\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 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 \(f \in \mathcal{E}^\infty(\mathbb{R}^2)\), where \(\mathcal{E}^\infty(\mathbb{R}^2)\) denotes the ring of smooth real-valued functions on \(\mathbb{R}^2\). Assume that the zero set
\[
Z(f):=\{(x,y)\in \mathbb{R}^2: f(x,y)=0\}
\]
is exactly the Hawaiian earring
\[
\mathcal H= \bigcup_{k=1}^{\infty} \left\{ (x,y) \in \mathbb{R}^2 : \left(x - \frac{1}{k}\right)^2 + y^2 = \frac{1}{k^2} \right\}.
\]
Under these assumptions, which statement about the behavior of \(f\) at the origin holds?

\(f\) is flat at the origin; equivalently,
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^\alpha\partial x_2^\beta}(0,0)=0
\qquad \text{for all } \alpha,\beta\in\mathbb{N}.
\]

Choices:
B.
\(f\) vanishes to finite order at the origin; equivalently, there exists an integer \(m\ge 1\) such that
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^\alpha\partial x_2^\beta}(0,0)=0
\quad \text{for all } \alpha+\beta<m,
\qquad \text{and} \qquad
\frac{\partial^{m}f}{\partial x_1^\alpha\partial x_2^\beta}(0,0)\neq 0
\]
for some \(\alpha,\beta\in\mathbb N\) with \(\alpha+\beta=m\).
C.
\(f\) vanishes at the origin together with all first-order partial derivatives; that is,
\[
f(0,0)=0,
\qquad
\frac{\partial f}{\partial x_1}(0,0)=\frac{\partial f}{\partial x_2}(0,0)=0.
\]
D.
\(f\) is flat only along the tangent directions determined by the circles of \(\mathcal H\) at the origin; equivalently, every directional derivative of every order of \(f\) along such tangent directions vanishes at \((0,0)\), but there may exist mixed partial derivatives
\[
\frac{\partial^{\alpha+\beta}f}{\partial x_1^\alpha\partial x_2^\beta}(0,0)\neq 0
\]
for some \(\alpha,\beta\in\mathbb N\).
E.
There exists an integer \(N\ge 1\) such that \(f\) satisfies a finite-order flatness estimate near the origin,
\[
|f(x,y)|\le C\,\operatorname{dist}\bigl((x,y),\mathcal H\bigr)^N,
\]
for some constant \(C>0\); equivalently, \(f\) need not be flat at the origin, but only vanish to sufficiently high finite order there.

all_higher_order_vanishing dropped to first-order vanishing only

from full jet flatness to directional-flatness-only

infinite-order flatness estimate replaced by one fixed finite exponent


\begin{thm}\label{Pointwise Semi-Classical OT} There exists $C>0$ and $p_1\in \N$ such that for any $p\geq p_1$, any $x\in X$, and any $v\in (A^p)_x$,  one can find a global holomorphic section $a=a_{p,x,v}\in H^0(X,A^p)$ with
\begin{equation}\label{OT bound}
    a(x)=v,\quad |v|_{(h_0^A)^p}\leq \sup_X|a|_{(h_0^A)^p}\leq \left(1+\frac{C}{p}\right)|v|_{(h_0^A)^p},
\end{equation}
where $C=C(n,X,\omega)$ depends only on dimension $n$, manifold $X$, and K\"ahler form $\omega=c_1(A,h_0^A)$.
\end{thm}

\begin{thm}\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\in \PSH(X,\theta)\cap L^\infty$, we have
\[
\lim_{k\to\infty}E_k(u)=E_\theta(u).
\]
\end{thm}

\begin{equation}\label{MA envelope}
		P_\theta(f):=usc(\sup\{u\in \PSH(X,\theta):u\leq f\})
	\end{equation}

Given an ample line bundle \(L\) over a complex projective manifold \(X\), endowed with the Kähler form \(\omega\) polarized by \(L\), a central theme in Kähler geometry—first articulated by Yau \cite[p.~139]{Yau}—is the approximation of transcendental geometric objects on \((X,\omega)\) by asymptotic algebraic data coming from the powers \(L^k:=L^{\otimes k}\). This is the philosophy of \emph{Kähler quantization}.

The initial proposal is to approximate the infinite-dimensional space
\[
   \mathcal{H}_\omega := \{\,\phi\in C^\infty(X) : \omega + dd^c \phi > 0\,\}
\]
of Kähler potentials by the finite-dimensional spaces \(\mathcal{H}_k\) of positive Hermitian forms on \(H^0(X,L^k)\). Via the Kodaira embedding, elements of \(\mathcal{H}_k\) correspond to Fubini–Study metrics and thus form a distinguished  subspace of \(\mathcal{H}_\omega\). This was established in the seminal works of Tian \cite{Tian}, Bouche \cite{Bouche}, Ruan \cite{Ruan}, Catlin \cite{Catlin}, Zelditch \cite{Zelditch}, Lu \cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\ref{on diagonal Bergman kernel}).

Donaldson later proposed that not only the metrics but also the \emph{geometry} of \(\mathcal{H}_\omega\) should be captured by the finite-dimensional geometries of \(\mathcal{H}_k\) \cite[p.~483]{DonaldsonI}. This principle has since been realized in various settings and has had deep impacts on problems concerning canonical metrics and stability; see, for example, \cite{PS,SongZelditchqu,CS,BerndtssonProb,DLR,DonaldsonI,DonaldsonScalII,RTZ,Zhang} among a rapidly expanding body of literature. We refer to \cite{Mama} for a detailed exposition of some classical developments in this direction.

A natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \(L\)  only holds in a weak sense.  The present note fits into this broader direction, establishing a quantization result for Monge--Amp\`ere energy in big and semipositive setting.

\subsection*{Statement of Main Result}
We now describe our results, referring to  Section~\ref{finite energy space and weak geodesic} and Section~\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive.  Let $h_0^L$ be a smooth metric on $L$ and let $\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\theta$ is a smooth semipositive $(1,1)$-form.

One of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with  the set of $\theta$-plurisubharmonic functions on $X$, denoted by $\PSH(X,\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\theta_u:=\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \cite{BEGZ}, for $u\in \PSH(X,\theta)$, one can define the \textit{non-pluripolar Monge--Amp\`ere measure} $\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by
\[
E_\theta:\PSH(X,\theta)\to [-\infty,\infty)
\]
the \textit{Monge--Amp\`ere energy} or \textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\`ere operator and plays an important role in the variational approach to canonical K\"ahler metrics.

On the other hand, let $K_X:=\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\in\mathbb N$, we consider the adjoint line bundle $L^k\otimes K_X$. For global sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$, locally written  as
\[
s_j=\sigma_j\, dz_1\wedge \cdots \wedge dz_n,\quad j=1,2,
\]
where $\sigma_j$ are local holomorphic sections of $L^k$, we set
\[
\langle s_1,s_2\ra_{(h_0^L)^k}:=\langle \sigma_1,\sigma_2\ra_{(h_0^L)^k}\left(\frac{i}{2}\right)^n
dz_1\wedge d\bar{z}_1\wedge \cdots \wedge dz_n\wedge d\bar{z}_n,
\]
yielding  a smooth volume form on $X$. For a function $u:X\to [-\infty,\infty)$, we define the \textit{Hilbert norm}
\begin{equation}\label{Hilbert norm}
		(s_1,s_2)_{Hilb_k(u)}:=\int_X \langle s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \mathbb N,
\end{equation}
for sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$ such that \eqref{Hilbert norm} is finite. We say $u$ is \textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\otimes K_X)$ for any $k\in \mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \textit{quantized Monge--Amp\`ere energy} by
\begin{equation}\label{quantized MA-intro}
    E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det Hilb_k(u)}{\det Hilb_k(0)}\right),
\end{equation}
which is first introduced by Donaldson \cite{DonaldsonScalII} (see also Berman–Boucksom \cite{BB10}).

Our main result is the  quantization of Monge--Amp\`ere energy in adjoint twisting.

The initial proposal is to approximate the infinite-dimensional space
\[
 \mathcal{H}_\omega := \{\,\phi\in C^\infty(X) : \omega + dd^c \phi > 0\,\}
\]
of Kähler potentials by the finite-dimensional spaces \(\mathcal{H}_k\) of positive Hermitian forms on \(H^0(X,L^k)\). Via the Kodaira embedding, elements of \(\mathcal{H}_k\) correspond to Fubini–Study metrics and thus form a distinguished subspace of \(\mathcal{H}_\omega\). This was established in the seminal works of Tian \cite{Tian}, Bouche \cite{Bouche}, Ruan \cite{Ruan}, Catlin \cite{Catlin}, Zelditch \cite{Zelditch}, Lu \cite{Lu}, among many others, using asymptotics of the Bergman kernel (cf. Theorem~\ref{on diagonal Bergman kernel}).

A natural question is whether this quantization picture extends beyond the ample case. This has become increasingly relevant in view of recent progress on canonical metrics in degenerate or singular settings \cite{BBJ,LTW,Li,DZ24,Dervan,Xu,PT25-1,PTT23,PT25-2}, where the positivity of \(L\) only holds in a weak sense. The present note fits into this broader direction, establishing a quantization result for Monge--Amp\`ere energy in big and semipositive setting.

\subsection*{Statement of Main Result}
We now describe our results, referring to Section~\ref{finite energy space and weak geodesic} and Section~\ref{Bergman Measures, Positivity of Direct Images, and Equilibrium} for details. Let $L$ be holomorphic line bundle over a complex projective manifold $X$ of dimension $n$. We assume that $L$ is big and semipositive. Let $h_0^L$ be a smooth metric on $L$ and let $\theta:=c_1(L,h_0^L)$ denote the Chern curvature of $h_0^L$ so that $\theta$ is a smooth semipositive $(1,1)$-form.

One of the difficulties in the degenerate setting is the lack of smooth potentials. Instead, we work with the set of $\theta$-plurisubharmonic functions on $X$, denoted by $\PSH(X,\theta)$. This parametrizes singular hermitian metrics on $L$ whose curvature $\theta_u:=\theta+dd^cu$ is positive in the current sense. Following the pioneering work of Boucksom et al. \cite{BEGZ}, for $u\in \PSH(X,\theta)$, one can define the \textit{non-pluripolar Monge--Amp\`ere measure} $\theta_u^n$, which is a Borel measure on $X$ that puts no mass on pluripolar sets. Moreover, we denote by
\[
E_\theta:\PSH(X,\theta)\to [-\infty,\infty)
\]
the \textit{Monge--Amp\`ere energy} or \textit{Aubin--Yau energy}. It is the primitive of the Monge--Amp\`ere operator and plays an important role in the variational approach to canonical K\"ahler metrics.

