\begin{thm} \label{thm:decaymap}
For all H\"older observables $v,w:\Sigma\to\R$, there is a constant
$C>0$ such that
$$
\left|\int_{\Sigma} v \cdot (w\circ g^n)\,d\mu_{\Sigma}-\int_{\Sigma}v\,d\mu_{\Sigma}\int_{\Sigma}w\,d\mu_{\Sigma}\right|
\leq C n^{-1}
\quad\text{for all $n\ge1$}.
$$
Moreover, for all H\"older observables $v,w:\Sigma\to\R$  supported in $\Sigma_0$,
\[
\int_{\Sigma} v \cdot (w\circ g^n)\,d\mu_{\Sigma}-\int_{\Sigma}v\,d\mu_{\Sigma}\int_{\Sigma}w\,d\mu_{\Sigma}\sim \bar\tau\sigma_R^2
n^{-1} \int_\Sigma v\,d\mu_\Sigma\, \int_\Sigma w\,d\mu_\Sigma \quad\text{for all $n\ge1$}.
\]
\end{thm}

\label{app:C}

For convenience, we list the Chernov axioms as stated in~\cite[Section 3]{LMM24}. A more in-depth discussion of the axioms, and where they differ from treatments in~\cite{Balint-Toth,Ch

\label{sec-intro}

It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.
Ergodicity with respect to volume was proved for surfaces in~\cite{Hopf39} and for general dimension in~\cite{Anosov67}. The Bernoulli property was shown in~\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\cite{DenkerPhilipp84,Ratner73}.
In major breakthroughs, exponential decay of correlations was established in~\cite{Dolgopyat98} for surfaces and~\cite{Liverani04} in general dimension.

In contrast, there are few results on statistical properties beyond the Bernoulli property for
geodesic flows on nonpositively curved manifolds when there exist points of zero curvature.
In~\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.
In particular, we considered a class of closed surfaces with one flat geodesic
and proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP
(with standard normalisation).
In this paper, we provide examples of geodesic flows on nonpositively curved surfaces
that satisfy a CLT and WIP with \emph{nonstandard normalisation}.

Our approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\cite{ChernovMarkarian}.
General results on statistical limit laws and decay of correlations seem infeasible,
but it is possible to analyse interesting classes of examples.
Moreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.
On the other hand, the prerequisites for the study of dispersing billiards are firmly established.
For geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.
A case in point is the smoothness of the foliations induced by the Green bundles and the necessity
to extend existing results such as those in~\cite{GerberNitica99,GerberWilkinson99}, see Section~\ref{sec:GW}.

\vspace{1ex}
The class of geodesic flows that we study in this paper is as follows.
Let $r\in[5,\infty)$ and fix $0<L<\ve_0$.
Define ${\mathcal R}$ to be the surface of revolution
with profile~$\xi$ given by
$\xi(s)=1$ for $|s|\leq L$ and $\xi(s)=1+(|s|-L)^r$ for $L\leq |s|\leq \ve_0$.
We form
a closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\beta_\pm$ bounding ${\mathcal R}$,
see Figure~\ref{fig:surface-intro}.
We call $S$ a \emph{surface with flat cylinder ${\mathcal C}=[-L,L]\times{\mathbb S}^1$}, where
${\mathbb S}^1={\mathbb R}/(2\pi{\mathbb Z})$.
Away from~${\mathcal C}$ the surface $S$ has negative curvature.

\begin{figure}[hbt!]
\centering
\def\svgwidth{11cm}
   \input{nonpositive_curvature-intro.pdf_tex}
\caption{Surface
with flat cylinder ${\mathcal C}$.
The region ${\mathcal R}$ between the two curves $\beta_\pm$ is a surface of revolution
with profile $\xi$.}
\label{fig:surface-intro}
\end{figure}

Let $g_t:M\to M$ denote the geodesic flow on the unit tangent bundle
$M=T^1S$, and let $\mu$ denote the normalised Riemannian volume on the
three-dimensional manifold $M$. Then $\mu$ is an ergodic $g_t$-invariant
probability measure \cite{Pesin77b}.

The flat cylinder ${\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to{\mathbb R}$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to{\mathbb R}$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

It is well-known that geodesic flows on negatively curved closed manifolds have very strong statistical properties.
Ergodicity with respect to volume was proved for surfaces in~\cite{Hopf39} and for general dimension in~\cite{Anosov67}. The Bernoulli property was shown in~\cite{OrnsteinWeiss73,Ratner74}. Statistical limit laws such as the central limit theorem (CLT), the weak invariance principle (WIP), also known as the functional CLT, and the almost sure invariance principle quickly followed~\cite{DenkerPhilipp84,Ratner73}.
In major breakthroughs, exponential decay of correlations was established in~\cite{Dolgopyat98} for surfaces and~\cite{Liverani04} in general dimension.

In contrast, there are few results on statistical properties beyond the Bernoulli property for
geodesic flows on nonpositively curved manifolds when there exist points of zero curvature.
In~\cite{LMM24}, we initiated the systematic study of such properties for geodesic flows.
In particular, we considered a class of closed surfaces with one flat geodesic
and proved polynomial decay of correlations and various statistical limit laws including the CLT and WIP
(with standard normalisation).
In this paper, we provide examples of geodesic flows on nonpositively curved surfaces
that satisfy a CLT and WIP with \emph{nonstandard normalisation}.

Our approach to studying geodesic flows on nonpositively curved manifolds resembles that for (semi)dispersing billiards~\cite{ChernovMarkarian}.
General results on statistical limit laws and decay of correlations seem infeasible,
but it is possible to analyse interesting classes of examples.
Moreover, whereas rigorous results for dispersing billiards have been restricted to planar billiards due to the complicated structure of singularities under iteration, there is in principle no such restriction for geodesic flows since they are smooth.
On the other hand, the prerequisites for the study of dispersing billiards are firmly established.
For geodesic flows on nonpositively curved manifolds, the corresponding prerequisites are still under development, so for the moment we focus on surfaces too.
A case in point is the smoothness of the foliations induced by the Green bundles and the necessity
to extend existing results such as those in~\cite{GerberNitica99,GerberWilkinson99}, see Section~\ref{sec:GW}.

\vspace{1ex}
The class of geodesic flows that we study in this paper is as follows.
Let $r\in[5,\infty)$ and fix $0<L<\ve_0$.
Define ${\mathcal R}$ to be the surface of revolution
with profile~$\xi$ given by
$\xi(s)=1$ for $|s|\leq L$ and $\xi(s)=1+(|s|-L)^r$ for $L\leq |s|\leq \ve_0$.
We form
a closed $C^4$ Riemannian surface $S$ of nonpositive curvature by isometrically gluing two negatively curved surfaces with boundary to the circles $\beta_\pm$ bounding ${\mathcal R}$,
see Figure~\ref{fig:surface-intro}.
We call $S$ a \emph{surface with flat cylinder ${\mathcal C}=[-L,L]\times{\mathbb S}^1$}, where
${\mathbb S}^1={\mathbb R}/(2\pi{\mathbb Z})$.
Away from~${\mathcal C}$ the surface $S$ has negative curvature.

Let $g_t:M\to M$ denote the geodesic flow on the unit tangent bundle
$M=T^1S$, and let $\mu$ denote the normalised Riemannian volume on the
three-dimensional manifold $M$. Then $\mu$ is an ergodic $g_t$-invariant
probability measure \cite{Pesin77b}.

The flat cylinder ${\mathcal C}$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to{\mathbb R}$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to{\mathbb R}$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

\begin{equation} \label{eq:alpha}
\alpha_0=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta, \qquad
\alpha_\pi=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta.
\end{equation}

\begin{figure}[hbt!]
\centering
\def\svgwidth{11cm}
   \input{nonpositive_curvature-intro.pdf_tex}
\caption{Surface
with flat cylinder $\cC$.
The region $\cR$ between the two curves $\beta_\pm$ is a surface of revolution
with profile $\xi$.}
\label{fig:surface-intro}
\end{figure}

The flat cylinder $\cC$ is foliated by two families of periodic orbits with clockwise/counterclockwise orientation.
Let $v:M\to\R$ be H\"older with $\int_M v\,d\mu=0$. We define $I_v=\alpha_0^2+\alpha_\pi^2$ where
$\alpha_0$ and $\alpha_\pi$ are the averages of $v$ over these two families of periodic orbits.
(The notation derives from the explicit formulas~\eqref{eq:alpha} given in Section~\ref{sec:WIP}.)
For $t>0$, define $v_t=\int_0^t v\circ g_s\,ds:M\to\R$.
Our first main result is a CLT for $v_t$ with standard normalisation $t^{1/2}$ or nonstandard normalisation $(t\log t)^{1/2}$ depending on whether $I_v=0$ or $I_v>0$.

\begin{rmk} As in~\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\"older observables $v$ with $\int_M v\,d\mu=I_v=0$. See for example the discussion in~\cite[End of Section~4]{HM07}.

\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}

\begin{rmk} \label{rmk:lower}
 The upper bound in Theorem~\ref{thm:decay} is sharp in the sense that if $v$ is H\"older with $\int_Mv\,d\mu=0$ and $I_v\neq0$, then
\[
t\int_M v \cdot (v\circ g_t)\,d\mu\not\to0
\quad\text{as $t\to\infty$}.
\]
This is a consequence of the nonstandard limit law in Theorem~\ref{thm:CLT}.
See~\cite[Corollary~1.3]{BalintGouezel06} or~\cite[Proposition~9.14]{BBM19}.
\end{rmk}

\begin{lemma}\label{lem:hyperbolicity}
Let $r\ge 5$, $\omega\in(1,r/2)$. The following are true:
\begin{description}
\item[{\rm (1)}] Lebesgue almost every $x\in\Sigma_0$ has an LSM/LUM $W^{s/u}_x$ for $f$.
\item[{\rm (2)}]
For all $x,\overline{x}\in \Sigma_{\rm in}$,
\begin{enumerate}
\item[{\rm (a)}] There is a continuous function $a:\Sigma_{\rm in}\to\R$ such that
$E^u_x$ is spanned by $\begin{bmatrix}a(x) \\ 1\end{bmatrix}$.
Moreover,
$|a(x)-a(\bar x)|\ll |\log d(x,\overline{x})|^{-\omega}$.
\item[{\rm (b)}] There is a $C^{1+{\rm Lip}}$ function $\Theta$ such that
$W^u_x$ is locally the graph $\{(\Theta(\psi),\psi)\}$ of $\Theta$.
\end{enumerate}
\item[{\rm (3)} Growth bounds:] {\color{white} a}
\begin{enumerate}
\item[{\rm (a)}] If $x\in\mfD_n^>$, then $\|df|_{E^u_x}\|\approx n^{3-\frac{2}{r}}$.
\item[{\rm (b)}] If $x\in\mfD_n^<$, then $\|df|_{E^u_x}\|\approx n^3$.
\end{enumerate}
\item[{\rm (4)} Distortion bounds:] If $x,\overline{x}\in\mfD_n^>$ or $x,\overline{x}\in\mfD_n^<$,
with $\overline{x}\in W^u_x$, then
$$
\big|\log \|df|_{E^u_x}\|-\log \|df|_{E^u_{\overline x}}\|\big| \ll
d(fx, f \overline{x})^\frac13.
$$
\item[{\rm (5)} Jacobian of holonomies:]
If $x,\overline{x}\in\mfD_n^>$ or $x,\overline{x}\in\mfD_n^<$,
with $\overline{x}\in W^s_x$, then
\[
\big|\log \|df|_{E^u_x}\|-\log \|df|_{E^u_{\overline{x}}}\|\big|\ll |\log d(x,\overline{x})|^{-\omega}.
\]
\end{description}
\end{lemma}

\begin{thm} \label{thm:JR}
Let $\{\alpha_i:i\in I\}\subset\R$ be a list of the values attained by $J$.
Let $\sigma_i\ge0$ be constants, $i\in I$, such that
$\sum_{i\in I}\sigma_i^2\in(0,\infty)$.
Assume that there are constants $C>0$, $\eps\in(0,1)$ such that
\begin{equation} \label{eq:JR1}
 \sum_{i\in I}|\mu_\Delta(R_0=n,\,J=\alpha_i)-2\sigma_i^2 n^{-3}|\le Cn^{-(3+\eps)}
\quad\text{for all $n\ge1$},
\end{equation}
and
\begin{equation} \label{eq:JR2}
\mu_\Delta(R_0=k,\,R_0\circ f_\Delta^n=\ell)\le Ck^{-(2+\eps)}\ell^{-(2+\eps)}
\quad\text{for all $k,\ell,n\ge1$}.
\end{equation}
Then $JR_0$ satisfies a nonstandard CLT with variance
$\sigma_J^2=\sum_{i\in I}\alpha_i^2\sigma_i^2\ge0$. That is,
$(n\log n)^{-1/2}\sum_{j=0}^{n-1}(JR_0)\circ f_\Delta^j\to_d N(0,\sigma_J^2)$.
\end{thm}

\begin{cor} \label{cor:RCLT}
$R:\Delta\to\Z^+$ satisfies a nonstandard CLT
on $(\Delta,f_\Delta,\mu_\Delta)$. That is,
$(n\log n)^{-1/2}(\sum_{j=0}^{n-1}R\circ f_\Delta^j-n\bar R)\to_d N(0,\sigma_R^2)$ where
$\bar R=\int_\Delta R\,d\mu_\Delta$.
\end{cor}

