\label{thm1}
    Let $\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\lambda\gg1$. We have
    \begin{equation}\label{i2}
        \| \psi \|_{L^6(X)} \lesssim_\varepsilon
        \lambda^{\frac{1}{6}-\frac{1}{144}+\varepsilon}.
    \end{equation}

\begin{align}\label{eq: HK L4}
    \| \psi \|_{L^4(X)}\lesssim_\epsilon \la^{\frac{3}{152}+\epsilon}.
\end{align}

\begin{align}\label{eq: Marshall L4}
    \| \psi \|_{L^4(X)}\lesssim_\epsilon \la^{\frac{1}{8}-\frac{1}{56}+\epsilon}.
\end{align}

\begin{theorem}\label{thm1}
    Let $\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\lambda\gg1$. We have
    \begin{equation}\label{i2}
        \| \psi \|_{L^6(X)} \lesssim_\epsilon
        \lambda^{\frac{1}{6}-\frac{1}{144}+\epsilon}.
    \end{equation}
\end{theorem}

\begin{proposition}\label{prop: bound for I_A}
    Let $\phi\in C^\infty(X)$ be such that $\|\phi\|_{L^{6/5}(X)}=1$.
     \begin{enumerate}
         \item For any $\gamma\in G$, $I_\sA(\lambda,\phi,\gamma)\lesssim_\epsilon\lambda^{1/3+\epsilon}$.
         \item Fix $0<\epsilon_0\ll\delta_1$. If $d(\gamma,e)\lesssim 1$ and $d(g_0^{-1}\gamma g_0,A)\geq \lambda^{-\delta_1+2\epsilon_0}$, then $I_\sA(\lambda,\phi,\gamma)\lesssim_{\epsilon_0,N} \lambda^{-N}$ for any $N>0$.
     \end{enumerate}
\end{proposition}

\begin{align}\label{eq: HK L6}
    \| \psi \|_{L^6(X)}\lesssim_\epsilon \la^{\frac{26}{171}+\epsilon}=\la^{\frac{1}{6}-\frac{5}{342}+\epsilon}.
\end{align}

\begin{equation}\label{qnusymbol}
A_\nu^\theta(x,\xi)=\phi(x) \, a_j^\theta(x,\xi) \, \tilde\beta\bigl(p(x,\xi)/\la\bigr),
\quad \nu =\theta j\in \theta \cdot {\mathbb Z}^{2}.
\end{equation}

\begin{equation}\label{1.2}
(\Delta_g+\la^2_j )e_{\la_j}=0, \quad
\text{and } \quad \int_M |e_{\la_j}(x)|^2 \, dx=1.
\end{equation}

\begin{proposition}\label{prop: amplification inequality}
    For any $\phi\in C^\infty(X)$, we have
    \[
    \left|\langle \sA\cT\psi,\phi\rangle_X\right|^2\leq\sum_{m,n\leq N} |\alpha_m  \alpha_n | \sum_{d|(m,n)} \frac{d}{\sqrt{mn}}\sum_{\gamma\in R(mn/d^2)} |I_\sA(\la,\phi,\gamma)|
    \]
\end{proposition}

\begin{align}\label{eq: Marshall geodesic bound}
    \sup_{\gamma\in\Pi} \| \psi|_\gamma \|_{L^2(\gamma)}\lesssim_\epsilon \lambda^{\frac{3}{14}+\epsilon}.
\end{align}

The purpose of this paper is to obtain improved $L^q$-norm estimates for eigenfunctions on an arithmetic hyperbolic surface for the critical exponent $q=6$, which in turn also yield improved estimates for all $q>2$ by interpolation.

Let $(M,g)$ be a  compact $n$-dimensional Riemannian manifold without boundary,
$\Delta_g$ be the
Laplace-Beltrami operator associated with the metric $g$ on $M$. We let $0=\la_0^2<\lambda^2_1\le \lambda^2_2\le \cdots$ denote the
eigenvalues labeled with respect to the multiplicity of
${-\Delta_g}$
and $e_{\la_j}$ the associated $L^2$-normalized eigenfunctions, that is,
\begin{equation}\label{1.2}
(\Delta_g+\lambda^2_j )e_{\la_j}=0, \quad
\text{and } \quad \int_M |e_{\la_j}(x)|^2 \, dx=1.
\end{equation}

In Sogge~\cite{sogge881} (see also Avakumovi\'c \cite{Avakumovic} and Levitan \cite{Levitan} when $p = \infty$), the universal bound for $q\ge2$
\begin{equation}\label{r1}
\|e_\lambda\|_{L^q(M)}\lesssim \lambda^{\delta(q,n)}\end{equation}
was obtained. Here $\delta(q,n)$ is given by
\begin{equation*}
            \delta(q,n) = \begin{dcases}
                \frac{n-1}{4} - \frac{n-1}{2q} \;&\text{ for }\;2\leq q\leq q_c,\\
                \frac{n-1}{2} - \frac{n}{q} \;\;\;\;&\text{ for }\;q_c\leq q\leq\infty,
            \end{dcases}
            \quad\text{ and }\quad q_c=\frac{2(n+1)}{n-1}.
\end{equation*}
The result also applies to quasimodes that are linear combinations of eigenfunctions whose eigenvalues lie in the unit interval $[\lambda,\lambda+1]$.
When $M$ is a compact locally symmetric space of higher rank, besides the Laplace-Beltrami operator, one can consider eigenfunctions of the full ring of differential operators on $M$. The general local bound was obtained by Sarnak \cite{sarnak2004letter} (for $q=\infty$) and Marshall \cite{Mar16HigerRank} (for any $q\ge2$ with some log losses at the corresponding critical exponents), which extends and improves the bound \eqref{r1}.

The estimate \eqref{r1} is sharp on the standard round sphere $\mathbb{S}^n$ by taking $e_\lambda$ to be the zonal functions or the highest weight spherical harmonics, which are point-focusing and geodesic-focusing eigenfunctions, respectively. Under additional geometric assumptions--such as nonpositive sectional curvature or on flat tori--one can improve \eqref{r1}. See e.g., the work of Bourgain-Demeter \cite[Section 2.3]{BoDe}, as well as  Germain-Myerson \cite{germain2022bounds}, and the reference therein for related results on the torus. On compact manifolds with non-positive sectional curvature,
$(\log\lambda)^{-\frac12}$ improvements to \eqref{r1} was obtained in B\'erard \cite{Berard} for $q=\infty$, and by Hassell and Tacy \cite{HassellTacy} for $q>q_c=\tfrac{2(n+1)}{n-1}$. The same improvement also holds, more generally, for spectral projection operators onto the shorter interval $[\lambda,\lambda+(\log\lambda)^{-1}]$. These results were further generalized under weaker dynamical assumptions in the recent work of Canzani-Galkowski \cite{CGGrowth}.

In the other direction, for $q\le q_c$ similar improvements were obtained in Huang--Sogge \cite{huang2024curvature}. More generally, they proved sharp quasimode estimates for logarithmic quasimodes, which were also used to characterize compact manifolds with constant curvature. See also Blair and Sogge \cite{blair2018concerning,SBLog} and Blair, Huang and Sogge \cite{blair2024improved} for related earlier work in this direction.

\subsection{\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}
The main result of the paper concerns improvements of \eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\mathbb{H}$ be the upper half plane and $X=\Gamma\backslash\mathbb{H}$ an arithmetic hyperbolic surface. Here $\Gamma\subset \operatorname{SL}(2,\mathbb{R})$ is taken to be the norm one units of a maximal order of an
indefinite quaternion division algebra over $\mathbb{Q}$ (see Section \ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\int_X|\psi(x)|^2dx=1$ and $\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\Delta_g$ and Hecke operators on $X$. Let $\lambda$ be the spectral parameter of $\psi$, so that $\Delta \psi+(\frac14+\lambda^2)\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\lambda$ is real and $\lambda\gg1$. Iwaniec and Sarnak \cite{IS95} made a Lindel\"of hypothesis type conjecture that $\|\psi\|_{L^\infty(X)}\lesssim_\varepsilon\lambda^{\varepsilon}$, which is strong and widely open. The conjecture implies that $\|\psi\|_{L^q(X)}\lesssim_\varepsilon\lambda^{\varepsilon}$ for any $q>2$.

For these $X$ and $\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \cite{IS95}. They showed that
\begin{align}\label{eq: IS bound}
    \|\psi\|_{L^\infty(X)} \lesssim_\varepsilon\lambda^{\frac{5}{12}+\varepsilon},
\end{align}
which is a $\lambda^{\frac1{12}}$ improvement over the uniform bound \eqref{r1} for $n=2$ and $q=\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \eqref{eq: IS bound} implies an improvement for  $ \| \psi \|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\infty$-norms of Hecke-Maass forms on other groups. For instance, see \cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.

For $q<6$, Marshall \cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\psi$ and obtained a similar $\lambda$-power improvement over the uniform result of Burq,  G{\'e}rard and Tzvetkov \cite{BGTrestr}. We let $\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound
\begin{align}\label{eq: Marshall geodesic bound}
    \sup_{\gamma\in\varPi} \| \psi|_\gamma \|_{L^2(\gamma)}\lesssim_\varepsilon \lambda^{\frac{3}{14}+\varepsilon}.
\end{align}
He reduced $L^2$-norm bounds for $\psi$ along $\gamma\in\varPi$ to bounds for various
Fourier coefficients along $\gamma$.
His improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.
Using a result of Blair and Sogge \cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \eqref{eq: Marshall geodesic bound} implies that
\begin{align}\label{eq: Marshall L4}
    \| \psi \|_{L^4(X)}\lesssim_\varepsilon \lambda^{\frac{1}{8}-\frac{1}{56}+\varepsilon}.
\end{align}
Moreover, \eqref{eq: Marshall L4} also implies an improvement over \eqref{r1} for all $2<q<6$ by interpolation. See \cite{hou2024restrictions,hou2025kakeya} for the extension of this method to improve bounds for Kakeya-Nikodym norms on other groups of rank one. For the higher rank group $\operatorname{SL}(3,\mathbb{R})$, \eqref{eq: Marshall geodesic bound} was extended by Marshall to $L^2$ maximal flat restrictions of Hecke-Maass forms. It would be interesting to see whether this result could be used to prove a power saving for the global $L^p$-norms of $\operatorname{SL}(3,\mathbb{R})$ Hecke Maass forms for small $p$ as in Blair-Sogge \cite{blair2015refined}.

It would be interesting to extend the above results and obtain a power saving over \eqref{r1} for the critical exponent $q=6$. A much stronger $L^4$-norm bound than \eqref{eq: Marshall L4} was obtained by Humphries and Khan \cite[Theorem 1.4]{HK25}:
\begin{align}\label{eq: HK L4}
    \| \psi \|_{L^4(X)}\lesssim_\varepsilon \lambda^{\frac{3}{152}+\varepsilon}.
\end{align}
They applied Parseval’s identity to $|\psi|^2$ and used the Watson–Ichino triple product formula to reduce bounds for $\| \psi \|_{L^4(X)}$ to bounds for certain moments of $L$-functions. By interpolating with the $L^\infty$-norm bound \eqref{eq: IS bound} of Iwaniec-Sarnak, Humphries and Khan got
\begin{align}\label{eq: HK L6}
    \| \psi \|_{L^6(X)}\lesssim_\varepsilon \lambda^{\frac{26}{171}+\varepsilon}=\lambda^{\frac{1}{6}-\frac{5}{342}+\varepsilon}.
\end{align}

On the noncompact quotient $\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, for a Hecke-Maass cusp form $\psi$ whose spectral parameter is $\lambda$, both $L^\infty$-norm bound \eqref{eq: IS bound} and $L^4$-norm bound \eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \eqref{eq: HK L6}. In \cite{ki20234}, Ki showed an essentially sharp upper bound $\|\psi\|_{L^4(\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H})}\lesssim_\varepsilon\lambda^{\varepsilon}$, by using the
Fourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\leq4$, and therefore implies a better $L^6$-norm bound
\begin{align*}
    \| \psi \|_{L^6(\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H})}\lesssim_\varepsilon \lambda^{\frac{1}{6}-\frac{1}{36}+\varepsilon}.
\end{align*}

The main result of the paper is the following sublocal $L^6$-norm bound in the \textit{compact} setting, which is weaker than \eqref{eq: HK L6} but is based on a completely different approach.

