/\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\ge 2$ for which the deviation exceeds $1/(\log x)^{1/2-\varepsilon}$.”

Finally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\subset(1,\infty)$ with $\int_0^\infty \mathds{1}_A(e^y)\,dy<\infty$ has zero natural density, via the bound
\[
0\le \limsup_{x\to\infty}\frac1x\int_1^x \mathds{1}_A(u)\,du \le \int_{\log L}^\infty \mathds{1}_A(e^y)\,dy
\]
(and “letting $L\to\infty$”).

To establish the results, the authors “rely on the explicit formula for prime counting functions,” but note that a direct application to the auxiliary Chebyshev function $\psi_{1/2}$ “would be inconclusive”: the error term would be $O(\log x)$ while the main term “is expected to be of order $O((\log\log\log x)^2)$.”

Instead, “our strategy is to relate $\pi_{1/2}$ to $\pi$ via summation by parts.” They show the following lemma.

\begin{lem}\label{ing}
    If \text{GRH} holds for all Dirichlet characters $\chi$ modulo $q$ satisfying $\langle t,\chi\rangle \ne 0$, then:\begin{enumerate}
        \item Uniformly for $Y\ge 1$: \[\bigl| G(Y;t) \bigr|\ll_t \log Y\,  .\]
        \item There exists $L\in \C$ such that uniformly for $Y \ge1$: \[ \int_{\log 2}^Y\frac{\Delta(y;t)}{y}\mathrm{d}y=L+O_t\left(\frac{\log Y}{Y}\right)\, .\]
        \item For all $\varepsilon>0 $, we have uniformly for $Y\ge 1$: \[\int_{\log 2}^Y \frac{|\Delta(y;t)|^2}{y^{1+\varepsilon}}\mathrm{d}y \ll_{\varepsilon,t} 1\, .\]
        \item For all $\varepsilon>0$, we have \[\lim_{Y\to \infty} \frac{1}{Y} \int_{\log 2}^Y\frac{|\Delta(u;t)|^2}{u^\varepsilon}\mathrm{d}u=0\, .\]
    \end{enumerate}
\end{lem}

Using the explicit formula for $\pi$, they then argue that “the integral term arising from this summation yields the main bias term”
\[
-\frac{\text{M}(q;a,b)}{2\varphi(q)}\log\log x + C + O\left(\frac{\log\log x}{\log x}\right).
\]
The “remaining non-integral term is essentially $(\pi(x;q,a)-\pi(x;q,b))/\sqrt{x}$.” Using that “the limiting distribution of $y\mapsto y(\pi(e^y;q,a)-\pi(e^y;q,b))/\exp(y/2)$ has a finite variance,” they “apply Markov's inequality to bound the measure of the set of $x\ge 2$ for which the deviation exceeds $1/(\log x)^{1/2-\varepsilon}$.”

Finally, since the estimates are “naturally formulated on a logarithmic scale,” they deduce existence of natural density by showing “the set of large deviations has finite logarithmic measure.” They use the principle that any Borel set $A\subset(1,\infty)$ with $\int_0^\infty \mathds{1}_A(e^y)\,dy<\infty$ has zero natural density, via the bound
\[
0\le \limsup_{x\to\infty}\frac1x\int_1^x \mathds{1}_A(u)\,du \le \int_{\log L}^\infty \mathds{1}_A(e^y)\,dy
\]
(and “letting $L\to\infty$”).

Question:
Let
\[
\pi_{1/2}(x;4,a)=\sum_{\substack{p\le x\\ p\equiv a\!\!\pmod 4}}\frac{1}{\sqrt p}
\]
for invertible residue classes $a\bmod 4$, where the sum is over primes $p\le x$. Assume the Generalized Riemann Hypothesis for the non-principal Dirichlet character modulo $4$. Consider
\[
\mathcal P_{1/2}(4;3,1)=\{x\ge 2:\ \pi_{1/2}(x;4,3)>\pi_{1/2}(x;4,1)\}.
\]
Which statement holds about this set, where its natural density means
\[
\lim_{X\to\infty}\frac{1}{X}\int_2^X \mathbf 1_{\mathcal P_{1/2}(4;3,1)}(u)\,du
\]
when the limit exists?

The natural density of $\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$.

