\begin{theorem} \label{thm homeo}
The map
\begin{equation*}
 \Psi_{\ge0}\colon Y_{\ge 0}\rightarrow X(\Sigma)_{\ge0}
 \quad ; \quad
 gB \mapsto [\Delta_{\varpi_1}(g),\ldots,\Delta_{\varpi_{\rkg}}(g);\qp{1}(g),\ldots,\qp{n}(g)]
\end{equation*}
is a homeomorphism satisfying
\begin{equation*}
\Psi_{\ge 0}(\RY{K}{J;>0}) = \Xp{K,J}
\end{equation*}
for $K\subseteq J \subseteq I$.
\end{theorem}

\begin{proposition}\label{prop description of YKJ}
For $K\subseteq J\subseteq I$, we have
\begin{align*}
 \RY{K}{J}
 = \left\{ x\dot{w}_J B\in G/B \ \left| \ x \in (U_J)^{e_J} \text{ and }
 \begin{cases}
 \Delta_{\varpi_i}(x\dot{w}_J)=0 \quad \text{if $i\in K$}, \\
 \Delta_{\varpi_i}(x\dot{w}_J)\ne0 \quad \text{if $i\in I-K$}
 \end{cases}
 \right.
 \right\},
\end{align*}
where $(U_J)^{e_J}=U_J\cap Z_{G_J}(e_J)$.
\end{proposition}

\begin{proposition}\label{prop restrict}
The morphism $\Psi \colon Y\to X(\fan)$ restricts to a continuous map
\begin{equation*}
\Psi_{\ge 0} \colon Y_{\ge 0}\to X(\fan)_{\ge 0}
\end{equation*}
which sends $\RY{K}{J;>0}$ to $\Xp{K,J}$ for $K\subseteq J \subseteq  I$.
\end{proposition}

\begin{proposition}\label{prop Delta is nonnegative}
For $x\in U_{\ge0}$ and $w\in W$, we have
\begin{equation*}
 \Delta_{\varpi_i}(x\dot{w}) \ge0
 \qquad (i\in I),
\end{equation*}
where $\Delta_{\varpi_i}$ is the function defined in \eqref{eq def of Delta}.
\end{proposition}

\begin{proposition}\label{prop description for R KI>0 2}
Let $K\subseteq J\subseteq I$. Then, we have
\begin{equation*}
\RY{K}{J;>0}=\left\{x\dot{w}_JB \ \left| \
\begin{matrix}
\text{
\rm $x\in (U_J)^{e_J}_{\ge0}$ and}
~\begin{cases}\Delta_{\varpi_i}(x\dot{w}_J)=0\quad\text{if}\quad i\in K\\
\Delta_{\varpi_i}(x\dot{w}_J)> 0\quad\text{if}\quad i\in J-K \\
\Delta_{\varpi_i}(x\dot{w}_J)=1 \quad\text{if}\quad i\in I-J
\end{cases}
\end{matrix}
\right.
\right\}.
\end{equation*}
\end{proposition}

\begin{proposition}\label{prop LR and our}
Let $i\in I$. For all $x\in U^e_{\ge0}$, we have
\begin{align*}
 \Delta_i([\dot{w}_0^{-1}x\dot{w}_0]) = (\Delta_{\varpi_i}(x\dot{w}_0))^{m_i}.
\end{align*}
\end{proposition}

\begin{theorem} \label{thm main}
    The totally nonnegative part of the Peterson variety is homeomorphic to the strongly dominant weight polytope as a cell-decomposed space. In particular, it is a regular CW-complex, which is homeomorphic to a cube.
\end{theorem}

\begin{proposition}\label{prop nonnengative stratum parametrization}
For $K\subseteq J\subseteq I$, the map
\begin{align*}
 (\R^{I})_{K,J;>0} \rightarrow \Xp{K,J}
 \quad ; \quad
 (x_1,\ldots,x_{\rkg}) \mapsto [x_1,\ldots,x_{\rkg}\hspace{2pt};\delta_1,\ldots,\delta_{\rkg}]
\end{align*}
is a bijection, where $(\delta_1,\ldots,\delta_{\rkg})\in\R^{I}$ is the element given by
\begin{align*}
\begin{cases}
 \delta_i=0 \quad (i\in I-J) \\
 \delta_i=1 \quad (i\in J) .
\end{cases}
\end{align*}
\end{proposition}

\label{sec Introduction}
Let $G$ be a semisimple algebraic group over $\mathbb C$ with a Borel subgroup $B\subset G$. The Peterson variety $Y$ is a certain remarkable subvarieties of the flag variety $G/B$, introduced by Dale Peterson \cite{peterson1997quantum} to realize quantum cohomology rings of all Langlands dual partial flag varieties geometrically. By using the geometry of $Y$, he discovered a connection of the quantum cohomology of those flag varieties with the homology of the affine Grassmannian $\mathcal{G} r_{G^{\vee}}$ of the Langlands dual group $G^{\vee}$, which was verified by Lam--Shimozono in \cite{lam2010quantum}. It is also closely related to the wonderful compactification of a certain unipotent subgroup of $G$ \cite{Bualibanu2017Peterson}. The geometry and topology of the Peterson variety have been studied extensively, see, for example, \cite{abe2024geometry,abe2023peterson,Bualibanu2017Peterson,Goldin2024positivity,gui2025structure,harada2015equivariant,kostant1996flag}.

The theory of total positivity for reductive algebraic groups was pioneered by Lusztig \cite{lusztig1994total} as a broad extension of the classical theory of Schoenberg and Gantmacher--Krein on total positivity for matrices. It has close connections to cluster algebras \cite{fomin2010total}, KP equations \cite{kodama2014kp,kodama2011kp}, Poisson geometry \cite{lu2020bott}, and the physics of scattering amplitudes \cite{williams2021positive}. Lusztig \cite{lusztig1994total,lusztig1998total} also defined the totally positive and the totally nonnegative parts of the flag variety, which naturally induces the corresponding definitions for subvarieties of the flag variety. Of particular interests are so-called {\it regularity theorems} on CW-complex structures on the totally non-negative parts of these varieties, see \cite{bao2024product,galashin2022regularity,hersh2014regular} for the most recent developments. Note that convex polytopes are prototypical examples of regular CW
complexes and the topology of a regular CW complex is completely determined by the combinatorial
structure of its associated cell closure poset \cite{bjorner1984posets}.

The interaction bewteen the Peterson variety and the total positivity was first studied by Rietsch. In \cite{rietsch2003totally}, she used Peterson's theory in type A to obtain a parametrization of totally nonnegative Toeplitz matrices. In \cite{Rietsch2006}, she gave a mirror construction of the totally nonnegative part of the Peterson variety $Y_{\geq 0}$ in type A and obtained its cell decomposition. She also showed in \cite{Rietsch2006} that the totally nonnegative part of the Peterson variety in type A is contractible. Based on the structure of the cells, she conjectured that as a cell decomposed space $Y_{\geq 0}$ is homeomorphic to a cube. In \cite{abe2025totally}, the first and the third authors of this paper gave a proof of Rietsch's conjecture in type A by using toric geometry closely related to the Peterson variety and concrete computations.

In this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\Delta_{\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\geq 0}$ (Proposition \ref{prop description for R KI>0 2}). Through a particular morphism $\Psi: Y \rightarrow X(\Sigma)$ constructed by the first and the third authors of this paper in \cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$, which sends the Richardson strata of $Y_{\geq 0}$ to the toric orbit strata of $X(\Sigma)_{\geq 0}$ (Proposition \ref{prop restrict}). We prove that $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ is actually a homeomorphism (Theorem \ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\Sigma)_{\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.

\begin{document}
\text{}\vspace{-10pt}
\title[Peterson variety and strongly dominant weight polytope]{Totally nonnegative Peterson variety \\and strongly dominant weight polytope}

In this paper, we study the totally nonnegative part of the Peterson variety in general Lie types. Now we summarize our main results. By intersecting the Bruhat and opposite Bruhat decompositions of the flag variety $G/B$ and the Peterson variety $Y$, one can obtain the Richardson stratification of $Y$. In Proposition \ref{prop description of YKJ}, we give a description of the Richardson strata of $Y$ in terms of certain functions $\Delta_{\varpi_i}$. Using the positivity of these functions on the totally nonnegative part (Proposition \ref{prop Delta is nonnegative}), we give a description of the Richardson strata of the totally nonnegative part $Y_{\geq 0}$ (Proposition \ref{prop description for R KI>0 2}). Through a particular morphism $\Psi: Y \rightarrow X(\Sigma)$ constructed by the first and the third authors of this paper in \cite{abe2023peterson}, the Peterson variety $Y$ is connected to a particular projective toric orbifold $X(\Sigma)$. We show that this morphism can be restricted to the nonnegative parts $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$, which sends the Richardson strata of $Y_{\geq 0}$ to the toric orbit strata of $X(\Sigma)_{\geq 0}$ (Proposition \ref{prop restrict}). We prove that $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ is actually a homeomorphism (Theorem \ref{thm homeo}). Since the torus orbit decomposition of the a projective toric variety gives rise to a cell decomposition of its nonnegative part (see Proposition \ref{prop nonnengative stratum parametrization} in our case), it follows that the Richardson stratification of $Y_{\geq 0}$ is actually a cell decomposition. Note that the moment map restricts to a cell-preserving homeomorphism between $X(\Sigma)_{\geq 0}$ and its moment polytope. Here, the moment polytope of $X(\Sigma)$ is the strongly dominant weight polytope, which is defined to be the intersection of the dominant Weyl chamber and a weight polytope associated with a regular weight. Since the strongly dominant weight polytope is proved to be combinatorially equivalent to a cube\footnote{Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between the strongly dominant weight polytope and the standard cube, which restricts to homeomorphisms between their facets (and hence all the faces).} \cite{burrull2023strongly}, we deduce the following main theorem of this paper, which conforms a conjecture of Rietsch\footnote{We thank Konstanze Rietsch for private communication of her conjecture.} for arbitrary Lie type.