\subsection{\texorpdfstring{$L^q$}{Lq}-norm bounds for Hecke-Maass forms}
The main result of the paper concerns improvements of \eqref{r1} for arithmetic eigenfunctions on hyperbolic surfaces. We let $\mathbb{H}$ be the upper half plane and $X=\Gamma\backslash\mathbb{H}$ an arithmetic hyperbolic surface. Here $\Gamma\subset \operatorname{SL}(2,\mathbb{R})$ is taken to be the norm one units of a maximal order of an
indefinite quaternion division algebra over $\mathbb{Q}$ (see Section \ref{sec: arith hyper sur} for the precise definitions) so that $X$ is compact. Let $\psi$ be an $L^2$-normalized Hecke–Maass form on $X$. Namely, $\int_X|\psi(x)|^2dx=1$ and $\psi$ is a joint eigenfunction of the Laplace-Beltrami operator $\Delta_g$ and Hecke operators on $X$. Let $\lambda$ be the spectral parameter of $\psi$, so that $\Delta \psi+(\frac14+\lambda^2)\psi=0$. As we are considering large-eigenvalue asymptotics, we will also assume that $\lambda$ is real and $\lambda\gg1$. Iwaniec and Sarnak \cite{IS95} made a Lindel\"of hypothesis type conjecture that $\|\psi\|_{L^\infty(X)}\lesssim_\varepsilon\lambda^{\varepsilon}$, which is strong and widely open. The conjecture implies that $\|\psi\|_{L^q(X)}\lesssim_\varepsilon\lambda^{\varepsilon}$ for any $q>2$.

For these $X$ and $\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \cite{IS95}. They showed that
\begin{align}\label{eq: IS bound}
 \|\psi\|_{L^\infty(X)} \lesssim_\varepsilon\lambda^{\frac{5}{12}+\varepsilon},
\end{align}
which is a $\lambda^{\frac1{12}}$ improvement over the uniform bound \eqref{r1} for $n=2$ and $q=\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \eqref{eq: IS bound} implies an improvement for $ \| \psi \|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\infty$-norms of Hecke-Maass forms on other groups. For instance, see \cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.

For $q<6$, Marshall \cite{Mar16} studied the related $L^2$ geodesic restriction problem for the Hecke-Maass form $\psi$ and obtained a similar $\lambda$-power improvement over the uniform result of Burq, G{\'e}rard and Tzvetkov \cite{BGTrestr}. We let $\varPi$ denote the space of all unit-length geodesic segments on $X$. Marshall obtained the uniform bound
\begin{align}\label{eq: Marshall geodesic bound}
 \sup_{\gamma\in\varPi} \| \psi|_\gamma \|_{L^2(\gamma)}\lesssim_\varepsilon \lambda^{\frac{3}{14}+\varepsilon}.
\end{align}
He reduced $L^2$-norm bounds for $\psi$ along $\gamma\in\varPi$ to bounds for various
Fourier coefficients along $\gamma$.
His improvements over the local bounds extend the technique of arithmetic amplification developed by Iwaniec-Sarnak.
Using a result of Blair and Sogge \cite[Theorem 1.1]{blair2015refined}, the geodesic restriction bound \eqref{eq: Marshall geodesic bound} implies that
\begin{align}\label{eq: Marshall L4}
 \| \psi \|_{L^4(X)}\lesssim_\varepsilon \lambda^{\frac{1}{8}-\frac{1}{56}+\varepsilon}.
\end{align}
Moreover, \eqref{eq: Marshall L4} also implies an improvement over \eqref{r1} for all $2<q<6$ by interpolation. See \cite{hou2024restrictions,hou2025kakeya} for the extension of this method to improve bounds for Kakeya-Nikodym norms on other groups of rank one. For the higher rank group $\operatorname{SL}(3,\mathbb{R})$, \eqref{eq: Marshall geodesic bound} was extended by Marshall to $L^2$ maximal flat restrictions of Hecke-Maass forms. It would be interesting to see whether this result could be used to prove a power saving for the global $L^p$-norms of $\operatorname{SL}(3,\mathbb{R})$ Hecke Maass forms for small $p$ as in Blair-Sogge \cite{blair2015refined}.

It would be interesting to extend the above results and obtain a power saving over \eqref{r1} for the critical exponent $q=6$. A much stronger $L^4$-norm bound than \eqref{eq: Marshall L4} was obtained by Humphries and Khan \cite[Theorem 1.4]{HK25}:
\begin{align}\label{eq: HK L4}
 \| \psi \|_{L^4(X)}\lesssim_\varepsilon \lambda^{\frac{3}{152}+\varepsilon}.
\end{align}
They applied Parseval’s identity to $|\psi|^2$ and used the Watson–Ichino triple product formula to reduce bounds for $\| \psi \|_{L^4(X)}$ to bounds for certain moments of $L$-functions. By interpolating with the $L^\infty$-norm bound \eqref{eq: IS bound} of Iwaniec-Sarnak, Humphries and Khan got
\begin{align}\label{eq: HK L6}
 \| \psi \|_{L^6(X)}\lesssim_\varepsilon \lambda^{\frac{26}{171}+\varepsilon}=\lambda^{\frac{1}{6}-\frac{5}{342}+\varepsilon}.
\end{align}

On the noncompact quotient $\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H}$, for a Hecke-Maass cusp form $\psi$ whose spectral parameter is $\lambda$, both $L^\infty$-norm bound \eqref{eq: IS bound} and $L^4$-norm bound \eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \eqref{eq: HK L6}. In \cite{ki20234}, Ki showed an essentially sharp upper bound $\|\psi\|_{L^4(\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H})}\lesssim_\varepsilon\lambda^{\varepsilon}$, by using the
Fourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\leq4$, and therefore implies a better $L^6$-norm bound
\begin{align*}
 \| \psi \|_{L^6(\operatorname{SL}(2,\mathbb{Z})\backslash\mathbb{H})}\lesssim_\varepsilon \lambda^{\frac{1}{6}-\frac{1}{36}+\varepsilon}.
\end{align*}

The main result of the paper is the following sublocal $L^6$-norm bound in the \textit{compact} setting, which is weaker than \eqref{eq: HK L6} but is based on a completely different approach.

For these $X$ and $\psi$, the first power-saving result was obtained by Iwaniec and Sarnak \cite{IS95}. They showed that
\begin{align}\label{eq: IS bound}
 \|\psi\|_{L^\infty(X)} \lesssim_\epsilon\lambda^{\frac{5}{12}+\epsilon},
\end{align}
which is a $\la^{\frac1{12}}$ improvement over the uniform bound \eqref{r1} for $n=2$ and $q=\infty$. In two-dimensional cases, the critical exponent is $q_c=6$, by interpolation, \eqref{eq: IS bound} implies an improvement for $ \| \psi \|_{L^q(X)}$ for all $q>6$. Their proof uses the technique known as arithmetic amplification. This approach has since been used by many authors to bound $L^\infty$-norms of Hecke-Maass forms on other groups. For instance, see \cite{BM13,Mar14,BHM16,BM16,BP16,BHM20} for results in the spectral aspect.

On the noncompact quotient $\SL(2,\BZ)\backslash\BH$, for a Hecke-Maass cusp form $\psi$ whose spectral parameter is $\la$, both $L^\infty$-norm bound \eqref{eq: IS bound} and $L^4$-norm bound \eqref{eq: HK L4} hold as well, so one still has the same $L^6$-norm bound \eqref{eq: HK L6}. In \cite{ki20234}, Ki showed an essentially sharp upper bound $\|\psi\|_{L^4(\SL(2,\BZ)\backslash\BH)}\lesssim_\epsilon\la^{\epsilon}$, by using the
Fourier-Whittaker expansions of Hecke-Maass cusp forms. This result justifies Iwaniec-Sarnak's conjecture for $2<q\leq4$, and therefore implies a better $L^6$-norm bound
\begin{align*}
 \| \psi \|_{L^6(\SL(2,\BZ)\backslash\BH)}\lesssim_\epsilon \la^{\frac{1}{6}-\frac{1}{36}+\epsilon}.
\end{align*}

\subsection{Proof strategy}
A main step in the proof of Theorem~\ref{thm1} is the following.
\begin{theorem}\label{thm2}
 Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\la$ be an $L^2$ normalized eigenfunction of $-\Delta_g$ with eigenvalue $\la^2$, satisfying \eqref{1.2} for some $\la\gg1$. Set $\theta_0=\la^{-\delta_1}$ with $0<\delta_1< \frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in \eqref{qnusymbol}. Assume that
\begin{equation}\label{i3}
 \sup_\nu\|A_\nu^{\theta_0} e_\la\|_{L^6(M)}\lesssim_\e \la^{\frac16-a{\delta_1}+\e},
 \end{equation}
 and
 \begin{equation}\label{i4}
 \|e_\la\|_{L^\infty(M)}\lesssim_\e \la^{\frac12-\delta_2+\e}
 \end{equation}
for some constant $a>0$ and $0<\delta_2<\frac{1}{2}$. Then
 \begin{equation}\label{i5}
 \|e_\la\|_{L^6(M)}\lesssim_\e \la^{\frac16-\frac23a{\delta_1}+\e}+\la^{\frac23(-\frac14+\frac53\delta_1)}+\la^{\frac16(1+\delta_1-2\delta_2)+\e}.
 \end{equation}
\end{theorem}
For each fixed $\nu$, the operator $A_\nu^{\theta_0}$ is microlocally supported in a $\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \ref{sec: microlocal decomposition}. The proof of Theorem~\ref{thm2} follows a strategy similar to that of \cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\infty$ estimate \eqref{i4}. We will prove Theorem~\ref{thm2} in Section \ref{sec: 2}.
The idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \cite{blair2024improved}.

\begin{theorem}\label{thm microlocal KN}
 Let $\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\lambda\gg1$. Set $\theta_0=\la^{-\delta_1}$ with $0<\delta_1<\frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in \eqref{qnusymbol}. We have
 \begin{equation*}
 \sup_\nu\|A_\nu^{\theta_0} \psi\|_{L^6(X)}\lesssim_\e \la^{\frac16-\frac{\delta_1}{12}+\e}.
 \end{equation*}
\end{theorem}

We let $\Delta_g$ be the Laplace-Beltrami operator on $\BH$ and $X$, which is induced from the standard hyperbolic Riemannian metric. Let $\psi\in L^2(X)$ be a Hecke-Maass form that is an eigenfunction of $\Delta_g$ and the operators $T_n$ with $(n,q)=1$. We let $\lambda(n)$ be the Hecke eigenvalues of $\psi$ and $\lambda$ be its spectral parameter, so that
\begin{equation}\label{def}
 \begin{aligned}
 &T_n\psi=\lambda(n)\psi,\\
 &\Delta_g\psi+(\frac{1}{4}+\lambda^2)\psi=0.
 \end{aligned}
\end{equation}
We assume that $\| \psi \|_{L^2(X)}=1$ with respect to the hyperbolic volume on $X$ and assume that $\lambda\gg1$. Note that because $\Delta_g$ and $T_n$ with $(n,q)=1$ are self-adjoint, we may assume that $\psi$ is real-valued. For functions on $X$, we will also think of them as functions on $G$ that are left $\Gamma$-invariant and right $K$-invariant.