In this section, we consider decay of correlations for the geodesic flow $g_t:M\to M$ and the global Poincar\'e map $g:\Sigma\to\Sigma$.
The general setup is the same as in~\cite[Section~7]{LMM24}.
We recall the basics here.\footnote{We caution that for example $\tau$ here was called $\sigma$ in~\cite{LMM24}.}
\begin{enumerate}[$\bullet$]
\item The flow $g_t$ and map $g$ are related by the identity
$g=g_h$ where $h:\Sigma\to\R^+$ is a H\"older roof function with $0<\inf h<\sup h<\infty$.
\item
The first return maps $g:\Sigma\to\Sigma$ and $f:\Sigma_0\to\Sigma_0$ are related by $f=g^R$ where $R:\Sigma_0\to\Z^+$ is a discrete first return time.
\item
The uniformly hyperbolic first return map $f:\Sigma_0\to\Sigma_0$ is modelled by a Young tower with exponential tails~\cite{Young98}: there is a subset $Y\subset\Sigma_0$ with product structure and bounded distortion, a probability measure $\mu_Y$ on $Y$, and a ``good'' return time $\tau:Y\to\Z^+$ constant along stable leaves with $\mu_Y(\tau>n)=O(e^{-cn})$ for some $c>0$, such that $f^\tau:Y\to Y$ is measure-preserving. Moreover, quotienting along stable leaves yields a full-branch Gibbs-Markov map $F:Z\to Z$ as in Section~\ref{sec:CLT}, and $\tau$ is constant on the associated partition elements.
\item
The global Poincar\'e map $g:\Sigma\to\Sigma$ is modelled by a subexponential Young tower over $g^{\varphi_*}=(g^R)^\tau=f^\tau:Y\to Y$ where the return time $\varphi_*:Y\to\Z^+$ is given by $\varphi_*=\sum_{\ell=0}^{\tau-1}R\circ f^\ell$.
\item
We can regard $f^\tau:Y\to Y$ as a (non-first return) Poincar\'e map for the flow $g_t$ with roof function $\varphi=h_{\varphi_*}=\sum_{\ell=0}^{\varphi_*-1}h\circ g^\ell$.
\end{enumerate}

\begin{equation} \label{eq:alpha}
\alpha_0=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta, \qquad
\alpha_\pi=\frac{1}{2L}\int_{\bbS^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta.
\end{equation}

\begin{ltheorem}[Nonstandard/Standard CLT] \label{thm:CLT}
If $I_v>0$, then $v$ satisfies a nonstandard CLT. That is,
$(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)$ as $t\to\infty$,
where $\sigma_v^2=b_SI_v>0$ and $b_S$ is a positive constant depending only on the surface $S$.

If $I_v=0$, then there exists $\sigma^2\ge0$, typically nonzero, such that
$v$ satisfies a standard CLT. That is,
$t^{-1/2}v_t \to_d N(0,\sigma^2)$ as $t\to\infty$.
\end{ltheorem}

\begin{ltheorem} \label{thm:decay}
 Let $v,\,w:M\to\R$ be sufficiently smooth observables. Then there is a constant
$C>0$ such that
$$
\left|\int_M v \cdot (w\circ g_t)\,d\mu-\int_Mv\,d\mu\int_Mw\,d\mu\right|
\leq C t^{-1}
\quad\text{for all $t>0$}.
$$
\end{ltheorem}

\begin{rmk} As in~\cite[Remark~4.6]{LMMsub}, the word ``typically'' is interpreted here in the very strong sense that $\sigma^2=0$ only within a closed subspace of infinite codimension amongst H\"older observables $v$ with $\int_M v\,d\mu=I_v=0$. See for example the discussion in~\cite[End of Section~4]{HM07}.

We note that the vanishing of $\int_Mv\,d\mu$ is ``without loss of generality'' since one can always centre the observable $v$ by considering $v-\int_Mv\,d\mu$.
The vanishing of $I_v$ is ``codimension two'' since $\alpha_0=\alpha_\pi=0$
may occur in generic two-parameter families. In contrast, the degenerate case $\sigma^2=0$
requires infinitely many coincidences beyond the constraint $I_v=0$.
\end{rmk}

In the case $I_v>0$, we also prove a nonstandard WIP (which implies the CLT).
Define the sequence of continuous functions
\[
W_n\in C[0,1], \qquad W_n(t)=(n\log n)^{-1/2}v_{nt},\quad n\ge1.
\]
Since each function $W_n$ depends on the initial condition in the probability space $(M,\mu)$, we
can regard $W_n$ as a random element in the metric space $(C[0,1],\|\;\|_\infty)$.

\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}

There are only a few situations in deterministic dynamical systems where
the nonstandard CLT/WIP is known to hold.
The first example was for intermittent maps $f$ satisfying $f(x)=x+bx^{3/2}$ at a neutral fixed point $x=0$; see~\cite{Gouezel04} for the nonstandard CLT and~\cite{DedeckerMerlevede09} for the nonstandard WIP.
For various billiard examples, namely the infinite horizon Lorentz gas, Bunimovich stadium and billiards with cusps, we refer to~\cite{BalintChernovDolgopyat11,BalintGouezel06,LMMsub,SzaszVarju07}.

Our proofs of Theorems~\ref{thm:CLT} and~\ref{thm:WIP} closely follows abstract arguments in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}, using the latter to reduce to a nonstandard CLT for a simpler class of observables, and using the former to establish this simplified nonstandard CLT.

Our final main result concerns the rate of decay of correlations for the flow $g_t$.

\begin{ltheorem} \label{thm:decay}
 Let $v,\,w:M\to{\mathbb R}$ be sufficiently smooth observables. Then there is a constant
$C>0$ such that
$$
\left|\int_M v \cdot (w\circ g_t)\,d\mu-\int_Mv\,d\mu\int_Mw\,d\mu\right|
\leq C t^{-1}
\quad\text{for all $t>0$}.
$$
\end{ltheorem}

\begin{rmk} An observable is ``sufficiently smooth'' if it is $C^k$ in the flow direction for some $k\ge0$ that depends only on the surface $S$. In
particular, if we choose the surface $S$ to be $C^\infty$ away from ${\mathcal C}$, then any observable that is $C^\infty$ on $M$ and flat near $\{s=\pm L\}$ is sufficiently smooth.

The analogous result for the infinite horizon Lorentz gas and Bunimovich stadia  was obtained in~\cite{BBM19}.
\end{rmk}

\begin{rmk} \label{rmk:lower}
 The upper bound in Theorem~\ref{thm:decay} is sharp in the sense that if $v$ is H\"older with $\int_Mv\,d\mu=0$ and $I_v\neq0$,  then
\[
t\int_M v \cdot (v\circ g_t)\,d\mu\not\to0
\quad\text{as $t\to\infty$}.
\]
This is a consequence of the nonstandard limit law in Theorem~\ref{thm:CLT}.
See~\cite[Corollary~1.3]{BalintGouezel06} or~\cite[Proposition~9.14]{BBM19}.
\end{rmk}

The remainder of this paper is organised as follows.
In Section \ref{sec:surfaces}, we review known facts about the geometry and dynamics of
geodesic flows on nonpositively curved surfaces, with special attention to surfaces with flat cylinder.
In Section \ref{sec:R-dynamics}, we make a systematic study of the dynamics of the geodesic flow on the surface of revolution ${\mathcal R}$.
In Section~\ref{sec:GW}, we prove sufficient regularity of the stable/unstable foliations
induced by the Green bundles, partly extending~\cite{GerberWilkinson99}.
In Section~\ref{sec:f}, we construct a uniformly hyperbolic first return map $f:\Sigma_0\to\Sigma_0$ satisfying the Chernov axioms~\cite{Chernov-1999}, showing that $f$ is modelled by a Young tower with exponential tails~\cite{Young98}.

In Section~\ref{sec:CLT}, we recall and slightly extend an abstract result on the nonstandard CLT due to~\cite{BalintChernovDolgopyat11}.
In Section~\ref{sec:WIP}, we prove Theorems~\ref{thm:CLT} and~\ref{thm:WIP}.
Finally, in Section~\ref{sec:decay} we prove Theorem~\ref{thm:decay} as well as a related result, Theorem~\ref{thm:decaymap}, for a suitable global Poincar\'e map $g$.
The Chernov axioms are stated for convenience in Appendix~\ref{app:C}.

\vspace{-2ex}
\paragraph{Notation}
We use ``big O'' and $\ll$ notation interchangeably, writing $a_n=O(b_n)$ or $a_n\ll b_n$
if there are constants $C>0$, $n_0\ge1$ such that
$a_n\le Cb_n$ for all $n\ge n_0$.
We write $a_n\approx b_n$ if $a_n\ll b_n$ and $b_n\ll a_n$, and
$a_n\sim b_n$ if $a_n/b_n\to1$.

Our proofs of Theorems~\ref{thm:CLT} and~\ref{thm:WIP} \emph{closely follows abstract arguments} in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}: using~\cite{LMMsub} to \emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\cite{BalintChernovDolgopyat11} to \emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove a nonstandard WIP (which implies the CLT), formulated via $W_n(t)=(n\log n)^{-1/2}v_{nt}$ and convergence $W_n\to_w W$ in $C[0,1]$ to Brownian motion with variance $\sigma_v^2$.

Our proofs of To prove the main theorem,  and
\begin{ltheorem}[Nonstandard WIP] \label{thm:WIP}
If $I_v>0$, then $v$ satisfies a nonstandard WIP. That is,
$W_n \to_w W$ in $C[0,1]$ as $n\to\infty$,
where $W$ is a Brownian motion with variance $\sigma_v^2$.
\end{ltheorem}
\emph{closely follows abstract arguments} in~\cite{BalintChernovDolgopyat11} and~\cite{LMMsub}: using~\cite{LMMsub} to \emph{reduce to a nonstandard CLT for a simpler class of observables}, and using~\cite{BalintChernovDolgopyat11} to \emph{establish this simplified nonstandard CLT}. In the case $I_v>0$, they also prove the preceding theorem (which implies the CLT), formulated via $W_n(t)=(n\log n)^{-1/2}v_{nt}$ and convergence $W_n\to_w W$ in $C[0,1]$ to Brownian motion with variance $\sigma_v^2$.

Question:
Let $S$ be a closed $C^4$ Riemannian surface of nonpositive curvature obtained by attaching negatively curved boundary pieces to a flat cylinder ${\mathcal C}=[-L,L]\times \mathbb S^1$, where $0<L<\varepsilon_0$, $r\in[5,\infty)$, and the cylinder comes from a surface of revolution with profile $\xi(s)=1$ for $|s|\le L$ and $\xi(s)=1+(|s|-L)^r$ for $L\le |s|\le \varepsilon_0$. Let $g_t:M\to M$ be the geodesic flow on the unit tangent bundle $M=T^1S$, let $\mu$ be the normalised Riemannian volume on $M$, and let $v:M\to\mathbb R$ be a H\"older observable with $\int_M v\,d\mu=0$. For $t>0$, define
\[
v_t=\int_0^t v\circ g_s\,ds.
\]
The flat cylinder is foliated by two families of periodic geodesics (clockwise and counterclockwise), and define
\[
\alpha_0=\frac1{2L}\int_{\mathbb S^1}\int_{-L}^L v(s,\theta,0)\,ds\,d\theta,
\qquad
\alpha_\pi=\frac1{2L}\int_{\mathbb S^1}\int_{-L}^L v(s,\theta,\pi)\,ds\,d\theta,
\]
with
\[
I_v=\alpha_0^2+\alpha_\pi^2.
\]
Which conclusion about the asymptotic distribution of $v_t$ holds under these hypotheses?

If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)\quad\text{as }t\to\infty,
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\sigma^2\ge 0$, typically nonzero, such that $v$ satisfies the standard central limit theorem:
\[
t^{-1/2}v_t \to_d N(0,\sigma^2)\quad\text{as }t\to\infty.
\]

Choices:
B.
If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t)^{-1/2}v_t \to_d N(0,\sigma_v^2)\quad\text{as }t\to\infty,
\]
where $\sigma_v^2=b_S I_v>0$ and $b_S$ is a positive constant depending only on the surface $S$. If $I_v=0$, then there exists $\sigma^2\ge 0$ such that
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma^2)\quad\text{as }t\to\infty.
\]
C.
If $I_v>0$, then $v$ satisfies a nonstandard central limit theorem:
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)\quad\text{as }t\to\infty,
\]
for some variance $\sigma_v^2>0$. If $I_v=0$, then there exists $\sigma^2\ge 0$ such that
\[
t^{-1/2}v_t \to_d N(0,\sigma^2)\quad\text{as }t\to\infty.
\]
D.
There exists a positive constant $b>0$, independent of the surface $S$ and the observable $v$, such that if $I_v>0$, then
\[
(t\log t)^{-1/2}v_t \to_d N(0,b I_v)\quad\text{as }t\to\infty.
\]
If $I_v=0$, then $v$ still satisfies the standard central limit theorem
\[
t^{-1/2}v_t \to_d N(0,\sigma^2)\quad\text{as }t\to\infty
\]
for some $\sigma^2\ge 0$.
E.
If $I_v>0$, then $v$ satisfies the stronger functional limit law
\[
(t\log t)^{-1/2}v_t \to_d N(0,\sigma_v^2)\quad\text{as }t\to\infty,
\]
with $\sigma_v^2=b_SI_v>0$, and if $I_v=0$, then necessarily $\sigma^2=0$, so that
\[
t^{-1/2}v_t \to_d 0\quad\text{as }t\to\infty.
\]

normalisation split between the cases $I_v>0$ and $I_v=0$

explicit identification $\sigma_v^2=b_S I_v$ with $b_S$ depending only on $S$

dependence of the variance constant on the surface $S$

the $I_v=0$ case allows arbitrary $\sigma^2\ge 0$, typically nonzero

The stem does not explicitly state the conclusion or the correct normalization/variance formula. It introduces the geometric setup and the quantity I_v, but the actual limit law must still be selected from the options.