Now we give some final remarks. Firstly we remark that the functions $\Delta_{\varpi_{i}}$ and $\qp{i}$ appearing in our map $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \cite[Remark 6.5]{lam2016total} and \cite[Remark~3.3.7]{rietsch2008mirror}. In \cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\Delta_{\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\geq 0}$ and $X(\Sigma)_{\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\Sigma)_{\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.

This paper is organized as follows. In Section \ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\geq 0}$ and give a description of its Richardson strata. In Section \ref{sec toric}, we study the toric orbifold $X(\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \ref{sec map}, we recall the definition of the morphism $\Psi: Y \rightarrow X(\Sigma)$, and reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting}). Sections \ref{sec splittings} and \ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.

Let $\lambda\in\WL=\Hom(T,\C^{\times})$ be a regular dominant weight. Namely, we have $\lambda=\sum_{i\in I} a_i \varpi_i$ for some positive coefficients $a_i\in\R_{>0}$ $(i\in I)$.
Regarding $\lambda$ as an element of $\mathfrak{t}^*_{\R}\coloneqq \WL\otimes_{\Z}\R$, the weight polytope associated to $\lambda$ is the convex hull of the $W$-orbit of $\lambda$ :
\begin{align*}
 \text{Conv}(W\cdot \lambda) \subseteq \mathfrak{t}^*_{\R}.
\end{align*}
For its relations to the irreducible representation $V_{\lambda}$ with highest weight $\lambda$, see e.g.\ \cite[Sect.~10.1]{hall2015gtm222}.
According to \cite{burrull2023strongly}, we define the \textit{strongly dominant weight polytope} $P^{\lambda}$ by the intersection
\begin{align*}
 P^{\lambda} \coloneqq \text{Conv}(W\cdot \lambda) \cap \sigma_+ \subseteq \mathfrak{t}^*_{\R},
\end{align*}
where
$
 \sigma_+ \coloneqq \{ a_1 \varpi_1 + \cdots + a_{\rkg} \varpi_{\rkg} \mid a_i\ge0 \ (i\in I) \}
$
is the (closed) dominant Weyl chamber in $\mathfrak{t}^*_{\R}$.
In loc.\ cit., $P^{\lambda}$ is proved to be a rational\footnote{This means that the vertices of $P^{\lambda}$ lie in $\WL\otimes_{\Z}\Q$.} combinatorial cube of dimension $n(=|I|)$. Topologically, this is equivalent to the existence of a (piecewise linear) homeomorphism between $P^\lambda$ and the standard $n(=|I|)$-cube, which restricts to homeomorphisms between their facets (and hence all the faces), e.g.\ \cite[Sect.\ 2.2]{ziegler1995lectures}. In particular, its combinatorial structure is independent of the choice of the regular dominant weight $\lambda$.
We review some facts about $P^{\lambda}$ from loc.\ cit.
We set
\begin{align*}
 &H_i \coloneqq \{ \mu \in \mathfrak{t}^*_{\R} \mid \alpha^{\vee}_i(\mu)= 0 \}, \\
 &H^{\lambda}_i \coloneqq \{ \lambda - \mu \in \mathfrak{t}^*_{\R} \mid \varpi^{\vee}_i(\mu)= 0 \}.
\end{align*}
Then the set of facets of $P^{\lambda}$ are given by $P^{\lambda}\cap H_i$ and $P^{\lambda}\cap H^{\lambda}_i$ for $i\in I$ with outward normal vectors $-\alpha^{\vee}_i$ $(i\in I)$ and $\varpi^{\vee}_i$ $(i\in I)$, respectively.
Moreover, for $K, J\subseteq I$, the intersection
\begin{align}\label{eq faces of SDWP}
 F_{K,J} \coloneqq P^{\lambda}\cap \bigcap_{j\in K} H_i \cap \bigcap_{j\notin J} H^{\lambda}_i
\end{align}
is non-empty if and only if $K\subseteq J$ (equivalently, $K\cap (I-J)=\emptyset$).
See \cite[Sect.~2]{burrull2023strongly} for details.
This means that the normal fan of $P^{\lambda}$ is precisely our fan $\Sigma$ given in Definition~\ref{def fan}.

Now we give some final remarks. Firstly we remark that the functions $\Delta_{\varpi_{i}}$ and $\qp{i}$ appearing in our map $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ are actually affine Schubert classes and quantum parameters in Peterson's theory, see \cite[Remark 6.5]{lam2016total} and \cite[Remark~3.3.7]{rietsch2008mirror}. In \cite[Theorem 7.3]{lam2016total}, Lam--Rietsch essentially used the functions $\Delta_{\varpi_{i}}$ (up to a certain power because they work on adjoint type while we work on the simply-connected case, see Proposition \ref{prop LR and our}) to give a parametrization of the totally nonnegative part of an affine piece of the Peterson variety. We remark that this parametrization is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. By analyzing a crucial connection with the affine Grassmannian and geometric Satake equivalence, Lam--Rietsch also showed that different notions of positivity---quantum Schubert and quantum parameter positivity, Lusztig's total positivity, and affine Schubert positivity---on an affine part of the Peterson variety all coincide. Finally, note that $Y_{\geq 0}$ and $X(\Sigma)_{\geq 0}$ are defined in quite different ways; the totally nonnegative part $Y_{\geq 0}$ is defined in terms of root datum of $G$ whereas $X(\Sigma)_{\geq 0}$ is defined in terms of semigroup algebras. It is quite remarkable that so many apparently different notions of positivity coincide in this setting.

This paper is organized as follows. In Section \ref{sec Notation}, we fix some notations which we use throughout this paper. In Section \ref{sec Richardson strata}, we recall the definition of the Peterson variety $Y$ and the construction of its Richardson strata. In Section \ref{sec Totally nonnegative Richardson strata}, we study the totally nonnegative part $Y_{\geq 0}$ and give a description of its Richardson strata. In Section \ref{sec toric}, we study the toric orbifold $X(\Sigma)$ whose moment polytope is the strongly dominant weight polytope. In Section \ref{sec map}, we recall the definition of the morphism $\Psi: Y \rightarrow X(\Sigma)$, and reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting}). Sections \ref{sec splittings} and \ref{sec LR in our setting} are devoted to give a proof of this key claim, which completes the proof of Theorem \ref{thm main}. In the Appendix, we provide proofs of two well-known lemmas for the reader because of the lack of suitable references.

\vspace{10pt}

\subsection{Acknowledgments}
We are grateful to Thomas Lam, Changzheng Li, and Zhi L$\ddot{\text{u}}$ for valuable discussions. We would particularly like to thank Konstanze Rietsch for telling us her conjecture. This research was partially done while the authors were visiting the  school of Mathematical Sciences in Fudan University.
The first author is supported in part by JSPS Grant-in-Aid for Scientific Research(C): 23K03102.
The second author is supported in part by NSFC:  12471309.
The third author is supported in part by NSFC:  11901218.

\vspace{30pt}

The post-theorem introduction indicates that the proof of Theorem~\ref{thm main} proceeds by constructing the morphism $\Psi: Y \rightarrow X(\Sigma)$ and then showing that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. The text says they "reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim (Theorem \ref{thm LR in our setting})", and that Sections \ref{sec splittings} and \ref{sec LR in our setting} "give a proof of this key claim, which completes the proof of Theorem \ref{thm main}." It is also remarked that a Lam--Rietsch parametrization using the functions $\Delta_{\varpi_i}$ is "important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$."

