In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\pi$ to be a set of primes. We write ${\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\rm I}_\pi (G)$ for the set of irreducible $\pi$-partial characters of $G$.

Gallagher's theorem \cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\chi\in {\rm Irr} (G)$ be such that $\chi_{N} = \theta \in {\rm Irr} (N)$, where $\chi_{N}$ is the restriction of $\chi$ to $N$. Then the characters $\beta\chi$ for $\beta\in {\rm Irr} (G/N)$ are irreducible and distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$.

The first goal in this paper is to prove the converse of this theorem:

\begin{abstract}
Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\chi \in {\rm Irr} (G)$ satisfies $\chi_N$ is irreducible, then $\chi \beta$ is irreducible and distinct for all $\beta \in {\rm Irr} (G/N)$. Furthermore, if $\theta = \chi_N$, then these are all of the irreducible constituents of $\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\pi$-partial characters and that a partial converse of that theorem is true.
\end{abstract}

The first goal in this paper is to prove the converse of this theorem:

We will see that we will obtain Theorem \ref{main1} as a corollary to the version for the Brauer character version of the theorem. The Brauer character version of Gallagher's theorem \cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\lhd G$ and let $\eta\in \IBr (G)$. If $\eta_{N} = \theta \in \IBr (N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$. Now we consider a partial converse of the Brauer character version of Gallagher's theorem.

\begin{introthm} \label{main2}
Let $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime. Suppose $\eta\in {\rm IBr}(G)$ and ${\rm IBr}(G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$, and suppose that $p\nmid |G/N|$ and the $\beta_{1}\eta, \cdots, \beta_{n}\eta$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible Brauer character of $N$.
\end{introthm}

\begin{cor} \label{bpi Gall,2inpi,|G|odd}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose $2 \in \pi$ or $|G|$ is odd. Suppose there exists a character $\chi \in {\rm B}_{\pi} (G)$ so that $\theta = \chi_N \in \irr (N)$. Then the map $\beta \mapsto \beta\chi$ is a bijection from ${\rm B}_{\pi} (G/N)$ to ${\rm B}_\pi (G \mid \theta)$.
\end{cor}

In fact, the partial converse of Gallagher's theorem will apply for sets of lifts of the $\pi$-partial characters. To see this we make the following definition we again only need that we have subsets of $\irr (G)$ and $\irr (N)$ that are in bijection with ${\rm I}_\pi (G)$ and ${\rm I}_\pi (N)$, and that irreducible constituents lie in this set. Thus, we prove the result in this generality so that it could be applied to any set of lifts.
\begin{cor}\label{lifts conv}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$ so that $G/N$ is a $\pi$-group. Assume there exist subsets $X_\pi (G) \subseteq \irr (G)$ and $X_\pi (N) \subseteq \irr (N)$ so that the map $\theta \mapsto \theta^o$ is a bijection from $X_\pi (G)$ to ${\rm I}_\pi (G)$ and from $X_\pi (N)$ to ${\rm I}_\pi (N)$, and assume for $\chi \in X_\pi (G)$ that the constituents of $\chi_N$ lie in $X_\pi (N)$. If there exist $\chi \in X_\pi (G)$ and $\irr (G/N) = \{ \gamma_1 = 1, \dots, \gamma_n \}$ that satisfy $(\gamma_1 \chi)^o, \dots, (\gamma_n \chi)^o$ are irreducible and distinct, then $\chi_N \in X_\pi (N)$.
\end{cor}

\begin{thm}
Let $\pi$ be a set of primes and let $G$ be a $\pi$-separable group. Let $N$ a normal subgroup of $G$. Let $\eta \in {\rm I}_\pi (G)$ and ${\rm I}_{\pi} (G/N) = \{\beta_{1}, \cdots, \beta_{n}\}$. Suppose that $|G/N|$ is divisible only by primes in $\pi$ and that the $\{\beta_{1}\eta, \cdots, \beta_{n}\eta\}$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible $\pi$-partial character of $N$.
\end{thm}

\begin{thm}\label{bpi conv}
 Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose there exist characters $\chi \in {\rm B}_{\pi} (G)$ and ${\rm B}_{\pi} (G/N) = \{\beta_1, \dots, \beta_n \}$. Assume that $G/N$ is a $\pi$-group and the $\{\beta_1 \chi, \dots, \beta_n \chi\}$ are distinct and irreducible Then $\chi_N$ is irreducible.
\end{thm}

\begin{introthm} \label{main1}
Let $G$ be a group and let $N$ be a normal subgroup of $G$.  Suppose ${\rm Irr} (G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$ and $\chi\in {\rm Irr} (G)$, and suppose that the $\beta_{1}\chi, \cdots, \beta_{n}\chi$ are distinct and irreducible.  Then $\chi_{N}$ is an irreducible character of $N$.
\end{introthm}

We will see that we will obtain Theorem \ref{main1} as a corollary to the version for the Brauer character version of the theorem.  The Brauer character version of Gallagher's theorem \cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\lhd G$ and let $\eta\in \IBr (G)$. If $\eta_{N} = \theta \in \IBr (N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$.  Now we consider a partial converse of the Brauer character version of Gallagher's theorem.

\begin{introthm} \label{main2}
Let $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime.  Suppose $\eta\in {\rm IBr}(G)$ and ${\rm IBr}(G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$, and suppose that $p\nmid |G/N|$ and the $\beta_{1}\eta, \cdots, \beta_{n}\eta$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible Brauer character of $N$.
\end{introthm}

We next consider Isaacs' $\pi$-theory, where $\pi$ is a set of primes.  For a $\pi$-separable group $G$, Isaacs has defined an analog of Brauer characters that are defined on the $\pi$-elements of $G$ that are called the $\pi$-partial characters of $G$.  We will write ${\rm I}_\pi (G)$ for the set of irreducible $\pi$-partial characters of $G$.  We will review the points of $\pi$-theory we need in Sections \ref{secn: pi theory} and \ref{secn: nucl}.  Interestingly, while Isaacs proved analogs of many other results for $\pi$-partial characters, he does not seem to have proved an analog of Gallagher's theorem.  We do that next.

\begin{introthm}\label{ipi Gall,gen}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose there exists a character $\zeta \in {\rm I}_{\pi} (G)$ so that $\xi = \zeta_N \in {\rm I}_\pi (N)$.  Then the map $\kappa \mapsto \kappa\zeta$ is a bijection from ${\rm I}_{\pi} (G/N)$ to ${\rm I}_\pi (G \mid \xi)$.
\end{introthm}

Now, in \cite{Isaacs1976}, there is a generalization of Gallagher's theorem (Theorem 6.16 of \cite{Isaacs1976}) of which Gallaher's theorem is obtained as a corollary.  We have not been able to obtain an analog of this generalization in cases, but we can obtain an analog when we add the extra assumption that $2 \in \pi$ or $|G|$ is odd.

\begin{introthm} \label{ipi pre,2inpi,|G|odd}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose $2 \in \pi$ or  $|G|$ is odd.  Assume there exist partial characters $\eta, \xi \in {\rm I}_\pi (N)$ so that $\eta$ is $G$-invariant, $\xi$ extends to $\zeta \in {\rm I}_\pi (G)$ and $\eta \xi \in {\rm I}_{\pi} (N)$.  Then the map $\kappa \mapsto \kappa \zeta$ is a bijection from ${\rm I}_\pi (G \mid \eta) \rightarrow {\rm I}_\pi (G \mid \eta \xi)$.
\end{introthm}

We close with a partial converse for the analog of Gallagher's theorem for $\pi$-partial characters.  We will show that it is not possible to have a full converse in this case.

\begin{introthm}\label{ipi conv}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.
Suppose there exists partial characters $\zeta \in {\rm I}_{\pi} (G)$ and ${\rm I}_{\pi} (G/N) = \{\kappa_1, \dots, \kappa_n \}$.  Assume that $G/N$ is a $\pi$-group and the $\kappa_1 \zeta, \dots, \kappa_n \zeta $ are distinct and irreducible  Then $\zeta_N$ is irreducible.
\end{introthm}

We will see that we will obtain Theorem~\ref{main1} as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\eta\in \IBr(G)$ and $\eta_N=\theta\in\IBr(N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$, and are all of the irreducible constituents of $\theta^G$; then it says: "Now we consider a partial converse" (stated as Theorem~\ref{main2}).