This is essentially a direct theorem-recall item: the hypotheses are spelled out in full and the task is to identify the theorem’s conclusion. It is very close to a restatement rather than an application or inference problem.

There is some discrimination required among subtly different conclusions (normalization, dependence on S, and the I_v=0 case), especially because one distractor is a weaker true-looking statement. However, the item mainly tests precise recall of a known result rather than genuine generative reasoning from the given data.

The distractors are mathematically plausible and target realistic failure modes: swapping the standard/nonstandard scalings, weakening the variance identification, asserting incorrect uniformity in constants, or forcing degeneracy in the I_v=0 case.

Well-constructed theorem-recall MCQ with strong distractors and no direct answer leakage, but it is largely tautological and only moderately tests reasoning.


In recent years the pluriclosed flow \cite{PCF, PCFReg} and generalized K\"ahler-Ricci flow \cite{apostolov2022generalized,StreetsSTB,GKRF} have been developed as a tool for understanding the geometry of complex, especially non-K\"ahler, manifolds \cite{barbaro2023bismut,barbaro2025global,barbaro2025pluriclosed,fino2024pluriclosed,fusi2024pluriclosed,garcia2023non, ye2024pluriclosed}. A natural class of non-K\"ahler manifolds are the Oeljeklaus-Toma (OT) manifolds \cite{oeljeklaus2005non}, whose geometry is linked to the structure of number fields, and which are natural higher dimensional generalizations of Inoue surfaces \cite{inoue1974surfaces}. In \cite{fusi2024pluriclosed} a complete description of the pluriclosed flow with left-invariant initial data on OT manifolds was obtained, in particular showing that the solution exists for all time and collapses after blowdown to a torus in the Gromov-Hausdorff sense. Moreover the blowdown on the universal cover converges in the Cheeger-Gromov sense to a soliton. It is natural to conjecture that these statements hold for arbitrary initial data (cf. Conjecture \ref{c:OT_PCF}). In this work we confirm some aspects of this conjecture.

The first main result is to establish the global existence of the flow:

\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

The first main result is to establish the global existence of the flow:

The expected qualitative behavior of the normalized pluriclosed flow on OT manifolds is captured by solutions with initial data the model metrics $\gw_{h}^{a,b}$. Straightforward computations show that the normalized pluriclosed flow with this initial data is
\begin{align}\label{e:normalized equation}
 \omega^{a,b}_h(t) = \sum_{i=1}^s \sqrt{-1}((1-e^{-t})\frac{3}{4}+e^{-t}a_i)\frac{1}{(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i + \sqrt{-1}e^{-t}b_i\operatorname{Im}w_i dz_i\wedge d\bar{z}_i.
\end{align}
Later, we shall denote the time-dependent normalized model metric starting with $\omega^{a,b}_h$ as $\omega^{a,b}_h(t)$. We shall write $\omega_h(t)$ for short if $a_i$, $b_i=1$, for all $1\leq i\leq s$.
Observe that for these model solutions the Chern torsion $T$ is uniformly bounded in time, i.e., $|T(t)|\leq C$. Moreover, it follows that
\begin{align*}
 \frac{\omega_h^{a,b}(t)}{t+1}\to \sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i,
\end{align*}
which can be considered as a degenerate metric on $X(K, U)$. As explained in \cite{fusi2024pluriclosed}, the blowdown manifolds converge in Gromov-Hausdorff sense to a torus $\mathbb T^s$ with a canonical flat metric $d(K, U)$ depending only on the algebraic field $K$ and the rank $s$ subgroup $U$. We recall the result here, noting that OT manifolds are compact solvmanifolds and the model metrics are left-invariant:
\begin{thm} \label{t:homogOT} (\cite{fusi2024pluriclosed}) Let $\omega_0$ be a left-invariant pluriclosed metric on an OT manifold $M$, then the normalized pluriclosed flow starting with $\omega_0$ converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{thm}

\noindent It is reasonable to conjecture that this behavior holds in the general case:
\begin{conj} \label{c:OT_PCF} Let $M = X(K,U)$ be an OT manifold of type $(s,s)$, then for any pluriclosed metric $\omega_0$, the normalized pluriclosed flow \ref{e:PCF_normalized} with initial metric $\omega_0$ exists on $[0, \infty)$, and converges to $(\mathbb{T}^s,d(K, U))$ in the Gromov-Hausdorff sense.
\end{conj}

We shall prove the first part of the conjecture in Section \ref{s:long time section}. For the second half of the conjecture, we have the following sufficiency condition.
\begin{prop}\label{p:GH convergence argument}
 Let $\omega(t)$ be the normalized pluriclosed flow solution on the OT manifold $X(K,U)$. Suppose that there exists a constant $C>0$ such that
 \begin{itemize}
 \item $C^{-1}\leq \operatorname{tr}_{\omega(t)}\omega_h(t)\leq C$,
 \item $\lim\limits_{t\to\infty}\operatorname{tr}_{\omega(t)}\omega_h(t) = 1$ for all $x\in X(K,U)$.
 \end{itemize}
 Then, we have
 \begin{align*}
 (X(K,U),\omega(t))\to (\mathbb{T}^s,d(K, U)),
\end{align*}
to the flat metric $d(K,U)$ in the Gromov-Hausdorff sense.
\end{prop}
\begin{proof} We give a brief sketch as the result is already essentially contained in e.g. \cite{fusi2024pluriclosed}, \cite{ZhengOT}. Note that for OT manifolds, there is a canonical fibration map:
 \begin{align*}
 F:X(K,U)\to \mathbb{T}^s
 \end{align*}
 where the fiber is diffeomorphic to $\mathbb{T}^{3s}$. In particular, $E_\mathbb{C}$ will be in the kernel of $dF$. Since the quotient of $\{z\}\times \mathbb{C}^s$ is dense in the $\mathbb{T}^{3s}$ fiber (See Section 2 of \cite{VerbiOT}), the degenerate metric $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$ will induce a metric on the base $\mathbb{T}^s$. Let $d(K,U)$ be the metric on $\mathbb{T}^s$ induced from $\sum_{i=1}^s \frac{3}{4(\operatorname{Im}w_i)^2}dw_i\wedge d\bar{w}_i$, which is flat.

Secondly, when $s = 1$, then $E_\mathbb{C}$ and $E_\mathbb{H}$ are holomorphic line bundles. Thus, the AM-GM inequality becomes equality in this case, and we can have the following lower bound of $\dot{\phi}$.
\begin{lemma}\label{l:potential derivative lower bound}
 On Inoue surface $S_M$, for the potential $\phi$, there exists a constant $C>0$, such that:
 \begin{align*}
 \dot{\phi}\geq -C.
 \end{align*}
\end{lemma}
\begin{proof}
 Note $s=1$. Choose a large constant $\Lambda>0$, consider the quantity $\dot{\phi} + \phi+\frac{1}{\Lambda}\phi$. Then
 \begin{align*}
 (\frac{\partial}{\partial t} - \Delta)(\dot{\phi}+\phi+\frac{1}{\Lambda}\phi) = 1 +\frac{1}{\Lambda}\dot{\phi}+ \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C} - \frac{1}{\Lambda}e^{-t}\operatorname{tr}_{g_\mathbb{H}}g^0_\mathbb{H} - (\frac{1}{\Lambda}(1-e^{-t})+1)\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}.
 \end{align*}
 Now, for $A>0$, a very large constant, such that $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi < -A$. By \eqref{e:potential eq}, notice that in this case $s=1$, we know:
 \begin{align*}
 \dot{\phi} +\phi = \log\frac{\det g_\mathbb{C}/c_1}{\det e^{-t}h_\mathbb{C}} - \log\frac{\det g_\mathbb{H}}{\det \frac{3}{4}h_\mathbb{H}} = -\log(c_1e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}) + \log (\frac{3}{4}\operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}).
 \end{align*}
 By Corollary \ref{c:Inoue C0}, in particular $|\phi|\leq C_0$. Then, there is a large constant $A' = A- C_0>0$ such that $\dot{\phi}\leq -A'$, and
 \begin{align*}
 \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}e^{-t}c_1 \operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C}\leq \frac{4}{3}e^{\frac{C_0}{\Lambda}-A}c_0e^{-t}\operatorname{tr}_{g_\mathbb{C}}g^0_\mathbb{C},
 \end{align*}
 for some constant $c_0$, only depends on the initial metric.
 By choosing $\Lambda$ large, such that $1-\frac{C_0}{\Lambda}>\frac{2}{3}$, fixed, when $A$ is large enough, $c_0e^{\frac{C_0}{\Lambda}-A} \leq \frac{1}{\Lambda}$. Thus, choose an $A = \frac{\Lambda}{2}$ large enough, such that $\dot{\phi}(t)\geq -\frac{1}{2}A$, for $t\in[0,T]$, where $T$ is large enough. Then for the first time $\dot{\phi}+\phi +\frac{1}{\Lambda}\phi = -A$, we have:
 \begin{align*}
 (\frac{\partial}{\partial t}-\Delta)(\dot{\phi} + \phi +\frac{1}{\Lambda}\phi)&\geq (\frac{1}{\Lambda } - \frac{C_1e^{-t}}{\Lambda}-c_0e^{\frac{C_0}{\Lambda}-A})e^{-t}\operatorname{tr}_{g_\mathbb{C}}h_\mathbb{C} + 1+ \frac{1}{\Lambda}\dot{\phi}\\
 &>0.
 \end{align*}
 By the maximum principle, the result follows.
\end{proof}
Combine with the potential derivative upper bound (Lemma \ref{l:potential derivative upper bound}), we have the following metric lower bound.
\begin{thm}\label{t:Inoue metric lower bound}
 On an Inoue surface $S_M$, for generalized K\"ahler metric $\omega_0$, there is a constant $C$, such that the normalized pluriclosed flow solution $\omega(t)$with initial data $\gw_0$ will satisfy
 \begin{align*}
 \omega(t)\geq C\omega_h(t),
 \end{align*}
 where $\omega_h(t)$ is the model flow.
\end{thm}
\begin{proof}
 By Lemma \ref{l:potential derivative upper bound} and Lemma \ref{l:potential derivative lower bound}, we have that:
 \begin{align*}
 -C\leq\dot{\phi} +\phi \leq C
 \end{align*}
 for some constant $C$. Now, from the equation \eqref{e:potential eq} and $s=1$, we have:
 \begin{align*}
 -C\leq \operatorname{tr}_{g_\mathbb{H}}h_\mathbb{H}/\operatorname{tr}_{g_\mathbb{C}}e^{-t}h_\mathbb{C}\leq C.
 \end{align*}
 Note that in the normalized pluriclosed flow model case, \eqref{e:normalized equation}, the $E_\mathbb{C}$ part will shrink at the rate of $e^{-t}$, while the $E_\mathbb{H}$ part will be equivalent to $h_\mathbb{H}$. Thus, by Corollary \ref{c:lower z}, we have the desired lower bound for $E_\mathbb{C}$.
\end{proof}
\begin{proof} [Proof of Theorem \ref{t:mainthm2}]
Combine Lemma \ref{l:C0 bound}, Corollary \ref{c:imporved C0}, Lemma \ref{l:potential derivative upper bound}, Corollary \ref{c:Inoue C0}, Lemma \ref{l:potential derivative lower bound}, and Theorem \ref{t:Inoue metric lower bound}, the results follow.
\end{proof}

\noindent The proof exploits natural class of background metrics arising from the homogeneous structure on OT-manifolds.  In the case of Inoue surfaces these are known as Tricerri metrics \cite{tricerri1982some}.  In the next theorem we give some refined estimates in the special case of generalized K\"ahler-Ricci flow (GKRF) \cite{GKRF}.  Oeljeklaus-Toma manifolds admit natural classes of generalized K\"ahler structures with split tangent bundle, and for such metrics the GKRF reduces to a scalar parabolic flow of Monge-Amp\`ere type \cite{StreetsSTB}.

\begin{thm} \label{t:mainthm2} For generalized K\"ahler-Ricci flow on an Oeljeklaus-Toma manifold
\begin{enumerate}
\item The scalar potential $\phi$ satisfies
\begin{align*}
    - C \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item Assuming there exists a Tricerri-type metric in $[\gw_0]$, we have
\begin{align*}
    - C e^{-t}(1 + t) \leq \phi \leq C e^{-t}(1 + t).
\end{align*}
\item On Inoue surfaces of type $S_M$, the estimate of item (2) holds. In addition:
\begin{align*}
    -C\leq \dot{\phi}\leq C.
\end{align*}
\item On Inoue surfaces of type $S_M$, we have:
\begin{align*}
        \omega(t)\geq C\omega_h(t),
\end{align*}
where $\omega_h(t)$ is the model flow with initial data the Tricerri metric $h$.
\end{enumerate}
\end{thm}

The proof of Theorem~\ref{t:mainthm1} is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

The proof of To prove the main theorem, is described only at a high level: it “exploits [a] natural class of background metrics arising from the homogeneous structure on OT-manifolds.” In the case of Inoue surfaces, “these are known as Tricerri metrics.”

\label{t:mainthm1} Fix $(M^{2n}, J)$ an OT manifold and $g_0$ a pluriclosed metric on $M$. The solution to pluriclosed flow with initial data $g_0$ exists on $[0, \infty)$.,

Question: Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\omega_0$ satisfies $\partial\bar\partial\omega_0=0$)?