The post-theorem introduction indicates that, in establishing the main theorem, the proof proceeds by constructing the morphism $\Psi: Y \rightarrow X(\Sigma)$ and then showing that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$. The text says they reduce the fact that it induces a homeomorphism $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$ to the key claim:

\begin{theorem}\label{thm LR in our setting}
The map
\begin{align*}
 U^e_{\ge0} \rightarrow \R^{I}_{\ge0}
 \quad ; \quad
 x \mapsto
 (\Delta_{\varpi_1}(x\dot{w}_0), \ldots, \Delta_{\varpi_{\rkg}}(x\dot{w}_0))
\end{align*}
is a homeomorphism.
 \end{theorem}

It is then explained that the subsequent parts of the paper give a proof of this key claim, which completes the proof of the main theorem. It is also remarked that a Lam--Rietsch parametrization using the functions $\Delta_{\varpi_i}$ is important in the proof of the bijectivity of $\Psi_{\geq 0}: Y_{\geq 0} \rightarrow X(\Sigma)_{\geq 0}$.

Question:
Let $G$ be a semisimple algebraic group over $\mathbb C$, let $B\subset G$ be a Borel subgroup, let $T\subset B$ be a maximal torus, and let $Y\subset G/B$ be the Peterson variety. Let $Y_{\ge 0}$ denote the totally nonnegative part of $Y$. Fix a regular dominant weight $\lambda=\sum_{i\in I} a_i\varpi_i\in \operatorname{Hom}(T,\mathbb C^{\times})$ with all $a_i>0$, let $W$ be the Weyl group, and define the strongly dominant weight polytope by
\[
P^{\lambda}=\operatorname{Conv}(W\cdot \lambda)\cap \sigma_+,
\qquad
\sigma_+=\Bigl\{\sum_{i\in I} b_i\varpi_i\mid b_i\ge 0\Bigr\}.
\]
Viewing $P^{\lambda}$ with its face decomposition, and interpreting “homeomorphic as a cell-decomposed space” to mean a homeomorphism that respects the cell decompositions, which existence statement holds?

There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.

Choices:
B. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $\operatorname{Conv}(W\cdot \lambda)$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to the full weight polytope $\operatorname{Conv}(W\cdot \lambda)$.
C. There exists a homeomorphism from $Y_{\ge 0}$ onto $P^{\lambda}$. Consequently, $Y_{\ge 0}$ is homeomorphic to a cube.
D. For every regular dominant weight $\lambda$, there exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ that is induced by the coordinate map $x\mapsto (\Delta_{\varpi_1}(x\dot w_0),\ldots,\Delta_{\varpi_{|I|}}(x\dot w_0))$. Consequently, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.
E. There exists a homeomorphism of cell-decomposed spaces from $Y_{\ge 0}$ onto $P^{\lambda}$ if and only if the chosen regular dominant weight $\lambda$ has rational coefficients $a_i\in \mathbb{Q}_{>0}$. Consequently, only for such $\lambda$ is $Y_{\ge 0}$ a regular CW-complex homeomorphic to a cube.

target polytope is the strongly dominant truncation, not the full weight polytope

drops the requirement that the homeomorphism respect the cell decompositions

confuses the Lam--Rietsch coordinate homeomorphism on $U^e_{\ge0}\cong \mathbb R^I_{\ge0}$ with the global cell-decomposed-space homeomorphism $Y_{\ge0}\to P^\lambda$

replaces independence of the combinatorial type from regular dominant $\lambda$ by a spurious rationality restriction


Question:
Let $G$ be a semisimple algebraic group over $\mathbb C$ with Borel subgroup $B\subset G$, and let $Y\subset G/B$ be the Peterson variety. Write $Y_{\ge 0}$ for its totally nonnegative part. If $\lambda=\sum_{i\in I} a_i\varpi_i$ is a regular dominant weight with all $a_i>0$, and if the strongly dominant weight polytope is defined by
\[
P^{\lambda}=\operatorname{Conv}(W\cdot \lambda)\cap \sigma_+,
\]
where $W$ is the Weyl group and $\sigma_+$ is the dominant Weyl chamber, which statement gives the correct topological and cell-decomposition description of $Y_{\ge 0}$?

$Y_{\ge 0}$ is homeomorphic to $P^{\lambda}$ as a cell-decomposed space; in particular, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to a cube.

Choices:
B. $Y_{\ge 0}$ is homeomorphic to $P^{\lambda}$ as a cell-decomposed space; in particular, $Y_{\ge 0}$ is a CW-complex whose cells are indexed by the nonempty intersections $F_{K,J}$ with $K\subseteq J$, but it need not be regular and is not in general homeomorphic to a cube.
C. $Y_{\ge 0}$ is homeomorphic to $P^{\lambda}$; in particular, $Y_{\ge 0}$ is homeomorphic to a cube.
D. $Y_{\ge 0}$ is homeomorphic to $P^{\lambda}$ for some choice of regular dominant weight $\lambda$, but there is no homeomorphism identifying the given cell decomposition on $Y_{\ge 0}$ with the face decomposition of $P^{\lambda}$; in particular, one cannot conclude that $Y_{\ge 0}$ is a regular CW-complex.
E. $Y_{\ge 0}$ is homeomorphic, as a cell-decomposed space, to the full weight polytope $\operatorname{Conv}(W\cdot \lambda)$; in particular, $Y_{\ge 0}$ is a regular CW-complex and is homeomorphic to that polytope.

cell-preserving homeomorphism to toric-orbit/face decomposition

as a cell-decomposed space / regular CW-complex conclusion

upgrade from mere homeomorphism to cell-decomposed homeomorphism via the map $\Psi_{\ge 0}$

target polytope is $P^{\lambda}=\operatorname{Conv}(W\cdot\lambda)\cap\sigma_+$, not the full weight polytope


\label{sec1}

Let $G$ be a compact Lie group, and $P$ a principal $G$-bundle over the $4$-dimensional sphere $S^4$. The $G$-gauge group over the base space $S^4$ is the topological group of $G$-bundle automorphisms of $P$. Suppose that $G$ is simply-connected and simple. Its classifying space $BG$ is $3$-connected and $\pi_4(BG)\cong \mathbb{Z}$. Let us denote the homotopy class of a map $f$ by the same symbol $f$. Then, the principal $G$-bundle $P$ is classified by a map $k\colon S^4 \to BG$ in $\pi_4(BG)\cong \mathbb{Z}$ and there are infinitely many isomorphism classes of principal $G$-bundles over $S^4$. In \cite{kono-1991}, Kono classified the homotopy types of $SU(2)$-gauge groups over $S^4$ and showed that there are six homotopy types of these groups.

Since then, many classification results on the homotopy types of gauge groups of low-rank simple Lie groups have been obtained. Among compact Lie groups, $SU(2)$ is the rank $1$ simply-connected simple Lie group. There are three simply-connected simple compact Lie groups of rank $2$: $SU(3)$, $Sp(2)$, and $G_2$. For $G=SU(3)$, Hamanaka and Kono \cite{hamanaka-kono-2006} proved that $\mathcal{G}_k\simeq \mathcal{G}_{k'}$ if and only if $(k, 24)=(k', 24)$, where $(m,n)$ is the greatest common divisor of $m$ and $n$. For $G=Sp(2)$, Theriault \cite{theriault-2010} showed $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if and only if $(k, 40)=(k', 40)$, where $\simeq_{(p)}$ means $p$-local homotopy equivalence. Then, for $G=G_2$, Kishimoto, Theriault and Tsutaya \cite{ktt-2017} showed that $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 168)=(k', 168)$ and if $\mathcal{G}_k \simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$, then $(k,84)=(k', 84)$. The discrepancy between $(k, 168)=(k',168)$ and $(k, 84)=(k', 84)$ has remained as an open problem.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

In this paper, we complete the classification for rank $2$ simple Lie groups up to $p$-local homotopy equivalence at any prime $p$ by proving the following.