On the other hand, let $K_X:=\Omega^{n,0}_X$ be the canonical bundle on $X$. For $k\in\mathbb N$, we consider the adjoint line bundle $L^k\otimes K_X$. For global sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$, locally written as
\[
s_j=\sigma_j\, dz_1\wedge \cdots \wedge dz_n,\quad j=1,2,
\]
where $\sigma_j$ are local holomorphic sections of $L^k$, we set
\[
\langle s_1,s_2\ra_{(h_0^L)^k}:=\langle \sigma_1,\sigma_2\ra_{(h_0^L)^k}\left(\frac{i}{2}\right)^n
dz_1\wedge d\bar{z}_1\wedge \cdots \wedge dz_n\wedge d\bar{z}_n,
\]
yielding a smooth volume form on $X$. For a function $u:X\to [-\infty,\infty)$, we define the \textit{Hilbert norm}
\begin{equation}\label{Hilbert norm}
 (s_1,s_2)_{Hilb_k(u)}:=\int_X \langle s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \mathbb N,
\end{equation}
for sections $s_1,s_2\in H^0(X,L^k\otimes K_X)$ such that \eqref{Hilbert norm} is finite. We say $u$ is \textit{admissible} if $Hilb_k(u)$ is finite on the whole space $H^0(X,L^k\otimes K_X)$ for any $k\in \mathbb N$. Clearly, any bounded function is admissible. For an admissible function $u$, we define the \textit{quantized Monge--Amp\`ere energy} by
\begin{equation}\label{quantized MA-intro}
 E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det Hilb_k(u)}{\det Hilb_k(0)}\right),
\end{equation}
which is first introduced by Donaldson \cite{DonaldsonScalII} (see also Berman–Boucksom \cite{BB10}).

Our main result is the quantization of Monge--Amp\`ere energy in adjoint twisting.

\begin{equation}
    \frac{d}{dt}\Big|_{t=0}E_\theta(P_\theta(u+tf))=\frac{1}{\vol(\{\theta\})}\int_Xf\theta_u^n.
\end{equation}
\end{lem}
\section{Bergman Measures, Positivity of Direct Images, and Equilibrium}\label{Bergman Measures, Positivity of Direct Images, and Equilibrium}
\subsection*{Bergman Measure and Quantized Monge--Amp\`ere Energy}
Let $V$ be a finite-dimensional complex vector space of dimension $N$. We denote $\H_V$ by the space of positive-definite hermitian inner products on $V$. For $H,H'\in \H_V$, there exists an $H$-hermitian endomorphism $A\in \End(V)$ such that
\begin{equation}\label{transfer map}
(s_1,s_2)_{H'}=(e^As_1,s_2)_H,\quad s_1,s_2\in V.
\end{equation}

\begin{equation}\label{Hilbert norm}
		(s_1,s_2)_{Hilb_k(u)}:=\int_X \la s_1,s_2\ra_{(h_0^L)^k}e^{-ku},\quad k\in \N,
\end{equation}

\begin{equation}\label{volume of a line bundle}
		\vol(\{L\})=\limsup_{k\to\infty}\frac{\dim_\C H^0(X,L^k)}{k^n/n!},
	\end{equation}
	which we  denote by $\vol(L)$. Hence, $L$ is big iff the right-hand side of \eqref{volume of a line bundle} is positive and furthermore $\limsup$ in \eqref{volume of a line bundle} is actually a limit by a Theorem of Fujita \cite{Fujita} (see also  \cite[\S 2.2C]{LazarsfeldI}).

\subsection*{Finite Energy Space and Weak Geodesics}\label{finite energy space and weak geodesic}
	Let $\theta$ be a closed, smooth $(1,1)$-form so that $\{\theta\}$ is big. If $u\in\PSH(X,\theta)$ has minimal singularity type, then we define \textit{Monge--Amp\`ere energy} by
	\begin{equation}\label{MA energy}
	E_\theta(u):=\frac{1}{(n+1)\vol(\{\theta\})}\sum_{j=0}^n\int_X (u-V_\theta) \theta_u^j\wedge \theta^{n-j}_{V_\theta}
	\end{equation}

\begin{equation}\label{on diagonal Bergman kernel}
K_p(x):=K_p(x,x)=\sum_{j=1}^{d_p}|S_j(x)|^2_{(h_0^A)^p}.
\end{equation}