\begin{proof}[Proof of Theorem \ref{thm microlocal KN}]
 By Lemma \ref{lem 3.10}, we may bound $\| \sA\psi\|_{L^6(X)}$ instead, and moreover by duality, it suffices to bound $\langle\sA\psi,\phi\rangle_X$ uniformly for any $\phi\in C^\infty(X)$ with $\|\phi\|_{L^{6/5}(X)}=1$.
 By Proposition \ref{prop: amplification inequality}, we have
 \begin{align}\label{eq: amplification inequality}
 \left|\langle \sA\cT\psi,\phi\rangle_X\right|^2\leq\sum_{m,n\leq N} |\alpha_m \alpha_n | \sum_{d|(m,n)} \frac{d}{\sqrt{mn}}\sum_{\gamma\in R(mn/d^2)} |I_\sA(\la,\phi,\gamma )|.
 \end{align}
 Fix $0<\epsilon_0\ll\delta_1$. By assuming the support of $k_\la$ and $\Omega$ are sufficiently small, and by Proposition \ref{prop: bound for I_A}, we only need to consider the terms in \eqref{eq: amplification inequality} with $d(g_0^{-1}\gamma g_0,e)\leq1$ and $d(g_0^{-1}\gamma g_0,A)\leq\lambda^{-\delta_1+2\epsilon_0}$. Lemma \ref{lem: Hecke return} gives
 \[
 M(g_0,n,\lambda^{-\delta_1 +2\epsilon_0})\ll_\epsilon n^\epsilon \lambda^{\epsilon_0+\epsilon}(n\lambda^{-\delta_1/2}+1),
 \]
 and so by the uniform estimate $|I_\sA(\la,\phi,\gamma )|\lesssim_\epsilon \la^{1/3+\epsilon}$ from Proposition \ref{prop: bound for I_A} we have
 \begin{align*}
 &\sum_{m,n\leq N} |\alpha_m\alpha_n| \sum_{d|(m,n)} \frac{d}{\sqrt{mn}}\sum_{\gamma\in R(mn/d^2)}\left| I_{\sA}(\lambda, \phi, \gamma ) \right|\\
 \lesssim& \sum_{m,n\leq N} |\alpha_m\alpha_n| \sum_{d|(m,n)} \frac{d}{\sqrt{mn}}\lambda^{1/3+\e} M(g_0,\frac{mn}{d^2},\lambda^{-\delta_1+2\epsilon_0})+O_{\epsilon_0,A}(\lambda^{-A})\\
 \lesssim&_\epsilon N^\epsilon\lambda^{\epsilon_0+\epsilon}\sum_{m,n\leq N} |\alpha_m\alpha_n| \sum_{d|(m,n)} \left(\frac{\sqrt{mn}}{d}\lambda^{1/3-\delta_1/2}+\frac{d}{\sqrt{mn}}\lambda^{1/3}\right) +O_{\epsilon_0,A}(\lambda^{-A}).
 \end{align*}
 As in \cite[p.310]{IS95}, we have
 \begin{align*}
 \sum_{m,n\leq N} \sum_{d|(m,n)}|\alpha_m\alpha_n|\frac{\sqrt{mn}}{d}\lesssim_\epsilon N^{1+\epsilon} \left( \sum_{n\leq N} |\alpha_n|\right)^2
 \end{align*}
 and
 \begin{align*}
 \sum_{m,n\leq N} \sum_{d|(m,n)}|\alpha_m\alpha_n|\frac{d}{\sqrt{mn}}&=\sum_{\substack{m,n\leq N\\(m,n)=1}} \sum_{ml,nl\leq N}\sum_{d|l}|\alpha_{ml}\alpha_{nl}|\frac{d}{l\sqrt{mn}}\\
 &\lesssim_\epsilon N^\epsilon\sum_{ml,nl\leq N}\left( \frac{|\alpha_{ml}|^2}{n} + \frac{|\alpha_{nl}|^2}{m} \right)\\
 &\lesssim_\epsilon N^\epsilon \sum_{n\leq N} |\alpha_n|^2.
 \end{align*}
 Hence from \eqref{eq: amplification inequality} we have
 \begin{align*}
 \left| \langle \sA\cT\psi,\phi\rangle_X \right|^2 \lesssim&_\epsilon N^\epsilon\lambda^{\epsilon_0+\epsilon}\left(N\lambda^{1/3-\delta_1/2}\left( \sum_{n\leq N} |\alpha_n|\right)^2+\lambda^{1/3}\sum_{n\leq N} |\alpha_n|^2\right) +O_{\epsilon_0,A}(\lambda^{-A}).
 \end{align*}
 Combining this with \eqref{eq: moment for alpha} and \eqref{eq: lower bound for amplifier eigenvalue} and choosing $\epsilon_0$ small gives
 \begin{align*}
 N^{1-\epsilon}\left|\langle \sA\psi,\phi\rangle_X \right|^2\lesssim&_\epsilon N^\epsilon\lambda^{\epsilon}\left(N^2\lambda^{1/3-\delta_1/2}+N^{1/2}\lambda^{1/3}\right)+O_{\epsilon,A}(\lambda^{-A}),
 \end{align*}
 which completes the proof by choosing $N=\lambda^{\delta_1/3}$.
\end{proof}

\subsection{Proof strategy}
A main step in the proof of Theorem~\ref{thm1} is the following.
\begin{theorem}\label{thm2}
    Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\lambda$   be an $L^2$ normalized eigenfunction of  $-\Delta_g$ with eigenvalue $\lambda^2$, satisfying \eqref{1.2} for some $\lambda\gg1$. Set $\theta_0=\lambda^{-\delta_1}$ with $0<\delta_1< \frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in \eqref{qnusymbol}. Assume that
\begin{equation}\label{i3}
        \sup_\nu\|A_\nu^{\theta_0} e_\lambda\|_{L^6(M)}\lesssim_\varepsilon \lambda^{\frac16-a{\delta_1}+\varepsilon},
    \end{equation}
    and
        \begin{equation}\label{i4}
        \|e_\lambda\|_{L^\infty(M)}\lesssim_\varepsilon \lambda^{\frac12-\delta_2+\varepsilon}
    \end{equation}
for some constant $a>0$ and $0<\delta_2<\frac{1}{2}$. Then
 \begin{equation}\label{i5}
        \|e_\lambda\|_{L^6(M)}\lesssim_\varepsilon \lambda^{\frac16-\frac23a{\delta_1}+\varepsilon}+\lambda^{\frac23(-\frac14+\frac53\delta_1)}+\lambda^{\frac16(1+\delta_1-2\delta_2)+\varepsilon}.
    \end{equation}
\end{theorem}
For each fixed $\nu$, the operator $A_\nu^{\theta_0}$ is microlocally supported in a $\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic; the precise definition will be given in Section \ref{sec: microlocal decomposition}. The proof of Theorem~\ref{thm2} follows a strategy similar to that of \cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\infty$ estimate \eqref{i4}. We will prove Theorem~\ref{thm2} in Section \ref{sec: 2}.
The idea of using improved microlocal $L^6$-norm estimate to derive improved global $L^6$-norm estimate is inspired by the recent work \cite{blair2024improved}.

Note that for a Hecke–Maass form $\psi$ on $X$, as in Theorem \ref{thm1}, the bound \eqref{i4} holds with $\delta_2=\tfrac{1}{12}$ by Iwaniec-Sarnak's $L^\infty$-norm bound \eqref{eq: IS bound}. If, in addition, \eqref{i3} holds for such $X$ and $\psi$ with $a=\frac1{12}$, then Theorem~\ref{thm1} follows immediately by choosing $\delta_1=\frac{3}{2}\delta_2=\tfrac{1}{8}$.

\begin{theorem}\label{thm microlocal KN}
    Let $\psi$ be a Hecke–Maass form on $X$ with spectral parameter $\lambda\gg1$. Set $\theta_0=\lambda^{-\delta_1}$ with $0<\delta_1<\frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in  \eqref{qnusymbol}. We have
    \begin{equation*}
        \sup_\nu\|A_\nu^{\theta_0} \psi\|_{L^6(X)}\lesssim_\varepsilon \lambda^{\frac16-\frac{\delta_1}{12}+\varepsilon}.
    \end{equation*}
\end{theorem}

We will prove Theorem \ref{thm microlocal KN} in Section \ref{sec 3}. This theorem can be viewed as the $L^6$-version of Theorem 1.1 and Lemma 3.9 in \cite{Mar16}, where Marshall used the amplification method to bound the $L^2$-mass of $\psi$ restricted to a geodesic segment with frequency localized. We give the amplification inequality in Proposition \ref{prop: amplification inequality} by integrating the amplified pretrace formula against a test function under the operator $A_\nu^{\theta_0}$. Let $\gamma$ be the geodesic segment centered in $A_\nu^{\theta_0}$.
Similar to usual amplification arguments, we have a counting problem and an analytic problem. The counting problem is to estimate the number of times a Hecke operator maps $\gamma$ back close to itself, which has been studied by Marshall \cite[Section 3.2]{Mar16}. Using the microlocal support of $A_\nu^{\theta_0}$, we can reduce the analytic problem to several known results from \cite{sogge881,Mar16,Mar16HigerRank}, which will be proved in Proposition \ref{prop: bound for I_A}.

\subsection{Generalizations}
In Theorem \ref{thm1}, the arithmetic assumption that $\psi$ is a Hecke eigenfunction is only used to invoke the bound Iwaniec-Sarnak \cite{IS95} and to prove Theorem \ref{thm microlocal KN}, both of which rely on arithmetic amplification. Hence, the maximal order in the theorem can be replaced with an Eichler order more generally, as in \cite{IS95}. Moreover, we use only unramified Hecke operators, and we may assume $\psi$ to be an eigenfunction under even fewer Hecke operators as long as the amplification argument can be adapted.

The strategy developed in this paper can also be generalized to study the $L^{q_c}$-norm problems for Hecke-Maass forms on compact locally symmetric spaces of rank one, as long as the corresponding $L^\infty$-norm problem and microlocal $L^{q_c}$ Kakeya-Nikodym problem can be solved by the arithmetic amplification technique; we will explore this direction in future work.
Theorem \ref{thm2} is expected to be generalized to Laplace-Beltrami eigenfunctions on compact Riemannian manifolds of any dimension, which allows us to control the $L^{q_c}$-norms by the $L^\infty$-norms and the microlocal Kakeya-Nikodym norms.
For example, when the manifold is taken to be a compact arithmetic hyperbolic 3-manifold (the critical exponent is $q_c=4$), the corresponding $L^\infty$-norm problem for Hecke-Maass forms is expected to be solved (see \cite{BHM16} in the cases of noncompact arithmetic
hyperbolic 3-manifolds with cusps). The corresponding microlocal $L^4$ Kakeya-Nikodym estimate should be established in a similar way as Theorem \ref{thm microlocal KN}, by modifying the proof for the usual $L^2$ Kakeya-Nikodym estimate in \cite{hou2024restrictions}.

\subsection{Notation}
Throughout the paper, the notation $A\lesssim B$ means that there is a positive constant $C$ such that $|A| \leq C B$, and $A\sim B$ means that $A\lesssim B\lesssim A$. We also use $A =O(B)$ to mean $A\lesssim B$. The notation  $A\gg B$ means there is a sufficiently large positive constant $C$ such that $A\ge CB$, and similarly for $A\ll B$.