We will see that we will obtain the main theorem as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\eta\in \IBr(G)$ and $\eta_N=\theta\in\IBr(N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$, and are all of the irreducible constituents of $\theta^G$; then it says: "Now we consider a partial converse" (stated as Theorem~\ref{main2}).,

Question: Let $G$ be a finite group and $N\trianglelefteq G$. Write ${\rm Irr}(H)$ for the set of irreducible complex characters of a finite group $H$. Suppose ${\rm Irr}(G/N)=\{\beta_1,\dots,\beta_n\}$, let $\chi\in {\rm Irr}(G)$, and assume that the characters $\beta_1\chi,\dots,\beta_n\chi$ on $G$ (where each $\beta_i$ is inflated from $G/N$ to $G$ and then multiplied pointwise with $\chi$) are pairwise distinct and irreducible. Which conclusion holds for the restriction $\chi_N$ of $\chi$ to $N$?

The restricted character $\chi_N$ is an irreducible character of $N$.

Choices:
B. The restricted character $\chi_N$ is a sum of pairwise distinct irreducible characters of $N$, each occurring with multiplicity $1$.
C. The restricted character $\chi_N$ has at least one irreducible constituent of $N$.
D. The induced character $(\chi_N)^G$ has irreducible constituents exactly $\beta_1\chi,\dots,\beta_n\chi$, each occurring with multiplicity $1$.
E. For every irreducible constituent $\theta$ of $\chi_N$, the characters $\beta_1\chi,\dots,\beta_n\chi$ are precisely the irreducible constituents of $\theta^G$.

irreducible-restriction conclusion weakened to multiplicity-free reducibility

dropped irreducibility of the whole restriction, keeping only existence of an irreducible constituent

converse conclusion replaced by full Gallagher constituent description for induction

hidden dependence on choosing the unique constituent of $\chi_N$; falsified by quantifying over every constituent


The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,
irreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are "irreducible" for the bi-action.\\
This result is well-known for groups of finite Morley rank \cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\
An interesting family of theories is finite-dimensional theories, in the sense of \cite{wagner2020dimensional}.
\begin{defn}
    A theory $T$ is said to be \emph{finite-dimensional} if there exists a function $\operatorname{dim}$ from the class of all interpretable subset in any model $\mathcal{M}$ of $T$ into $\omega\cup\{-\infty\}$ such that for any $\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,
\begin{itemize}
    \item If $a,a'$ have same type over $\emptyset$, $\operatorname{dim}(\phi(x,a))=\operatorname{dim}(\phi(x,a'))$
    \item $\operatorname{dim}(\emptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
    \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
    \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
    \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
    \item $\Gamma,\Delta$ sharply commute;
    \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
    \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
    There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
    Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
    \begin{itemize}
        \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
        \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
        \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
        \item $A_0=\Kat(\Gamma,\Delta)$.
    \end{itemize}
    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

mptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
 \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
 \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
 \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
 \item $\Gamma,\Delta$ sharply commute;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
 There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
 Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
 \begin{itemize}
 \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
 \item $A_0=\Kat(\Gamma,\Delta)$.
 \end{itemize}
 Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

\begin{lemma}\label{boundedind}
    Let $G$ be a definable group of finite dimension and $\{H_i\}_{i\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\omega$ such that there is no $J=\{j_1,...,j_n\}\subseteq I$ of cardinality $n$ such that $|\bigcap_{i=1}^kH_{j_i}:\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\leq n-1$.
\end{lemma}

\def\notind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0
\hbox to 0pt{\mathchardef\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}
\hbox to 0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\end{theorem}
\begin{proof}
 We prove the base case. Given $\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\Delta$, that therefore is finite.\\
 Being the katakernel finite, there exists only boundedly many elements for $\Delta$ or $\Delta$ is unbounded (and then the proof follows using the previous idea).\\
 Then we can take a representative with maximal image and we have to prove that
\end{proof}
\begin{theorem}

\begin{proof}
Let $n=\dim(G)$. We will work in a sufficiently saturated structure $\mathfrak{M}$ in which $G$ is definable and assume that $\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\in \omega$, there exists $g_1,...,g_N$ such that $|\bigcap_{j\leq k-1} \phi(\mathfrak{M},g_j)/\bigcap_{j\leq k} \phi(\mathfrak{M},g_j)|\geq k$. Therefore, the partial type given by the formulas
$$\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\in \bigcap_{j=1}^i \phi(\mathfrak{M},x_j)\wedge a^i_j{a^i_k}^{-1}\not\in \phi(\mathfrak{M},x_j)\wedge \forall_{i\leq N}\ \phi(\mathfrak{M},x_i)\leq G(\mathfrak{M})$$
for all $k<\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\{H_i\}_{i\leq N}$ such that $|\bigcap_{i<j}H_i/\bigcap_{i\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.
\end{proof}
If we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\widetilde{\mathfrak{M}}_c$-property.
\begin{defn}
    A group $G$ is \emph{hereditarily-$\widetilde{\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers
    $$C_H(a_0/N)\geq C_H(a_0,a_1/N)\geq...\geq C_H(a_0,a_1,...,a_n/N)\geq...$$
    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.
\end{defn}
The notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.
\begin{defn}
\begin{itemize}
    \item  Let $G$ be a group and $A, B\leq G$ subgroups. $A$ \emph{almost contains} $B$, denoted $A\apprge B$, if $|B:B\cap A|$ is finite.
    \item  If $A\apprle B$ and $B\apprle A$, $A$ and $B$ are \emph{commensurable}.
    \item Given a family of subgroups $\{G_i\}_{i\in I}$ in $G$, the family $\{G_i\}_{i\in I}$ is \emph{uniformly commensurable} if there exists $n<\omega$ such that $|G_i/G_i\cap G_j|\leq n$ for every $i,j\in I$.
\end{itemize}
  \end{defn}

Assume towards a contradiction that the conclusion of Theorem~\ref{boundedind} fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. Hence the desired bounds $n,d<\omega$ exist.

Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. This completes the proof of the main theorem.

Question: Let $G$ be a definable group of finite dimension, and let $\{H_i\}_{i\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all $H_i$ are given by instances of a single defining formula). Which statement holds about finite chains of successive intersections of members of this family?

There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$. Equivalently, for any $n$ distinct indices $j_1,\dots,j_n$, at least one successive index
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|
\]
is at most $d$.