Rescaling $w=\sqrt{k}z$ on the standard shrinking polydisc gives the uniform approximation
\[
k\Phi_\epsilon(w/\sqrt{k})
=\Phi_\epsilon^0(w)+\rho_k+\epsilon\widetilde\rho_k,
\qquad
\rho_k,\widetilde\rho_k\to0.
\]
Repeating the submean-value argument used for Theorem~\ref{Berman's local Morse inequality} then yields
\[
\frac{n!|s|^2(x)}{k^n\|s\|^2}
\le
e^{\rho_k+\epsilon\widetilde\rho_k}
\frac{n!dV_n}{(2\pi)^n\prod_{j=1}^n\int_0^{\log k} e^{-(\lambda_j+\epsilon)r^2}r\,dr}.
\]
Since the quadratic model satisfies $\lambda_j+\epsilon\ge\epsilon>0$, the Gaussian integrals in the denominator behave exactly as in the proof of Theorem~\ref{Berman's local Morse inequality}, with $\lambda_j$ replaced by $\lambda_j+\epsilon$.
Using the identity
\[
(\theta+\epsilon\omega)^n
= \frac{n!}{\pi^n}\!\left(\prod_{j=1}^n (\lambda_j+\epsilon)\right)dV_n,
\]
the same computation as in the untwisted case yields
\[
\frac{n!|s|^2(x)}{k^n\|s\|^2}
\;\le\;
e^{\rho_k+\epsilon\widetilde\rho_k}
\Biggl(\prod_{j=1}^n
 \frac{1}{1-e^{-(\lambda_j+\epsilon)(\log k)^2}}\Biggr)
(\theta+\epsilon\omega)^n(x)\leq {(1-e^{-\epsilon(\log k)^2})^n}(\theta+\epsilon\omega)^n(x),
\]
since $\lambda_j(x)+\epsilon\ge\epsilon$ for all $j$ and $x\in X$. \eqref{epsilon twisted local holomorphic morse} then follows from extremal characterization \eqref{extremal for twisted Bergman measure}.
\end{proof}
\section{Proof of Theorem \ref{quantization of MA energy} and Corollary \ref{convergence of Bergman measures}}
Recall that $X$ is a compact projective manifold of dimension $n$, $L$ be a big and semipositive line bundle over $X$ with fixed smooth metric $h_0^L$ on $L$ with Chern curvature form $\theta:=c_1(L,h^L_0)\geq 0$. Let us first deduce Corollary \ref{convergence of Bergman measures} from Theorem \ref{quantization of MA energy}.
\begin{proof}[Proof of Corollary \ref{convergence of Bergman measures}] For $f\in C^0(X)$ and $t\in \R$, $u+tf$ is bounded and thus is admissible. We set
\[
f_k(t):=E_k(u+tf),\quad g(t):=E_\theta(P_\theta(u+tf)).
\]
Notice that $u+tf\geq P_\theta(u+tf)\geq m:=\inf_X(u+tf)>-\infty$ by definition of envelope and thus, $P_\theta(u+tf)\in \PSH(X,\theta)\cap L^\infty$. By Theorem \ref{quantization of MA energy}, we obtain that for any $t\in \R$,
\[
\liminf_{k\to\infty}f_k(t)\geq \lim E_k(P_\theta(u+tf))=g(t),\quad \lim_{k\to\infty}f_k(0)=\lim_{k\to\infty}E_k(u)=E_\theta(u)=g(0).
\]
By Lemma \ref{variational formula for QMA along affine path}, $f_k(t)$ is concave and is differentiable at $t=0$:
\[
f_k'(0)=\int_X f\beta_{L^k,u}.
\]
On the other hand, by Lemma \ref{variation formula of MA energy}, $g(t)$ is differentiable at $t=0$ and
\[
g'(0)=\int_Xf\frac{\theta_u^n}{\vol(\{\theta\})}.
\]
The result follows from the elementary lemma proved in \cite[Lemma 3.1]{BBWN}.
\end{proof}
\begin{rmk}
 In fact, the arguments in Theorem \ref{upper bound estimate} below shows that $\lim_{k\to\infty}f_k(t)=g(t)$.
\end{rmk}

We now begin with the upper bound in Theorem \ref{quantization of MA energy} which holds for any finite-energy potentials.
\begin{thm}\label{upper bound estimate} For $u\in \E^1(X,\theta)$, $\limsup_{k\to\infty}E_k(u)\leq E_\theta(u)$.
\end{thm}
\begin{proof} The proof goes exactly as in \cite[Theorem 3.5]{BF14} and \cite[Theorem A]{BB10}. Since $u$ is upper semi-continuous, we can find a sequence $\phi_j\in C^\infty(X)$ such that $\phi_j\downarrow u$. Clearly, $\phi_j$ is admissible and $E_k(u)\leq E_k(\phi_j)$ by Lemma \ref{properties of Hilb and QMA} (ii). By Corollary \ref{Quantization of MOnge-Ampere energy for smooth functions},
\[
\limsup_{k\to\infty}E_k(u)\leq \lim_{k\to\infty}E_k(\phi_j)=E_\theta(P_\theta(\phi_j)),\quad \forall j\in \N.
\]
On the other hand, by definition of envelope, $u\leq P_\theta(\phi_j)\leq \phi_j$ and thus $P_\theta(\phi_j)\downarrow u$. Then
\[
\lim_{k\to\infty}E_k(u)\leq\lim_{j\to\infty}E_\theta(P_\theta(\phi_j))=E_\theta(u),
\]
by continuity of Monge--Amp\`ere energy along decreasing sequences.
\end{proof}
\begin{rmk} The proof actually shows that Theorem \ref{upper bound estimate} holds for any big and nef line bundle $L$.
\end{rmk}
\begin{proof}[Proof of Theorem \ref{quantization of MA energy}] It remains to prove that for any $u\in \PSH(X,\theta)\cap L^\infty$, we have
\begin{equation}\label{lower bound on energy}
 \liminf_{{k\to\infty}}E_k(u)\geq E_\theta(u).
\end{equation}
Following \cite{BF14}, we consider the following functional on $\E^1(X,\theta)$:
\[
\mathcal{F}_k(u):=E_k(u)-E_\theta(u),\quad u\in \E^1(X,\theta).
\]
By Lemma \ref{Properties on Energy and E1} (iii) and Lemma \ref{properties of Hilb and QMA} (ii), $\F_k$ is continuous along decreasing sequences in $\E^1(X,\theta)$. Also, by definition \eqref{MA energy} and Lemma \ref{properties of Hilb and QMA} (iii), $\F_k(u+C)=\F_k(u)$ for $C\in \R$. For fixed $u\in \PSH(X,\theta)\cap L^\infty$, we may assume that $u\leq 0$ by addition of a constant. By Lemma \ref{minimal singular/finite energy geodesic} (iii), $\dot{u}_0\in L^\infty(X)$.