If $f\in L^1(\mathbb{R}^n)$, the Fourier transform $\hat{f}$ of $f$ in this paper is defined as
\begin{align*}
    \hat{f}(\xi)=\int_{\mathbb{R}^n} f(x)e^{-i\langle x,\xi\rangle} dx.
\end{align*}
Here $x=(x_i),\xi=(\xi_i)\in\mathbb{R}^n$ and $\langle x,\xi\rangle=\sum_i x_i\xi_i$.
If $f\in\mathcal{S}(\mathbb{R}^n)$ is a Schwartz function, the Fourier inversion formula reads
\[
    f(x) = (2\pi)^{-n}\int_{\mathbb{R}^n}\hat{f}(\xi)e^{i\langle x,\xi\rangle} d\xi.
\]
We say a symbol $p(x,\xi)$ is in the class $S^m_{\rho,\delta}$ if\[
|\partial_x^\beta\partial^\alpha_\xi p(x,\xi)|\lesssim_{\alpha,\beta} \langle\xi\rangle^{m-\rho|\alpha|+\delta|\beta|},
\]
where $\langle\xi\rangle=(1+|\xi|^2)^{1/2}$ is the Japanese bracket.
The pseudo-differential operator $p(x,D)$ is defined by the integral
\[
p(x,D)f(x)=(2\pi)^{-n}\int p(x,\xi)\hat{f}(\xi) e^{i\langle x,\xi\rangle}d\xi=(2\pi)^{-n}\iint p(x,\xi)f(y) e^{i\langle x-y,\xi\rangle}d\xi dy.
\]
We say $P$ is the pseudo-differential operator with the compound symbol $p(x,y,\xi)$ if
\[Pf(x)=(2\pi)^{-n}\iint p(x,y,\xi)f(y) e^{i\langle x-y,\xi\rangle}d\xi dy.\]

\subsection{Acknowledgements}
The authors would like to thank Simon Marshall for helpful comments.  The second author was supported in part by the Simons Foundation and NSF (DMS-2452860).
\bigskip

A stated proof strategy for Theorem~\ref{thm1} is:

- Reduce Theorem~\ref{thm1} to Theorem~\ref{thm2}: “A main step in the proof of Theorem~\ref{thm1} is the following” theorem (Theorem~\ref{thm2}), which turns (i) a microlocal $L^6$ bound of the form \eqref{i3} for $A_\nu^{\theta_0}e_\lambda$ and (ii) an improved $L^\infty$ bound \eqref{i4} into a global $L^6$ bound \eqref{i5}. Here, “for each fixed $\nu$, the operator $A_\nu^{\theta_0}$ is microlocally supported in a $\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”

- Establish Theorem~\ref{thm2} by “a strategy similar to that of \cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\infty$ estimate \eqref{i4}.”

- For Hecke–Maass forms $\psi$ on $X$ (as in Theorem~\ref{thm1}), use that \eqref{i4} holds “with $\delta_2=\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\infty$-norm bound.” Then prove the needed microlocal estimate \eqref{i3} via Theorem~\ref{thm microlocal KN}, which gives
  \[\sup_\nu\|A_\nu^{\theta_0} \psi\|_{L^6(X)}\lesssim_\varepsilon \lambda^{\frac16-\frac{\delta_1}{12}+\varepsilon},\]
  i.e. \eqref{i3} “with $a=\frac1{12}$.”

- Conclude Theorem~\ref{thm1} by parameter choice: “If, in addition, \eqref{i3} holds for such $X$ and $\psi$ with $a=\frac1{12}$, then Theorem~\ref{thm1} follows immediately by choosing $\delta_1=\frac{3}{2}\delta_2=\tfrac{1}{8}$.”

- Proof approach for Theorem~\ref{thm microlocal KN} (supplying \eqref{i3}): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\nu^{\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\nu^{\theta_0}$, we can reduce the analytic problem to several known results from \cite{sogge881,Mar16,Mar16HigerRank}.”

A stated proof strategy for the main theorem is:

- Reduce the main theorem to the following auxiliary theorem.

\begin{theorem}\label{thm2}
    Let $(M,g)$ be a compact $2$-dimensional Riemannian manifold without boundary. Let $e_\la$   be an $L^2$ normalized eigenfunction of  $-\Delta_g$ with eigenvalue $\la^2$, satisfying \eqref{1.2} for some $\la\gg1$. Set $\theta_0=\la^{-\delta_1}$ with $0<\delta_1< \frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in \eqref{qnusymbol}. Assume that
\begin{equation}\label{i3}
        \sup_\nu\|A_\nu^{\theta_0} e_\la\|_{L^6(M)}\lesssim_\e \la^{\frac16-a{\delta_1}+\e},
    \end{equation}
    and
        \begin{equation}\label{i4}
        \|e_\la\|_{L^\infty(M)}\lesssim_\e \la^{\frac12-\delta_2+\e}
    \end{equation}
for some constant $a>0$ and $0<\delta_2<\frac{1}{2}$. Then
 \begin{equation}\label{i5}
        \|e_\la\|_{L^6(M)}\lesssim_\e \la^{\frac16-\frac23a{\delta_1}+\e}+\la^{\frac23(-\frac14+\frac53\delta_1)}+\la^{\frac16(1+\delta_1-2\delta_2)+\e}.
    \end{equation}
\end{theorem}

Here, “for each fixed $\nu$, the operator $A_\nu^{\theta_0}$ is microlocally supported in a $\theta_0$-neighborhood of a geodesic segment and is frequency-localized to directions near that geodesic.”

- Establish the auxiliary theorem by “a strategy similar to that of \cite{huang2024curvature}, relying on bilinear techniques from harmonic analysis together with the improved $L^\infty$ estimate
\begin{equation}\label{i4}
        \|e_\la\|_{L^\infty(M)}\lesssim_\e \la^{\frac12-\delta_2+\e}
    \end{equation}.”

- For Hecke–Maass forms $\psi$ on $X$ (as in the main theorem), use that the preceding estimate holds “with $\delta_2=\tfrac{1}{12}$ by Iwaniec-Sarnak’s $L^\infty$-norm bound.” Then prove the needed microlocal estimate
\begin{equation}\label{i3}
        \sup_\nu\|A_\nu^{\theta_0} e_\la\|_{L^6(M)}\lesssim_\e \la^{\frac16-a{\delta_1}+\e},
    \end{equation}
via the following theorem.

\begin{theorem}\label{thm microlocal KN}
    Let $\psi$ be a HeckeMaass form on $X$ with spectral parameter $\lambda\gg1$. Set $\theta_0=\la^{-\delta_1}$ with $0<\delta_1<\frac12$, and let $A_\nu^{\theta_0}$ be the pseudo-differential operator defined in  \eqref{qnusymbol}. We have
    \begin{equation*}
        \sup_\nu\|A_\nu^{\theta_0} \psi\|_{L^6(X)}\lesssim_\e \la^{\frac16-\frac{\delta_1}{12}+\e}.
    \end{equation*}
\end{theorem}

This gives
\[\sup_\nu\|A_\nu^{\theta_0} \psi\|_{L^6(X)}\lesssim_\varepsilon \lambda^{\frac16-\frac{\delta_1}{12}+\varepsilon},\]
that is, the microlocal bound above “with $a=\frac1{12}$.”

- Conclude the main theorem by parameter choice: “If, in addition, the microlocal bound above holds for such $X$ and $\psi$ with $a=\frac1{12}$, then the main theorem follows immediately by choosing $\delta_1=\frac{3}{2}\delta_2=\tfrac{1}{8}$.”

- Proof approach for the preceding microlocal theorem (supplying the microlocal bound): it “can be viewed as the $L^6$-version” of Marshall’s microlocal restriction results; the authors “give the amplification inequality … by integrating the amplified pretrace formula against a test function under the operator $A_\nu^{\theta_0}$.” As in “usual amplification arguments,” there is “a counting problem” (how often a Hecke operator maps the relevant geodesic segment $\gamma$ back close to itself) and “an analytic problem,” and “using the microlocal support of $A_\nu^{\theta_0}$, we can reduce the analytic problem to several known results from \cite{sogge881,Mar16,Mar16HigerRank}.”

Question:
Let \(\mathbb H\) be the upper half-plane, and let \(X=\Gamma\backslash \mathbb H\) be a compact arithmetic hyperbolic surface, where \(\Gamma\subset \mathrm{SL}(2,\mathbb R)\) is the group of norm-one units of a maximal order in an indefinite quaternion division algebra over \(\mathbb Q\). Let \(\psi\) be an \(L^2\)-normalized Hecke--Maass form on \(X\), meaning that \(\int_X |\psi(x)|^2\,dx=1\), \(\psi\) is a joint eigenfunction of the Laplace--Beltrami operator and the Hecke operators, and its spectral parameter \(\lambda\) is defined by
\[
\Delta \psi + \left(\tfrac14+\lambda^2\right)\psi=0,
\]
with real \(\lambda\gg 1\). Which of the following quantitative estimates holds for the global \(L^6\)-norm of \(\psi\)?

For every \(\varepsilon>0\),
\[
\|\psi\|_{L^6(X)}\lesssim_{\varepsilon} \lambda^{\frac16-\frac1{144}+\varepsilon}.
\]
Equivalently, there is a constant \(C_{\varepsilon}\) such that \(\|\psi\|_{L^6(X)}\le C_{\varepsilon}\lambda^{\frac16-\frac1{144}+\varepsilon}\).

Choices:
B.
For every \(\varepsilon>0\),
\[
\|\psi\|_{L^6(X)}\lesssim_{\varepsilon} \lambda^{\frac16-\frac1{72}+\varepsilon}.
\]
Equivalently, there is a constant \(C_{\varepsilon}\) such that \(\|\psi\|_{L^6(X)}\le C_{\varepsilon}\lambda^{\frac16-\frac1{72}+\varepsilon}\).
C.
For every \(\varepsilon>0\),
\[
\|\psi\|_{L^6(X)}\lesssim_{\varepsilon} \lambda^{\frac16+\varepsilon}.
\]
Equivalently, there is a constant \(C_{\varepsilon}\) such that \(\|\psi\|_{L^6(X)}\le C_{\varepsilon}\lambda^{\frac16+\varepsilon}\).
D.
There exists \(\varepsilon_0>0\) such that
\[
\|\psi\|_{L^6(X)}\lesssim \lambda^{\frac16-\frac1{144}+\varepsilon_0}.
\]
Equivalently, there is a constant \(C\) such that \(\|\psi\|_{L^6(X)}\le C\lambda^{\frac16-\frac1{144}+\varepsilon_0}\).
E.
For every \(\varepsilon>0\),
\[
\|\psi\|_{L^6(X)}\lesssim_{\varepsilon} \lambda^{\frac16-\frac5{342}+\varepsilon}.
\]
Equivalently, there is a constant \(C_{\varepsilon}\) such that \(\|\psi\|_{L^6(X)}\le C_{\varepsilon}\lambda^{\frac16-\frac5{342}+\varepsilon}\).

optimized exponent from choosing \(\delta_1=\tfrac18\) with \(a=\tfrac1{12}\)

drop the power-saving term \(-\tfrac1{144}\) from the final exponent

confusing the interpolated Humphries--Khan exponent with the main theorem's sharper exponent


\label{thm}
	Let $s>0, \lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\mapsto(u,b)$ for the equations \eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C\left([0,T],H^{s}(\mathbb{R}^d)\right)\times C\left([0,T],H^{s}(\mathbb{R}^d)\right)$.

  More precisely, for any $\gamma>0$ and arbitrary constant magnetic field ${\bf B_0}\in \mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
	\begin{itemize}
		\item the solutions satisfy
\begin{align*}
			(u^{\pm 1,\lambda}, b^{\pm 1,\lambda} - {\bf B_0}) \in C([0,T], H^{s}(\mathbb{R}^{d})) \times C([0,T], H^{s}(\mathbb{R}^{d}));
\end{align*}
		\item  the initial data satisfy
\begin{equation}\label{nonuniform-initial-bound}
	\begin{array}{l}
							\|u^{+1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{+1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma, \\[2mm]
			\|u^{-1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{-1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma;
	\end{array}
\end{equation}
		\item and the non-uniform continuity is characterized by
		\begin{itemize}
			\item at initial time $t=0$,
			\begin{align}\label{nonuniform-initial-diff}
				\lim_{\lambda\to\infty} \|u^{+1,\lambda}(0,\cdot) - u^{-1,\lambda}(0,\cdot)\|_{H^{s}} = 0,
			\end{align}
			\item for evolution times $t>0$,
			\begin{align}\label{nonuniform-low-bound-sint}
				\lim_{\lambda \to\infty} \|u^{+1,\lambda}(t) - u^{-1,\lambda}(t)\|_{H^{s}} \geq c\gamma |\sin t|,
			\end{align}
		\end{itemize}
		where $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.
	\end{itemize}

\begin{theorem}\label{thm}
	Let $s>0, \lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\mapsto(u,b)$ for the equations \eqref{mhd} is non-uniformly continuous from a bounded subset in $H^{s}(\mr^d)\times H^{s}(\mr^d)$ into $C\left([0,T],H^{s}(\mr^d)\right)\times C\left([0,T],H^{s}(\mr^d)\right)$.