Choices:
B.
There exists a natural number $d<\omega$ such that for every natural number $n<\omega$, no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$.
C.
There exists a natural number $n<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\text{ is infinite}
\]
for every $k\le n-1$.
D.
There exist natural numbers $n,d<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ and every $k\le n-1$,
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.
\]
E.
There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|H_{j_k}:H_{j_k}\cap H_{j_{k+1}}\right|>d
\]
for every $k\le n-1$.

simultaneous existence of both bounds n and d

replace uniform finite bound by mere finiteness of some drop

at least one bounded drop versus all drops bounded

successive intersections replaced by pairwise intersection indices


\begin{thm}\label{XiaFock}
		We have
		\begin{equation*}
			C^{*} (\mathcal{WL})=\mathcal{T}^{(1)}.
		\end{equation*}
	\end{thm}

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation}\label{Fnorm1}
		\left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
	\end{equation}
	Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation*}
		\left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
	\end{equation*}
	The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

	Let \(d\mu\) be the Gaussian measure on \(\mathbb{C}^n\), \(n\ge 1\). In terms of the standard volume measure \(dV\) on \(\mathbb{C}^n\), it is given by
	\begin{equation*}
		d\mu(z) = \pi^{-n} e^{-|z|^2} dV(z)~.
	\end{equation*}

	 The Fock space \(H^{2}(\mathbb{C}^n,d\mu)\) is defined to be the subspace of the (Hilbert-) Lebesgue space \(L^{2}(\mathbb{C}^n,d\mu)\) consisting of entire functions. Notice that \(H^{2}(\mathbb{C}^n,d\mu) = F^{2}_{1} \). The symbol \(K_z\) denotes the reproducing kernel and the symbol \(k_z\) denotes the normalized reproducing kernel for \(H^{2}(\mathbb{C}^n,d\mu)\). That is,
	\begin{equation*}
		K_z(\zeta) = e^{\langle \zeta, z \rangle}, \quad k_{z}(\zeta) = e^{\langle \zeta, z\rangle}e^{-\frac{|z|^2}{2}}~, \quad z,\zeta \in \mathbb{C}^n.
	\end{equation*}

	In \cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \(\mathbb{C}^n\) that the norm closure of \(\{T_{f}: f \in L^{\infty}(B,dv) \}\) coincides with the \(C^{*}\)-algebra of weakly localized operators. Also, he stated in \cite[Section 4 ]{xia2015localization}  that  the analogue of \cite[Theorem 1.5]{xia2015localization} on the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \(H^{2}(\mathbb{C}^n,d\mu)\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.

	\begin{defn}
		For \(f \in L^{\infty}(\mathbb{C}^n,dV) \), the \textbf{Toeplitz operator} \(T_{f}\) is defined by the formula
		\begin{equation*}
			T_{f}h = P(fh)~,\quad h \in H^{2}(\mathbb{C}^n,d\mu)~,
		\end{equation*}
		where \(P \colon L^{2}(\mathbb{C}^n,d\mu) \rightarrow H^{2}(\mathbb{C}^n,d\mu)\) is the orthogonal projection.
	\end{defn}

	The \textbf{standard lattice} in \(\mathbb{C}^n\) is denoted by
	\begin{equation*}
		\mathbb{Z}^{2n} = \{(m_{1}+il_{1}, \dots, m_{n}+il_{n}): m_{1}, l_{1}, \dots, m_{n},l_{n} \in \mathbb{Z}\}~.
	\end{equation*}
	We fix an orthonormal set \(\{e_{u}: u \in \mathbb{Z}^{2n}\}\) in \(H^{2}(\mathbb{C}^n,d\mu)\). We let \(S\) denote the \textbf{fundamental unit cube in }\(\mathbb{C}^{n}\). That is,
	\begin{equation*}
		S = \{(x_{1}+iy_{1}, \dots, x_{n}+iy_{n}) : x_{1},y_{1}, \dots , x_{n},y_{n} \in [0,1)\}.
	\end{equation*}
	With \(\mathbb{Z}^{2n}\) and \(S\), we have
	\begin{equation*}
		\cup_{u \in \mathbb{Z}^{2n}} \{S+u\} = \mathbb{C}^{n} = \cup_{u \in \mathbb{Z}^{2n}} \{u-S\} ~,
	\end{equation*}
	which is a tiling of the space, meaning that there is no overlap between \(S+u\) and \(S+v\) for \(u \neq v\) in \(\mathbb{Z}^{2n}\) (resp. between \(u-S\) and \(v-S\) for \(u \neq v \in \mathbb{Z}^{2n}\) ).

	\begin{defn}
		Let \(\mathcal{T}^{(1)}\) denote the norm closure of \(\{T_{f}: f\in L^{\infty}(\mathbb{C}^n,dV)\}\) in \(\mathcal{B}(H^2(\mathbb{C}^n,d\mu))\) with respect to the operator norm. That is
		\begin{equation*}
			\mathcal{T}^{(1)} = \{B : \lim_{k \to \infty } \norm{B-T_{b_{k}}}=0 , b_{k} \in L^{\infty}(\mathbb{C}^{n},dV)\}.
		\end{equation*}
	\end{defn}

	\begin{defn}
		We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
		\begin{equation}\label{R1}
			\langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
		\end{equation}
		holds for all \(z,w \in \mathbb{C}^n\).
	\end{defn}
	The inner product here is with respect to \(d\mu\).

	We define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \cite{xia2013localization}.
	\begin{defn}
		A bounded linear operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be XZ-\textbf{sufficiently localized} if there exist constants \(2n<\beta<\infty\) and \(0<C<\infty\) such that
		\begin{equation}\label{SL1}
			|\langle Bk_{z},k_{w}\rangle| \le \frac{C}{(1+|z-w|)^{\beta}}
		\end{equation}
		for all \(z,w\in \mathbb{C}^n\).
	\end{defn}

		\begin{defn}
		An admissible operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be \textbf{weakly localized} if it satisfies the following four conditions
		\begin{equation*}
			\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_{z}, k_{w} \rangle| dV(w) <\infty, \quad \sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) <\infty
		\end{equation*}

		and
		\begin{equation*}
			\lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle Bk_{z}, k_{w} \rangle| dV(w) =0, \quad \lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) =0~.
		\end{equation*}
	\end{defn}

	\begin{exa}\label{Ex1}
		If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
		\begin{equation*}
			|\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
		\end{equation*}
		for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
	\end{exa}

	\begin{proof}
		For each \(z, w, \xi \in \mathbb{C}^n\), we have
		\begin{equation*}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
		\end{equation*}
		By the triangular inequality we have
		\begin{equation*}
			|z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
		\end{equation*}
		It follows that
		\begin{eqnarray}\label{kzw}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray}
		Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
		\begin{equation}\label{Toepfgh}
			\langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{equation}
		In fact,
		\begin{eqnarray*}
			\langle T_{f}g, h \rangle &=&  \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle  h, K_{\xi}  \rangle} d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{eqnarray*}
		Therefore,
		\begin{eqnarray*}
			|\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
			f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right|  \\
			&\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}  dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
			&=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
			&=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray*}
		This completes the proof.
	\end{proof}

	 \begin{pro}
	 	Any XZ-sufficiently localized operator is weakly localized.
	 \end{pro}
	\begin{proof}
	Let \(B\) be a XZ-sufficiently localized operator, then  for all \(z\in \mathbb{C}^n\), we have
	\begin{eqnarray*}
		\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) &\le& \frac{C}{\pi^n} \int_{\mathbb{C}^n} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& \frac{Cc_n}{\pi^n} \int_{0}^{\infty} \frac{r^{2n-1}}{(1+r)^{\beta}} dr \\
		&=& C_n \left[ \int_{0}^{1} \frac{r^{2n-1}}{(1+r)^{\beta}} dr  + \int_{1}^{\infty } \frac{r^{2n-1}}{(1+r)^{\beta}} dr\right] \\
		&\le & C_n \left[ \int_{0}^{1} \frac{1}{(1+r)^{\beta}} dr +  \int_{1}^{\infty } \frac{r^{2n-1}}{r^{\beta}} dr\right] \\
		&=& C_n \left\{\frac{1}{\beta-1}\left(1-\frac{1}{2^{\beta}-1}\right) + \frac{1}{\beta-2n}\right\} \\
		&:=& C(n,\beta)~,
	\end{eqnarray*}
	which does not depend on \(z\). Therefore,
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) \le C(n,\beta)  <\infty~.
	\end{equation*}
	And in the same way, we also have
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^*k_z, k_w \rangle| dV(w) \le C(n,\beta) <\infty~.
	\end{equation*}
	We now show that \(B\) satisfies the third condition of weakly localized operator.