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$ if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$ if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$ if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k \to G \to {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps, and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

\begin{proposition}\label{proposition:2.1}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{8}(S^{6})=\mathbb{Z}/2\{ \eta_{6}^{2}\}$, $\pi_{11}(S^{6})=\mathbb{Z}\{ \Delta(\iota_{13})\}$, $\pi_{14}(S^{6})_{{(2)}}=\mathbb{Z}/8\{\bar{\nu}_6 \} \oplus \mathbb{Z}/2\{ \varepsilon_6 \}$,
\item[{\rm (2)}] $\pi_{14}(S^{8})=\mathbb{Z}/2\{ \nu_{8}^{2} \}$, $\pi_{14}(S^{9})=\{0\}$, $\pi_{14}(S^{10})=\{0\}$.
\item[{\rm (3)}] $\eta_{n}\nu_{n+1}\simeq 0$ for $n\geq 5$.
\end{itemize}
\end{proposition}

\begin{proposition}[Mimura] \label{proposition:2.2}
Let $i\colon SU(3)\to G_2$ be the inclusion map and
$p\colon G_2\to S^6=G_2/SU(3)$ the projection map.
By $\langle \alpha \rangle$, we denote an element in $\pi_i(G_2)$ such that $p_*(\langle \alpha\rangle)\simeq \alpha$.
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(G_{2})=\mathbb{Z}\{ \langle 2 \Delta(\iota_{13})\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\}$.
\item[{\rm (2)}] $\pi_{14}(G_{2})_{(2)}=\mathbb{Z}/8\{ \langle \bar{\nu}_{6}+\varepsilon_{6}\rangle\} \oplus \mathbb{Z}/2\{ i_{*}([\nu_{5}^{2}])\circ \nu_{11}\}$
\end{itemize}
\end{proposition}

\begin{theorem}[{\oshima} \cite{oshima-2005}*{Theorem 2.1}] \label{theorem:3.1}
There is a map $\gamma \colon S^{11}\to G_2$ such that
\begin{itemize}
\item[{\rm (1)}]
$p\circ \gamma\simeq 2\Delta(\iota_{13})$,
\item[{\rm (2)}] $\langle i_3, \gamma\rangle\in \pi_{14}(G_2)$ has order $21=3\cdot 7$.
\end{itemize}
\end{theorem}

\begin{proposition}\label{proposition:3.2}
Let $i_{8}\colon S^8\to P^9(2)$ be the inclusion map of the bottom cell. Then, the following hold.
\begin{itemize}
\item[{\rm (1)}] ${i_8}_{*}\colon \pi_{14}(S^8)\to \pi_{14}(P^9(2))$ is an isomorphism.
\item[{\rm (2)}] For maps $f\colon S^{14}\to S^8$, and $g\colon S^{8}\to S^{6}$,
$g \circ f \simeq 0$ in $\pi_{14}(S^6)$.
\item[{\rm (3)}] If a map $f\colon S^{14}\to S^{6}$ factors through $P^9(2)$, then $f\simeq 0$ in $\pi_{14}(S^{6})$.
\end{itemize}
\end{proposition}

\begin{proposition}\label{proposition:3.3}
The following holds.
\begin{itemize}
\item[{\rm (1)}] $\pi_{11}(C)\simeq \mathbb{Z}\{ \bar{i}_{6}\circ \Delta(\iota_{13})\} \oplus \mathbb{Z}\{ \bar{s}\}$,
\item[{\rm (2)}] $\mathop{\mathrm{Ker}} \bar{p}_{6\; *} \colon \pi_{11}(C)\to \pi_{11}(S^6)$ is generated by $w\simeq a \bar{i}_{6}\circ \Delta(\iota_{13})+b \bar{s}$ where $a\in \mathbb{Z}$ and $b=1$ or $a$ is odd and $b=2$.
\item[{\rm (3)}] $j\circ q\circ \gamma\simeq cw$ where $c\in \{ \pm 1, \pm 2\}$.
\item[{\rm (4)}] $p_{11}\circ q\circ \gamma \simeq bc (\iota_{11})$ where $bc \in \{ \pm 1, \pm 2, \pm 4\}$.
\end{itemize}
\end{proposition}

\begin{theorem}\label{theorem:1.1}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$. The following holds.
\begin{itemize}
\item[{\rm (1)}]
If $\mathcal{G}_k\simeq \mathcal{G}_{k'}$, then $(k, 84)=(k', 84)$ and
\item[{\rm (2)}]  $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}$ at any prime $p$ if $(k, 84)=(k', 84)$.
\end{itemize}
\end{theorem}

Theorem~\ref{theorem:1.1} (2) is equivalent to the following local form. \begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

Theorem~\ref{theorem:1.1} (1) was established in \cite{ktt-2017}*{Theorem 1.2 (a)} and Proposition~\ref{proposition:1.2} (2), (3), (4) were established in \cite{ktt-2017}*{Propositions 4.6 and 3.2}. In this paper, we prove Proposition~\ref{proposition:1.2} (1) and complete the proof of Theorem~\ref{theorem:1.1} assuming the results of \cite{ktt-2017}.

First, we recall the basic method for determining the homotopy types of $G$-gauge groups over $S^4$, which relates their homotopy types to Samelson products. We begin with the following homotopy fiber sequence of mapping spaces.
\[
\Omega {\mathrm{Map}} (S^4, BG)_k  \to G \to  {\mathrm{Map}}_* (S^4, BG)_k \to {\mathrm{Map}} (S^4, BG)_k \stackrel{\mathrm{ev}}{\longrightarrow} BG,
\]
where $\mathrm{Map} (S^4, BG)_k$ is the connected component of the space of continuous maps containing the map $k\colon S^4\to BG$, $\mathrm{Map}_* (S^4, BG)_k$ is its subspace consisting of base point preserving maps,  and $\mathrm{ev}$ is the evaluation map.
There is a homotopy equivalence  $\Omega_0^3 G\simeq \mathrm{Map}_{*}(S^4, BG)_0 \simeq \mathrm{Map}_{*}(S^4, BG)_k$ and Gottlieb \cite{gottlieb-1972} showed that the classifying space of the gauge group $B\mathcal{G}_k$ is homotopy equivalent to the mapping space $ \mathop{\mathrm{Map}} (S^4, BG)_k$. Therefore, we have the following fiber sequence.
\[
\mathcal{G}_k  \to G \stackrel{\partial_k}{\longrightarrow} \Omega_0^3 G \to B\mathcal{G}_k \to BG.
\]
Thus, the $G$-gauge group $\mathcal{G}_k$ is homotopy equivalent to the homotopy fiber of the map $\partial_k$. Let $i_3\colon S^3 \to G$ be the inclusion map of the bottom cell. Then, Lang \cite{lang-1973} proved that the map $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3, 1\rangle$ where $1$ is the identity map of $G$. By the linearity of the Samelson product, we have $\langle k\cdot i_3, 1\rangle\simeq k \cdot \langle i_3, 1\rangle$. Furthermore, in \cite{theriault-2010}, Theriault showed that $\mathcal{G}_k$ is $p$-locally homotopy equivalent to $\mathcal{G}_{k'}$ at all prime $p$ if $(k, m)=(k',m)$ where $m$ is the order of the Samelson product $\langle i_3, 1\rangle$.

For an abelian group $A$, we denote by $A_{(p)}$ its localization at the prime $p$. If $A$ is finite, then $A_{(p)}$ is the $p$-primary subgroup of $A$. By \cite{ktt-2017}*{Corollary 6.6}, the order of $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at least $4$. Therefore, to show Proposition~\ref{proposition:1.2} (1), it suffices to prove the following proposition.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

As for the order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$, together with \cite{ktt-2017}*{Lemma 3.1, Proposition 4.5 and Corollary 6.6}, by Proposition~\ref{proposition:1.3}, we have the following theorem.

\begin{theorem}\label{theorem:1.4}
The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]$ is $84$.
\end{theorem}

Next, we recall some properties of a space $V:=G_2/SU(2)$, which played an important role in \cite{ktt-2017} and is crucial in this paper. Let $q\colon G_2\to V$ be the obvious projection map. By \cite{ktt-2017}*{Lemmas 7.1 and 7.2}, the Samelson product $3 \cdot \langle i_3, 1\rangle\colon \Sigma^3 G_2\to G_2$ factors through $\Sigma^3 q \colon \Sigma^3 G_2 \to \Sigma^3 V$.
Let us write this factorization as follows.
 \[
3\cdot \langle i_3, 1\rangle\simeq \psi\circ \Sigma^3 q.
\]
Let $P^{n+1}(2)$ be the mapping cone of the degree $2$ map $\times 2 \colon S^n\to S^n$.
The $6$-skeleton of $V=S^5\cup e^6\cup e^{11}$ is $P^6(2)$ and, by \cite{ktt-2017}{Lemma 7.4}, there is a homotopy equivalence $\Sigma^3 V\simeq P^9(2) \vee S^{14}$.  Let \[
p_{11}\colon V \to S^{11}
\]
 be the pinch map to the top cell, collapsing the $6$-skeleton to the base point. We denote by
 \[
 i_6\colon P^6(2) \to V
 \]
  the inclusion map. Then, the above homotopy equivalence provides maps $s'\colon \Sigma^3 V \to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ such that the identity map of $\Sigma^3 V$ is homotopic to
  \[
  \Sigma^3 i_6 \circ s' + s'' \circ \Sigma^3 p_{11}.
  \] The following proposition is what we prove in the rest of this paper.