The pluriclosed flow with initial metric $g_0$ exists for all forward time; equivalently, its solution is defined on the entire interval $[0,\infty)$.

Choices:
B. The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\to\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$.
C. The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$.
D. There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$.
E. For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure.

adds asymptotic Gromov--Hausdorff convergence not asserted in the theorem

dropped the global-in-time conclusion $[0,\infty)$ to mere short-time existence on a maximal interval

replaces infinite-time existence by a uniform manifold-dependent finite existence time

confuses use of homogeneous background metrics in the proof with the actual initial-data/general-flow statement

The stem does not state the global-in-time conclusion outright; it only asks which existence statement is valid. The equivalence defining pluriclosed is background, not a hint toward the correct option.

The item is close to a theorem-recall question: it asks for the correct existence statement attached to a specific geometric setting. However, it is not a pure verbatim restatement because the options vary in maximal-time behavior, quantifiers, and added convergence claims.

Some reasoning is needed to distinguish global existence from merely local existence and from stronger unsupported claims. Still, the question mainly tests recognition of the theorem rather than substantial derivation or synthesis.

The distractors are plausible and mathematically meaningful: finite-time maximal existence, merely local existence, incorrect quantifier dependence, and an overstrong convergence/uniqueness statement. They reflect common failure modes in interpreting PDE existence theorems.

A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward recall of a known result than genuine generative reasoning.


Question: Which statement holds for every Oeljeklaus--Toma (OT) complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$ (that is, the associated fundamental $(1,1)$-form $\omega_0$ satisfies $\partial\bar\partial\omega_0=0$)?

Choices:
B. The pluriclosed flow with initial metric $g_0$ exists for all forward time and, for every such initial metric, converges as $t\to\infty$ in the Gromov--Hausdorff sense to a flat torus canonically associated to $M$.
C. The pluriclosed flow with initial metric $g_0$ exists on some maximal time interval $[0,T)$ with $T>0$.
D. There exists a time $T=T(M,J)>0$, depending only on the OT complex manifold, such that for every pluriclosed Hermitian metric $g_0$ on $M$, the pluriclosed flow with initial metric $g_0$ exists on $[0,T]$.
E. For every OT complex manifold $(M^{2n},J)$ and every pluriclosed Hermitian metric $g_0$ on $M$, the normalized pluriclosed flow with initial metric $g_0$ exists on $[0,\infty)$ and is generated by one of the homogeneous background metrics arising from the OT-manifold structure.

The stem does not explicitly reveal the correct conclusion, and it does not contain strong linguistic cues that single out choice A. The correct answer must be identified from the mathematical content of the options.

The item is close to a direct theorem-recall question: choice A is essentially the theorem-level statement. However, the presence of stronger, weaker, and proof-confusion alternatives makes it more than a pure verbatim restatement.

Some reasoning is needed to distinguish the exact asserted result from stronger claims (B, E) and weaker or altered existence statements (C, D). Still, the question primarily tests recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: B is an overstrengthening, C is a weaker true statement, D alters the time range in a subtle way, and E confuses proof machinery with the theorem statement. These reflect realistic failure modes.

A solid MCQ with strong distractors and no answer leakage, but it mainly tests theorem recall/precision rather than deeper generative reasoning.


\label{main_theorem}
	Let $u\in C(0,T;L^2(\Omega,\mathbb{R}^N))\cap L^q(0,T;W^{1,q}(\Omega,\mathbb{R}^N))\cap L^\infty(\Omega_T)$ be a weak solution to (\ref{eq}) in $\Omega_T$.
	There exist constants $\ep_0=\ep_0(\mathit{data})$ and $c=c(\mathit{data} , \|a\|_{L^\infty(\Om_T)})$ such that for every $Q_{4r}(z_0)\subset\Om_T$ with $r\in (0,1)$ and $\epsilon\in(0,\ep_0)$,
	\begin{align*}
    \begin{split}
        &\iint_{Q_{r}(z_0)}H(z,|\nabla u|)^{1+\epsilon}\,dz\\
	&\le c\left(\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^p+\|a\|_{L^\infty(\Om_T)}\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^q +1\right)^{1+\frac{q\epsilon}{p}}\\
int_{Q_{4r}(z_0)}H(z,|F|)^{1+\epsilon} \,dz\right)^{1+\frac{q}{2}}\,.
    \end{split}
\end{align*}

\begin{proposition}\label{p_est_vitali}
Suppose \ref{p1} and \ref{p2}.
	There exist constants $c=c(\data)$ and $\theta_0=\theta_0(n,p)\in(0,1)$, such that for any $\theta\in(\theta_0,1)$ we have
	\begin{align*}
			\begin{split}
			    \iint_{Q_{2\cv\rho}^\la(z_0)}H(z,|\na u|)\,dz
			&\le c\La^{1-\theta}\iint_{Q_{2\rho}^\la(z_0)\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz\\
            &\qquad+c\iint_{Q_{2\rho}^\la(z_0)\cap \Theta(c^{-1}\La)}H(z,|F|) \,dz\,.
			\end{split}
	\end{align*}
    Here, $\kappa$ is defined in \eqref{K_and_kappa} and it appears in \ref{p4}.
 \end{proposition}

\begin{lemma}\label{sobolev_lem}
	Let $B_{\rho}(x_0)\subset\RR^n$, $\sig,s,r\in[1,\infty)$ and $\vartheta\in(0,1)$ such that
	\[
		-\frac{n}{\sig}\le \vartheta\left(1-\frac{n}{s}\right)-(1-\vartheta)\frac{n}{r}\,.
	\]
	Then for every $v\in W^{1,s}(B_{\rho}(x_0))$ there exists a constant $c=c(n,\sig)$ such that
	\[
nt_{B_{\rho}(x_0)}\frac{|v|^\sig}{\rho^\sig}\,dx
nt_{B_{\rho}(x_0)}\frac{|v|^r}{\rho^r}\,dx\right)^\frac{(1-\vartheta)\sig}{r}\,.
	\]
\end{lemma}

\subsection{Solutions}\label{ssec:sol}
The notion of weak solutions to \eqref{eq} is defined as follows.
\begin{definition}	A map $u:\Om_T\longrightarrow\RR^N$ satisfying
 \[
			u\in C(0,T;L^2(\Om,\RR^N))\cap L^q(0,T;W^{1,q}(\Om,\RR^N))
	\]
is a weak solution to \eqref{eq}, if
for every $\varphi\in C_0^\infty(\Omega_T,\RR^N)$ it holds
\begin{align*}
\begin{split}
    &\iint_{\Om_T}(-u\cdot\varphi_t+\mA(z,\na u)\cdot \na \varphi)\,dz\\
    &=\iint_{\Om_T}(|F|^{p-2}F\cdot\na \varphi+a(z)|F|^{p-2}F\cdot \na\varphi)\,dz\,.
\end{split}
\end{align*}
\end{definition}

 We note that to our best knowledge that despite the existence of weak solutions to~\eqref{eq} is expected in the full scope of our current inhnomogeneous in time and space study~\eqref{range_pq}, it is not yet established. We refer to~\cite{cgzg,C-b} for existence to problems with more general growth, but covering probably not sharp regularity with respect to the time variable, and \cite{bgs-discontinuous} for the existence to related problems, that do not fully embrace our case, but remarkably relax demanded time regularity.  At this place we shall stress that in the steady case, due to~\cite{eslemi}, if for $u$ being a weak solution or a local minimizer to related variational functionals, $H(x,\nabla u)\in L^1$, $a\in C^{0,\alpha}$ and $p$ and $q$ satisfy some closeness condition governed by $\alpha$, then $u\in L^q$. Surprisingly, a counterpart of such a result in a parabolic setting is still missing. Nonetheless, we expect it holds true and we continue our work for $L^q(0,T;W^{1,q}(\Om,\RR^N))$-solutions. As for the parabolic approaches under the natural energy regime  $H(z,\nabla u)\in L^1$, let us refer to~\cite{KIM2025110738},

\section{Energy estimates}\label{sec:energy-estimates}
In this section, we provide energy estimates. The first estimate is the Caccioppoli inequality.
\begin{lemma}\label{caccio_lem}
	Let $u$ be a weak solution to \eqref{eq} in $\Omega_T$. Then for every $Q_{R,\ell}(z_0)\subset\Omega_T$, with $R,\ell>0$, and for $r\in [R/2,R)$ and $\tau^2\in[\ell^2/2^2,\ell^2)$, there exists a constant $c=c(n,p,q,\nu,L)$, such that
	\begin{align*}
		\begin{split}
int_{Q_{r,\tau}(z_0)}(|\na u|^p+a(z)|\na u|^q)\,dz\\
int_{Q_{R,\ell}(z_0)}\left(\frac{|u-u_{Q_{R,\ell}(z_0)}|^p}{(R-r)^p}+a(z)\frac{|u-u_{Q_{R,\ell}(z_0)}|^q}{(R-r)^q}\right)\,dz\\
int_{Q_{R,\ell}(z_0)} (|F|^p+a(z)|F|^q)  \,dz\,.
		\end{split}
	\end{align*}
\end{lemma}

\begin{split}
             \iint_{G_{2\cv\rho}^\la(z_0)}H(z,|\na u|)\,dz
         &\le 2\kappa^{n+2}c\La^{1-\theta}\iint_{G_{2\rho}^\la(z_0)\cap \Psi((4c)^{-1/\theta}\La)}H(z,|\na u|)^{\theta }\,dz\\
         &\qquad+2\kappa^{n+2}c\iint_{G_{2\rho}^\la(z_0)\cap \Theta((4c)^{-1}\La)}H(z,|F|) \,dz\,.
         \end{split}
     \end{align*}
     Replacing the constant $c$ above with $(4\kappa^{n+2}c)^\frac{1}{\theta_0}$, the proof is completed.
\end{proof}

\section{Proof of the main result}\label{sec:main-proof}
\subsection{Stopping time argument}\label{stopping time argument}
In this section, we prove that conditions \ref{p1}-\ref{p2} and \ref{q1}-\ref{q2} are satisfied under our regime. First of all, for any $\rho>0$ and $\hbar>1$, we denote the constant in Lemma~\ref{caccio_lem} for $r=\rho$, $R=2\rho$, $\tau^2=\hbar^{2-p}\rho^2$ and $\ell^2=\hbar^{2-p}(2\rho)^2$ by
\begin{align}\label{def_const_caccio}
    \ccc=\ccc(n,p,q,\nu,L)\,.
\end{align}
Let $r\in(0,1)$ and suppose $Q_{4r}(z_0)\subset\Om_T$. We define $\la_0$ and $\La_0$ as
\begin{align}\label{def_la}
		\begin{split}
		    \la^p_0
int_{Q_{4r}(z_0)} H(z,|F|)\,dz \right)^\frac{p}{2} +1
		\end{split}

\begin{lemma}\label{q_reverse_lem}
Suppose \ref{q1} and \ref{q2}.
	There exist constants $c=c(n,N,p,q,\nu,L,\ca)$ and $\theta_0=\theta_0(n,p,q)\in(0,1)$ such that for any $\theta\in(\theta_0,1)$ there holds
	\begin{align*}
int_{G_{\rho}^\la(z_0)}H(z,|\na u|)\,dz
int_{G_{2\rho}^\la(z_0)}H(z,|F|) \,dz\,.
	\end{align*}
\end{lemma}

\begin{lemma}\label{p_reverse_lem}
Suppose \ref{p1} and \ref{p2}.
	There exist constants $c=c(\data)$ and $\theta_0=\theta_0(n,p)\in(0,1)$ such that for any $\theta\in(\theta_0,1)$ there holds
	\begin{align*}
int_{Q_{\rho}^\la(z_0)}H(z,|\na u|)\,dz
int_{Q_{2\rho}^\la(z_0)}H(z,|F|)\,dz\,.
	\end{align*}
\end{lemma}

\begin{theorem}\label{main_theorem}
	Let $u\in C(0,T;L^2(\Om,\RR^N))\cap L^q(0,T;W^{1,q}(\Om,\RR^N))\cap L^\infty(\Omega_T)$ be a weak solution to \eqref{eq} in $\Omega_T$.
	There exist constants $\ep_0=\ep_0(\mathit{data})$ and $c=c(\mathit{data} , \|a\|_{L^\infty(\Om_T)})$ such that for every $Q_{4r}(z_0)\subset\Om_T$ with $r\in (0,1)$ and $\ep\in(0,\ep_0)$,
	\begin{align*}
    \begin{split}
        &\iint_{Q_{r}(z_0)}H(z,|\na u|)^{1+\epsilon}\,dz\\
	&\le c\left(\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^p+\|a\|_{L^\infty(\Om_T)}\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^q +1\right)^{1+\frac{q\epsilon}{p}}\\
int_{Q_{4r}(z_0)}H(z,|F|)^{1+\ep} \,dz\right)^{1+\frac{q}{2}}\,.
    \end{split}
\end{align*}
\end{theorem}

\begin{proposition}
    \label{q_est_vitali}
Suppose \ref{q1} and \ref{q2}.
	There exist constants $\theta_0=\theta_0(n,p,q)\in(0,1)$ and $c=c(n,N,p,q,\nu,L,\ca)$  such that for any $\theta\in(\theta_0,1)$ we have
	\begin{align*}
			\begin{split}
			    \iint_{G_{2\cv\rho}^\la(z_0)}H(z,|\na u|)\,dz
			&\le c\La^{1-\theta}\iint_{G_{2\rho}^\la(z_0)\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz \\
            &\qquad+c\iint_{G_{2\rho}^\la(z_0)\cap \Theta(c^{-1}\La)}H(z,|F|)\,dz\,.
			\end{split}
	\end{align*}
    Here, $\kappa$ is defined in \eqref{K_and_kappa} and it appears in \ref{q5},
\end{proposition}

Parabolic equations with double-phase growth represent a natural parabolic counterpart of models describing materials with heterogeneous hardening, composite media, or diffusion processes with switching intensities. Mathematical description of them have attracted considerable attention over the past decade~\cite{mira,C-b,H-book}. Even for strongly nonlinear problems, one typically expects solutions to exhibit regularity beyond that guaranteed by mere membership in the energy space; see, for instance,~\cite{ElMe,Str,KiLe-Duke}. Recently, there has been remarkable progress in the regularity theory for parabolic double-phase problems, such as \[u_t-\dv(|\nabla u|^{p-2}\nabla u+a(x,t)|\nabla u|^{q-2}\nabla u)=-\dv(|F|^{p-2}F+a(x,t)|F|^{q-2}F)\,. \] In a short time, it has led to several deep and influential regularity results including~\cite{WMS,WS,sen2025lipschitzregularityparabolicdouble,Sen2025,MR4926910,K-CZ,KKM,KIM2025110738, buskr,ssy}. Some results in a more refined framework are also available,
 see \cite{oh2025gradient,MR4926910,MR4774133,Sen2025}. Nevertheless, the theory is still far from being fully understood.