	Let \(z\in \mathbb{C}^n\), using the change of variables \(\zeta=z-w\) and the spherical coordinates, it holds
	\begin{eqnarray*}
		\int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) &\le& C \int_{|z-w|\ge r} \frac{1}{(1+|z-w|)^{\beta}} dV(w) \\
		&=& C \int_{|\zeta|\ge r} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{(1+\rho)^{\beta}} d \rho \\
		&\le& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{\rho^{\beta}} d \rho =  C c_n \int_{\rho\ge r} \frac{1}{\rho^{\beta-2n+1}} d \rho \\
		&=& \frac{Cc_n}{\beta-2n}\frac{1}{r^{\beta-2n}}~.
	\end{eqnarray*}
	It follows that,
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
	Using the duality relation (\ref{R1}) of an admissible operator and similarly like with \(B\), we also have:
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle B^*k_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
\end{proof}

	We denote by \(\mathcal{WL}\) the collection of weakly localized operators on \(H^{2}(\mathbb{C}^n,d\mu)\). We recall the following definition from \cite{zhu2007}.

		\begin{defn}
		A \textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \(\left\lVert xy \right\rVert \le \left\lVert x \right\rVert\left\lVert y \right\rVert\).
		A \(C^{*}\)\textbf{-algebra} is a Banach algebra \(\mathcal{A}\) with an involution \(x \mapsto x^{*}\) on it satisfying the following conditions:
		\begin{enumerate}[label=$\blacktriangleright$]
			\item \(x^{**}=x\) for all vectors \(x \in \mathcal{A} \).
			\item \((ax + by)^{*} = \bar{a}x^{*} + \bar{b}y^{*} \) for all vectors \(x,y \in \mathcal{A}\) and \(a,b \in \mathbb{C}\).
			\item \((xy)^{*}=y^{*}x^{*}\) for all vectors \(x,y \in \mathcal{A}\).
			\item \(\norm{xx^{*}} = \left\lVert x \right\rVert^{2}\) for all vector \(x \in \mathcal{A}\).
		\end{enumerate}
	\end{defn}

	\begin{defn}
		A Banach algebra \(\mathcal{A}\) is called a \textbf{\(\star\)-algebra} if for every \(A\in \mathcal{A}\), we have \(A^{*} \in \mathcal{A} \).
	\end{defn}

	\begin{defn}
		We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
	\end{defn}

	We will prove the following main result.

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation}\label{Fnorm1}
 \left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
 \end{equation}
 Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation*}
 \left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
 \end{equation*}
 The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

\begin{defn}
 We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
 \begin{equation}\label{R1}
 \langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
 \end{equation}
 holds for all \(z,w \in \mathbb{C}^n\).
 \end{defn}
 The inner product here is with respect to \(d\mu\).

\begin{exa}\label{Ex1}
 If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
 \begin{equation*}
 |\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
 \end{equation*}
 for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
 \end{exa}

\begin{proof}
 For each \(z, w, \xi \in \mathbb{C}^n\), we have
 \begin{equation*}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
 \end{equation*}
 By the triangular inequality we have
 \begin{equation*}
 |z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
 \end{equation*}
 It follows that
 \begin{eqnarray}\label{kzw}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray}
 Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
 \begin{equation}\label{Toepfgh}
 \langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{equation}
 In fact,
 \begin{eqnarray*}
 \langle T_{f}g, h \rangle &=& \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle h, K_{\xi} \rangle} d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{eqnarray*}
 Therefore,
 \begin{eqnarray*}
 |\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
 f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right| \\
 &\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
 &=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
 &=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray*}
 This completes the proof.
 \end{proof}

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

We will prove the following main result.

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\C^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\C^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

\begin{defn}\label{normstar}
 For an entire function \(h\) in \(\C^{n}\), we write
 \begin{equation*}
 \norm{h}_{*} = \left(\int_{\C^{n}} |h(\zeta)|^{2} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta)\right)^{\frac{1}{2}}~.
 \end{equation*}
 We denote by \(\mathcal{H}_{*}\) the collection of entire functions \(h\) on \(\C^n\) satisfying \(\norm{h}_{*}<\infty\).
 \end{defn}
 \begin{rem}
 The norm \(\norm{\cdot}_{*}\) is equivalent to the norm on the Fock space \(F^{2}_{1/2}\) given in relation (\ref{Fnorm1}) which is an Hilbert space. More precisely, \(\norm{\cdot}_{*} = (2\pi)^{n/2} \norm{\cdot}_{F^{2}_{1/2}}\). This ensures the continuity of \(\norm{\cdot}_{*}\).
 \end{rem}
 In what follows, we will use the operator \(U_{z}\) defined by
 \begin{equation*}
 U_{z}f(w) = f(z-w) k_{z}(w)~, \quad f \in H^{2}(\C^n, d\mu)~.
 \end{equation*}

\begin{pro}\label{proYz}
 Suppose \(\{c_{u}: u \in \Z^{2n}\}\) are complex numbers satisfying the condition \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. Then for each \(z\in \C^{n}\), the operator
 \begin{equation}\label{Yz}
 Y_{z} = \sum_{u \in \Z^{2n}} c_{u} k_{u-z} \otimes k_{u-z}
 \end{equation}
 is bounded on \(H^{2}(\C^n, d\mu)\). Moreover, the map \(z \mapsto Y_{z}\) from \(\C^{n}\) to \(\mathcal{B}(H^{2}(\C^{n},d\mu))\) is continuous with respect to the operator norm.
 \end{pro}
 \begin{proof}
 Let \(\{c_{u} : u \in \Z^{2n}\}\) be complex numbers such that \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. For \(z\in \C^{n}\), we define
 \begin{equation*}
 A_{z} = \sum_{u \in \Z^{2n}} c_{u} (U_{u}k_{z}) \otimes e_{u} \quad \text{ and } \quad B_{z} = \sum_{u \in \Z^{2n}}(U_{u}k_{z})\otimes e_{u}~.
 \end{equation*}
 Using the relation (\ref{Uu2}), we have \(Y_{z} = A_{z}B^{*}_{z}\). Set \(h_{u} = c_{u}k_{z}\), then using Definition \ref{normstar}, we have
 \begin{equation*}
 \norm{h_{u}}_{*} = |c_{u}|\norm{k_{z}}_{*} = (2\pi)^{n/2} e^{\frac{|z|^2}{2}} |c_{u}|~.
 \end{equation*}
 Therefore, \(\sup_{u\in \Z^{2n}} \norm{h_{u}}_{*} < \infty\). Applying Lemma \ref{LemUh} to \(h_{u} = c_{u}k_{z}\), we see that \(\norm{A_{z}} \le C \sup_{u\in \Z^{2n}}|c_{u}|\). Thus each \(A_{z}\) is bounded. Since \(B_z\) is just a special case of \(A_{z}\) (with \(c_{u}=1\) for all \(u \in \Z^{2n}\)), then it is also bounded and hence \(Y_{z} = A_{z}B^{*}_{z}\) is bounded.