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

\begin{proof}[Proof of Proposition~\ref{proposition:1.3}] We consider the following decomposition in $[\Sigma^3 G_2, G_2]$. \[ 3 \cdot \langle i_3, 1\rangle\simeq \psi \circ \Sigma^3 i_6\circ s' \circ \Sigma^3 q +\psi \circ s'' \circ \Sigma^3 (p_{11}\circ q). \]
By \cite{ktt-2017}*{Lemma 5.2 (a) and (b)}, the order of the identity map of $P^9(2)$ is $4$. Hence, we have
$4 (\Sigma^3 i_6\circ s' )\simeq 0$. Therefore, by Proposition~\ref{proposition:1.5}, the order of $ \langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.  \end{proof}

In what follows, we use the symbol $p$ to express the map $p\colon G_2\to S^6\simeq G_2/SU(3)$ only. When we need to indicate the generator of a cyclic group, we write $\mathbb{Z}/m\{ a\}$ for the cyclic group of order $m$ generated by $a$.

This paper is organized as follows. In Section \ref{sec2}, we collect some facts on homotopy groups of $G_2$ and $S^6$ and reduce the problem on the $2$-primary homotopy group $\pi_{14}(G_2)_{(2)}$ to that of $\pi_{14}(S^6)_{(2)}$. In Section \ref{sec3}, we prove Proposition~\ref{proposition:1.5}.

To complete Theorem~\ref{theorem:1.1} (2) it suffices to prove the local statement Proposition~\ref{proposition:1.2}. The paper reduces this to proving Proposition~\ref{proposition:1.2} (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so “to show Proposition~\ref{proposition:1.2} (1), it suffices” to prove Proposition~\ref{proposition:1.3}: the order is at most $4$.

The proof sketch for Proposition~\ref{proposition:1.3} is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by Proposition~\ref{proposition:1.5}, namely $4(\psi\circ s'')\simeq 0$ in $\pi_{14}(G_2)_{(2)}$. Combining these shows “the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$,” hence Proposition~\ref{proposition:1.2}(1), and therefore Theorem~\ref{theorem:1.1}(2) via the Samelson-product/Thériault criterion.

(They additionally note that with Proposition~\ref{proposition:1.3} and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

To establish the main theorem (2) it suffices to prove the local statement

\begin{proposition}\label{proposition:1.2}
Let $\mathcal{G}_k$ be the gauge group of a principal $G_2$-bundle over $S^4$ whose classifying map is $k\in \mathbb{Z}\cong \pi_{4}(BG_2)$.
Then, we have
\begin{itemize}
\item[{\rm (1)}] $\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}$  if $(k, 4)=(k', 4)$,
\item[{\rm (2)}]
$\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}$  if $(k, 3)=(k', 3)$,
\item[{\rm (3)}] $\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}$  if $(k, 7)=(k', 7)$
and
\item[{\rm (4)}] $\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'} \simeq _{(p)} G_2 \times \Omega_0^4 G_2$ for $p\not=2,3,7$.
\end{itemize}
\end{proposition}

The paper reduces this to proving the preceding proposition (1) (the $2$-local case), since (2),(3),(4) are cited from \cite{ktt-2017}.

The method recalled is: using Gottlieb’s identification $B\mathcal{G}_k\simeq \mathrm{Map}(S^4,BG)_k$ and the resulting fiber sequence
\[
\mathcal{G}_k\to G\xrightarrow{\partial_k} \Omega_0^3 G\to B\mathcal{G}_k\to BG,
\]
so $\mathcal{G}_k$ is the homotopy fiber of $\partial_k$. By Lang, $\partial_k$ is the triple adjoint of the Samelson product $\langle k\cdot i_3,1\rangle$, hence by linearity $\langle k\cdot i_3,1\rangle\simeq k\cdot\langle i_3,1\rangle$. Theriault’s result is then invoked: $\mathcal{G}_k\simeq_{(p)}\mathcal{G}_{k'}$ for all primes $p$ if $(k,m)=(k',m)$ where $m$ is the order of $\langle i_3,1\rangle$.

For $G=G_2$ at $p=2$, \cite{ktt-2017} gives that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at least $4$, so to show the preceding proposition (1), it suffices to prove the following.

\begin{proposition}\label{proposition:1.3}  The order of the Samelson product $\langle i_3, 1\rangle$ in $[\Sigma^3 G_2, G_2]_{(2)}$ is at most $4$.
\end{proposition}

The proof sketch for the preceding proposition is given: using that $3\cdot\langle i_3,1\rangle$ factors through $\Sigma^3q\colon \Sigma^3G_2\to \Sigma^3V$ (with $V=G_2/SU(2)$), write
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3q.
\]
With the homotopy decomposition $\Sigma^3V\simeq P^9(2)\vee S^{14}$, choose maps $s'\colon \Sigma^3V\to P^9(2)$ and $s''\colon S^{14}\to \Sigma^3V$ so that
\[
\mathrm{id}_{\Sigma^3V}\simeq \Sigma^3i_6\circ s'\; +\; s''\circ \Sigma^3p_{11}.
\]
Then
\[
3\cdot\langle i_3,1\rangle\simeq \psi\circ \Sigma^3i_6\circ s'\circ \Sigma^3q\; +\; \psi\circ s''\circ \Sigma^3(p_{11}\circ q).
\]
Using \cite{ktt-2017} (Lemma 5.2(a),(b)) that “the order of the identity map of $P^9(2)$ is $4$,” one gets $4(\Sigma^3i_6\circ s')\simeq 0$. The remaining summand is controlled by

\begin{proposition} \label{proposition:1.5}
We have  $4(\psi\circ s'')\simeq 0$  in $\pi_{14}(G_2)_{(2)}$. \end{proposition}

Combining these shows that the order of $\langle i_3,1\rangle$ in $[\Sigma^3G_2,G_2]_{(2)}$ is at most $4$, hence the preceding proposition (1), and therefore the main theorem (2) via the Samelson-product/Thériault criterion.

(They additionally note that with the proposition above and results of \cite{ktt-2017}, the global order is computed as $84$ (Theorem~\ref{theorem:1.4}).)

Question: Let \(\mathcal{G}_k\) denote the gauge group of a principal \(G_2\)-bundle over \(S^4\) whose classifying map represents \(k\in \pi_4(BG_2)\cong \mathbb Z\). Write \((m,n)\) for the greatest common divisor of integers \(m\) and \(n\), write \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\) for homotopy equivalence, and write \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for \(p\)-local homotopy equivalence. Which statement holds for integers \(k\) and \(k'\)?

If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).

Choices:
B. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,168)=(k',168)\). Moreover, if \((k,168)=(k',168)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).
C. If \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq_{(2)} \mathcal{G}_{k'}\), \(\mathcal{G}_k\simeq_{(3)} \mathcal{G}_{k'}\), and \(\mathcal{G}_k\simeq_{(7)} \mathcal{G}_{k'}\).
D. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,84)=(k',84)\), then \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\).
E. If \(\mathcal{G}_k\simeq \mathcal{G}_{k'}\), then \((k,84)=(k',84)\). Moreover, if \((k,4)=(k',4)\), then \(\mathcal{G}_k\simeq_{(p)} \mathcal{G}_{k'}\) for every prime \(p\).


\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) associated with a kernel satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
	Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_1(\mN)}.
	\]
We refer to see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $ \left\| \cdot \right\| _{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $ \left\| \cdot \right\| _{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.

\begin{align}\label{Hormander condition}
	\sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| \dif x < \infty,
    \end{align}

\begin{align} \label{L2-integral regularity condition}
	\sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d  \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 \dif x \Big)^{\frac{1}{2}}
	<\infty,
    \end{align}

\begin{theorem}\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) associated with a kernel satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
	Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_1(\mN)}.
	\]
We refer to see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for the definitions of the noncommutative maximal norm $\norm{\cdot}_{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $\norm{\cdot}_{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.
    \end{theorem}

Calder\'on-Zygmund operators, a well-known class of singular integral operators, were introduced by Calder\'on and Zygmund \cite{Calderon1952On} in their seminal 1952 work.
The kernel of such an operator satisfies specific conditions, including the size and regularity conditions, the latter of which was refined by H\"ormander \cite{Hormander1960Estimates} in 1960.