  More precisely, for any $\gamma>0$ and arbitrary constant magnetic field ${\bf B_0}\in \mr^d$,  there exists two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
	\begin{itemize}
		\item the solutions satisfy
\begin{align*}
			(u^{\pm 1,\lambda}, b^{\pm 1,\lambda} - {\bf B_0}) \in C([0,T], H^{s}(\mathbb{R}^{d})) \times C([0,T], H^{s}(\mathbb{R}^{d}));
\end{align*}
		\item  the initial data satisfy
\begin{equation}\label{nonuniform-initial-bound}
	\begin{array}{l}
							\|u^{+1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{+1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma, \\[2mm]
			\|u^{-1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{-1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma;
	\end{array}
\end{equation}
		\item and the non-uniform continuity is characterized by
		\begin{itemize}
			\item at initial time $t=0$,
			\begin{align}\label{nonuniform-initial-diff}
				\lim_{\lambda\to\infty} \|u^{+1,\lambda}(0,\cdot) - u^{-1,\lambda}(0,\cdot)\|_{H^{s}} = 0,
			\end{align}
			\item for evolution times $t>0$,
			\begin{align}\label{nonuniform-low-bound-sint}
				\lim_{\lambda \to\infty} \|u^{+1,\lambda}(t) - u^{-1,\lambda}(t)\|_{H^{s}} \geq c\gamma |\sin t|,
			\end{align}
		\end{itemize}
		where $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.
	\end{itemize}
\end{theorem}

\,\,\,\,\,\,\,\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):
\begin{align}\label{mhd}
	\begin{cases}
	\partial_t u+u \cdot \nabla u+\nabla p=b\cdot \nabla b,\\[1mm]
\partial_t b-\Delta b+u\cdot \nabla b=b\cdot\nabla u,  \\[1mm]
\text{div}\, u=\text{div}\, b=0, \\[1mm]
u(x, 0)=u_0(x),\,\, b(x, 0)=b_0(x),
	\end{cases}
\end{align}
where $u(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}^d$,
$b(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}^d$,
and $p(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}$
denote the velocity field, magnetic field, and pressure field, respectively,
with $x \in \mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\equiv0$, \eqref{mhd} reduces to the classical Euler equations.

The resistive MHD equations provide a fundamental framework for modeling key plasma phenomena where finite electrical resistivity plays a critical role, particularly in astrophysical magnetic reconnection processes that govern energy release in solar flares and magnetospheric activity, as well as in the geodynamo mechanisms responsible for generating and sustaining planetary magnetic fields \cite{priest-2000,roberts1967introduction}.

 There have been a large number of mathematical studies on the  well-posedness theory for the incompressible MHD equations under various assumptions on velocity dissipation and magnetic diffusion.
Previous results on local well-posedness can be found in \cite{sermange-1983,jiu-2006-local,li-local-nonresistive-2017-adv,chemin-local-nonresistive-2016-adv,fefferman-local-nonressitive-2014-jfa,fefferman-local-nonressitive-2017-arma}. The pioneering work of Bardos, Sulem and Sulem \cite{full-not-bardos} and Lin and Zhang \cite{lin-2014-GlobalSmallSolutions} on perturbation theory near constant background magnetic fields first revealed the stabilizing role of such equilibrium configurations in the MHD equations. Physical interpretations of this stabilization effect are discussed in \cite{background-phy1,background-phy2}. These foundational results have inspired extensive research on global well-posedness and stability of the MHD equations, as documented in \cite{lei-bkm-2009-dcds,full-1-duvaut-lions,sermange-1983,cao-wu-1-mix,cao-wu-2-mix,full-not-bardos,lin-2014-GlobalSmallSolutions,chen-2022-3dmhd-Diophant,deng-zhang-2018-decay,panGlobalClassicalSolutions2018,ren2014global,zhang-2016-non-resistive-global-jde,xie-2024-cvpde-dio,wei--global-resistive-2020-cmr,zhou-zhu-glbal-symmetry-jmp-2018,ye-global-resistive-2022-acta} and references therein.

  In recent years, ill-posedness theory  for the incompressible MHD equations has also been invetigated. For the non-resistive MHD equations  which contain velocity dissipation but  no magnetic diffusion, Chen, Nie and Ye \cite{chen-ye-sharp-ill-nonresistive-2024-jfa}  established sharp strong ill-posedness results that provide a striking contrast to the local well-posedness theory developed by Fefferman et al. \cite{fefferman-local-nonressitive-2017-arma}. For the resistive case \eqref{mhd}, Wu and Zhao \cite{wu-zhao-mild-resistive-2023-IMRN}  obtained mild ill-posedness results near the background magnetic field $(1,0)$ in $\mathbb{R}^2$.

It is noted that the concept of ill-posedness manifests rather strongly in many fundamental cases. As evidenced by \cite{chen-ye-sharp-ill-nonresistive-2024-jfa,wu-zhao-mild-resistive-2023-IMRN} and other works of PDE systems, such behavior typically occurs in critical or supercritical function spaces, while  some problems remain unresolved to this day. These substantial difficulties have motivated researchers to consider relaxed notions of ill-posedness by examining weaker properties.
 For certain equations, the solution operator may exhibit non-uniform continuity properties under stronger topological frameworks, which provides meaningful insights into the refined continuity structure of solution mappings and can be seen a kind of instability of the solutions or ill-posedness the equations.

In this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:
\begin{definition}\label{define-nonuniform}
	Let $X$ be a Banach space, and consider the Cauchy problem:
	\begin{align*}
		\begin{cases}
			\partial_t v = N(v), \\
			v(0) = v_0,
		\end{cases}
	\end{align*}
	where $N$ is a (possibly nonlinear) differential operator. The \textbf{data-to-solution map} $\Phi_t \colon X \to X$ (for fixed $t > 0$) defined by $\Phi_t(v_0) = v(t)$ is said to be \textbf{non-uniformly continuous} on $X$ if the following holds:

	\noindent
	\textbf{Sequential Formulation:} \\
	For every $t > 0$, there exists $\epsilon_0 > 0$ and sequences $\{v_{1,n}(0)\}, \{v_{2,n}(0)\} \subset X$ such that:
	\begin{align*}
		\lim_{n \to \infty} \| v_{1,n}(0) - v_{2,n}(0) \|_X = 0,
	\end{align*}
	but
	\begin{align*}
		\limsup_{n \to \infty} 	\| \Phi_t(v_{1,n}(0)) - \Phi_t(v_{2,n}(0)) \|_X \geq \epsilon_0.
	\end{align*}

	\noindent
	\textbf{$\delta$-$\epsilon$ Formulation (Equivalent):} \\
	For every $t > 0$, there exists $\epsilon_0 > 0$ such that for any $\delta > 0$, one can find initial data $v_1(0), v_2(0) \in X$ satisfying:
	\begin{align*}
		\| v_1(0) - v_2(0) \|_X < \delta,
	\end{align*}
	but
	\begin{align*}
		\| \Phi_t(v_1(0)) - \Phi_t(v_2(0)) \|_X \geq \epsilon_0.
	\end{align*}
\end{definition}

In \cite{nonuniform-2010-cmp-himonas}, Himonas and Misiołek first  proved non-uniform continuity of the data-to-solution map  on the  incompressible Euler equations in both $H^s(\mathbb{R}^d)$ ($s > 0$) and $H^r(\mathbb{T}^d)$ ($r \in \mathbb{R}$) with $d = 2,3$, which was later extended by Li and Bourgain \cite{nonuniform-2019-cmp-bourgain-li} through Galilean boost techniques to the endpoint case $s \geq 0$ in $H^s(\mathbb{R}^d)$, where they further proved the stronger property of nowhere uniform continuity.
For the non-resistive MHD equations with only velocity dissipation, recent work by Li, Yin, and Zhu \cite{nonuniforem-2023-adv-li-yin-zhu} demonstrated non-uniform continuity in $H^s(\mathbb{R}^d)$ for $s > \frac{d}{2}$.

  We will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$. In comparison with the work by Li, Yin, and Zhu \cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\mathbb{R}^d)$ for $s > \frac{d}{2}$,  our results hold for the resistive MHD equations and  in  $H^s(\mathbb{R}^d)$ for all $s>0$ and  $d=2,3$.  Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.

Our main results can be stated as follows:

\,\,\,\,\,\,\,\, In this paper, we consider the Cauchy problem for the incompressible MHD equations with only magnetic diffusion (hereinafter called the resistive MHD equations):
\begin{align}\label{mhd}
 \begin{cases}
 \partial_t u+u \cdot \nabla u+\nabla p=b\cdot \nabla b,\\[1mm]
\partial_t b-\Delta b+u\cdot \nabla b=b\cdot\nabla u, \\[1mm]
\text{div}\, u=\text{div}\, b=0, \\[1mm]
u(x, 0)=u_0(x),\,\, b(x, 0)=b_0(x),
 \end{cases}
\end{align}
where $u(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}^d$,
$b(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}^d$,
and $p(x,t) \colon \mathbb{R}^d \times \mathbb{R}_+ \to \mathbb{R}$
denote the velocity field, magnetic field, and pressure field, respectively,
with $x \in \mathbb{R}^d~(d=2,3)$ and $t>0$. Clearly, when $b\equiv0$, \eqref{mhd} reduces to the classical Euler equations.

In this paper, we are concerned with the non-uniform continuity properties of the data-to-solution map for the resistive MHD equations. We begin by precisely defining the notion of non-uniform continuity as follows:
\begin{definition}\label{define-nonuniform}
 Let $X$ be a Banach space, and consider the Cauchy problem:
 \begin{align*}
 \begin{cases}
 \partial_t v = N(v), \\
 v(0) = v_0,
 \end{cases}
 \end{align*}
 where $N$ is a (possibly nonlinear) differential operator. The \textbf{data-to-solution map} $\Phi_t \colon X \to X$ (for fixed $t > 0$) defined by $\Phi_t(v_0) = v(t)$ is said to be \textbf{non-uniformly continuous} on $X$ if the following holds:

\noindent
 \textbf{Sequential Formulation:} \\
 For every $t > 0$, there exists $\epsilon_0 > 0$ and sequences $\{v_{1,n}(0)\}, \{v_{2,n}(0)\} \subset X$ such that:
 \begin{align*}
 \lim_{n \to \infty} \| v_{1,n}(0) - v_{2,n}(0) \|_X = 0,
 \end{align*}
 but
 \begin{align*}
 \limsup_{n \to \infty} \| \Phi_t(v_{1,n}(0)) - \Phi_t(v_{2,n}(0)) \|_X \geq \epsilon_0.
 \end{align*}

\noindent
 \textbf{$\delta$-$\epsilon$ Formulation (Equivalent):} \\
 For every $t > 0$, there exists $\epsilon_0 > 0$ such that for any $\delta > 0$, one can find initial data $v_1(0), v_2(0) \in X$ satisfying:
 \begin{align*}
 \| v_1(0) - v_2(0) \|_X < \delta,
 \end{align*}
 but
 \begin{align*}
 \| \Phi_t(v_1(0)) - \Phi_t(v_2(0)) \|_X \geq \epsilon_0.
 \end{align*}
\end{definition}

We will prove in this paper the non-uniform continuity of the data-to-solution map for the resistive MHD equations in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. In comparison with the work by Li, Yin, and Zhu \cite{nonuniforem-2023-adv-li-yin-zhu} which is for non-resistive MHD equations with only velocity dissipation and in $H^s(\mathbb{R}^d)$ for $s > \frac{d}{2}$, our results hold for the resistive MHD equations and in $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which shows that the strong magnetic field may provide the stabilization effect but no help for the uniform continuity of the data-to-solution map.