\begin{lem}\label{Lim=0}
 Let \(B \in \mathcal{WL} \), then for every \(z,w \in \C^n \), we have
 \begin{equation*}
 \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle B k_{u-z}, k_{v-w} \rangle| =0 \quad \text{ and } \quad \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle k_{u-z}, Bk_{v-w}\rangle| = 0~.
 \end{equation*}
 \end{lem}
 \begin{proof}
 By \cite[Lemma 2.32]{zhu2012}, for any entire function \(f\) on \(\C^n\), we have
 \begin{equation*}
 \left|f(z)e^{-\frac{\alpha}{2}|z|^2}\right|^p \le C \int_{B(z,\delta)} |f(w)e^{-\frac{\alpha}{2}|w|^2}|^p dV(w) \quad \text{ for } z \in \C^n.
 \end{equation*}
 Hence for \(\alpha=p=1\) and \(\delta\) small such that the balls \(\{B(v-w,\delta): v \in \Z^{2n} \}\) are mutually disjoint, we have
 \begin{equation*}
 |\langle B k_{u-z}, k_{v-w} \rangle| = |Bk_{u-z} (v-w) |e^{-\frac{|v-w|^2}{2}} \le C \int_{B(v-w,\delta)} |Bk_{u-z}(\zeta) | e^{-\frac{|\zeta|^2}{2}} dV(\zeta).
 \end{equation*}
 Indeed, for \(\delta < \frac{1}{2} \), the balls \(\{B(v-w,\delta) : v \in \Z^{2n} \}\) are mutually disjoint. Otherwise, there would exist \(v,v' \in \Z^{2n} \) such that \(v \neq v' \), and a point \(\xi\) such that \(\xi \in B(v-w,\delta) \cap B(v'-w,\delta) \). In other words:
 \begin{equation*}
 |v-w-\xi| < \delta \quad \text{ and } \quad |v'-w - \xi| < \delta~.
 \end{equation*}
 This implies that
 \begin{equation*}
 |v-v'|=|(v-w-\xi)-(v'-w-\xi)|\le |v-w-\xi| + |v'-w-\xi| < \delta + \delta= 2 \delta < 1~.
 \end{equation*}
 That is \(|v-v'|<1\). This contradicts the well-known fact that \(|v-v'|\ge 1\). This result actually implies that there exists \(N\in \N\) such that each \(\zeta\in \C^n \) belongs to at most \(N\) balls in \(\{B(v-w,\delta): v\in \Z^{2n}\}\). That is \(\sum_{v \in \Z^{2n}} \chi_{B(v-w,\delta)}(\zeta) \le N \) for each \(\zeta \in \C^n \).

\begin{proof}[\textbf{Proof of Proposition \ref{EwBEzD} }]
 From (\ref{EwBEz}), we have
 \begin{equation*}
 E_{w} BE_{z} = \frac{1}{\pi^{2n}} \sum_{u,v \in \Z^{2n}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 Thus for any \(R>0\), we can write \(E_{w}BE_{z} = V_{R} + W_{R}\), where
 \begin{equation*}
 V_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|\le R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z} \quad \text{ and }
 \end{equation*}
 \begin{equation*}
 \quad W_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|>R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 To complete the proof, it suffices to prove that:
 \begin{enumerate}[label=(\alph*)]
 \item \label{itm:first1} \(\lim_{R\to \infty} \norm{W_{R}}=0\).
 \item \label{itm:second1} \(V_{R} \in \) span(\(\mathcal{D}_{0}\)) for every \(R>0\).
 \end{enumerate}

\begin{pro}\label{DoT1}
 We have \(\mathcal{D}_{0} \subset \mathcal{T}^{(1)}\).
 \end{pro}
 To establish the proof of this proposition, we will need the next three propositions.
 \begin{pro}\label{YzT1}
 Suppose that \(\{c_{u}: u \in \Z^{2n}\}\) is a bounded set of complex coefficients. Then for each \(z \in \C^{n}\), the operator \(Y_{z}\) defined in (\ref{Yz}) belongs to \(\mathcal{T}^{(1)}\).
 \end{pro}
 \begin{proof}
 \begin{enumerate}[label=(\alph*)]
 \item Let us first show that \(Y_{0} \in \mathcal{T}^{(1)}\). We have \(|u-v|\ge 1\) for all \(u \neq v \in \Z^{2n}\). Hence \(B(u,\frac{1}{2}) \cap B(v,\frac{1}{2}) = \emptyset \) for \(u\neq v\). For each \(0 < \varepsilon < \frac{1}{2}\), define the operator
 \begin{equation*}
 A_{\varepsilon} = \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} Y_{z} dV(z) .
 \end{equation*}
 From Proposition \ref{proYz}, we have the norm continuity of the map \(z\mapsto Y_{z}\) and it implies that
 \begin{equation*}
 \lim_{\varepsilon \to 0} \norm{Y_{0}-A_{\varepsilon}} = 0.
 \end{equation*}
 This comes from the fact that
 \begin{eqnarray*}
 \norm{Y_{0}-A_{\varepsilon}} &=& \norm{\frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \le \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} \norm{Y_{0}-Y_{z}} ~dV(z)
 \end{eqnarray*}
 and \(\lim_{z\to 0}\norm{Y_{z}-Y_{0}}=0\) .

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

To establish the proof of Theorem~\ref{XiaFock}, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of Theorem~\ref{XiaFock}."

To establish the proof of the main theorem, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of the main theorem."

Question: Let $H^{2}(\mathbb{C}^{n},d\mu)$ be the classical Fock space, where $d\mu(z)=\pi^{-n}e^{-|z|^{2}}\,dV(z)$. Let $\mathcal{WL}$ denote the class of weakly localized operators on $H^{2}(\mathbb{C}^{n},d\mu)$, and let $C^{*}(\mathcal{WL})$ be the $C^{*}$-algebra generated by $\mathcal{WL}$. Let $\mathcal{T}^{(1)}$ denote the Toeplitz algebra on this Fock space, i.e. the norm-closed algebra generated by Toeplitz operators $T_{f}$ with bounded measurable symbols $f$. Which statement holds about these two operator algebras?

They are equal: $C^{*}(\mathcal{WL})=\mathcal{T}^{(1)}$.

Choices:
B. One only has a strict inclusion $C^{*}(\mathcal{WL})\subsetneq \mathcal{T}^{(1)}$.
C. There is a containment $C^{*}(\mathcal{WL})\subseteq \mathcal{T}^{(1)}$.
D. They coincide only after taking compact-operator quotients, namely $C^{*}(\mathcal{WL})/\mathcal{K}=\mathcal{T}^{(1)}/\mathcal{K}$, but not necessarily before quotienting.
E. They have the same strong-operator closure, but $C^{*}(\mathcal{WL})$ and $\mathcal{T}^{(1)}$ need not be equal as norm-closed algebras.


\begin{theorem}\label{thm-Serrin}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{theorem}

\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

When \(f=0\), equation \eqref{Meq-gradient} becomes \(\Delta u=0\). Cheng-Yau \cite{CY1975} proved that any positive or bounded solution from above or below to \eqref{Meq-gradient} is a constant on Riemannian manifolds with nonnegative Ricci curvature.

For \(q=0\), the equation \eqref{Meq-gradient} becomes
\begin{equation}\label{Meq-q-0}
-\Delta u =f(u)\quad\text{in }M^n.
\end{equation}
For more general nonlinearity \(f(x,u)\), with additional assumptions imposed on \(f\), Gidas-Spruck \cite[Theorem 6.1]{GS1981} proved that solutions of equation \eqref{Meq-q-0} must be constant. In the Euclidean case, Serrin-Zou \cite{SZ2002} studied the \(m\)-Laplacian case. By focusing on \(m=2\), they proposed the subcritical condition on \(f\), that is
\[f(u)\geqslant0,\quad (p-1)f(u)-uf'(u)\geqslant0\]
for \(u>0, 1<p<\frac{2n}{n-2}\). Under the subcritical condition on \(f\), they obtained a Liouville theorem of equation \eqref{Meq-q-0} in \(\mathbb R^n\), that is if one of following condition holds:

(i) If \(n=2, 3\) and \(f\) is subcritical,

(ii) If \(n\geqslant 4\), \(f(u)\geqslant Cu\) for sufficiently large \(u\), and \(f\) is subcritical,

\noindent then equation \eqref{Meq-q-0} admits only constant solutions. For more Liouville theorems involving equation \eqref{Meq-q-0}, see \cite{L2025, L2025-1, LZ2003, MW2024, Z2025} and the references therein.

A typical case for equation \eqref{Meq-q-0} is
\begin{equation}\label{Meq-q-0-ty}
-\Delta u =u^p\quad\text{in }M^n.
\end{equation}
This equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \(1<p<\frac{n+2}{n-2}\) on a complete Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci tensor. Grigor’yan-Sun \cite{GS2014} proved that equation \eqref{Meq-q-0-ty} admits no positive solution under condition \(\operatorname{vol}(B_R(x_0))\leqslant CR^{\frac{2p}{p-1}}\ln^{\frac{1}{p-1}}R\), corresponding with the Serrin subcritical case \(p<\frac{n}{n-2}\) in dimension \(n\), when \(p>1\) on a connected geodesically complete Riemannian manifold.