The standard nonconvolution type Calder\'on-Zygmund operator is defined as
    \[
    Tf(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \operatorname{supp} f,
    \]
where $K:\R^d \times \R^d \setminus  \left\{ (x,x): x\in\R^d \right\}  \rightarrow \C$
is a standard nonconvolution type Calder\'on-Zygmund kernel, which is a binary function satisfying
    \begin{itemize}
	\item \textbf{Size condition:} there exists $C_1>0$ such that
	   \begin{align} \label{Size condition}
       |K(x,y)| \leq \frac{C_1}{|x-y|^d} ;
       \end{align}
	\item  \textbf{Lipschitz regularity condition:} there exists  $\gamma\in [0,1]$ and $C_2>0$ such that
	   \begin{align} \label{Lipschitz regularity condition}
	   |K(x,y)-K(x,z)|+|K(y,x)-K(z,x)| \leq \frac{C_2 |y-z|^{\gamma}}{|x-y|^{d+\gamma}}, \quad\text{if}\  |x-y|\geq 2|y-z|,
	   \end{align}
    \end{itemize}
where $x$, $y$, $z\in\R^d$.
The Lipschitz regularity condition (\ref{Lipschitz regularity condition}) can be relaxed to certain variants of smoothness conditions (see \cite{Grafakos2019Alimited, Suzuki2021TheCalderon, Watson1990Weighted}).
In the following, all Calder\'on-Zygmund kernels under consideration are of nonconvolution type.

Inspired by operator algebras, harmonic analysis, noncommutative geometry, and quantum probability (see \cite{Cadilhac2018Weak, Chen2013Harmonic, Junge2014Smooth, Junge2018Noncommutative, Mei2007Operator, Mei2009Pseudo-localization, Mei2017Free, Mei2022Mikhlin, Parcet2009Pseudo-localization}), noncommutative harmonic analysis has become an exciting field in recent years.
However, generalizing the Calder\'on-Zygmund decomposition, a key technique in harmonic analysis first introduced in \cite{Calderon1952On}, to the noncommutative setting is challenging.

In 2009, Parcet \cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \cite{Cuculescu1971Martingales}.
With this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\ref{Lipschitz regularity condition})
(Cadilhac later gave a shorter proof in \cite{Cadilhac2018Weak}).
Notably,  the H\"ormander condition
    \begin{align}\label{Hormander condition}
	\sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| {\rm d} x < \infty,
    \end{align}
 which is weaker than (\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\'on-Zygmund theory.
It is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\"ormander condition (\ref{Hormander condition})?
This question remains open.

A step forward was made in 2022 by Cadilhac et al. \cite{Cadilhac2022Spectral}.
They developed a more effective noncommutative Calder\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\'on-Zygmund operators if the kernel $K$ satisfies
    \begin{align} \label{L2-integral regularity condition}
	\sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d  \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\rm d} x \Big)^{\frac{1}{2}}
	<\infty,
    \end{align}
 where $\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.
 More generally, for $1\leq q\leq\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if
    \begin{align} \label{sum delta < infty}
	\sum_{m\geq 1} \delta_q(m) <\infty,
    \end{align}
where $\delta_q(m)$ is defined as
     \begin{align*}
	 \sup_{{\begin{subarray}{c} y\in \R^d,\ R>0 \\ |v|<R \end{subarray}}} & \Big( (2^m R)^{d(q-1)} \int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x\Big)^{\frac{1}{q}},
    \end{align*}
for $m\geq 1$.
Roughly speaking, the H\"ormander condition (\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\ref{sum delta < infty}) with $q=1$, while \eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\ref{sum delta < infty}) with $q=2$.
The fact $\delta_1(m)\lesssim_d \delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\"ormander condition (\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\ref{Lipschitz regularity condition}).
By applying the noncommutative Calder\'on-Zygmund decomposition in  \cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\'on-Zygmund operators under the $L_2$-integral regularity condition.
We refer the reader to \cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\'on-Zygmund operators.

In this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}): the $L_q$-mean H\"ormander condition ($1\leq q\leq\infty$), defined by
    \begin{align} \label{Hq < infty}
	\sum_{m\geq 1} H_q (m) < \infty,
    \end{align}
where $H_q(m)$ is defined as
    \begin{align*}	 \sup_{{\begin{subarray}{} y\in \R^d, \\ R>0 \end{subarray}}} & \Big(\frac{(2^m R)^{d(q-1)}}{|B_R|} \int_{|v|\leq R}\int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x {\rm d} v\Big)^{\frac{1}{q}},     \end{align*}
for $m\geq 1$.
Here, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.
It is not hard to see that $H_q(m) \lesssim_d \delta_q(m)$, meaning the $L_q$-mean H\"ormander condition (\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}).
In what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\"ormander condition:
    \begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
where $H_2(m)$ is defined as in (\ref{Hq < infty}) with $q=2$.

For two smoothness conditions $A$ and $B$, we write $A\prec B$ to mean that $A$ is weaker than $B$.
Thus, we have the following conclusion: \newline
    \begin{center}
    \scalebox{0.94}{
    $\begin{array}{c}
        \begin{array}{c}
            \textbf{\large $L_2$-mean H\"ormander} \\
            \textbf{\large condition} \\
        \end{array}
        $\scalebox{1.3}{$\prec$}$
        \begin{array}{c}
            \textbf{\large $L_2$-integral regularity} \\
            \textbf{\large condition} \\
            \end{array}
        $\scalebox{1.3}{$\prec$}$
        \begin{array}{c}
            \textbf{\large Lipschitz regularity} \\
            \textbf{\large condition}\\
        \end{array}
    \end{array}$
    } \newline
    \end{center}
This naturally leads to a question: can the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}) still guarantee the noncommutative weak type $(1,1)$ boundedness of the maximal Calder\'on-Zygmund operators?
This paper gives a positive answer to this question.

Before stating the main result, we introduce some notions.
Let $\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\tau$.
Let $\mN = L_{\infty}(\R^d)\bar\otimes \mM$ be the tensor von Neumann algebra with the tensor trace  $\phi = \int\otimes \tau$.
Denote the noncommutative $L_p$ space associated with $(\mN, \phi)$ by $L_p(\mN)$ for $1\leq p\leq\infty$ (see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).
The noncommutative Calder\'on-Zygmund operator $T$ with a kernel $K$ is defined as
    \begin{align}\label{def of Tf}
	T f(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \overrightarrow{\operatorname{supp}}\ f,
    \end{align}
where $f\in L_1(\mN)\bigcap L_{\infty}(\mN)$ is compactly supported and measurable.
Here, $\overrightarrow{\text{supp}}\ f$ denotes the support of $f$ as an operator-valued function in $\R^d$ rather than the support projection as an element in a von Neumann algebra.
For any $\epsilon>0$, the associated truncated singular integral $T_{\epsilon}f$ is defined as
    \begin{align}\label{def of T epsilon f}
	T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) {\rm d} y.
    \end{align}

The main result
is stated as follows.

In 2009, Parcet \cite{Parcet2009Pseudo-localization} rigorously constructed a kind of noncommutative Calder\'on-Zygmund decomposition by using Cuculescu's maximal weak type $(1,1)$ estimates for noncommutative martingales \cite{Cuculescu1971Martingales}.
With this tool, Parcet established the weak type $(1,1)$ estimates for operator-valued Calder\'on-Zygmund singular integrals with the kernel satisfying the Lipschitz regularity condition (\ref{Lipschitz regularity condition})
(Cadilhac later gave a shorter proof in \cite{Cadilhac2018Weak}).
Notably, the H\"ormander condition
 \begin{align}\label{Hormander condition}
 \sup_{y\in\R^d,|v|>0} \int_{|x-y|\geq 2|v|} |K(x,y+v)-K(x,y)| {\rm d} x < \infty,
 \end{align}
 which is weaker than (\ref{Lipschitz regularity condition}), is already sufficient for weak type $(1,1)$ boundedness in classical Calder\'on-Zygmund theory.
It is therefore natural to ask whether the noncommutative weak type $(1,1)$ boundedness still holds under the H\"ormander condition (\ref{Hormander condition})?
This question remains open.