Our main results can be stated as follows:

\begin{align}\label{mhd}
	\begin{cases}
	\partial_t u+u \cdot \nabla u+\nabla p=b\cdot \nabla b,\\[1mm]
\partial_t b-\Delta b+u\cdot \nabla b=b\cdot\nabla u,  \\[1mm]
\text{div}\, u=\text{div}\, b=0, \\[1mm]
u(x, 0)=u_0(x),\,\, b(x, 0)=b_0(x),
	\end{cases}
\end{align}

\section{Preliminaries}\label{sec:2}
\subsection{Auxiliary Analysis Tools}\label{sec:2.1}
In this subsection, we introduce two key lemmas needed later. The first one is
\begin{lemma}\cite{nonuniform-2010-cmp-himonas}\label{phi-cmp-2010}
 Let $\sigma \geq 0, \delta\geq 0, a\in \mr$ and $\lambda\gg 1$. For any Schwartz function $\psi\in \mathcal{S}(\mr)$, it holds that
 \begin{align*}
 \lambda^{\delta/2}\left\|\psi\right\|_{L^2(\mathbb{R})}\leq\left\|\psi\left(\frac{\cdot}{\lambda^\delta}\right)\right\|_{H^\sigma(\mathbb{R})}\leq\lambda^{\delta/2}\left\|\psi\right\|_{H^\sigma(\mathbb{R})},
 \end{align*}
and \begin{align}\label{622}
 \left\|\psi\left(\frac{\cdot}{\lambda^\delta}\right)\cos(\lambda\cdot-a)\right\|_{H^\sigma(\mathbb{R})}\simeq\lambda^{\sigma+\delta/2}\|\psi\|_{L^2(\mathbb{R})}.
\end{align}
Moreover, \eqref{622} holds true if $\cos(\lambda \cdot-a)$ is replaced by $\sin(\lambda \cdot-a)$.
\end{lemma}

\subsection{Approximate Solutions Scheme}\label{sec:3.1}
We begin by constructing two sequences of approximate solutions $b^{w,\lambda}$ and $u^{w,\lambda}$ with $w=\pm 1$, where $b^{w,\lambda}$ contains low frequencies only and $u^{w,\lambda}$ contain both high and low frequencies. More precisely, we set
\begin{align}\label{u-low+high}
 u^{w,\lambda}(x,t)=u^{l,w,\lambda}(x,t)+u^{h,w,\lambda}(x,t).
\end{align}
The high-frequency components $u^{h,w,\lambda}$ are constructed as
\begin{align}\label{u-high}
 u^{h,w,\lambda}(x,t)=\begin{cases}
 \nabla^\perp \phi^{h,w,\lambda}(x,t)=(\partial_{x_2}\phi^{h,w,\lambda}(x,t),-\partial_{x_1}\phi^{h,w,\lambda}(x,t)), &~d=2,\\[3mm]
 \left(\nabla^\perp \phi^{h,w,\lambda}(x,t),0 \right)=(\partial_{x_2}\phi^{h,w,\lambda}(x,t),-\partial_{x_1}\phi^{h,w,\lambda}(x,t),0), &~d=3.
 \end{cases}
\end{align}
In \eqref{u-high}, the function $\phi^{h,w,\lambda}(x,t)$ is defined as
\begin{align}\label{phi-high}
 \phi^{h,w,\lambda}(x,t)=\begin{cases}
 \lambda^{-s-\delta-1}\phi(\dfrac{x_1}{\lambda^\delta})\phi(\dfrac{x_2}{\lambda^\delta})\sin (\lambda x_2-wt),&~d=2,\\[3mm]
 \lambda^{-s-\delta-1}\phi(\dfrac{x_1}{\lambda^\delta})\phi(\dfrac{x_2}{\lambda^\delta})\sin (\lambda x_2-wt)\phi(x_3),&~d=3,
 \end{cases}
\end{align}
where $s > 0, \lambda>0, w=\pm 1$, the parameter $\delta > 0$ is to be specified later and the smooth function $\phi \in C_c^\infty(\mathbb{R})$ satisfies $\text{supp}\,\phi \subset [-2,2]$ and $\phi(x) \equiv 1$ on $|x| < 1$.

Following similar estimates established in \eqref{u,b-low-t-local-bound} and \eqref{u-high-bound}, we can obtain
\begin{align}\label{4.5}
 \|\bar{u}^{l,w,\lambda}(t)\|_{H^m} + \|\bar{b}^{w,\lambda}(t)\|_{H^{m}}
 \leq C\lambda^{-1+\delta},
\end{align}
and
\begin{align}\label{4.6}
 \|\bar{u}^{h,w,\lambda}(t)\|_{H^r}\leq C\lambda^{r-s},\quad \|\bar{u}^{h,w,\lambda}(t)\|_{L^\infty}\leq C\lambda^{-s-\delta},\quad \|\nabla \bar{u}^{h,w,\lambda}(t)\|_{L^\infty}\leq C\lambda^{-s+1-\delta}
\end{align}
for any $0 \leq t \leq T\leq 1$, where $[0,T]$ denotes the existence interval of the solution guaranteed by Lemma~\ref{local-well-pri}.
\subsection{Error Estimates for the Approximate Solutions}\label{sec:4.2}
Following the approach in subsection \ref{sec:3.2} and based on system \eqref{mhd-B}, we construct a perturbed system as follows
\begin{align}\label{4.9}
 \begin{cases}
 \partial_t \bar{u}^{w,\lambda}+\bar{u}^{w,\lambda} \cdot \nabla \bar{u}^{w,\lambda}+\nabla p^{w,\lambda}=\bar{b}^{w,\lambda}\cdot \nabla \bar{b}^{w,\lambda}+{\bf B_0}\cdot\nabla \bar{b}^{w,\lambda}+E_1^{w,\lambda},\\[1mm]
 \partial_t \bar{b}^{w,\lambda}-\Delta \bar{b}^{w,\lambda}+\bar{u}^{w,\lambda}\cdot \nabla \bar{b}^{w,\lambda}=\bar{b}^{w,\lambda}\cdot\nabla \bar{u}^{w,\lambda}+{\bf B_0}\cdot\nabla \bar{u}^{w,\lambda}+F_1^{w,\lambda}, \\[1mm]
 \diver\, \bar{u}^{w,\lambda}=\diver\, \bar{b}^{w,\lambda}=0,
 \end{cases}
\end{align}
where
\begin{align*}
 E_1^{w,\lambda}&=\partial_t \bar{u}^{h,w,\lambda}+\bar{u}^{l,w,\lambda}\cdot\nabla \bar{u}^{h,w,\lambda}+\bar{u}^{h,w,\lambda}\cdot\nabla \bar{u}^{h,w,\lambda}+\bar{u}^{h,w,\lambda}\cdot\nabla \bar{u}^{l,w,\lambda},\\\nonumber
 F_1^{w,\lambda}&=\bar{u}^{h,w,\lambda}\cdot\nabla \bar{b}^{w,\lambda}-\bar{b}^{w,\lambda}\cdot\nabla \bar{u}^{h,w,\lambda}-{\bf B_0}\cdot \nabla \bar{u}^{h,w,\lambda}\\
 &\define\diver \tilde{F}_1^{w,\lambda}-{\bf B_0}\cdot \nabla \bar{u}^{h,w,\lambda}.
\end{align*}
The perturbed system \eqref{4.9} is analogous to \eqref{mhd-low+high}. The directional derivative of the high-frequency component along the background magnetic field ${\bf B_0} = (B_1, B_2)$ is
\begin{align*}
 {\bf B_0} \cdot \nabla \bar{u}^{h,w,\lambda} = B_1\partial_{x_1}\bar{u}^{h,w,\lambda} + B_2\partial_{x_2}\bar{u}^{h,w,\lambda}.
\end{align*}
Remarkably, we observe that the oscillatory part in $\bar{u}^{h,w,\lambda}$ satisfies
\begin{align*}
 {\bf B_0} \cdot \nabla \left[\phi\left(\frac{B_1x_2 - B_2x_1}{\lambda^\delta}\right) \cos\left(\lambda(B_1x_2 - B_2x_1) - wt\right)\right] = 0,
\end{align*}
which leads to the following simplified expression
\begin{align*}
 & {\bf B_0}\cdot \nabla \left(\lambda^{-s-\delta}\phi\left(\dfrac{x_1}{\lambda^\delta}\right)
 \phi\left(\dfrac{B_1x_2-B_2x_1}{\lambda^\delta}\right)\cos \left(\lambda \left(B_1x_2-B_2x_1\right)-wt\right)\right)\\
 &=B_1\lambda^{-s-2\delta}\phi'\left(\frac{x_1}{\lambda^{\delta}}\right) \phi\left(\dfrac{B_1x_2-B_2x_1}{\lambda^\delta}\right)\cos \left(\lambda \left(B_1x_2-B_2x_1\right)-wt\right)
\end{align*}
Therefore, following analogous arguments as \eqref{4.3}, we can obtain
\begin{align}\label{4.7}
 \left\| {\bf B_0}\cdot\nabla \bar{u}^{h,w,\lambda}\right\|_{L^2}\leq C\lambda^{-s-\delta}.
\end{align}
Combining the estimates \eqref{4.5}, \eqref{4.6} with Lemma \ref{E,F-error-estimate}, we deduce that
\begin{align}\label{4.8}
 \|E_1^{w,\lambda}(t)\|_{L^2}\leq C\lambda^{-\sigma_{s,\delta}},\quad \|\tilde{F}_1^{w,\lambda}(t)\|_{L^2}\leq C\lambda^{-s-1},
\end{align}
where \begin{align*}
 \sigma_{s,\delta}=\min\{s+1-\delta,2s-1+\delta\}.
\end{align*}
\subsection{Exact Solutions}\label{sec:4.3}
Let $(\bar{u}_{w,\lambda},\bar{b}_{w,\lambda})$ be the unique solution to system \eqref{mhd-B} with the initial data $\left(\bar{u}^{w,\lambda}(x,0),\bar{b}^{w,\lambda}(x,0)\right)$, satisfying
\begin{align*}
 \begin{cases}
 \partial_t \bar{u}_{w,\lambda}+\bar{u}_{w,\lambda} \cdot \nabla \bar{u}_{w,\lambda}+\nabla p_{w,\lambda}=\bar{b}_{w,\lambda}\cdot \nabla \bar{b}_{w,\lambda}+{\bf B_0}\cdot \nabla \bar{b}_{w,\lambda},\\[1mm]
 \partial_t \bar{b}_{w,\lambda}-\Delta \bar{b}_{w,\lambda}+\bar{u}_{w,\lambda}\cdot \nabla \bar{b}_{w,\lambda}=\bar{b}_{w,\lambda}\cdot\nabla \bar{u}_{w,\lambda}+{\bf B_0}\cdot \nabla \bar{u}_{w,\lambda}, \\[1mm]
 \diver\, \bar{u}_{w,\lambda}=\diver\, \bar{b}_{w,\lambda}=0,
 \end{cases}
\end{align*}
It follows from \eqref{4.9} that
\begin{align*}
 \begin{cases}
 \partial_t \bar{u}^{w,\lambda}+\bar{u}^{w,\lambda} \cdot \nabla \bar{u}^{w,\lambda}+\nabla p^{w,\lambda}=\bar{b}^{w,\lambda}\cdot \nabla \bar{b}^{w,\lambda}+{\bf B_0}\cdot\nabla \bar{b}^{w,\lambda}+E_1^{w,\lambda},\\[1mm]
 \partial_t \bar{b}^{w,\lambda}-\Delta \bar{b}^{w,\lambda}+\bar{u}^{w,\lambda}\cdot \nabla \bar{b}^{w,\lambda}=\bar{b}^{w,\lambda}\cdot\nabla \bar{u}^{w,\lambda}+{\bf B_0}\cdot\nabla \bar{u}^{w,\lambda}+\diver \tilde{F}_1^{w,\lambda}-{\bf B_0}\cdot \nabla \bar{u}^{h,w,\lambda}, \\[1mm]
 \diver\, \bar{u}^{w,\lambda}=\diver\, \bar{b}^{w,\lambda}=0.
\end{cases}
\end{align*}
The two systems share same initial conditions, which are
\begin{align*}
 \bar{u}_{w,\lambda}(x, 0)&=\bar{u}^{w,\lambda}(x,0)=\nabla^\perp\left( \bar{\phi}^{l,w,\lambda}(x)+\bar{\phi}^{h,w,\lambda}(x,0)\right),\\[1mm]
 \bar{b}_{w,\lambda}(x, 0)&=\bar{b}^{w,\lambda}(x,0)=\nabla^\perp\bar{\phi}^{l,w,\lambda}(x)
\end{align*}

\begin{remark}
	The stabilizing role of background magnetic fields in the MHD equations has been well studied in previous works \cite{lin-2014-GlobalSmallSolutions,full-not-bardos}. Our analysis reveals that despite the stabilization effect of the  background magnetic fields, the data-to-solution map for the resistive MHD equations \eqref{mhd} maintains its non-uniform continuity property for arbitrary non-zero constant fields ${\bf B_0}\neq {\bf 0}$.
\end{remark}

To simplify the notations, we write the perturbed system as follows
\begin{align}\label{mhd-B}
	\begin{cases}
		\partial_t u+u \cdot \nabla u+\nabla p=b\cdot \nabla b+{\bf B_0}\cdot\nabla b,\\[1mm]
		\partial_t b-\Delta b+u\cdot \nabla b=b\cdot\nabla u+{\bf B_0}\cdot\nabla u,  \\[1mm]
		\text{div}\, u=\text{div}\, b=0, \\[1mm]
		u(x, 0)=u_0(x),\,\, b(x, 0)=b_0(x),
	\end{cases}
\end{align}
where we denote $b - {\bf B_0}$ by $b$ for convenience.