In the critical case  \(p=\frac{n+2}{n-2}\), the equation \eqref{Meq-q-0-ty} becomes
\[-\Delta u =u^{\frac{n+2}{n-2}}\quad\text{in }M^n, \]
which is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \(\mathbb R^n\) has been successfully resolved, as can be seen in \cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form
\begin{equation}\label{solution-q-0}
u(x)=\Big(\frac{\sqrt{n(n-2)}\lambda}{1+(\lambda|x-x_0|)^2}\Big)^{\frac{n-2}{2}}.
\end{equation}
On generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \(\mathbb R^n\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \cite{FMM2023} got the rigidity result under the assumption \(u=O(r^{-\frac{n-2}{2}})\) at infinity. It is worth mentioning that Catino-Monticelli \cite{CM2022} obtained the rigidity result and classified it with \(n=3\), while \(n\geqslant4\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \cite{CFP2024} extended these results to \(n=4,5\) without any finite energy condition by the \(P\)-function method. Catino-Monticelli-Roncoroni \cite{CMR2023}, Ou \cite{O2025}, and Sun-Wang \cite{SW2025} extended the classification results on manifolds to \(m\)-Laplacian case.

Now, we return to the case with the gradient term. A crucial prototype of semilinear equation \eqref{Meq-gradient} is the case \(f(u)=u^p\), i.e.,
\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}
There are three important curves of \((p,q)\) for this equation, namely the homogeneous curve \(A_0(p,q)=0\), the first critical curve \(A_1(p,q)=0\), and the second critical curve \(A_2(p,q)=0\), with
\[A_0(p,q)=p+q-1,\quad A_1(p,q)=(n-2)p+(n-1)q-n,\]
\[A_2(p,q)=(n-2)p+(n-1)q-\Big(n+\frac{2-q}{1-q}\Big),\quad0\leqslant q<1.\]
As shown in Figure \ref{fig-1}, we mark these three curves with \(A_0\), \(A_1\), and \(A_2\) respectively. We introduce
\[2_*(q)=\frac{2-q}{n-2}(n-1),\quad 2^*(q)=\frac{2-q}{n-2}(n+\frac{q}{1-q}),\]
thus the first subcritical range \(A_1(p,q)<0\) is equivalent with \(p<2_*(q)-1\), and the second subcritical range \(A_2(p,q)<0\) is equivalent with \(p<2^*(q)-1\) when \(0\leqslant q<1\).

In the special case \(p=0\), equation \eqref{eq-u^p} becomes \(-\Delta u=|\nabla u|^q\), called as the Hamilton-Jacobi equation. For this equation, Lions \cite{L1985} obtained a Liouville theorem for \(q>1\) in \(\mathbb R^n\). Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite{BV-GH-V2014} established the gradient estimate to \(m\)-Laplacian case and obtained some Liouville theorems on complete noncompact manifolds, which satisfy a lower bound estimate on the Ricci curvature and sectional curvature.

In the Euclidean case, there has already been a great deal of work on equation \eqref{eq-u^p}. In the first subcritical case \(p<2_*(q)-1\), any supersolution to \eqref{eq-u^p} must be constant, see \cite{BP-GM-Q2016, BV-GH-V2019, CHZ2022, CM1997, F2009, MP2001}. Those past proofs are usually composed of three cases split by the homogeneous curve \(A_0(p,q)=0\), which is avoided in our proof of Theorem \ref{thm-Serrin}. Besides, for \(q\geqslant2\) and \(p\geqslant0\), Filippucci-Pucci-Souple \cite{FPS2020} obtained that any positive bounded solution to equation \eqref{eq-u^p} is constant. Later, Bidaut-V\'{e}ron \cite{BV2021} extended this result to the \(m\)-Laplace equation for \(q\geqslant m\) and established a Liouville theorem with \(p\geqslant 0\).

Now we concentrate on the second subcritical range \(p<2^*(q)-1\) with \(0\leqslant q<1\) for equation \eqref{eq-u^p} in \(\mathbb R^n\). For \(0<q<2\) and \(p>0\), Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite{BV-GH-V2019} conjectured that any solution to \eqref{eq-u^p} must be constant in the second critical range. They gave a positive answer to the left of the curve \(G(p,q)=0\) in Figure \ref{fig-1}, where
\[G(p,q)=\big((n-1)^2q+n-2\big)p^2+c(q)p-nq^2,\]
\[c(q)=n(n-1)q^2-(n^2+n-1)q-n-2.\]
Besides, they also showed that the second critical curve is optimal by proving the existence of a nonconstant solution in the second supercritical range \(A_2(p,q)>0\). Using Bernstein's technique, Chang-Hu-Zhang \cite{CHZ2022} derived a Liouville theorem in \(m\)-Laplacian case, and the left of the curve \(C\) in Figure \ref{fig-1} shows their range when \(m=2\). An exciting result was obtained by Ma-Wu \cite{MW2023}. With the help of integral identities, they completely solved the conjecture in \cite{BV-GH-V2019} when \(0<q<\frac{1}{n-1}\), and extended the range \(G(p,q)<0\) in \cite{BV-GH-V2019} to \(H(p,q)<0\) when \(q\geqslant\frac{1}{n-1}\), where
\[H(p,q)=p^2+\Big(\frac{n-1}{n-2}q-\frac{n^2-3}{(n-2)^2}\Big)p+\frac{1-(n-1)q}{(n-2)^2}.\]

\begin{figure}[htbp]\small
\centering
\includegraphics[width=0.8\textwidth]{images/diagram.jpg}
\caption{some parameter ranges of equation \eqref{eq-u^p} in \(\mathbb R^n\) when \(n=5\)}
\label{fig-1}
\end{figure}

Return to the Riemannian manifold case. Sun-Xiao-Xu \cite{SXX2022} obtained some Liouville theorems of equation \eqref{eq-u^p} under the growth of the geodesical ball and some range \((p,q)\in\mathbb R^2\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \cite{HHW2023} obtained that any \(C^1\) smooth positive solution of equation \eqref{eq-u^p} is a constant when \(p+q<\frac{n+3}{n-1}\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \cite{HHW2023} and \cite{SXX2022} studied \(m\)-Laplacian case.

Thus, research on Liouville theorems for equation \eqref{eq-u^p} on manifolds is relatively scarce, let alone equation \eqref{Meq-gradient} with general semilinear term \(f(u)\), which motivates us to study equation \eqref{Meq-gradient} on manifolds. First, we consider supersolutions to \eqref{Meq-gradient}.

\begin{definition}
We say that \(u\) is a supersolution to \eqref{Meq-gradient} if \(u\in H_{\mathrm{loc}}^1(M^n)\cap C(M^n)\) satisfies
\begin{align}\label{weak-supsol}
\int_{M^n} \nabla u\cdot\nabla\varphi\geqslant\int_{M^n}f(u)|\nabla u|^q\varphi,\quad\varphi\in C_c^\infty(M^n),~\varphi\geqslant 0.
\end{align}
\end{definition}

\begin{remark}
By an approximation argument, the test function \(\varphi\) may be chosen from \(H^1(M^n)\) with compact support.
\end{remark}

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

A typical case for equation \eqref{Meq-q-0} is
\begin{equation}\label{Meq-q-0-ty}
-\Delta u =u^p\quad\text{in }M^n.
\end{equation}
This equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \(1<p<\frac{n+2}{n-2}\) on a complete Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci tensor. Grigor’yan-Sun \cite{GS2014} proved that equation \eqref{Meq-q-0-ty} admits no positive solution under condition \(\operatorname{vol}(B_R(x_0))\leqslant CR^{\frac{2p}{p-1}}\ln^{\frac{1}{p-1}}R\), corresponding with the Serrin subcritical case \(p<\frac{n}{n-2}\) in dimension \(n\), when \(p>1\) on a connected geodesically complete Riemannian manifold.