In this work, we focus on gradient higher integrability, which plays a decisive role in the analysis of finer regularity of weak solutions. Solutions to double phase problems are expected be regular provided the closeness condition on the exponents is controlled by the regularity of the weight $a$, which broadens under a priori knowledge about the regularity of $u$. This is observed in the elliptic case~\cite{comi-bdd,bacomi-cv,ok-na}, but its parabolic counterpart remains largely unexplored. We study the regularity to parabolic double phase problems for a priori bounded solutions, where the evolution brings deep complications absent in the elliptic situation. Although related results are available for problems of similar type~\cite{BoDu-nonst,DiBFr,KiLe-Duke,KKM}, the structure of the system considered here prevents an application of the existing techniques. What is more, once we finished the first draft of our manuscript we learned about~\cite{kim2025boundedsolutionsinterpolativegap} where the method cannot embrace the presence of the non-zero right-hand side. Henceforth, we deliver a substantially new approach.

Let us present in detail our result. We shall consider weak solutions to the parabolic double-phase system
\begin{align}\label{eq}
 u_t-\dv\mathcal{A}(z,\nabla u)=-\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\quad \text{in $\Omega_T=\Omega\times(0,T)$\,,}
\end{align}
 where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $T>0$. Here,
we assume $\mathcal{A}(z,\nabla u):\Omega_T\times \mathbb{R}^{Nn}\longrightarrow \mathbb{R}^{Nn}$ with $N\ge1$ is a Carath\'eodory vector field satisfying the following structure assumptions: there exist constants $0<\nu\le L<\infty$ such that
 \begin{align}\label{str}
 \mathcal{A}(z,\xi)\cdot \xi\ge \nu(|\xi|^p+a(z)|\xi|^q)
 \quad\text{and}\quad
 |\mathcal{A}(z,\xi)|\le L(|\xi|^{p-1}+a(z)|\xi|^{q-1}),
 \end{align}
for almost every $z\in \Omega_T$ and every $\xi\in \mathbb{R}^{Nn}$. Throughout the rest of the paper, we use the notation for $z\in\Omega_T$ and $s\geq 0$ \begin{equation}
 \label{H-def}
H(z,s)=s^p+a(z) s^q\,.
\end{equation}

We focus of gradient higher integrability of the solutions in the spirit of \cite{KiLe-Duke}. We prove it for a priori bounded $u$, $a \geq 0$ and $a\in \mathcal{C}^{\alpha,\frac\alpha 2}(\Omega_T)$ for some $\alpha \in (0,1]$ and
\begin{align} \label{range_pq}
2\le p <\infty, \quad p<q\le p+\alpha\,.
\end{align}
Here $a\in \mathcal{C}^{\alpha,\frac\alpha 2}(\Omega_T)$ means that $a\in L^{\infty}(\Omega_T)$ and there exist constants $c_a>0$, such that
\begin{align}
 \label{def_holder}
 |a(x,t)-a(y,s)|\le c_a\left(\max\{ |x-y|^\alpha,|t-s|^\frac{\alpha}{2} \}\right)
\end{align}
for every $(x,y)\in\Omega$ and $(t,s)\in (0,T)$.

The main result of this paper is the following higher integrability estimate for the gradient of a weak solution to (\ref{eq}) assuming that the forcing term $H(z,|F|)\in L^\gamma$,
where $\gamma=\tfrac{n+p}{p}$. We denote the constant $c=c(\mathit{data})$ if $c$ depends on the following
 \begin{align*}
 \begin{split}
 \mathit{data}=&(n,N,p,q,\nu,L,c_a,\|u\|_{L^\infty(\Om_T)},\|H(z,|F|)\|_{L^\gamma(\Om_T)})\,.
 \end{split}
\end{align*} Now we state our main result.

\begin{align}\label{eq}
	u_t-\dv\mA(z,\na u)=-\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\quad \text{in $\Omega_T=\Omega\times(0,T)$\,,}
\end{align}

\subsection{Proof of Theorem~\ref{main_theorem}}
Recall we have set $r\le r_1<r_2\le 2r$ with $r\in(0,1)$.
For selected subcollection $\mathcal{G}$ in Lemma~\ref{vitali_lem}, we simply write
\[
 \mathcal{G}=\{ \mQ_i \}_{1\le i \le \infty}\,,
\]
where $\mQ_i=\mQ(w_i)$ for some $w_i\in \Psi(\La,r_1)$. Depending on the intrinsic geometry of $\mQ_i\in \mathcal{G}$, it follows by Proposition~\ref{p_est_vitali} and Proposition~\ref{q_est_vitali} that for any $i\in \mathbb{N}$
\begin{align*}
 \begin{split}
 \iint_{\cv\mQ _{i}}H(z,|\na u|)\,dz
 &\le c\La^{1-\theta}\iint_{\mQ _i\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz\\
 &\qquad + c\iint_{U_i\cap \Theta(c^{-1} \la) } H(z,|F|)\,dz\,,
 \end{split}
\end{align*}
where $c=c(\data)>1$ is a constant and fixed $\theta\in (0,1)$.
Employing disjointedness and covering property in Lemma~\ref{vitali_lem}, we get
\begin{align*}
 \begin{split}
 \iint_{\Psi(\La,r_1)}H(z,|\na u|)\,dz
&\le\sum_{i=1}^\infty\iint_{\cv\mQ _{i}}H(z,|\na u|)\,dz\\
&\le c\La^{1-\theta}\sum_{i=1}^\infty\iint_{\mQ _i\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz\\
&\qquad +c \sum_{i=1}^\infty\iint_{\mQ _i\cap \Theta(c^{-1}\La)}H(z,|F|)\,dz\\
&\le c\La^{1-\theta}\iint_{\Psi(c^{-1}\La,r_2)}H(z,|\na u|)^\theta\,dz\\
&\qquad+ c\iint_{\Theta(c^{-1}\La,r_2)}H(z,|F|)\,dz\,.
 \end{split}
\end{align*}
On the other hand, since we have
\[
\iint_{\Psi(c^{-1}\La,r_1)\setminus \Psi(\La,r_1)}H(z,|\na u|)\,dz
\le\La^{1-\theta}\iint_{\Psi(c^{-1}\La,r_2)}H(z,|\na u|)^{\theta}\,dz\,,
\]
we deduce that
\begin{align}\label{covering_est_1}
\begin{split}
 \iint_{\Psi(c^{-1}\La,r_1)}H(z,|\na u|)\,dz
 &\le c\La^{1-\theta}\iint_{\Psi(c^{-1}\La,r_2)}H(z,|\na u|)^\theta\,dz\\
 &\qquad +c \iint_{\Theta(c^{-1}\La,r_2)}H(z,|F|)\,dz\,,
\end{split}
\end{align}
where $c=c(data)>1$ is a constant.
We now take $k\in\mathbb N$ and consider
\[
H(z,|\na u|)_k=\min\{H(z,|\na u|),k\}
\]
and
\[
\Psi_k(\La,\rho)=\{z\in Q_{\rho}(z_0):H(z,|\na u(z)|)_k>\La\}\,.
\]
It is easy to see that if $\La>k$, then $\Psi_k(\La,\rho)=\emptyset$, and if $\La\le k$, then $\Psi_k(\La,\rho)=\Psi(\La,\rho)$. Therefore, along with these notations, \eqref{covering_est_1} becomes
\begin{align*}
 \begin{split}
 \iint_{\Psi_k(c^{-1}\La,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz
 &\le c\La^{1-\theta}\iint_{\Psi_k(c^{-1}\La,r_2)}H(z,|\na u|)^\theta\,dz\\
 &\qquad +c \iint_{\Theta(c^{-1}\La,r_2)}H(z,|F|)\,dz
 \end{split}
\end{align*}
for any
\[
\La>(4\ccc)^{q}\left(\frac{8\cv r}{r_2-r_1}\right)^{\frac{q^2}{p}+\frac{q(n+2)}{2} }\La_0\,.
\]
Denoting
\[
 \La_1= c^{-1}(4\ccc)^q\left(\frac{8\cv r}{r_2-r_1}\right)^{\frac{q^2}{p}+\frac{q(n+2)}{2}}\La_0\,,
\]
for any $\La>\La_1$ we obtain
\begin{align*}
 \begin{split}
 &\iint_{\Psi_k(\La,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\\
 &\le c\La^{1-\theta}\iint_{\Psi_k(\La,r_2)}H(z,|\na u|)^\theta\,dz+c\iint_{\Theta(\La,2r)}H(z,|F|)\,dz\,.
 \end{split}
\end{align*}

Let $\ep\in(0,1)$ will be determined later. We multiply the inequality above by $\La^{\ep-1}$ and integrate over $(\La_1,\infty)$. Then, we get
\begin{align*}
 \begin{split}
 \mathrm{I}&=\int_{\La_1}^{\infty}\La^{\ep-1}\iint_{\Psi_k(\La,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\,d\La\\
 &\le c\int_{\La_1}^{\infty}\La^{\ep-\theta}\iint_{\Psi_k(\La,r_2)}H(z,|\na u|)^\theta\,dz\,d\La + c \int_{\La_1}^{\infty}\La^{\ep-1} \iint_{\Theta(\La,2r)}H(z,|F|)\,dz \, d\La \\
 &= \mathrm{II} + \mathrm{III}\,.
 \end{split}
\end{align*}
Applying Fubini's theorem to $\mathrm{I}$, it follows that
\begin{align*}
 \begin{split}
 \mathrm{I}
 &=\frac{1}{\ep}\iint_{\Psi_k(\La_1,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\qquad-\frac{1}{\ep}\La_1^\ep\iint_{\Psi_k(\La_1,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\,.
 \end{split}
\end{align*}
Meanwhile, since we have
\begin{align*}
 \begin{split}
 &\iint_{Q_{r_1}(z_0)\setminus \Psi_k(\La_1,r_1)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\le \La_1^{\ep}\iint_{Q_{2r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\,,
 \end{split}
\end{align*}
we obtain
\begin{align*}
 \begin{split}
 \mathrm{I}\ge& \frac{1}{\ep}\iint_{Q_{r_1}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\qquad-\frac{2}{\ep}\La_1^\ep\iint_{Q_{2r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\,.
 \end{split}
\end{align*}
Similarly, by Fubini's theorem, we have
\[
 \mathrm{II}
 \le\frac{1}{1-\theta+\ep}\iint_{Q_{r_2}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta \,dz
\]
and
\[
\mathrm{III}\le \frac{1}{\ep}\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz\,.
\]
Combining the estimates above, we obtain
\begin{align*}
 \begin{split}
 &\iint_{Q_{r_1}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\le \frac{c\ep}{1-\theta+\ep}\iint_{Q_{r_2}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta \,dz\\
 &\qquad+c\La_1^\ep\iint_{Q_{2r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz \\
 &\qquad + c\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz
 \end{split}
\end{align*}
for $c=c(\data)$ and $\theta=\theta(\data)\in(0,1)$. We choose $\ep_0=\ep_0(\data)\in(0,1)$ so that for any $\ep\in(0,\ep_0)$,
\[
 \frac{c\ep}{1-\theta+\ep}\le\frac{1}{2}\,.
\]
Then, there holds
\begin{align*}
 \begin{split}
 &\iint_{Q_{r_1}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\le \frac{1}{2}\iint_{Q_{r_2}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta \,dz\\
 &\qquad+c\La_1^\ep\iint_{Q_{2r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz \\
 &\qquad+ c\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz\,.
 \end{split}
\end{align*}
Recalling $r\le r_1<r_2\le 2r$, we conclude from Lemma~\ref{tech_lem} that
\begin{align*}
 \begin{split}
 &\iint_{Q_{r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta+\ep}H(z,|\na u|)^\theta\,dz\\
 &\le c\La_0^\ep\iint_{Q_{2r}(z_0)}\left(H(z,|\na u|)_k\right)^{1-\theta}H(z,|\na u|)^\theta\,dz\\
 &\qquad+ c\iint_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz\,.
 \end{split}
\end{align*}
Letting $k$ go to $\infty$ and recalling \eqref{def_la} and $\eqref{def_La}$, we have
\begin{align*}
 \begin{split}
int_{Q_{r}(z_0)}H(z,|\na u|)^{1+\epsilon}\,dz\\
 &\le c\left(\frac{\|u\|^p_{\infty}}{r^p} +\|a\|_{\infty} \frac{\|u\|^q_{\infty}}{r^q} +
int_{Q_{4r}(z_0)}H(z,|F|)\,dz \right)^\frac{p}{2} +1\right)^{\frac{q\epsilon}{p}}\\
int_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz\,,
 \end{split}
\end{align*}
where we abbreviated $\| \cdot \|_{\infty}= \| \cdot \|_{L^\infty(\Om_T)}$.
Finally, applying Lemma~\ref{caccio_lem} with $Q_{2r}(z_0)$ and $Q_{4r}(z_0)$, we obtain
\begin{align*}
 \begin{split}
int_{Q_{r}(z_0)}H(z,|\na u|)^{1+\epsilon}\,dz\\
 &\le c\left(\frac{\|u\|^p_{\infty}}{r^p} +\|a\|_{\infty} \frac{\|u\|^q_{\infty}}{r^q} +
int_{Q_{4r}(z_0)}H(z,|F|)\,dz \right)^\frac{p}{2} +1\right)^{\frac{q\epsilon}{p}}\\
int_{Q_{4r}(z_0)}H(z,|F|)\,dz \right)\\
int_{Q_{2r}(z_0)}H(z,|F|)^{1+\ep} \,dz\,,
 \end{split}
\end{align*}
where $c=c(\data,\|a\|_{L^\infty(\Om_T)})$.
Since $p\ge2$, we apply Young's inequality to the term with exponent $2$ on the right hand side in order to have
\begin{align*}
 \begin{split}
int_{Q_{r}(z_0)}H(z,|\na u|)^{1+\epsilon}\,dz
 &\le c\left(\frac{\|u\|^p_{\infty}}{r^p} +\|a\|_{\infty}\frac{\|u\|^q_{\infty}}{r^q} +1\right)^{1+\frac{q\epsilon}{p}} \\
int_{Q_{4r}(z_0)}H(z,|F|)^{1+\ep} \,dz\right)^{1+\frac{q}{2}}\,.
 \end{split}
\end{align*}
The proof is completed. \qed