A step forward was made in 2022 by Cadilhac et al. \cite{Cadilhac2022Spectral}.
They developed a more effective noncommutative Calder\'on-Zygmund decomposition without the off-diagonal term of the good function, which allowed them to prove the noncommutative weak type $(1,1)$ estimates for Calder\'on-Zygmund operators if the kernel $K$ satisfies
 \begin{align} \label{L2-integral regularity condition}
 \sum_{m\geq 1} \sup_{{\begin{subarray}{c} y\in Q \\ Q\ \text{dyadic} \end{subarray}}} \Big( 2^{md}\ell(Q)^d \int_{2^m\ell(Q)\leq|x-c(Q)|\leq2^{m+1}\ell(Q)} |K(x,y)-K(x,c(Q))|^2 {\rm d} x \Big)^{\frac{1}{2}}
 <\infty,
 \end{align}
 where $\ell(Q)$ and $c(Q)$ stand for the length and the center of the dyadic cube $Q$, respectively.
 More generally, for $1\leq q\leq\infty$, we say that a kernel $K$ satisfies the $L_q$-integral regularity condition if
 \begin{align} \label{sum delta < infty}
 \sum_{m\geq 1} \delta_q(m) <\infty,
 \end{align}
where $\delta_q(m)$ is defined as
 \begin{align*}
 \sup_{{\begin{subarray}{c} y\in \R^d,\ R>0 \\ |v|<R \end{subarray}}} & \Big( (2^m R)^{d(q-1)} \int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x\Big)^{\frac{1}{q}},
 \end{align*}
for $m\geq 1$.
Roughly speaking, the H\"ormander condition (\ref{Hormander condition}) corresponds to the $L_1$-integral regularity condition i.e. (\ref{sum delta < infty}) with $q=1$, while \eqref{L2-integral regularity condition} corresponds to the $L_2$-integral regularity condition, i.e., (\ref{sum delta < infty}) with $q=2$.
The fact $\delta_1(m)\lesssim_d \delta_2(m)$ shows that the $L_2$-integral regularity condition is stronger than the H\"ormander condition (\ref{Hormander condition}), but weaker than the Lipschitz regularity condition (\ref{Lipschitz regularity condition}).
By applying the noncommutative Calder\'on-Zygmund decomposition in \cite{Cadilhac2022Spectral}, Hong, the first author, and Xu \cite{hong2023maximal} established the weak type (1,1) boundedness for the noncommutative maximal truncated Calder\'on-Zygmund operators under the $L_2$-integral regularity condition.
We refer the reader to \cite{Cadilhac2022Spectral, Lai2024Noncommutative, Mei2009Pseudo-localization} for more results on weak (1,1) boundedness theory for noncommutative Calder\'on-Zygmund operators.

In this paper, we study a kernel smoothness condition that is slightly weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}): the $L_q$-mean H\"ormander condition ($1\leq q\leq\infty$), defined by
 \begin{align} \label{Hq < infty}
 \sum_{m\geq 1} H_q (m) < \infty,
 \end{align}
where $H_q(m)$ is defined as
 \begin{align*} \sup_{{\begin{subarray}{} y\in \R^d, \\ R>0 \end{subarray}}} & \Big(\frac{(2^m R)^{d(q-1)}}{|B_R|} \int_{|v|\leq R}\int_{2^mR\leq |x-y|\leq 2^{m+1}R} |K(x,y+v)-K(x,y)|^q {\rm d} x {\rm d} v\Big)^{\frac{1}{q}}, \end{align*}
for $m\geq 1$.
Here, $B_R$ denotes the ball centered at the origin with radius $R>0$, and $|B_R|$ denotes its volume.
It is not hard to see that $H_q(m) \lesssim_d \delta_q(m)$, meaning the $L_q$-mean H\"ormander condition (\ref{Hq < infty}) is indeed weaker than the $L_q$-integral regularity condition (\ref{sum delta < infty}).
In what follows, we focus on the case $q=2$, i.e., the $L_2$-mean H\"ormander condition:
 \begin{align}\label{L2 mean Hormander condition}
 \sum_{m\geq 1} H_2 (m) < \infty,
 \end{align}
where $H_2(m)$ is defined as in (\ref{Hq < infty}) with $q=2$.

Before stating the main result, we introduce some notions.
Let $\mM$ be a von Neumann algebra equipped with a normal semifinite faithful ($n. s. f. $, for short) trace $\tau$.
Let $\mN = L_{\infty}(\R^d)\bar\otimes \mM$ be the tensor von Neumann algebra with the tensor trace $\phi = \int\otimes \tau$.
Denote the noncommutative $L_p$ space associated with $(\mN, \phi)$ by $L_p(\mN)$ for $1\leq p\leq\infty$ (see Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms} for details).
The noncommutative Calder\'on-Zygmund operator $T$ with a kernel $K$ is defined as
 \begin{align}\label{def of Tf}
 T f(x) = \int_{\R^d} K(x,y) f(y) {\rm d} y, \quad x\notin \overrightarrow{\operatorname{supp}}\ f,
 \end{align}
where $f\in L_1(\mN)\bigcap L_{\infty}(\mN)$ is compactly supported and measurable.
Here, $\overrightarrow{\text{supp}}\ f$ denotes the support of $f$ as an operator-valued function in $\R^d$ rather than the support projection as an element in a von Neumann algebra.
For any $\epsilon>0$, the associated truncated singular integral $T_{\epsilon}f$ is defined as
 \begin{align}\label{def of T epsilon f}
 T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) {\rm d} y.
 \end{align}

The main result
is stated as follows.

\textbf{Step 1.}
Let $T_{\epsilon}$ be a noncommutative Calder\'on-Zygmund truncated operator with a complex kernel $K$ (defined as in (\ref{def of T epsilon f})).
We decompose $K$ into its real and imaginary parts: $\Re(K)$ and $\Im(K)$.
The truncated operators associated with $\Re(K)$ and $\Im(K)$ are then given by
 \begin{align*}
 \Re(T_{\epsilon})f(x) = \int_{|x-y|>\epsilon} \Re(K)(x,y) f(y) \dif y
 \end{align*}
and
 \begin{align*}
 \Im(T_{\epsilon})f(x) = \int_{|x-y|>\epsilon} \Im(K)(x,y) f(y) \dif y.
 \end{align*}
\begin{lemma}\label{reduction to real kernel}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator (defined as in (\ref{def of Tf})) whose kernel $K$ satisfies (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $T_{\epsilon}$ be defined as in (\ref{def of T epsilon f}).
 Then,
 \begin{enumerate}[(i)]
 \item Both $\Re(K)$ and $\Im(K)$ satisfy (\ref{Size condition}) and (\ref{L2 mean Hormander condition}).
 \item If $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $p_0\in (1,\infty)$, then, both $(\Re(T_{\epsilon}))_{\epsilon>0}$ and $(\Im(T_{\epsilon}))_{\epsilon>0}$ are of strong type $(p_0,p_0)$.
 \item If $(\Re(T_{\epsilon}))_{\epsilon>0}$ and $(\Im(T_{\epsilon}))_{\epsilon>0}$ are of weak type $(1,1)$, so is $(T_{\epsilon})_{\epsilon>0}$.
 \end{enumerate}
\end{lemma}
By this lemma, we may assume that the kernel $K$ is real throughout this paper.

\begin{lemma}\label{reduction to lacunary sequence}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator (defined as in (\ref{def of Tf})) whose kernel $K$ satisfies (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $T_{\epsilon}^{i_{\epsilon}}$ be defined as in (\ref{T epsilon,i epsilon f(x)}).
 Then
 \begin{enumerate}[(i)]
 \item $(T_{\epsilon}^{i_{\epsilon}})_{\epsilon>0}$ is of strong type $(p,p)$ for $1<p\leq\infty$.
 \item $(T_{\epsilon}^{i_{\epsilon}})_{\epsilon>0}$ is of weak type $(1,1)$.
 \end{enumerate}
 \end{lemma}

The preceding two lemmas imply that Theorem \ref{Main theorem} follows directly from the weak type (1,1) boundedness of $S_i$ given in (\ref{Sif(x)}).
Thus, it remains to prove
\begin{theorem}\label{Second theorem}
 Let $T$ be a noncommutative Calder\'on-Zygmund operator defined as in (\ref{def of Tf}) with the real kernel $K$ satisfying (\ref{Size condition}) and the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
 Let $S_i$ be defined as in (\ref{Sif(x)}).
 Then, the sequence of operators $(S_i)_{i\in\Z}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
 \[
 \norm{(S_i f)_{i\in\Z}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim \norm{f}_{L_1(\mN)}.
 \]
 More precisely, for any $f\in L_1(\mN)$ and $\lambda>0$, there exists a projection $e\in\mN$ such that
 \[
 \sup_{i\in\Z}\norm{e S_if e}_{L_{\infty(\mN)}} \leq \lambda \quad\text{and}\quad \phi(e^{\bot})\lesssim \lambda^{-1} \norm{f}_{L_1(\mN)} .
 \]
\end{theorem}