In the critical case \(p=\frac{n+2}{n-2}\), the equation \eqref{Meq-q-0-ty} becomes
\[-\Delta u =u^{\frac{n+2}{n-2}}\quad\text{in }M^n, \]
which is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \(\mathbb R^n\) has been successfully resolved, as can be seen in \cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form
\begin{equation}\label{solution-q-0}
u(x)=\Big(\frac{\sqrt{n(n-2)}\lambda}{1+(\lambda|x-x_0|)^2}\Big)^{\frac{n-2}{2}}.
\end{equation}
On generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \(\mathbb R^n\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \cite{FMM2023} got the rigidity result under the assumption \(u=O(r^{-\frac{n-2}{2}})\) at infinity. It is worth mentioning that Catino-Monticelli \cite{CM2022} obtained the rigidity result and classified it with \(n=3\), while \(n\geqslant4\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \cite{CFP2024} extended these results to \(n=4,5\) without any finite energy condition by the \(P\)-function method. Catino-Monticelli-Roncoroni \cite{CMR2023}, Ou \cite{O2025}, and Sun-Wang \cite{SW2025} extended the classification results on manifolds to \(m\)-Laplacian case.

Return to the Riemannian manifold case. Sun-Xiao-Xu \cite{SXX2022} obtained some Liouville theorems of equation \eqref{eq-u^p} under the growth of the geodesical ball and some range \((p,q)\in\mathbb R^2\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \cite{HHW2023} obtained that any \(C^1\) smooth positive solution of equation \eqref{eq-u^p} is a constant when \(p+q<\frac{n+3}{n-1}\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \cite{HHW2023} and \cite{SXX2022} studied \(m\)-Laplacian case.

\begin{remark}
By an approximation argument, the test function \(\varphi\) may be chosen from \(H^1(M^n)\) with compact support.
\end{remark}

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

\begin{equation}\label{Meq-q-0}
-\Delta u =f(u)\quad\text{in }M^n.
\end{equation}

\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}

In both cases, \(\mathbf E=0\) holds in \(Z^c\) by letting \(R\to\infty\). For \(x\in\mathring Z\), \(\nabla u(x)=0\) yields \(\mathbf E=0\). By the continuity of \(\mathbf E\), we immediately conclude that \(\mathbf E=0\) on \(M\). Together with Lemma \ref{critical-thm-pf}, the proof is complete.
\qedsections/intro.tex0000664000000000000000000004363015111220774013270 0ustar rootroot\section{Introduction}\label{sec-introduction}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature.
Consider a class of semilinear equations with a gradient term
\begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
where \(\Delta\) is the Laplace-Beltrami operator, \(\nabla\) is the gradient operator, and \(f:\mathbb R_+\to\mathbb R_+\) is continuous.
The equation \eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \cite{BBCP2000} ), the porous media equation (see e.g., \cite{P2018}).

Under an integrable condition on \(f\), we obtain the following Liouville theorem by Serrin's technique.

When \(\liminf_{v\to0^+}f(v)>0\), the condition \eqref{cond-Serrin} trivially holds.

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\[\liminf_{v\to0^+}f(v)>0.\]
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{corollary}

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Then any positive supersolution to \eqref{eq-u^p} must be constant if \(p<2_*(q)-1\),
\end{corollary}

\begin{theorem}\label{thm-subcritical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Assume that \(f\in C^1(\mathbb R_+;\mathbb R_+)\) satisfies the following conditions:
\begin{equation}\label{cond-sub-1}
f(u)\geqslant u^{\beta_0},~u>N,\quad\text{for some }\beta_0\in(1-q,2^*(q)-1)\text{ and }N>0,
\end{equation}
\begin{equation}
uf'(u)\leqslant\alpha_0 f(u),\quad\text{for some }\alpha_0\in[\beta_0,2^*(q)-1).
\end{equation}
Then any positive \(C^3\) solution to \eqref{Meq-gradient} must be constant if \(p<2^*(q)-1\).
\end{theorem}

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Then any positive \(C^3\) solution to \eqref{eq-u^p} must be constant.
\end{corollary}

\begin{theorem}\label{thm-critical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\in\{3,4,5\}\) with nonnegative Ricci curvature, \(0<q<\frac{1}{n-1}\) and \(f(u)=u^{2^*(q)-1}\) in \eqref{Meq-gradient}. Let \(u\in C^3(M)\) be a positive solution to \eqref{Meq-gradient}, then either \(u\) is constant, or \((M^n, g)\) is isometric to \(\mathbb R^n\) and \(u\) is the form given by \eqref{solution}.
\end{theorem}

\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}

\begin{equation}\label{eq-u^p}
-\Delta u = u^p|\nabla u|^q\quad\text{in }M^n.
\end{equation}

\begin{equation}\label{solution}
U_{\lambda,x_0}(x):=\Big(\frac{(n+\frac{q}{1-q})^{\frac{1}{2-q}}(n-2)^{\frac{1-q}{2-q}}\lambda}{1+(\lambda|x-x_0|)^{\frac{2-q}{1-q}}}\Big)^{\frac{(n-2)(1-q)}{2-q}},\quad\lambda>0,~x_0\in\mathbb R^n.
\end{equation}

When \(\liminf_{v\to0^+}f(v)>0\), the condition \eqref{cond-Serrin} trivially holds.

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\[\liminf_{v\to0^+}f(v)>0.\]
Then any positive supersolution to \eqref{Meq-gradient} must be constant.
\end{corollary}

Theorem \ref{thm-Serrin} recovers the first subcritical range \(p<2_*(q)-1\) by taking \(f(u)=u^p\).

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Then any positive supersolution to \eqref{eq-u^p} must be constant if \(p<2_*(q)-1\),
\end{corollary}

For solutions to \eqref{Meq-gradient}, we concentrate on the cases \(0<q<\frac{1}{n-1}\). We establish the following Liouville theorem. Our proof also works for the case \(q=0\), which recovers the \(m=2\) case in Serrin-Zou \cite{SZ2002} in \(M=\mathbb R^n\).

\begin{theorem}\label{thm-subcritical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Assume that \(f\in C^1(\mathbb R_+;\mathbb R_+)\) satisfies the following conditions:
\begin{equation}\label{cond-sub-1}
f(u)\geqslant u^{\beta_0},~u>N,\quad\text{for some }\beta_0\in(1-q,2^*(q)-1)\text{ and }N>0,
\end{equation}
\begin{equation}
uf'(u)\leqslant\alpha_0 f(u),\quad\text{for some }\alpha_0\in[\beta_0,2^*(q)-1).
\end{equation}
Then any positive \(C^3\) solution to \eqref{Meq-gradient} must be constant if \(p<2^*(q)-1\).
\end{theorem}

Theorem \ref{thm-subcritical} recovers the second subcritical range \(p<2^*(q)-1\) by taking \(f(u)=u^p\).

\begin{corollary}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature, and \(0<q<\frac{1}{n-1}\). Then any positive \(C^3\) solution to \eqref{eq-u^p} must be constant.
\end{corollary}

By taking \(M=\mathbb R^n\), this corollary recovers the case \(0<q<\frac{1}{n-1}\) in \cite[Theorem 1.3]{MW2023}, where Ma and Wu solved the conjectured raised in \cite{BV-GH-V2019}.