We now explain the main ideas of the proof of  Theorem \ref{thm}. It is recalled that  Himonas and Misiołek  \cite{nonuniform-2010-cmp-himonas} introduced a frequency decomposition to construct  approximate solutions of the two-dimensional incompressible Euler equations. Denote
 \begin{align}\label{1.1}
	u^{\pm1,\lambda}(x,t)=u^{l,\pm1,\lambda}(x,t)+u^{h,\pm1,\lambda}(x,t),
\end{align}
where
\begin{itemize}
	\item[(i)] the low-frequency components $u^{l,\pm1,\lambda}$ are  obtained by solving the Euler equations with  low-frequency initial data;
	\item[(ii)] the high-frequency components $u^{h,\pm1,\lambda}$ are explicitly constructed by using  oscillatory profiles, with frequency parameter $\lambda$ controlling the spatial concentration,
	\begin{align}\label{1.2}
		u^{h,\pm1,\lambda}(x,t)=\nabla^\perp \left(\lambda^{-\delta-s-1}\phi\left(\dfrac{x_1}{\lambda^\delta}\right)\phi\left(\dfrac{x_2}{\lambda^\delta}\right)\sin (\lambda x_2\mp t)\right).
	\end{align}
Here, $\lambda>0, \max\{1-s,0\}<\delta < 1$ and $\phi\in C_c^\infty(\mathbb{R})$ with $\text{supp}\,\phi \subset [-2,2]$ and $\phi(x) \equiv 1$ on $|x| < 1$.
\end{itemize}

It is required to modify the construction of the approximate solutions in the presence of the magnetic field. To the resistive MHD equations, due to the diffusive nature of the magnetic field evolution (versus the purely transport equations of the Euler equations ), we restrict high-frequency perturbations to the velocity field alone and  maintain low-frequency components for the magnetic field. More precisely, in the case $\mathbf{B_0} = \mathbf{0}$ (no background magnetic field) and $d=2$,  we set
\begin{itemize}
	\item the low-frequency pairs $(u^{l,\pm1,\lambda}, b^{\pm1,\lambda})$ by solving the resistive MHD equations \eqref{mhd} (equivalently, system \eqref{mhd-B} with $\mathbf{B_0} = \mathbf{0}$) with low-frequency initial data;
	\item the high-frequency velocity components $u^{h,\pm1,\lambda}$ by retain their profiles as in  \eqref{1.2}.
\end{itemize}

In the three-dimensional case $(d = 3)$, we modify  the high-frequency velocity components as
	\begin{align}\label{1.3}
	u^{h,\pm1,\lambda}(x,t)=\begin{pmatrix}
		\partial_{x_1}\\[1mm]
		-\partial_{x_2}\\[1mm]
		0
	\end{pmatrix} \left(\lambda^{-\delta-s-1}\phi\left(\dfrac{x_1}{\lambda^\delta}\right)\phi\left(\dfrac{x_2}{\lambda^\delta}\right)\sin (\lambda x_2\mp t)\phi(x_3)\right).
\end{align}

The error terms induced by the  high-frequency  $u^{h,\pm1,\lambda}$   can be written as
\begin{align}\label{1.4}
	E^{\pm1,\lambda}&=\partial_t u^{h,\pm1,\lambda}+u^{l,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{l,\pm1,\lambda},\\\label{1.5}
	F^{\pm1,\lambda}&=u^{h,\pm1,\lambda}\cdot\nabla b^{\pm1,\lambda}-b^{\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda},
\end{align}
respectively.

Direct estimation of \eqref{1.5} will lead to uncontrolled error terms. Our alternative approach consists of three key steps:
\begin{enumerate}
	\item {\bf Divergence reformulation:} We express $F^{\pm1,\lambda} =\text{div}\,\, \tilde{F}^{\pm1,\lambda}$ and establish estimates for $\tilde{F}^{\pm1,\lambda}$;
	\item {\bf Integration by parts:} In $L^2$ inner product computations, we transfer the divergence operator to $b^{\pm1,\lambda}$ via integration by parts;
	\item {\bf Diffusion cancellation:} The resulting terms are precisely canceled by exploiting the magnetic diffusion term $\Delta b^{\pm1,\lambda}$.
\end{enumerate}

For non-zero background fields $\mathbf{B_0}=(B_1,B_2) \neq \mathbf{0}$, we introduce the coordinate transformation
\begin{align*}
	x_2 \mapsto B_1x_2 - B_2x_1,
\end{align*}
which maintains all norm estimates while achieving exact cancellation of the dominant high-frequency linear term $\mathbf{B_0} \cdot \nabla u^{h,\pm 1,\lambda}$ appearing in the error analysis of $F^{\pm1,\lambda}$.

A key observation is that the directional derivative exhibits a better decay:
\begin{align*}
	\left\|\mathbf{B_0} \cdot \nabla u^{h,\pm 1,\lambda}\right\|_{L^2}\leq C\lambda^{-s-\delta} \ll \left\|u^{h,\pm 1,\lambda}\right\|_{L^2}\leq C\lambda^{-s}, \text{~for~}\lambda \to \infty,
\end{align*}
where $\delta>0$ represent the improved decay rate.

	 The paper is organized as follows. section \ref{sec:2} is the preliminary analytical framework, in which subsection \ref{sec:2.1} is about key Lemmas and subsection \ref{sec:2.2} is on the local well-posedness theory. The core  strategy of the proof is implemented in section \ref{sec:3} (zero background field case, $\mathbf{B_0} = \mathbf{0}$) and section  \ref{sec:4} (non-zero background field case, $\mathbf{B_0} \neq \mathbf{0}$), following the approach as follows: construction of approximate solutions in subsections \ref{sec:3.1} and \ref{sec:4.1}, precise estimation of approximation errors in subsections \ref{sec:3.2} and \ref{sec:4.2}, rigorous construction of exact solutions in subsections \ref{sec:3.3} and \ref{sec:4.3}, finishing the proof of Theorem in the case $\mathbf{B_0} = \mathbf{0}$ and $\mathbf{B_0} \neq \mathbf{0}$ in subsections \ref{sec:3.4} and \ref{sec:4.4}, respectively.

{\bf Notations:}
\begin{enumerate}
\item {\bf Function Spaces:} Throughout this work, let $X$ denote a Banach space equipped with norm $\|\cdot\|_X$. Since all function spaces considered here are defined on $\mathbb{R}^d$ for $d = 2,3$, we will suppress the domain $\mathbb{R}^d$ in our notation unless otherwise specified.
	\item {\bf Differential Operators:} For $x \in \mathbb{R}^2$, we define the perpendicular gradient operator as $\nabla^\perp := (\partial_{x_2}, -\partial_{x_1})$.
	\item {\bf Joint Norms:} Given functions $f, g \in X(\mathbb{R}^d)$, we define their joint norm by
\begin{align*}
	\|f(\cdot),g(\cdot)\|_{X(\mathbb{R}^d)}=	\|f(\cdot)\|_{X(\mathbb{R}^d)}+	\|g(\cdot)\|_{X(\mathbb{R}^d)}
\end{align*}
adopting this concise notation for simplicity.
\end{enumerate}

To prove Theorem~\ref{thm}, the authors adapt the “frequency decomposition” method of Himonas–Misio\l{}ek. They decompose the velocity as
\[
 u^{\pm1,\lambda}(x,t)=u^{l,\pm1,\lambda}(x,t)+u^{h,\pm1,\lambda}(x,t),
\]
where (i) the low-frequency parts $u^{l,\pm1,\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\pm1,\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\lambda$ (in 2D, profile as in \eqref{1.2}; in 3D, modified as in \eqref{1.3}).

For resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\pm1,\lambda}, b^{\pm1,\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\pm1,\lambda}$ retains the oscillatory profile.

They then estimate the approximation errors induced by $u^{h,\pm1,\lambda}$, written as
\[
E^{\pm1,\lambda}=\partial_t u^{h,\pm1,\lambda}+u^{l,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{l,\pm1,\lambda},
\]
\[
F^{\pm1,\lambda}=u^{h,\pm1,\lambda}\cdot\nabla b^{\pm1,\lambda}-b^{\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}.
\]
Since “direct estimation of \eqref{1.5} will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\pm1,\lambda}=\mathrm{div}\,\tilde F^{\pm1,\lambda}$” and estimate $\tilde F^{\pm1,\lambda}$; (2) “transfer the divergence operator to $b^{\pm1,\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\Delta b^{\pm1,\lambda}$.”

For nonzero constant background fields $\mathbf{B_0}\neq\mathbf{0}$, they “introduce the coordinate transformation $x_2\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\mathbf{B_0}\cdot\nabla u^{h,\pm1,\lambda}$” in the error analysis of $F^{\pm1,\lambda}$. They also note the key decay bound
\[
\|\mathbf{B_0}\cdot\nabla u^{h,\pm1,\lambda}\|_{L^2}\le C\lambda^{-s-\delta}\ll \|u^{h,\pm1,\lambda}\|_{L^2}\le C\lambda^{-s}\quad (\lambda\to\infty),
\]
with $\delta>0$ giving “improved decay.”

Finally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\mathbf{B_0}=\mathbf{0}$ and $\mathbf{B_0}\neq\mathbf{0}$.

To prove the main theorem, the authors adapt the “frequency decomposition” method of Himonas–Misio\l{}ek. They decompose the velocity as
\[
 u^{\pm1,\lambda}(x,t)=u^{l,\pm1,\lambda}(x,t)+u^{h,\pm1,\lambda}(x,t),
\]
where (i) the low-frequency parts $u^{l,\pm1,\lambda}$ come from solving the (appropriate) equations with low-frequency initial data, and (ii) the high-frequency parts $u^{h,\pm1,\lambda}$ are “explicitly constructed by using oscillatory profiles,” with frequency parameter $\lambda$ (in 2D, profile as in
\begin{align}\label{1.2}
		u^{h,\pm1,\lambda}(x,t)=\nabla^\perp \left(\lambda^{-\delta-s-1}\phi\left(\dfrac{x_1}{\lambda^\delta}\right)\phi\left(\dfrac{x_2}{\lambda^\delta}\right)\sin (\lambda x_2\mp t)\right).
	\end{align}
; in 3D, modified as in \eqref{1.3}).

For resistive MHD, “due to the diffusive nature of the magnetic field evolution,” they “restrict high-frequency perturbations to the velocity field alone and maintain low-frequency components for the magnetic field”: the low-frequency pair $(u^{l,\pm1,\lambda}, b^{\pm1,\lambda})$ is obtained by solving the resistive MHD system with low-frequency initial data, while $u^{h,\pm1,\lambda}$ retains the oscillatory profile.