Choices:
B. The natural density of $\mathcal P_{1/2}(4;3,1)$ exists and is equal to $\tfrac12$.
C. The upper natural density of $\mathcal P_{1/2}(4;3,1)$ is equal to $1$.
D. The logarithmic density of $\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$, but the natural density need not exist.
E. There exists a subset $S\subseteq \mathcal P_{1/2}(4;3,1)$ with natural density $1$.

main bias forces eventual dominance toward residue class $3$ rather than symmetry

existence of the full natural density limit

upgrade from logarithmic-scale control to natural density via finite logarithmic measure of exceptions

replacing density of the whole set by density of a dense subset

The stem defines the objects carefully and asks which existence statement holds, but it does not explicitly reveal or strongly hint at the correct density conclusion.

The item is essentially asking for the exact theorem-level conclusion about the specific set \(\mathcal{P}_{1/2}(4;3,1)\). This makes it close to a direct restatement rather than a problem requiring selection among independently derived consequences.

There is some reasoning pressure because the options distinguish natural density, upper natural density, logarithmic density, and strict versus non-strict inequalities. However, the question mainly tests recall/recognition of the precise result rather than substantial derivation.

The distractors are mathematically plausible and target realistic confusions: symmetry suggesting density \(1/2\), weakening to upper density, mixing logarithmic and natural density, and incorrectly upgrading a density-one non-strict inequality statement to strict inequality on the original set.

A mathematically careful MCQ with strong distractors and no answer leakage, but it is largely a theorem-recall item and only moderately tests generative reasoning.


Question:
Let
\[
\pi_{1/2}(x;4,a)=\sum_{\substack{p\le x\\ p\equiv a\!\!\pmod 4}}\frac{1}{\sqrt p}
\]
for invertible residue classes $a\bmod 4$, where the sum is over primes $p\le x$. Assume the Generalized Riemann Hypothesis for the non-principal Dirichlet character modulo $4$. Consider
\[
\mathcal P_{1/2}(4;3,1)=\{x\ge 2:\ \pi_{1/2}(x;4,3)>\pi_{1/2}(x;4,1)\}.
\]
Which statement holds about this set, where its natural density means
\[
\lim_{X\to\infty}\frac{1}{X}\int_2^X \mathbf 1_{\mathcal P_{1/2}(4;3,1)}(u)\,du
\]
when the limit exists?

Choices:
B. The natural density of $\mathcal P_{1/2}(4;3,1)$ exists and is equal to $\tfrac12$.
C. The upper natural density of $\mathcal P_{1/2}(4;3,1)$ is equal to $1$.
D. The logarithmic density of $\mathcal P_{1/2}(4;3,1)$ exists and is equal to $1$, but the natural density need not exist.
E. There exists a subset $S\subseteq \mathcal P_{1/2}(4;3,1)$ with natural density $1$.

The stem does not state or strongly hint at the correct conclusion; it only defines the weighted prime race and asks which density statement is true under GRH.

This is essentially a direct specialization of a theorem-sized statement: once the relevant result is known, the correct option is just the theorem instantiated at modulus 4 and weight 1/2.

There is some pressure to distinguish between natural density, upper natural density, and logarithmic density, and to choose the strongest valid conclusion. However, the item mainly tests recognition/application of a known result rather than genuine multi-step reasoning.

The distractors are plausible and mathematically meaningful: symmetry-based confusion (1/2), a weaker true statement, density-type confusion, and a dense-subset wildcard. They reflect common failure modes and are clearly distinct.

A mathematically coherent MCQ with strong distractors and no obvious answer leakage, but it is largely a theorem-recall/specialization item rather than a genuinely generative reasoning question.


\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 \(u\).

The stem does not explicitly state the conclusion or reproduce the exact final estimate; it only gives the hypotheses and asks which bound follows. There is no direct answer leakage from the stem itself.

This is essentially a theorem-recall item: the question asks for the exact quantitative local boundedness estimate under a list of hypotheses. The correct option is basically the theorem statement rather than a derived or competing conclusion.

Some reasoning is needed to distinguish subtle variants involving the exponent, tail scaling, the small-scale restriction, and the inverse relation with g_{B_r}. However, the task is still mostly recognition of the exact statement, not genuine generative mathematical reasoning.

The distractors are mathematically plausible and target realistic errors: wrong scaling exponent, omission of part of the theorem, incorrect dependence on the Hölder data/smallness scale, and confusion between G_{B_r}, g_{B_r}, and their inverses.

A solid recall/recognition MCQ with strong distractors, but it is largely a direct theorem-statement identification problem rather than a genuinely generative reasoning task.