Now, for fixed $\epsilon\in \Q^+$ and an positive line bundle $(A,h_0^A)$ with curvature $\omega$, we set $\omega_\epsilon:=\theta+\epsilon\omega$. By Theorem \ref{Bergman kernel comparison} and \eqref{epsilon twisted local holomorphic morse},
\[
\frac{1}{N_k}B_{L^k,0}\leq N_k^{-1}\left(1+\frac
{C}{\epsilon k}\right)^2B_{k,\epsilon}\leq \frac{k^n/n!}{N_k}\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\omega_\epsilon^n
\]
As $\dot{u}_0\leq 0$, we obtain
\begin{equation}\label{key estimate}
\F_k(u)\geq \int_X\dot{u}_0\left(\left(1+\frac{C}{\epsilon k}\right)^2\frac{k^n/n!}{N_k}\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\omega_\epsilon^n-\frac{1}{\vol L}\theta^n\right).
\end{equation}
Taking $k\to \infty$, we see that for fixed $\epsilon\in \Q^+$, since $\rho_k,\td{\rho}_k\to 0$ and from \eqref{volume of a line bundle} (since $L$ is big):
\[
1+\frac{C}{\epsilon k}\to 1,\quad \frac{k^n/n!}{N_k}\to \frac{1}{\vol L},\quad\frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}\to 1.
\]
As a result, we obtain for fixed $\epsilon\in \Q^+$,
\[
\liminf_{k\to\infty}\F_k(u)\geq \frac{1}{\vol L}\int_X \dot{u}_0\left(\omega_\epsilon^n-\theta^n\right)=\frac{1}{\vol L}\sum_{j=1}^n\int_X\dot{u}_0\theta^{n-j}\wedge (\epsilon \omega)^j.
\]
Taking $\epsilon\to 0+$, we obtain the result.
\end{proof}\begin{rmk}\label{pitfall for finite energy potentials}
The argument proving \eqref{lower bound on energy} fails for general
\(u\in \mathcal{E}^1(X,\theta)\), since one cannot in general guarantee that the
initial tangent \(\dot u_0\) of the weak geodesic \(t\mapsto u_t\) in
\(\PSH(X,\theta)\) is integrable with respect to \(\omega_\epsilon^n\) in
\eqref{key estimate}, even for arbitrarily small \(\epsilon>0\). One obstruction comes from a counterexample of Di Nezza
\cite[Example 4.5]{DN15}, which shows that for any \(\epsilon>0\),
\[
\mathcal{E}^1(X,\theta)\not\subset \mathcal{E}^1(X,\omega_\epsilon),
\]
Indeed, let \(u\in \mathcal{E}^1(X,\theta)\setminus
\mathcal{E}^1(X,\omega_\epsilon)\) with \(u\le0\). From
\(\PSH(X,\theta)\subset \PSH(X,\omega_\epsilon)\) and envelope definition of the weak geodesic, the weak geodesic
\(v_t\) joining \(0\) and \(u\) in \(\PSH(X,\omega_\epsilon)\) satisfies
\[
v_t \le u_t,\qquad t\in[0,1],
\]
by the envelope construction. Since \(u\notin \mathcal{E}^1(X,\omega_\epsilon)\), one can shows that
\[
\int_X \dot v_0\,\omega_\epsilon^n = -\infty.
\]
As \(u_t\) and \(v_t\) agree at the endpoints,
we get $
\dot u_0 \le \dot v_0 \le 0$. This implies that $\dot{u}_0\notin L^1(X,\omega_\epsilon^n)$ since
\[
\int_X \dot u_0\,\omega_\epsilon^n
 \;\le\; \int_X \dot v_0\,\omega_\epsilon^n
 \;=\; -\infty,
\]
Thus, a necessary condition for the integrability of \(\dot u_0\) is that
\(u\in \mathcal{E}^1(X,\omega_\epsilon)\). To author's current knowledge, we do not aware of any general sufficient condition.
\end{rmk}
\printbibliography

A result such as Theorem~\ref{quantization of MA energy} was raised as an open problem in Berman–Freixas i Montplet \cite[Remark~3.6]{BF14}, and our theorem provides a partial answer in the big and semipositive setting.
When $L$ is ample and no adjoint twisting is present, Donaldson \cite{DonaldsonScalII} first established the convergence for smooth Kähler potentials; this was later extended to continuous potentials by Berman–Boucksom \cite[Theorem~A]{BB10} and to all finite-energy potentials by Darvas–Lu–Rubinstein \cite{DLR}.
In the adjoint setting, Berman–Freixas i Montplet \cite[Theorem~3.5]{BF14} proved the quantization of energy for finite-energy potentials, and subsequently Darvas–Xia \cite{DX22} obtained the result for arbitrary twisting.
In the big, untwisted case, Berman–Boucksom \cite[Theorem~A]{BB10} showed convergence for all continuous metrics.
More recently, Darvas–Xia \cite{DX24} proved a “partial quantization’’ theorem for continuous metrics under arbitrary twisting on a pseudoeffective line bundle.