They then estimate the approximation errors induced by $u^{h,\pm1,\lambda}$, written as
\begin{align}\label{1.4}
	E^{\pm1,\lambda}&=\partial_t u^{h,\pm1,\lambda}+u^{l,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda}+u^{h,\pm1,\lambda}\cdot\nabla u^{l,\pm1,\lambda},\\\label{1.5}
	F^{\pm1,\lambda}&=u^{h,\pm1,\lambda}\cdot\nabla b^{\pm1,\lambda}-b^{\pm1,\lambda}\cdot\nabla u^{h,\pm1,\lambda},
\end{align}
Since “direct estimation of the equation above will lead to uncontrolled error terms,” their alternative consists of three steps: (1) “express $F^{\pm1,\lambda}=\mathrm{div}\,\tilde F^{\pm1,\lambda}$” and estimate $\tilde F^{\pm1,\lambda}$; (2) “transfer the divergence operator to $b^{\pm1,\lambda}$ via integration by parts” in $L^2$ inner products; (3) use “diffusion cancellation,” where “the resulting terms are precisely canceled by exploiting the magnetic diffusion term $\Delta b^{\pm1,\lambda}$.”

For nonzero constant background fields $\mathbf{B_0}\neq\mathbf{0}$, they “introduce the coordinate transformation $x_2\mapsto B_1x_2-B_2x_1$,” which “maintains all norm estimates” and yields “exact cancellation of the dominant high-frequency linear term $\mathbf{B_0}\cdot\nabla u^{h,\pm1,\lambda}$” in the error analysis of $F^{\pm1,\lambda}$. They also note the key decay bound
\[
\|\mathbf{B_0}\cdot\nabla u^{h,\pm1,\lambda}\|_{L^2}\le C\lambda^{-s-\delta}\ll \|u^{h,\pm1,\lambda}\|_{L^2}\le C\lambda^{-s}\quad (\lambda\to\infty),
\]
with $\delta>0$ giving “improved decay.”

Finally, they describe the proof strategy as: “construction of approximate solutions,” “precise estimation of approximation errors,” “rigorous construction of exact solutions,” and “finishing the proof” in the cases $\mathbf{B_0}=\mathbf{0}$ and $\mathbf{B_0}\neq\mathbf{0}$.

\label{thm}
	Let $s>0, \lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\mapsto(u,b)$ for the equations
\begin{align}\label{mhd}
	\begin{cases}
	\partial_t u+u \cdot \nabla u+\nabla p=b\cdot \nabla b,\\[1mm]
\partial_t b-\Delta b+u\cdot \nabla b=b\cdot\nabla u,  \\[1mm]
\text{div}\, u=\text{div}\, b=0, \\[1mm]\nu(x, 0)=u_0(x),\,\, b(x, 0)=b_0(x),
	\end{cases}
\end{align}
 is non-uniformly continuous from a bounded subset in $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C\left([0,T],H^{s}(\mathbb{R}^d)\right)\times C\left([0,T],H^{s}(\mathbb{R}^d)\right)$.

  More precisely, for any $\gamma>0$ and arbitrary constant magnetic field ${\bf B_0}\in \mathbb{R}^d$,  there exists two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
	\begin{itemize}
		\item the solutions satisfy
\begin{align*}
			(u^{\pm 1,\lambda}, b^{\pm 1,\lambda} - {\bf B_0}) \in C([0,T], H^{s}(\mathbb{R}^{d})) \times C([0,T], H^{s}(\mathbb{R}^{d}));
\end{align*}
		\item  the initial data satisfy
\begin{equation}\label{nonuniform-initial-bound}
	\begin{array}{l}
						\|u^{+1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{+1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma, \\[2mm]
			\|u^{-1,\lambda}(0,\cdot)\|_{H^{s}} + \|b^{-1,\lambda}(0,\cdot) - {\bf B_0}\|_{H^{s}} \leq \gamma;
	\end{array}
\end{equation}
		\item and the non-uniform continuity is characterized by
		\begin{itemize}
			\item at initial time $t=0$,
			\begin{align}\label{nonuniform-initial-diff}
				\lim_{\lambda\to\infty} \|u^{+1,\lambda}(0,\cdot) - u^{-1,\lambda}(0,\cdot)\|_{H^{s}} = 0,
			\end{align}
			\item for evolution times $t>0$,
			\begin{align}\label{nonuniform-low-bound-sint}
				\lim_{\lambda \to\infty} \|u^{+1,\lambda}(t) - u^{-1,\lambda}(t)\|_{H^{s}} \geq c\gamma |\sin t|,
			\end{align}
		\end{itemize}
		where $c = c(s,d) > 0$ is a constant depending only on  $s$ and $d$.
	\end{itemize}

Existential–Universal
Asymptotic or Limit

Question:
For the incompressible resistive MHD system on $\mathbb{R}^d$ with $d\in\{2,3\}$,
\[
\begin{cases}
\partial_t u+u\cdot\nabla u+\nabla p=b\cdot\nabla b,\\
\partial_t b-\Delta b+u\cdot\nabla b=b\cdot\nabla u,\\
\operatorname{div}u=\operatorname{div}b=0,\\
u(x,0)=u_0(x),\quad b(x,0)=b_0(x),
\end{cases}
\]
let the data-to-solution map send initial data $(u_0,b_0)$ to the corresponding solution $(u,b)$. Fix $s>0$ and $T>0$, and view this map as acting from a bounded subset of $H^s(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ into $C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d))$. Which statement holds, including the explicit sequential form of non-uniform continuity around an arbitrary constant background magnetic field ${\bf B}_0\in\mathbb{R}^d$?

The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C([0,T],H^{s}(\mathbb{R}^d))\times C([0,T],H^{s}(\mathbb{R}^d))$. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$, indexed by $\lambda\to\infty$, such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
with initial bounds
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\qquad
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
and such that
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for every evolution time $t>0$ (in particular, $t\in(0,T]$),
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}\ge c(s,d)\,\gamma\,|\sin t|,
\]
where $c(s,d)>0$ depends only on $s$ and $d$.

Choices:
B.
The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C([0,T],H^{s}(\mathbb{R}^d))\times C([0,T],H^{s}(\mathbb{R}^d))$. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$, indexed by $\lambda\to\infty$, such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
with the same initial bounds as in the theorem, and such that
\[
\lim_{\lambda\to\infty}\Big(\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-b^{-1,\lambda}(0)\|_{H^s}\Big)=0,
\]
while for every evolution time $t>0$,
\[
\lim_{\lambda\to\infty}\Big(\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}+\|b^{+1,\lambda}(t)-b^{-1,\lambda}(t)\|_{H^s}\Big)\ge c(s,d)\,\gamma\,|\sin t|,
\]
where $c(s,d)>0$ depends only on $s$ and $d$.
C.
The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C([0,T],H^{s}(\mathbb{R}^d))\times C([0,T],H^{s}(\mathbb{R}^d))$. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$, indexed by $\lambda\to\infty$, such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
with initial bounds
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\qquad
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
and such that
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0.
\]
In particular, the data-to-solution map fails to be uniformly continuous on that bounded subset.
D.
The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C([0,T],H^{s}(\mathbb{R}^d))\times C([0,T],H^{s}(\mathbb{R}^d))$. More precisely, there exists a constant $c=c(s,d)>0$ such that for every $\gamma>0$ one can choose two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ with the property that for every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$,
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
with the same initial bounds, and such that
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for every evolution time $t>0$,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}\ge c\,\gamma\,|\sin t|.
\]
E.
The data-to-solution map is non-uniformly continuous from a bounded subset of $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C([0,T],H^{s}(\mathbb{R}^d))\times C([0,T],H^{s}(\mathbb{R}^d))$. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$, indexed by $\lambda\to\infty$, such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
with the same initial bounds, and such that
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for every evolution time $t\in[0,T]$,
\[
\liminf_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}\ge c(s,d)\,\gamma,
\]
where $c(s,d)>0$ depends only on $s$ and $d$.

velocity-only high-frequency instability promoted to joint $(u,b)$ separation

dropped the explicit positive-time lower bound $\ge c\gamma|\sin t|$ and retained only failure of uniform continuity

quantifier order in dependence on ${\bf B}_0$

time-dependent lower bound $|\sin t|$ replaced by uniform-in-time positive separation on $[0,T]$


Question:
Consider the incompressible resistive MHD system on $\mathbb{R}^d$ with $d\in\{2,3\}$:
\[
\begin{cases}
\partial_t u+u\cdot\nabla u+\nabla p=b\cdot\nabla b,\\[1mm]
\partial_t b-\Delta b+u\cdot\nabla b=b\cdot\nabla u,\\[1mm]
\operatorname{div}u=\operatorname{div}b=0,\\[1mm]
u(x,0)=u_0(x),\quad b(x,0)=b_0(x).
\end{cases}
\]
Fix $s>0$ and $T>0$, and consider the data-to-solution map
\[(u_0,b_0)\mapsto (u,b)
\]
from a bounded subset of $H^s(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ into $C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d))$. Here one allows perturbations around an arbitrary constant background magnetic field ${\bf B}_0\in\mathbb{R}^d$, so that $b-{\bf B}_0\in H^s(\mathbb{R}^d)$. As the parameter $\lambda\to\infty$, which explicit asymptotic statement holds and gives the non-uniform continuity of this data-to-solution map?

The map is non-uniformly continuous. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
and their initial data satisfy
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
\[
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma.
\]
Moreover,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for evolution times $t>0$,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}\ge c\,\gamma\,|\sin t|,
\]
where $c=c(s,d)>0$ depends only on $s$ and $d$.

Choices:
B.
The map is non-uniformly continuous. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
and their initial data satisfy
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
\[
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma.
\]
Moreover,
\[
\lim_{\lambda\to\infty}\Big(\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-b^{-1,\lambda}(0)\|_{H^s}\Big)=0,
\]
while for evolution times $t>0$,
\[
\lim_{\lambda\to\infty}\Big(\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}+\|b^{+1,\lambda}(t)-b^{-1,\lambda}(t)\|_{H^s}\Big)\ge c\,\gamma\,|\sin t|,
\]
where $c=c(s,d)>0$ depends only on $s$ and $d$.
C.
The map is non-uniformly continuous. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
and their initial data satisfy
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
\[
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma.
\]
Moreover,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for evolution times $t>0$,
\[
\limsup_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}>0.
\]
D.
The map is non-uniformly continuous. More precisely, there exists $\gamma>0$ and a constant $c=c(s,d)>0$ such that for every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$ one can find two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ with
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
whose initial data satisfy the same bounds by $\gamma$, and
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0.
\]
In addition, for every evolution time $t\in(0,T]$,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(t)-u^{-1,\lambda}(t)\|_{H^s}\ge c\,\gamma\,|\sin t|.
\]
E.
The map is non-uniformly continuous. More precisely, for every $\gamma>0$ and every constant magnetic field ${\bf B}_0\in\mathbb{R}^d$, there exist two sequences of solutions $(u^{+1,\lambda},b^{+1,\lambda})$ and $(u^{-1,\lambda},b^{-1,\lambda})$ such that
\[
(u^{\pm1,\lambda},\, b^{\pm1,\lambda}-{\bf B}_0)\in C([0,T],H^s(\mathbb{R}^d))\times C([0,T],H^s(\mathbb{R}^d)),
\]
and their initial data satisfy
\[
\|u^{+1,\lambda}(0)\|_{H^s}+\|b^{+1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma,
\]
\[
\|u^{-1,\lambda}(0)\|_{H^s}+\|b^{-1,\lambda}(0)-{\bf B}_0\|_{H^s}\le \gamma.
\]
Moreover,
\[
\lim_{\lambda\to\infty}\|u^{+1,\lambda}(0)-u^{-1,\lambda}(0)\|_{H^s}=0,
\]
while for evolution times $t>0$,
\[
\lim_{\lambda\to\infty}\|b^{+1,\lambda}(t)-b^{-1,\lambda}(t)\|_{H^s}\ge c\,\gamma\,|\sin t|,
\]
where $c=c(s,d)>0$ depends only on $s$ and $d$.

instability asserted in the magnetic component as well as velocity

dropped the explicit lower bound $c\gamma|\sin t|$ and kept only positive limsup separation

quantifier order on $\gamma$ changed from for every $\gamma>0$ to there exists $\gamma>0$

placed the asymptotic lower bound on $b^{+1,\lambda}-b^{-1,\lambda}$ instead of on the velocity difference