For the rest of the paper, whenever we say that $u$ is a weak solution of (\ref{eq}) in $\Omega_T$, we assume that $u\in C(0,T;L^2(\Omega,\mathbb{R}^N))\cap L^q(0,T;W^{1,q}(\Omega,\mathbb{R}^N))\cap L^\infty(\Omega_T)$. See Section~\ref{ssec:sol} for more comments.

\begin{remark}[Sharpness]\rm
    The question on the optimality of a range of parameters for multiple regularity results, was a topic studied in depth on elliptic problems~\cite{Zhikov86,eslemi}, providing that under a priori boundedness assumption~(\ref{range_pq}) is actually sharp, cf.~\cite{comi,comi-bdd,bacomi-cv,fomami,bcdfm}. From this point of view, the current paper is a natural spin-off of~\cite{WMS}, where  gradient higher integrability of the solutions to~(\ref{eq}) was proven under no extra a priori regularity of a solution itself.
\end{remark}

{\bf Methods.} The main idea is typical for the double phase problems -- we use the phase separation to regions when the contribution of $a$ is small or dominating. As much as employing the exit time reasoning is expected, a key point is to exploit a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument, where the stopping time depends on the energy of $\nabla u$ and of $F$. We face several new challenges in the proof of reverse H\"older-type result (in Section~\ref{sec:rev-hold}) comparing to the previous analysis. In fact, the phase separation in  \cite{KKM}, i.e., in the case of non a priori bounded solutions, relied on the  comparison of two values $\lambda^p$ and $a(z_0)\lambda^q$ by using just the energy of the gradient of the solutions. In contrast, the phase analysis here is formulated on integrated quantities, allowing us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa. Then the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems.   We shall point out that the analysis requires delicate interplay between the homogeneity of the scaling factors of cubes and their radii.

A major challenge is obtaining a variant of reverse H\"older inequality within the intrinsic and locally varying scaling, that forms the analytic backbone for the higher integrability proved in Theorem~\ref{main_theorem}. We prove the reversed H\"older inequality  adapted to the intrinsic geometry distinguishing {\it $p$-intrinsic case}, when our problem is a perturbed $p$-Laplace evolution and a perturbation is controlled by  $\|u\|_{L^\infty}$. Namely, for some $K=K(\mathit{data})>1$ and relevant cube $Q_\rho$ it holds
\[K\geq (\|u\|_{\infty}/\rho)^{q-p}\sup_{cQ_{\rho}} a\,.\]
On the other hand, when this condition fails, we deal with the {\it $(p,q)$-intrinsic case}, when our problem resembles the $q$-Laplace problem. The reversed H\"older inequality is provided  for $p$-phase in Lemma~\ref{p_reverse_lem} and for $(p,q)$-phase in Lemma~\ref{q_reverse_lem}.

This separation is effective in the presence of $F$ under the sharp range of phase parameters~(\ref{range_pq}). The proof is possible due to Lemma~\ref{lemma_decay} yielding a key decay property enabling the datum $F$ to be nonzero.

The arguments are closed with use of the Vitali covering theorem, see Lemma~\ref{vitali_lem}, and consequences of the reverse H\"older inequality provided for  $p$-phase in Proposition~\ref{p_est_vitali} and for $(p,q)$-phase in Proposition~\ref{q_est_vitali}. \textcolor{black}{Let us stress that since intrinsic geometries may vary from point to point, the standard Vitali covering lemma cannot be directly applied. The required comparability of the scaling factors is ensured by the Hölder continuity of the coefficient $a$ in Lemma~\ref{la_comp_lem}, which allows us to establish a modified Vitali covering lemma with a scaled covering constant.} The admissible range $q\le p+\alpha$ is precisely what guarantees that the oscillation of $a(\cdot)$ over the cylinders is below the threshold needed for the proof to be valid, see Section~\ref{stopping time argument}.\newline

{\bf Organization. } In Section~\ref{sec:prelim} we present notation and basic definitions. Section~\ref{sec:energy-estimates} provides a priori estimates, while Section~\ref{sec:rev-hold} is devoted to the reverse H\"older estimate. The main proof is concluded in Section~\ref{sec:main-proof}.\newline

{\bf Methods.} The proof of Theorem~\ref{main_theorem} follows the typical double phase strategy: one uses \emph{phase separation} into regions where the contribution of $a$ is small or dominating. A key ingredient is an \emph{exit time} (stopping time) argument, exploiting “a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument,” where “the stopping time depends on the energy of $\nabla u$ and of $F$.”

Compared to earlier work, the phase analysis is “formulated on integrated quantities,” which “allow[s] us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa.” Then “the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems,” requiring a “delicate interplay between the homogeneity of the scaling factors of cubes and their radii.”

The analytic backbone is a \emph{reverse H\"older inequality} “within the intrinsic and locally varying scaling,” adapted to the intrinsic geometry and split into:
\begin{itemize}
\item the \emph{$p$-intrinsic case}, when the problem is a perturbed $p$-Laplace evolution and the perturbation is controlled by $\|u\|_{L^\infty}$; for some $K>1$ and relevant cube $Q_\rho$ one has
\[K\ge (\|u\|_{\infty}/\rho)^{q-p}\sup_{cQ_{\rho}} a\,;\]
\item the \emph{$(p,q)$-intrinsic case}, when the above condition fails and the problem resembles the $q$-Laplace problem.
\end{itemize}
The corresponding reverse H\"older inequalities are proved “for $p$-phase in Lemma~\ref{p_reverse_lem} and for $(p,q)$-phase in Lemma~\ref{q_reverse_lem}.”

This separation “is effective in the presence of $F$ under the sharp range of phase parameters~(\ref{range_pq}),” and “is possible due to Lemma~\ref{lemma_decay} yielding a key decay property enabling the datum $F$ to be nonzero.”

Finally, “the arguments are closed with use of the Vitali covering theorem,” via Lemma~\ref{vitali_lem} and the consequences of the reverse H\"older inequality in Proposition~\ref{p_est_vitali} (for $p$-phase) and Proposition~\ref{q_est_vitali} (for $(p,q)$-phase). Since “intrinsic geometries may vary from point to point,” a standard Vitali lemma cannot be applied directly; “comparability of the scaling factors is ensured by the H\"older continuity of the coefficient $a$ in Lemma~\ref{la_comp_lem},” yielding a modified Vitali covering. The range “$q\le p+\alpha$” is stated to guarantee the needed small oscillation of $a(\cdot)$ over cylinders for the stopping-time/Vitali argument to work.

{\bf Methods.} To prove the main theorem, one follows the typical double phase strategy: one uses \emph{phase separation} into regions where the contribution of $a$ is small or dominating. A key ingredient is an \emph{exit time} (stopping time) argument, exploiting “a delicate relation between the radii of the exit time balls and the level at which we perform the exit time argument,” where “the stopping time depends on the energy of $\nabla u$ and of $F$.”

Compared to earlier work, the phase analysis is “formulated on integrated quantities,” which “allow[s] us to absorb the $q$-phase into the $p$-phase inside appropriate intrinsic cylinders and vice versa.” Then “the energy control (including the Caccioppoli type inequality) can be considered as perturbation corresponding estimates known for the $p$-Laplace systems,” requiring a “delicate interplay between the homogeneity of the scaling factors of cubes and their radii.”

The analytic backbone is a \emph{reverse H\"older inequality} “within the intrinsic and locally varying scaling,” adapted to the intrinsic geometry and split into:
\begin{itemize}
\item the \emph{$p$-intrinsic case}, when the problem is a perturbed $p$-Laplace evolution and the perturbation is controlled by $\|u\|_{L^\infty}$; for some $K>1$ and relevant cube $Q_\rho$ one has
\[K\ge (\|u\|_{\infty}/\rho)^{q-p}\sup_{cQ_{\rho}} a\,;\]
\item the \emph{$(p,q)$-intrinsic case}, when the above condition fails and the problem resembles the $q$-Laplace problem.
\end{itemize}
We first record the corresponding reverse H\"older inequalities.
\begin{lemma}\label{p_reverse_lem}
Suppose \ref{p1} and \ref{p2}.
\tThere exist constants $c=c(\data)$ and $\theta_0=\theta_0(n,p)\in(0,1)$ such that for any $\theta\in(\theta_0,1)$ there holds
\t\begin{align*}
int_{Q_{\rho}^\la(z_0)}H(z,|\na u|)\,dz
int_{Q_{2\rho}^\la(z_0)}H(z,|F|)\,dz\,.
\t\end{align*}\t
\end{lemma}
\begin{lemma}\label{q_reverse_lem}
Suppose \ref{q1} and \ref{q2}.
\tThere exist constants $c=c(n,N,p,q,\nu,L,\ca)$ and $\theta_0=\theta_0(n,p,q)\in(0,1)$ such that for any $\theta\in(\theta_0,1)$ there holds
\t\begin{align*}
int_{G_{\rho}^\la(z_0)}H(z,|\na u|)\,dz
int_{G_{2\rho}^\la(z_0)}H(z,|F|) \,dz\,.
\t\end{align*}\t
\end{lemma}

This separation “is effective in the presence of $F$ under the sharp range of phase parameters
\begin{align} \label{range_pq}
2\le p <\infty, \quad p<q\le p+\alpha\,.
\end{align}
,” and “is possible due to” the following decay property, which yields a key decay estimate enabling the datum $F$ to be nonzero.
\begin{lemma}\label{lemma_decay}
   Suppose \ref{p1} and \ref{p2}. Then there exists a constant $\cc=\cc(\data)\ge K^\frac{1}{p}$
   such that
   \[
   \rho\le \cc \la^{-1}\,.
   \]
\end{lemma}

Finally, “the arguments are closed with use of the Vitali covering theorem,” via the following Vitali-type selection.
\begin{lemma}\label{vitali_lem}
    Let $\mathcal{F}$ be defined as above and $\kappa$ is in \eqref{K_and_kappa}, Then there exists a pairwise disjoint countable subcollection $\mathcal{G}$ of $\mathcal{F}$ such that for any $U(w)\in \mathcal{F}$, there exists $U(v)\in \mathcal{G}$ with
    \[
    U(w)\subset \kappa U(v)\,.
    \]
\end{lemma}
Together with the consequences of the reverse H\"older inequalities encoded in the Vitali-estimate propositions for the $p$-phase and $(p,q)$-phase:
\begin{proposition}\label{p_est_vitali}
Suppose \ref{p1} and \ref{p2}.
\tThere exist constants $c=c(\data)$ and $\theta_0=\theta_0(n,p)\in(0,1)$, such that for any $\theta\in(\theta_0,1)$ we have
\t\begin{align*}
\t\t\t\begin{split}
\t\t\t    \iint_{Q_{2\cv\rho}^\la(z_0)}H(z,|\na u|)\,dz
\t\t\t&\le c\La^{1-\theta}\iint_{Q_{2\rho}^\la(z_0)\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz\\
            &\qquad+c\iint_{Q_{2\rho}^\la(z_0)\cap \Theta(c^{-1}\La)}H(z,|F|) \,dz\,.
\t\t\t\end{split}
\t\end{align*}
    Here, $\kappa$ is defined in \eqref{K_and_kappa} and it appears in \ref{p4}.
 \end{proposition}
\begin{proposition}
    \label{q_est_vitali}
Suppose \ref{q1} and \ref{q2}.
\tThere exist constants $\theta_0=\theta_0(n,p,q)\in(0,1)$ and $c=c(n,N,p,q,\nu,L,\ca)$  such that for any $\theta\in(\theta_0,1)$ we have
\t\begin{align*}
\t\t\t\begin{split}
\t\t\t    \iint_{G_{2\cv\rho}^\la(z_0)}H(z,|\na u|)\,dz
\t\t\t&\le c\La^{1-\theta}\iint_{G_{2\rho}^\la(z_0)\cap \Psi(c^{-1}\La)}H(z,|\na u|)^\theta\,dz \\
            &\qquad+c\iint_{G_{2\rho}^\la(z_0)\cap \Theta(c^{-1}\La)}H(z,|F|)\,dz\,.
\t\t\t\end{split}
\t\end{align*}
    Here, $\kappa$ is defined in \eqref{K_and_kappa} and it appears in \ref{q5},
\end{proposition}
Since “intrinsic geometries may vary from point to point,” a standard Vitali lemma cannot be applied directly; “comparability of the scaling factors is ensured” by the following comparability statement.
\begin{lemma}\label{la_comp_lem}
    Suppose $U(w), U(v)\in \mathcal{F}$ with $U(w)\cap U(v)\ne \emptyset$ and $l_w\le 2l_v$. Then for $\cc\ge 2$ defined in Lemma~\ref{p_decay_lem}, we have
    \[
    \lav\le (2(1+\ca+10\cc\ca))^\frac{1}{p}\law\,.
    \]
    Moreover, if $U(v)$ is $(p,q)$-intrinsic, then we also have
    \[
    \law\le (2(1+\ca+10\cc\ca))^\frac{1}{p}\lav\,.
    \]
\end{lemma}
The range “$q\le p+\alpha$” in \eqref{range_pq} is stated to guarantee the needed small oscillation of $a(\cdot)$ over cylinders for the stopping-time/Vitali argument to work.