Lastly, we consider the second critical case \(f(u) = u^{2^*(q)-1}\). Bidaut-V\'{e}ron, Huidobro, and V\'{e}ron \cite[Theorem D]{BV-GH-V2019} found an explicit 1-parameter family of positive radial solutions in \(\mathbb R^n\)
\[u_c(r)=c(Kc^{\frac{(2-q)^2}{(n-2)(1-q)}}+r^{\frac{2-q}{1-q}})^{-\frac{(n-2)(1-q)}{2-q}},\quad c>0,~K=K(n,q)>0.\]
By inserting them into equation \eqref{Meq-gradient} with \(f(u) = u^{2^*(q)-1}\), we rewrite the solutions as
\begin{equation}\label{solution}
U_{\lambda,x_0}(x):=\Big(\frac{(n+\frac{q}{1-q})^{\frac{1}{2-q}}(n-2)^{\frac{1-q}{2-q}}\lambda}{1+(\lambda|x-x_0|)^{\frac{2-q}{1-q}}}\Big)^{\frac{(n-2)(1-q)}{2-q}},\quad\lambda>0,~x_0\in\mathbb R^n.
\end{equation}
For \(q=0\), the solution \eqref{solution} is equivalent to \eqref{solution-q-0}. The rigidity of manifolds has been rigorously established in low-dimensional settings for \(q=0\), but for \(0<q<1\), the classification of equation \eqref{Meq-gradient} in the second critical case \(f(u) = u^{2^*(q)-1}\) on \(\mathbb R^n\) is open, let alone the rigidity of manifolds.

With the help of the invariant tensor \(\operatorname{\mathbf E}\) and its corresponding pseudo-invariant function \(l\) (i.e., \(\nabla l\) is only composed of terms involving the invariant tensor \(\operatorname{\mathbf E}\)), we obtain the rigidity for the ambient manifold and classify positive solutions of the second critical equation, \eqref{Meq-gradient} with \(f(u)=u^{2^*(q)-1}\), without any assumption in a low-dimensional setting. To our knowledge, this is the first rigidity result in the second critical case of equations with gradient terms.

\begin{theorem}\label{thm-critical}
Let \((M^n, g)\) be a complete, connected, and noncompact Riemannian manifold of dimension \(n\in\{3,4,5\}\) with nonnegative Ricci curvature, \(0<q<\frac{1}{n-1}\) and \(f(u)=u^{2^*(q)-1}\) in \eqref{Meq-gradient}. Let \(u\in C^3(M)\) be a positive solution to \eqref{Meq-gradient}, then either \(u\) is constant, or \((M^n, g)\) is isometric to \(\mathbb R^n\) and \(u\) is the form given by \eqref{solution}.
\end{theorem}

Differential identities are crucial in studying properties of an elliptic equation. However, establishing differential identities usually relies on some geometric background, such as the Bochner formula. Thus, finding a proper differential identity may be challenging for a sophisticated equation. In \cite{MW2024}, Ma-Wu (the third author of this paper) developed the invariant tensor technique, a powerful method of finding vector fields and differential identities without any geometric instructions, to study semilinear equations \(-\Delta u=f(u)\) on \(M^n\), which gave a new proof of Bidaut-V\'{e}ron and V\'{e}ron's rigidity result in \cite{BV-V1991}. Using the new technique, Ma-Ou-Wu \cite{MOW2025} reconstructed Jerison-Lee's identity to answer the question raised by Jerison-Lee \cite{JL1988}. Besides, Ma-Ou-Wu \cite{MOW2023} verified the subcritical rigidity conjecture raised by Wang \cite{W2022} via this new technique with sophisticated computations. Recently, Guo-Zhang \cite{GZ2025} utilised the invariant tensor technique to give a new proof of semilinear equations again. The invariant tensor technique is also widely applied to classify quasilinear cases or involving gradient terms, see \cite{LWY2025, Z2024}.

Although we concentrate only on the case \(0<q<\frac{1}{n-1}\), we make the first attempt to successfully extend the invariant tensor technique to equations with gradient terms. The pseudo-invariant function \(l\), which plays a crucial role in the classification process, is found by differential invariance. However, it becomes difficult to find \(l\) through variable transformations as previously done. The authors have extended the invariant tensor technique to the equation \eqref{Meq-gradient} to \(m\)-Laplace in \cite{WZ}, where a wider range of \(q\) is studied with more sophisticated computations.

Throughout this paper, we assume constant \(R>1\), \(\varepsilon>0\) small enough, and \(C>0\) are constants independent of \(R\). Let \(B_R\) be the geodesic ball centered at some fixed point with radius \(R\). Unless otherwise specified, we employ the summation convention for repeated indices from \(1\) to \(n\). In proofs involving manifolds, we choose a local frame, \(\varphi_i\) denotes the covariant derivative of the function \(\varphi\). Let \(\eta\) be a smooth cutoff function supported in \(B_{2R}\) satisfying \(\eta\equiv 1\) in \(B_R\), \(0\leqslant\eta\leqslant 1\), and \(|\nabla\eta|\leqslant CR^{-1}\).

This paper is structured as follows. In Section 2, we prove Theorem \ref{thm-Serrin} by using Serrin's technique with a delicate test function \(b(u)\eta^n\) and a three-term Young's inequality. In Section 3, using the invariant tensor technique, we establish a key differential identity and necessary propositions to prove Theorem \ref{thm-subcritical}. With the help of this differential identity, the proof of Theorem \ref{thm-critical}, classification of solutions in the critical case, is finished in Section 4.sections/Serrin.tex0000664000000000000000000000411015111220774013365 0ustar  rootroot

To prove Theorem~\ref{thm-Serrin}, the paper uses “Serrin's technique with a delicate test function $b(u)\eta^n$ and a three-term Young's inequality” (carried out in Section 2).

To prove the main theorem, the paper uses “Serrin's technique with a delicate test function $b(u)\eta^n$ and a three-term Young's inequality” (carried out later).

\label{thm-Serrin}
Let \((M^n, g)\) be a complete, connected, and noncompact  Riemannian manifold of dimension \(n\geqslant3\) with nonnegative Ricci curvature. Assume that \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfies
\begin{equation}\label{cond-Serrin}
\int_0^1[f(v)]^{-\frac{1}{2_*(q)-1}}\mathrm dv<+\infty.
\end{equation}
Then any positive supersolution to \begin{equation}\label{Meq-gradient}
-\Delta u = f(u)|\nabla u|^q\quad\text{in }M^n,
\end{equation}
must be constant.,

Question:
Let \((M^n,g)\) be a complete, connected, noncompact Riemannian manifold of dimension \(n\ge 3\) with nonnegative Ricci curvature. Let \(f\in C(\mathbb R_+;\mathbb R_+)\) satisfy
\[
\int_0^1 [f(v)]^{-\frac{1}{2_*(q)-1}}\,dv<\infty,
\]
and consider the equation with gradient term
\[
-\Delta u=f(u)|\nabla u|^q\quad \text{in }M^n,
\]
where \(\Delta\) is the Laplace--Beltrami operator and \(\nabla u\) is the gradient of \(u\). If \(u\) is a positive supersolution, meaning \(u>0\) and \(-\Delta u\ge f(u)|\nabla u|^q\) on \(M^n\), which conclusion holds?

Every positive supersolution \(u\) is constant on \(M^n\).

Choices:
B. Every positive supersolution \(u\) is constant on \(M^n\) provided additionally that \(0<q<\frac{1}{n-1}\); without this restriction on \(q\), the conclusion need not hold.
C. Every bounded positive supersolution \(u\) is constant on \(M^n\).
D. There exists a constant \(C=C(M,n,q,f)>0\) such that every positive supersolution \(u\) satisfies \(u\equiv C\) on \(M^n\).
E. Every positive \(C^1\) solution \(u\) of \(-\Delta u=f(u)|\nabla u|^q\) on \(M^n\) is constant, but the same conclusion is not asserted for positive supersolutions satisfying only \(-\Delta u\ge f(u)|\nabla u|^q\).

no subcritical range restriction on q in the Serrin argument

dropped the full class of positive supersolutions to the smaller class of bounded positive supersolutions

constancy depends on the individual supersolution, not a universal constant determined only by the data

Serrin test-function method is stated for supersolutions, not merely for exact C^1 solutions