Theorem \ref{quantization of MA energy} has an immediate application to quantization of non-pluripolar measure. For an admissible $u$, we fix a basis $\{S_1,\dots,S_{N_k}\}$ of $H^0(X,L^k\otimes K_X)$ with $N_k=\dim_\mathbb C H^0(X,L^k\otimes K_X)$. Then we define \textit{Bergman measure} and its normalization by
	\begin{equation}\label{Bergman measure}
	B_{L^k,u}(x)=\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_u^L)^k}=\sum_{j=1}^{N_k}|S_j(x)|^2_{(h_0^L)^k}e^{-ku(x)},\quad \beta_{L^k,u}=\frac{1}{N_k}B_{L^k,u}.
	\end{equation}
Coupling with  a standard trick in Berman--Boucksom--Witt Nystr\"om \cite{BBWN}, Theorem \ref{quantization of MA energy} yields the following weak convergence of normalized Bergman measures to non-pluripolar measures
\begin{cor}\label{convergence of Bergman measures}  Let $u\in \PSH(X,\theta)\cap L^\infty(X)$. Then $\beta_{L^k,u}$ converges weakly to non-pluripolar measure $\frac{1}{\vol L}\theta_{u}^n$ as $k\to\infty$, where $\vol L$ is the volume of the line bundle $L$, see \eqref{volume of a line bundle}.
\end{cor}
Our approach refines the arguments in \cite{BF14} as follows. Fix an ample line bundle $A$ on $X$ and $\varepsilon\in \mathbb Q^+$. We prove a comparison Theorem for Bergman measure to $L$ and to small ample twist $L\otimes A^\varepsilon$ with almost optimal asymptotic bound (Theorem \ref{Bergman kernel comparison}). This allows us to apply Berman's local Morse inequality \cite[Proposition 2.5]{BF14} to the twisted bundle with explicit dependence on $\varepsilon$, see  \eqref{epsilon twisted local holomorphic morse}. Combining this with the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures, and the convexity of the quantized Monge–Ampère energy along weak geodesics,  we can carry out the proof of Theorem \ref{quantization of MA energy} for bounded potentials following standard argument as in \cite[Theorem A]{BB10} and \cite[Theorem 3.5]{BF14}. However, as we will explain in Remark \ref{pitfall for finite energy potentials}, our method does not presently extend to arbitrary finite-energy potentials, due to integrability issues for initial tangent vectors of weak geodesics.

As this work was being finalized, Siarhei Finski kindly informed us of his related work \cite{FinskiBig}, in which he quantizes the Darvas–Mabuchi $d_p$ distances on big line bundles and for potentials of the type $P_\theta(f)$ (defined below \ref{MA envelope}), where $f$ is continuous. In particular, for $p=1$ and $\theta$ semipositive, his result implies the version of Theorem~\ref{quantization of MA energy} for such potentials,  a subset of the class $\PSH(X,\theta)\cap L^\infty$. Our setting and approach differ from his in two main aspects. His quantization is formulated using the untwisted section spaces $H^0(X,L^k)$, whereas we work in the adjoint setting. This allows us to apply the Berndtsson–P\u{a}un positivity theorem directly. Additionally, he reduces the problem to the ample case via passing to certain birational models over $X$, while our reduction is by twisting $L^k\otimes K_X$ by an ample line bundle on $X$ as explained above.

\subsection*{Organization}
We briefly describe the structure of the paper.

In Section~2 we collect the necessary background from pluripotential theory,  following \cite{BEGZ,BBGZ,DDL18,DDL18L1}. We also recall the basic variational formulas for \(E_\theta\).

Section~3 is devoted to Bergman measures and quantized Monge--Amp\`ere energies in the adjoint setting. We prove basic properties of $B_{L^k,u}$ and $E_k(u)$ and discuss their variational formula. Next, we recall the positivity of direct images à la Berndtsson–P\u{a}un \cite{BerndtssonPositivity1,BerndtssonPositivity2,BP08,PT18} and the relation with convexity of $E_k$ along weak geodesics. Finally, we recall Berman’s local holomorphic Morse inequalities and the convergence of Bergman measures at equilibrium, adapting the arguments of \cite{Ber09,BB10} to our adjoint framework.

Section 4 is the new input in our argument.  We prove the the comparison Theorem  (Theorem \ref{Bergman kernel comparison}).    This follows from choosing peak sections with asymptotically optimal $L^\infty$-control from a pointwise semi-classical Ohsawa--Takegoshi type extension theorem for powers of an ample line bundle (Theorem \ref{Pointwise Semi-Classical OT}). We also discuss the relation of Theorem \ref{Pointwise Semi-Classical OT} with recent work of Finski \cite[Theorem 1.10]{Finski1}. We also present a  local Morse inequality in ample twisting setting.