\label{main_theorem}
	Let $u\in C(0,T;L^2(\Omega,\mathbb{R}^N))\cap L^q(0,T;W^{1,q}(\Omega,\mathbb{R}^N))\cap L^\infty(\Omega_T)$ be a weak solution to
	\begin{align}\label{eq}
\tu_t-\dv\mA(z,\na u)=-\dv(|F|^{p-2}F+a(z)|F|^{q-2}F)\quad \text{in $\Omega_T=\Omega\times(0,T)$\,,}
\end{align}
	in $\Omega_T$.
	There exist constants $\ep_0=\ep_0(\mathit{data})$ and $c=c(\mathit{data} , \|a\|_{L^\infty(\Om_T)})$ such that for every $Q_{4r}(z_0)\subset\Om_T$ with $r\in (0,1)$ and $\epsilon\in(0,\ep_0)$,
	\begin{align*}
    \begin{split}
        &\iint_{Q_{r}(z_0)}H(z,|\nabla u|)^{1+\epsilon}\,dz\\
	&\le c\left(\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^p+\|a\|_{L^\infty(\Om_T)}\left( \frac{\|u\|_{L^\infty(\Om_T)}}{r} \right)^q +1\right)^{1+\frac{q\epsilon}{p}}\\
int_{Q_{4r}(z_0)}H(z,|F|)^{1+\epsilon} \,dz\right)^{1+\frac{q}{2}}\,.
    \end{split}
\end{align*},

Existential–Universal
Inequality or Bound

Question:
Let \(\Omega\subset \mathbb R^n\) be a bounded domain with \(n\ge 2\), let \(T>0\), and write \(\Omega_T=\Omega\times(0,T)\). Fix \(N\ge 1\). Assume \(a\ge 0\) and \(a\in C^{\alpha,\alpha/2}(\Omega_T)\) for some \(\alpha\in(0,1]\), meaning that for some \(c_a>0\),
\[
|a(x,t)-a(y,s)|\le c_a\max\{|x-y|^\alpha,|t-s|^{\alpha/2}\}
\]
for all relevant points, and assume
\[
2\le p<\infty,\qquad p<q\le p+\alpha.
\]
Define
\[
H(z,s)=s^p+a(z)s^q .
\]
Let \(\mathcal A:\Omega_T\times\mathbb R^{Nn}\to\mathbb R^{Nn}\) be a Carath\'eodory vector field such that for some \(0<\nu\le L<\infty\),
\[
\mathcal A(z,\xi)\cdot \xi\ge \nu\big(|\xi|^p+a(z)|\xi|^q\big),
\qquad
|\mathcal A(z,\xi)|\le L\big(|\xi|^{p-1}+a(z)|\xi|^{q-1}\big)
\]
for a.e. \(z\in\Omega_T\) and every \(\xi\in\mathbb R^{Nn}\). Assume also that \(H(z,|F|)\in L^\gamma(\Omega_T)\) with \(\gamma=(n+p)/p\), and that
\[
u\in C(0,T;L^2(\Omega,\mathbb R^N))\cap L^q(0,T;W^{1,q}(\Omega,\mathbb R^N))\cap L^\infty(\Omega_T)
\]
is a weak solution of
\[
u_t-\operatorname{div}\mathcal A(z,\nabla \nu)=-\operatorname{div}\big(|F|^{p-2}F+a(z)|F|^{q-2}F\big)
\quad\text{in }\Omega_T.
\]
For a parabolic cylinder \(Q_{4r}(z_0)\subset \Omega_T\) with \(r\in(0,1)\), which quantitative estimate holds for \(H(z,|\nabla \nu|)\)?

There exist constants \(\epsilon_0=\epsilon_0(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})>0\) and \(c=c(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)},\|a\|_{L^\infty(\Omega_T)})\) such that for every \(\epsilon\in(0,\epsilon_0)\),
\[
\iint_{Q_r(z_0)} H(z,|\nabla \nu|)^{1+\epsilon}\,dz
\le c\left(\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^p+\|a\|_{L^\infty(\Omega_T)}\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^q+1\right)^{1+\frac{q\epsilon}{p}}
+ c\left(\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\epsilon}\,dz\right)^{1+\frac q2}.
\]

Choices:
B.
There exist constants \(\epsilon_0=\epsilon_0(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})>0\) and \(c=c(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)},\|a\|_{L^\infty(\Omega_T)})\) such that for every \(\epsilon\in(0,\epsilon_0)\),
\[
\iint_{Q_r(z_0)} H(z,|\nabla \nu|)^{1+\epsilon}\,dz
\le c\left(\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^p+\|a\|_{L^\infty(\Omega_T)}\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^q+1\right)^{1+\frac{q\epsilon}{p}}
+ c\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\epsilon}\,dz.
\]
C.
There exist constants \(\epsilon_0=\epsilon_0(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})>0\) and \(c=c(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)},\|a\|_{L^\infty(\Omega_T)})\) such that for every \(\epsilon\in(0,\epsilon_0)\),
\[
\iint_{Q_r(z_0)} H(z,|\nabla \nu|)^{1+\epsilon}\,dz
\le c\left(\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^p+\|a\|_{L^\infty(\Omega_T)}\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^q+1\right)^{1+\frac{q\epsilon}{p}}
+ c\left(\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\epsilon}\,dz\right).
\]
D.
There exist constants \(\epsilon_0=\epsilon_0(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})>0\) and \(c=c(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)},\|a\|_{L^\infty(\Omega_T)})\) such that for every \(\epsilon\in(0,\epsilon_0)\) and every parabolic cylinder \(Q_{4r}(z_0)\subset \Omega_T\) with \(r>0\),
\[
\iint_{Q_r(z_0)} H(z,|\nabla \nu|)^{1+\epsilon}\,dz
\le c\left(\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^p+\|a\|_{L^\infty(\Omega_T)}\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^q+1\right)^{1+\frac{q\epsilon}{p}}
+ c\left(\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\epsilon}\,dz\right)^{1+\frac q2}.
\]
E.
There exist constants \(\epsilon_0=\epsilon_0(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})>0\) and \(c=c(n,N,p,q,\nu,L,c_a,\|\nu\|_{L^\infty(\Omega_T)},\|H(z,|F|)\|_{L^\gamma(\Omega_T)})\) such that for every \(\epsilon\in(0,\epsilon_0)\),
\[
\iint_{Q_r(z_0)} H(z,|\nabla \nu|)^{1+\epsilon}\,dz
\le c\left(\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^p+\|a\|_{L^\infty(\Omega_T)}\left(\frac{\|\nu\|_{L^\infty(\Omega_T)}}{r}\right)^q+1\right)^{1+\frac{q\epsilon}{p}}
+ c\left(\iint_{Q_{4r}(z_0)} H(z,|F|)^{1+\epsilon}\,dz\right)^{1+\frac q2}.
\]

forcing-term exponent after Young inequality

drop the stronger outer exponent on the forcing term

radius restriction r\in(0,1) tied to intrinsic scaling and exit-time geometry

dependence of the constant on \|a\|_{L^\infty} introduced in the final Caccioppoli step

The stem does not explicitly reveal the correct option or quote the exact conclusion. It gives the full hypothesis set of a theorem, but the distinguishing features of the right answer (such as the precise forcing-term exponent and dependence structure of constants) are not stated in the stem.

This is essentially a direct theorem-recall item: the stem lists the hypotheses and asks for the guaranteed estimate. The task is mainly to recognize the exact stated conclusion rather than infer among genuinely different mathematical outcomes.

Some reasoning is needed to compare subtle differences in exponents, radius restrictions, and parameter dependence, but the item mostly rewards memory/recognition of the theorem statement rather than generative mathematical reasoning from the assumptions.

The distractors are plausible and mathematically close to the target statement: they alter the forcing-term exponent, weaken the estimate, change the radius range, or tamper with dependence of constants. These align with realistic failure modes in recalling technical regularity results.

Technically well-constructed with strong distractors and little direct answer leakage, but it is largely a theorem-restatement/recognition question rather than a strong test of generative reasoning.


The study of zero sets of smooth functions and ideals in the ring
$\mathcal{E}(\Omega)$ of infinitely differentiable functions on an open set
$\Omega \subset \mathbb{R}^n$ lies at the intersection of real analytic geometry,
singularity theory, and differential topology.
A central tool in this context is the {\L}ojasiewicz inequality, which provides a
quantitative relation between a function and its gradient, and plays a fundamental
role in the analysis of subanalytic sets, stratifications, and resolution of singularities.

A finitely generated ideal $I = (f_1, \ldots, f_k)\mathcal{E}(\Omega)$ is said
to be a {\L}ojasiewicz ideal if there exists a function $g \in I$ that satisfies
a {\L}ojasiewicz inequality with respect to
$
Z(I) := \{ x \in \Omega \mid f_1(x) = f_2(x) = \cdots = f_k(x) = 0 \}.
$
See the definition below.\\
A classical result of Ren\'e Thom \cite{thom1967some} asserts that if $I$ is a finitely generated Lojasiewicz ideal,
then its zero set $Z(I)$ contains an open dense subset of smooth points.\\ In analytic geometry this is expected: analytic ideals naturally enforce
geometric regularity on their zero sets.
However, in the smooth category such regularity properties are more subtle, since
arbitrary smooth functions may exhibit behavior impossible in the analytic setting.

This raises a natural question:
\emph{does the converse hold?}
Does the presence of an open dense set of smooth points in $Z(I)$ imply that
the ideal $I$ is \L{}ojasiewicz?
The purpose of the present work is to examine this question and to identify
mechanisms by which such a converse can fail.\\
To illustrate this phenomenon, we focus on a classical topological example:
the \emph{Hawaiian earring}, defined as the union of the circles
\[
C_n = \left\{ (x_1,x_2) \in \mathbb{R}^2 :
\left(x_1 - \frac{1}{n}\right)^2 + x_2^2 = \frac{1}{n^2} \right\}, \qquad n \ge 1.
\]
This compact set consists of infinitely many circles tangent at the origin and
accumulating there.
It is well known that the Hawaiian earring has highly pathological local
topology at the origin: it is not locally contractible, not semianalytic, and
cannot arise as the zero set of any nontrivial real analytic function.

We consider smooth functions with real values $f \in \mathcal{E}^\infty(\mathbb{R}^2)$ whose zero
set $Z(f)$ is \emph{exactly} the Hawaiian earring.
Such functions exist by Whitney's extension theorem, see theorem 3.1, Ch IV, \cite{Tougeron1972}, yet they exhibit extremely
degenerate behavior at the origin.
Our first main result states that any such function must vanish to infinite order.

This raises a natural question:
\emph{does the converse hold?}
Does the presence of an open dense set of smooth points in $Z(I)$ imply that
the ideal $I$ is \L{}ojasiewicz?
The purpose of the present work is to examine this question and to identify
mechanisms by which such a converse can fail.\\
To illustrate this phenomenon, we focus on a classical topological example:
the \emph{Hawaiian earring}, defined as the union of the circles
\[
C_n = \left\{ (x_1,x_2) \in \mathbb{R}^2 :
\left(x_1 - \frac{1}{n}\right)^2 + x_2^2 = \frac{1}{n^2} \right\}, \qquad n \ge 1.
\]
This compact set consists of infinitely many circles tangent at the origin and
accumulating there.
It is well known that the Hawaiian earring has highly pathological local
topology at the origin: it is not locally contractible, not semianalytic, and
cannot arise as the zero set of any nontrivial real analytic function.