Finally, in Section~5 we combine these ingredients to prove Theorems~\ref{quantization of MA energy} and Corollary  \ref{convergence of Bergman measures} and conclude with obstacles to aribtrary finite-energy potentials in Remark \ref{pitfall for finite energy potentials}.
\subsection*{Acknowledgment}  This work was supported by NSF Grant DMS-2405274 and the Hauptman Fellowship. The author is deeply grateful to his advisor, Tamás Darvas, for his constant support and guidance. The author also wishes to thank Siarhei Finski for his enlightening lecture and discussions at the summer school at the Rényi Institute, Budapest, and for kindly sharing his preprint \cite{FinskiBig} along with related insights.

Fix an ample line bundle $A$ on $X$ and $\varepsilon\in\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\otimes A^\varepsilon$ with an almost optimal asymptotic bound (Theorem~\ref{Bergman kernel comparison}); (ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\varepsilon$ (see \eqref{epsilon twisted local holomorphic morse}); (iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to “carry out the proof of Theorem~\ref{quantization of MA energy} for bounded potentials following standard argument as in \cite[Theorem A]{BB10} and \cite[Theorem 3.5]{BF14}.” The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics” (Remark~\ref{pitfall for finite energy potentials}).

Fix an ample line bundle $A$ on $X$ and $\varepsilon\in\mathbb Q^+$. The approach is to (i) prove a comparison theorem for Bergman measures for $L$ versus the small ample twist $L\otimes A^\varepsilon$ with an almost optimal asymptotic bound. We first prove the following theorem.

\begin{thm}\label{Bergman kernel comparison} For $\epsilon\in \Q^+$ and admissible $u$, and for sufficiently divisible $k\gg 1$, we have
\[
B_{L^k,u}\leq \left(1+\frac{C}{\epsilon k}\right)^2B_{(L\otimes A^{\epsilon})^k,u},
\]
where $C=C(n,X,\omega)$ depends only on dimension $n$, manifold $X$, and K\"ahler form $\omega=c_1(A,h_0^A)$.
\end{thm}

(ii) use this to apply Berman's local Morse inequality to the twisted bundle with explicit dependence on $\varepsilon$, namely
\begin{equation}\label{epsilon twisted local holomorphic morse}
    \frac{n!}{k^n}B_{k,\epsilon}\leq \frac{e^{\rho_k+\epsilon\td{\rho}_k}}{(1-e^{-\epsilon(\log k)^2})^n}(\theta+\epsilon\omega)^n,
\end{equation}

(iii) combine this with “the standard variational relations between the Monge–Ampère energy and non-pluripolar measures, the quantized energy and Bergman measures,” together with “the convexity of the quantized Monge–Ampère energy along weak geodesics,” to carry out the proof of the main theorem for bounded potentials following the standard argument as in Berman–Boucksom, Theorem A in \cite[Theorem A]{BB10} and Berman–Fujita, Theorem 3.5 in \cite[Theorem 3.5]{BF14}.

The method is noted not to extend to arbitrary finite-energy potentials because of “integrability issues for initial tangent vectors of weak geodesics.” More precisely:

\begin{rmk}\label{pitfall for finite energy potentials}
The argument proving \eqref{lower bound on energy} fails for general
\(u\in \mathcal{E}^1(X,\theta)\), since one cannot in general guarantee that the
initial tangent \(\dot u_0\) of the weak geodesic \(t\mapsto u_t\) in
\(\PSH(X,\theta)\) is integrable with respect to \(\omega_\epsilon^n\) in
\eqref{key estimate}, even for arbitrarily small \(\epsilon>0\). One obstruction comes from a counterexample of Di Nezza
\cite[Example 4.5]{DN15}, which shows that for any \(\epsilon>0\),
\[
\mathcal{E}^1(X,\theta)\not\subset \mathcal{E}^1(X,\omega_\epsilon),
\]
Indeed, let \(u\in \mathcal{E}^1(X,\theta)\setminus
\mathcal{E}^1(X,\omega_\epsilon)\) with \(u\le0\). From
\(\PSH(X,\theta)\subset \PSH(X,\omega_\epsilon)\) and envelope definition of the weak geodesic, the weak geodesic
\(v_t\) joining \(0\) and \(u\) in \(\PSH(X,\omega_\epsilon)\) satisfies
\[
v_t \le u_t,\qquad t\in[0,1],
\]
by the envelope construction. Since \(u\notin \mathcal{E}^1(X,\omega_\epsilon)\), one can  shows that
\[
\int_X \dot v_0\,\omega_\epsilon^n = -\infty.
\]
As \(u_t\) and \(v_t\) agree at the endpoints,
we get $
\dot u_0 \le \dot v_0 \le 0$. This implies that $\dot{u}_0\notin L^1(X,\omega_\epsilon^n)$ since
\[
\int_X \dot u_0\,\omega_\epsilon^n
   \;\le\; \int_X \dot v_0\,\omega_\epsilon^n
   \;=\; -\infty,
\]
Thus, a necessary condition for the integrability of \(\dot u_0\) is that
\(u\in \mathcal{E}^1(X,\omega_\epsilon)\). To author's current knowledge, we do not aware of any  general sufficient condition.
\end{rmk}

\label{quantization of MA energy} If $L$ is a big and semipositive line bundle on $X$, then for any  $u\in \PSH(X,\theta)\cap L^\infty$, we have
\[
\lim_{k\to\infty}E_k(u)=E_\theta(u).
\],