We then formulate a general geometric criterion for flatness, valid for arbitrary
collections of smooth arcs tangent at a point with unbounded curvature or
infinitely varying radius of osculation.
This leads to a \emph{degenerate \L{}ojasiewicz inequality} adapted to situations
where the classical inequality necessarily fails.

\begin{theo}
Let $\Gamma = \bigcup\limits_{n \ge 1} \gamma_n$ be a union of smooth embedded arcs
meeting at a common point $p$ with curvature tending to $\infty$ or oscillating
without bound.
If $f \in \mathcal{E}^\infty(\Omega)$ vanishes on $\Gamma$, then $f$ satisfies a flatness
estimate of the form
\[
|f(x)| \le C_N\, d(x,\Gamma)^N,
\qquad \text{for all } N \ge 1 \text{ and all } x \text{ near } p,
\]
for suitable constants $C_N>0$.
In particular, $f$ is flat at $p$.
\end{theo}

\begin{rem}
In this case, for any system of generators $g_1,\ldots,g_p$ of $I$,
the functions $\sum\limits_{j=1}^p g_j^2$ and $\sum\limits_{j=1}^p |g_j|$ both satisfy
the {\L}ojasiewicz inequality with respect to $Z(I)$.
\end{rem}
It should be noted that the property of an ideal to be {\L}ojasiewicz is a
local one: if an ideal is {\L}ojasiewicz on an open set $U$, then the induced
ideal on any smaller open subset is also {\L}ojasiewicz.\\
Any analytic ideal is {\L}ojasiewicz, as this follows from the fundamental
{\L}ojasiewicz inequality for analytic functions \cite{Lojasiewicz1965}. Moreover, any finitely
generated ideal that is closed in $\mathcal{E}^\infty(\Omega)$ is also a
{\L}ojasiewicz ideal, see Corollary 4.4, Ch V, \cite{Tougeron1972}. Indeed, by Whitney's spectral theorem, a smooth function
belongs to a closed finitely generated ideal $I=(f_1,\ldots,f_k)$ if and only if,
for every point $x\in\Omega$, its Taylor expansion at $x$ lies in the ideal
generated by the Taylor expansions $T_x f_1,\ldots,T_x f_k$ in the formal power
series algebra $\mathbb{R}[[X-x]]$.
However, it should be emphasized that there exist {\L}ojasiewicz ideals
which are not closed, in dimension $n >1$, see Examples 4.8, Ch V, \cite{Tougeron1972}.
\subsection{Smooth points of the locus of zeros}
\begin{defn}[Smooth point]
Let $E \subset \mathbb{R}^n$ be closed set.
A point $x \in E$ is said to be smooth if, in a neighborhood of $x$,
the set $E$ coincides with a $k$-dimensional embedded submanifold of $\mathbb{R}^n$,
where $k$ is its local dimension at $x$ and may vary with $x$.
\end{defn}
\begin{Ex}
 The \emph{Hawaiian earring},
 defined as the union of circles
\[
\bigcup_{n=1}^{\infty} \left\{(x,y)\in \mathbb{R}^2 : \left(x-\frac{1}{n}\right)^2 + y^2 = \frac{1}{n^2} \right\},
\]
is smooth at every point except the origin, where infinitely many circles accumulate.
Hence, the set of smooth points of the Hawaiian earring is dense in $E= \bigcup_{n=1}^{\infty} \left\{(x,y)\in \mathbb{R}^2 : \left(x-\frac{1}{n}\right)^2 + y^2 = \frac{1}{n^2} \right\}$.

\begin{theo}
Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^\infty$ function.
Assume that $f$ vanishes on the Hawaiian earring
\[
\mathcal H=\bigcup_{n\ge1} C_n, \qquad
C_n=\left\{(x,y):\left(x-\tfrac1n\right)^2+y^2=\tfrac1{n^2}\right\}.
\]
Then $f$ is flat at the origin:
\[
\frac{\partial^{\alpha_1 +\alpha_2 f}}{\partial x_1^{\alpha_1} \partial x^{\alpha_2}_2}(0,0)
 = 0
\qquad \text{for all } \alpha =(\alpha_1,\alpha_2) \in\N^2.
\]
\end{theo}

Thus there is no nonzero homogeneous term in the Taylor expansion of $f$ at $(0,0)$,
so all derivatives of $f$ at the origin vanish.
Hence $f$ is flat at $(0,0)$.
\section{Łojasiewicz Inequality and the Hawaiian Earring}
\begin{theo}
Let $f \in C^\infty(\mathbb{R}^2)$ and assume its zero set is the Hawaiian earring
\[
\mathcal H:= \bigcup_{k=1}^{\infty} \left\{ (x,y) \in \mathbb{R}^2 : \left(x - \frac{1}{k}\right)^2 + y^2 = \frac{1}{k^2} \right\}.
\]
Then $f$ is flat at the origin and does \emph{not} satisfy a Łojasiewicz inequality with respect to $H$ at $0$; that is, there exist no constants $C>0$ and $\theta>0$ such that
\[
|f(x)| \ge C \, \mathrm{dist}(x,H)^\theta
\]
for all $x$ in a neighborhood of $0$.
\end{theo}
{\bf{Proof.}}
Since $f \in C^\infty$ vanishes on the Hawaiian earring, which has tangent directions at $0$ dense in the unit circle $S^1$, Theorem 4 implies that $f$ is flat at the origin:
\[
D^\alpha f(0) = 0 \quad \text{for all multi-indices } \alpha.
\]
Let $x \in \mathbb{R}^2$ be close to $0$. Consider the sequence of circles in $H$ with radii $r_k = 1/k$ and centers $c_k = (1/k,0)$.
Choosing $k \sim 1/\|x\|$, the circle $C_k$ closest to $x$ has radius $r_k \sim \|x\|$.
Hence, for points near $0$,
\[
\mathrm{dist}(x,H) \sim \|x\|.
\]
Assume, for contradiction, that there exist constants $C>0$ and $\theta>0$ such that
\[
|f(x)| \ge C \, \mathrm{dist}(x,H)^\theta
\]
for all $x$ near $0$.

Choosing $N > \theta$ gives
\[
|f(x)| \ll \mathrm{dist}(x,H)^\theta
\]
for $x$ sufficiently close to $0$, which contradicts the assumed Łojasiewicz inequality.
Therefore, the ideal generated by
 the function $f$ in the the ring $\mathcal{E}^\infty(\R^2)$ is not a
 Lojasiewicz ideal.
\section{{Jet-Determination of Smooth Functions via Families of Arcs}}
\begin{theo}
Let $\mathcal A = \{\gamma_i\}_{i \in I}$ be a family of smooth arcs
\[
\gamma_i : (-\varepsilon,\varepsilon) \to \mathbb{R}^n, \qquad \gamma_i(0)=0, \quad \gamma_i \not\equiv \text{constant}.
\]
For each arc, denote its $m$-jet at $0$ by
\[
j^m \gamma_i(0) = (\gamma_i'(0), \gamma_i''(0), \dots, \gamma_i^{(m)}(0)).
\]
The following assertions are equivalent:
\begin{enumerate}
\item[(A)] \textbf{Jet–Determination Property:} For every function $f \in C^\infty(\mathbb{R}^n)$,
\[
f(\gamma_i(s)) \equiv 0 \ \text{for all } i \quad \Longrightarrow \quad D^\alpha f(0)=0 \ \text{for all multi-indices } \alpha.
\]
\item[(B)] \textbf{Jet–Nondegeneracy Condition:} For every integer $m \ge 1$, there exists no nonzero homogeneous polynomial $P_m$ of degree $m$ such that
\[
P_m(j^m \gamma_i(0))=0 \qquad \text{for all } i \in I.
\]
\end{enumerate}
\end{theo}
{\bf{Proof.}}
\textbf{(A) $\implies$ (B).} \\Suppose (A) holds and, for contradiction, that (B) fails for some $m$.
Then there exists a nonzero homogeneous polynomial $P_m$ of degree $m$ such that
\[
P_m(j^m \gamma_i(0)) = 0 \quad \text{for all } i.
\]

Then $E_n$ is jet-determining at $0$: if $f \in C^\infty(\mathbb{R}^n)$ vanishes on $E_n$, then $f$ is flat at the origin:
\[
D^\alpha f(0) = 0 \quad \text{for all multi-indices } \alpha.
\]
\end{cor}

The proof relies on purely geometric considerations: any nonzero term of the
Taylor expansion of $f$ at $(0,0)$ would define an algebraic curve of finite
order, and no such curve can contain infinitely many smooth arcs with tangency
and curvature behavior comparable to the circles $C_n$.
Thus the geometry of the Hawaiian earring forces infinite-order vanishing.

We then formulate a general geometric criterion for flatness, valid for arbitrary
collections of smooth arcs tangent at a point with unbounded curvature or
infinitely varying radius of osculation.
This leads to a \emph{degenerate \L{}ojasiewicz inequality} adapted to situations
where the classical inequality necessarily fails.

\begin{theo}
Let $\Gamma = \bigcup\limits_{n \ge 1} \gamma_n$ be a union of smooth embedded arcs
meeting at a common point $p$ with curvature tending to $\infty$ or oscillating
without bound.
If $f \in \mathcal{E}^\infty(\Omega)$ vanishes on $\Gamma$, then $f$ satisfies a flatness
estimate of the form
\[
|f(x)| \le C_N\, d(x,\Gamma)^N,
\qquad \text{for all } N \ge 1 \text{ and all } x \text{ near } p,
\]
for suitable constants $C_N>0$.
In particular, $f$ is flat at $p$.
\end{theo}

This result demonstrates that certain pathological smooth zero sets necessarily
force infinite-order degeneracy, and therefore cannot arise from \L{}ojasiewicz
ideals.
In particular, the ideal generated by a function defining the Hawaiian earring is
never \L{}ojasiewicz, despite the fact that its zero set is smooth away from a
single point.

The proof is described as relying on “purely geometric considerations”: if the Taylor expansion of $f$ at $(0,0)$ had “any nonzero term,” it would “define an algebraic curve of finite order,” but “no such curve can contain infinitely many smooth arcs with tangency and curvature behavior comparable to the circles $C_n$.” Hence, “the geometry of the Hawaiian earring forces infinite-order vanishing,” i.e., flatness at the origin.

The introduction then says it “formulate[s] a general geometric criterion for flatness” for “collections of smooth arcs tangent at a point with unbounded curvature or infinitely varying radius of osculation,” yielding a “degenerate \L{}ojasiewicz inequality” (a flatness estimate $|f(x)|\le C_N d(x,\Gamma)^N$) in settings “where the classical inequality necessarily fails.”

Let $f \in \mathcal{E}^\infty(\mathbb{R}^2)$ be such that $Z(f)$ is the Hawaiian earring. Then $f$ is flat at the origin:
\[
\frac{\partial^{\alpha +\beta}f}{\partial x_1^\alpha \partial x_2^\beta}(0,0)
 = 0
\qquad \text{for all } \alpha,\beta \in\mathbb{N}.
\],

Question:
Let
\[
\mathcal H:= \bigcup_{k=1}^{\infty} \left\{ (x,y) \in \mathbb{R}^2 : \left(x-\frac{1}{k}\right)^2+y^2=\frac{1}{k^2} \right\}
\]
be the Hawaiian earring, and let \(Z(f)=\{(x,y)\in\mathbb R^2: f(x,y)=0\}\) denote the zero set of a smooth function \(f\in \mathcal E^{\infty}(\mathbb R^2)=C^{\infty}(\mathbb R^2)\). Which statement holds for every smooth function \(f\) such that \(Z(f)=\mathcal H\)?

The function \(f\) is flat at the origin; equivalently, for every pair of integers \(\alpha,\beta\ge 0\),
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0.
\]

Choices:
B.
There exists an integer \(N\ge 1\) such that \(f\) vanishes to order at least \(N\) at the origin; equivalently,
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0
\qquad\text{for all }\alpha,\beta\in\mathbb N\text{ with }\alpha+\beta\le N,
\]
but not necessarily for all orders.
C.
For every pair of integers \(\alpha,\beta\ge 0\), if \(\alpha+\beta=1\) then
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^{\alpha}\partial x_2^{\beta}}(0,0)=0.
\]
D.
There exist constants \(C>0\) and \(\theta>0\) such that, for all \((x,y)\) sufficiently close to \((0,0)\),
\[
|f(x,y)|\ge C\,\mathrm{dist}\bigl((x,y),\mathcal H\bigr)^{\theta}.
\]
E. The function \(f\) is real-analytic in a neighborhood of the origin; equivalently, if its Taylor expansion at \((0,0)\) has any nonzero term, then that finite-order algebraic curve contains the arcs of \(\mathcal H\) near the origin.

infinite-order vanishing replaced by finite-order vanishing

dropped all higher-order derivative vanishing, keeping only first-order consequences

classical Lojasiewicz lower bound in a setting where only degenerate flatness estimates are compatible

finite-order algebraic-curve obstruction turned into analyticity near the origin

The stem does not explicitly state or strongly hint that flatness at the origin is the required conclusion. It only specifies the Hawaiian earring as the zero set and asks for a universal consequence.

The question is close to a theorem-recall item: the correct choice is essentially the main conclusion associated with this zero-set configuration. However, it is not completely tautological because the options include weaker true and plausible false alternatives that must be distinguished.

The item demands substantial reasoning or theorem-level understanding: one must infer infinite-order vanishing from the geometry of the Hawaiian earring and reject tempting alternatives such as finite-order vanishing, mere first-order vanishing, a Łojasiewicz bound, or analyticity.

The distractors are strong and mathematically meaningful. B is a near-miss, C is a weaker true statement, D reflects a common confusion with analytic/geometric inequalities, and E exploits a plausible but false analyticity intuition.

A strong MCQ with little answer leakage and excellent distractors. Its main weakness is that the correct option closely mirrors the underlying theorem, so it leans somewhat toward theorem recognition rather than fully novel inference.