Question:
Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and choose a smooth Hermitian metric $h_0^L$ on $L$ with semipositive Chern curvature $\theta:=c_1(L,h_0^L)$. For $u\in \PSH(X,\theta)\cap L^\infty(X)$, where $\PSH(X,\theta)$ denotes the $\theta$-plurisubharmonic functions (so $\theta+dd^c u\ge 0$ in the sense of currents), define for each $k\in\mathbb N$ the Hilbert norm on $H^0(X,L^k\otimes K_X)$ by
\[
(s_1,s_2)_{\mathrm{Hilb}_k(u)}:=\int_X \langle s_1,s_2\rangle_{(h_0^L)^k}e^{-ku},
\]
and the quantized Monge--Amp\`ere energy by
\[
E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det \mathrm{Hilb}_k(u)}{\det \mathrm{Hilb}_k(0)}\right),
\]
where $N_k=\dim_\mathbb C H^0(X,L^k\otimes K_X)$. Let $E_\theta(u)$ denote the Monge--Amp\`ere (Aubin--Yau) energy on $\PSH(X,\theta)$, i.e. the primitive of the non-pluripolar Monge--Amp\`ere operator. Under these assumptions, which statement about $E_k(u)$ holds as $k\to\infty$?

One has
\[
\lim_{k\to\infty} E_k(u)=E_\theta(u).
\]

Choices:
B.
One has
\[
\limsup_{k\to\infty} E_k(u)\le E_\theta(u),
\]
and in fact equality holds only when the semipositive form \(\theta\) is K\"ahler.
C.
One has
\[
\limsup_{k\to\infty} E_k(u)\le E_\theta(u).
\]
D.
For every \(u\in \PSH(X,\theta)\cap L^\infty(X)\), there exists a subsequence \(k_j\to\infty\) such that
\[
\lim_{j\to\infty} E_{k_j}(u)=E_\theta(u),
\]
and the convergence is uniform in \(u\) over bounded subsets of \(\PSH(X,\theta)\cap L^\infty(X)\).
\]
E.
If \(L\) is big and semipositive, then for any \(u\in \PSH(X,\theta)\cap \mathcal E^1(X,\theta)\), one has
\[
\lim_{k\to\infty} E_k(u)=E_\theta(u).
\]

full limit equality weakened to upper-bound asymptotics

pointwise convergence upgraded to uniform-in-u subsequential convergence

boundedness hypothesis replaced by finite-energy class


Question:
Let $X$ be a complex projective manifold, let $L$ be a big and semipositive holomorphic line bundle on $X$, and fix a smooth Hermitian metric $h_0^L$ on $L$ with curvature form $\theta:=c_1(L,h_0^L)$. For $u\in \mathrm{PSH}(X,\theta)\cap L^\infty(X)$, where $\mathrm{PSH}(X,\theta)$ denotes the functions satisfying $\theta+dd^c u\ge 0$ in the sense of currents, let $K_X$ be the canonical bundle and, for each $k\in\mathbb N$, define the Hilbert form on $H^0(X,L^k\otimes K_X)$ by
\[
(s_1,s_2)_{\mathrm{Hilb}_k(u)}:=\int_X \langle s_1,s_2\rangle_{(h_0^L)^k}e^{-ku},
\]
where $\langle s_1,s_2\rangle_{(h_0^L)^k}$ is the natural pointwise Hermitian pairing induced by $(h_0^L)^k$. Writing $N_k:=\dim_\mathbb C H^0(X,L^k\otimes K_X)$, define the quantized Monge--Amp\`ere energy
\[
E_k(u):=-\frac{1}{kN_k}\log\left(\frac{\det \mathrm{Hilb}_k(u)}{\det \mathrm{Hilb}_k(0)}\right).
\]
Let $E_\theta(u)$ denote the Monge--Amp\`ere (Aubin--Yau) energy of $u$. As $k\to\infty$, which limiting statement holds?

For every $u\in \mathrm{PSH}(X,\theta)\cap L^\infty(X)$, one has \[\lim_{k\to\infty} E_k(u)=E_\theta(u).\]

Choices:
B. For every $u\in \mathrm{PSH}(X,\theta)$ such that $\mathrm{Hilb}_k(u)$ is finite for all $k$, one has \[\lim_{k\to\infty} E_k(u)=E_\theta(u).\]
C. For every $u\in \mathrm{PSH}(X,\theta)\cap L^\infty(X)$, the sequence $E_k(u)$ converges as $k\to\infty$, and its limit is finite.
D. There exists a constant $C>0$, depending only on $(X,L,h_0^L)$, such that for every $u\in \mathrm{PSH}(X,\theta)\cap L^\infty(X)$ and every $k\in\mathbb N$, one has \[\bigl|E_k(u)-E_\theta(u)\bigr|\leq \frac{C}{k}.\]
E. For every $u\in \mathrm{PSH}(X,\theta)\cap L^\infty(X)$, one has \[\lim_{k\to\infty} E_k(u)=E_{\theta+\omega}(u),\] where $\omega$ is any K\"ahler form on $X$.

boundedness hypothesis replaced by mere admissibility/finite Hilbert norms

identification of the limit with $E_\theta(u)$

non-uniform asymptotic comparison upgraded to uniform $O(1/k)$ rate for all bounded potentials

auxiliary $\varepsilon$-twisted form used in comparison argument substituted into the limiting energy