\subsection{Estimate for the good function (\ref{estimate of good function}) }
For any $i\in\Z$, it follows from the definitions of $S_i$ and $T_{\epsilon}^{i_{\epsilon}}$ ((\ref{Sif(x)}) and (\ref{T epsilon,i epsilon f(x)})) that there exists $\epsilon_i > 0$ such that
 \[ S_i = S_{\epsilon_i}- T_{\epsilon_i}^i. \]
Under the assumptions of Theorem \ref{Main theorem} and Lemma \ref{reduction to lacunary sequence} $(i)$, both $(S_{\epsilon_i})_{i \in \mathbb{Z}}$ and $(T_{\epsilon_i}^i)_{i \in \mathbb{Z}}$ are of strong type $(p_0, p_0)$.
Therefore, from the above equality, $(S_i)_{i \in \mathbb{Z}}$ is also of strong type $(p_0, p_0)$.
In addition, since $g$ is positive (by Theorem \ref{nc CZ decomposition with convolution}) and the kernel $K$ is real-valued (as stated in Section \ref{Section: Two reductions}), the operator $S_i g$ is self-adjoint.
These two facts imply that for any $i\in\Z$, we can find a positive element $a\in\mN$ such that for any $i\in\Z$,
 \[
 -a \leq S_i g \leq a \quad\text{and}\quad \norm{a}_{L_{p_0}(\mN)}\lesssim \norm{g}_{L_{p_0}(\mN)}.
 \]
Taking $e_1 = \chi_{(0,\lambda]}(a)$. Then we have
 \[
 -\lambda \leq -e_1 a e_1 \leq e_1 S_i g e_1 \leq e_1 a e _1 \leq \lambda.
 \]
Finally, applying the Chebyshev inequality, the H\"older inequality, along with the boundedness of $g$ in Lemma \ref{properties of nc CZ decomposition with convolution} $(i)$, we obtain
 \begin{align*}
 \phi(e_1^{\bot})
 \leq \lambda^{-p_0} \norm{a}_{L_{p_0}(\mN)}^{p_0}
 \leq \lambda^{-p_0} \norm{g}_{L_{\infty}(\mN)}^{p_0-1} \norm{g}_{L_1(\mN)}
 \lesssim \lambda^{-1} \norm{f}_{L_1(\mN)},
 \end{align*}
which yields (\ref{estimate of good function}).

To estimate $F_1$.
We first observe that by the supports of $h_{r_{n-j}}$ and $\chi_Q$, the noncommutative $L_1$-norm in $F_1$ equals
 \begin{align*} \Big\|\int_Q \Big(\int_{|u-y|\leq r_{n-j}} (K_j(x,y)-K_j(x,c(Q))) h_{r_{n-j}}(y-u)\dif y \Big)p_Q f(u) q_Q\dif u \Big\|_{L_1(\mM)}.
 \end{align*}
By using Lemma \ref{a useful lemma for bad function} (in Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms}), the above estimate is bounded by a product
 \begin{align*}
 & \Big\| \Big(\int_{Q} \Big(\int_{B_{r_{n-j}}(u)} \abso{K_i(x,y)-K_i(x,c(Q))} |h_{r_{n-j}}(y-u)|\dif y \Big)^2 p_Q f(u)p_Q \dif u\Big)^{^{\frac{1}{2}}}\Big\|_{L_{1}(\mM)} \\
 & \cdot \Big\|\Big(\int_{Q} q_Q f(u) q_Q \dif u\Big)^{\frac{1}{2}} \Big\|_{L_{\infty}(\mM)}.
 \end{align*}
For the second term, an application of \eqref{q_Q f_Q q_Q < lambda q_Q} shows that
 \begin{align*} \Big\|\Big(\int_{Q} q_Q f(u) q_Q \dif u\Big)^{\frac{1}{2}} \Big\|_{L_{\infty}(\mM)} \lesssim (|Q|\lambda)^{\frac{1}{2}}.
 \end{align*}
Hence,
 \begin{align}
 F_1
 \lesssim &\sum_{n\geq 2} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}} (|Q|\lambda)^{\frac{1}{2}} \notag\\
 & \label{xxxyyy}\frac{1}{|B_{r_{n-j}}|}\int_{\R^d} \Big\|\int_{Q} \Big(\int_{B_{r_{n-j}}(u)} \abso{K_i(x,y)-K_i(x,c(Q))} \dif y \Big)^2 p_Q f(u)p_Q \dif u\Big\|_{L_{\frac{1}{2}}(\mM)}^{\frac{1}{2}} \dif x.
 \end{align}
The same argument as in (\ref{B(c(Q))}) allows us to majorize (\ref{xxxyyy}) by
 \begin{align*}
 \frac{1}{|B_{r_{n-j-1}}|}\int_{\R^d} \Big\|\int_{Q} \Big(\int_{B_{r_{n-j-1}}(c(Q))} \abso{K_j(x,y)-K_j(x,c(Q))} \dif y \Big)^2 p_Q f(u)p_Q \dif u\Big\|_{L_{\frac{1}{2}}(\mM)}^{\frac{1}{2}} \dif x.
 \end{align*}
By Lemma \ref{the second lemma for bad function} $(i)$ and the $L_2$-mean H\"ormander condition \eqref{L2 mean Hormander condition}, we obtain
 \begin{align*}
 F_1 &\lesssim \sum_{n\geq 2} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}}
 \lbracket{ \big(H_2^2(n-1) + H_2^2(n) + 2^{-2(n-1)}\big) \tau(p_Q) \phi(fp_Q \chi_{Q}) }^{\frac{1}{2}} (|Q| \lambda)^{\frac{1}{2}} \\
 &\lesssim \sum_{n\geq 2} \lbracket{2H_2(n-1)+2^{-n+1}} \sum_{j\leq n-1} \sum_{Q\in\mQ_{n-j}}\big(\tau(p_Q) \phi(fp_Q \chi_{Q}) |Q| \lambda \big)^{\frac{1}{2}} \\
 &\lesssim \sum_{m\geq 1} \sum_{Q\in\mQ_m} \big( \tau(p_Q) \phi(fp_Q \chi_{Q}) |Q| \lambda \big)^{\frac{1}{2}}.
 \end{align*}
Finally, applying the Cauchy-Schwarz inequality twice, we estimate $F_1$ as follows:
 \begin{align*}
 F_1 \lesssim \Big(\sum_{m=1}^{\infty} \lambda \phi(p_m)\Big)^{\frac{1}{2}}
 \Big(\sum_{m=1}^{\infty} \phi(f p_m)\Big)^{\frac{1}{2}}
 = \big(\lambda \phi(1-q)\big)^{\frac{1}{2}} \big(\phi(f(1-q))\big)^{\frac{1}{2}}
 &\lesssim \norm{f}_{L_1(\mN)}.
 \end{align*}

Our main strategy in this paper is to develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).
The previous methods are insufficient in this context.
On one hand, Parcet's decomposition \cite{Parcet2009Pseudo-localization} requires the stronger Lipschitz regularity condition (\ref{Lipschitz regularity condition}) and may not be directly applied to the noncommutative maximal operator.
On the other hand, while the decomposition by Cadilhac et al. \cite{Cadilhac2022Spectral} can deal with the $L_2$-integral regularity condition (\ref{sum delta < infty}) with $q=2$, it does not cover our target.
To address this gap, we introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.

This paper is organized as follows.
We begin by presenting our key tool: the refined noncommutative Calder\'on-Zygmund decomposition (Theorem \ref{nc CZ decomposition with convolution}) in Section \ref{Section: Noncommutative C-Z decomposition}.
Next, Section \ref{Section: Two reductions} starts the proof of the main result (Theorem \ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem \ref{Second theorem}).
With this reduction established, Section \ref{Proof of the main theorem} is devoted to proving Theorem \ref{Second theorem}.
Finally, some related background concepts are included in Appendix \ref{Appendix: Noncommutative Lp spaces and maximal norms}.

Now, we conclude this section by listing the notation required for our later analysis.\\

\textbf{Notation}
\begin{itemize}
	\item $\mQ_n$: the set of all dyadic cubes of side length $2^{-n}$, $n\in\Z$.
	\item $Q_{x,n}$: the unique cube in $\mQ_n$ containing $x\in\R^d$.
	\item $\mQ$: the set of all standard dyadic cubes in $\R^d$, that is, $\mQ = \bigcup\limits_n\mQ_n$.
	\item $c(Q)$: the center of the cube $Q$.
	\item $lQ$: the concentric cube sharing the center of  $Q$ such that its length is $l$ times the length of $Q$.
	\item $B_r(x)$: the ball with $x\in\R^d$ as its center and $r>0$ as its radius, and the center $x$ will be omitted when it is the origin.
	\item $|E|$: the volume of set $E\subset\R^d$.
	\item $[k]$: the integer part of $k$.

\end{itemize}

To prove Theorem~\ref{Main theorem}, the paper’s strategy is to “develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition (\ref{L2 mean Hormander condition}).” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”

The proof of Theorem~\ref{Main theorem} then proceeds via the following stated structure: (1) present the refined decomposition (Theorem~\ref{nc CZ decomposition with convolution}); (2) in Section~\ref{Section: Two reductions}, “start the proof of the main result (Theorem~\ref{Main theorem}), reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\ref{Second theorem} in Section~\ref{Proof of the main theorem}.

To prove the main theorem, the paper’s strategy is to “develop a refined noncommutative Calder\'on-Zygmund decomposition, adapted to kernels satisfying the $L_2$-mean H\"ormander condition
\begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
.” Since “previous methods are insufficient,” the authors “introduce a slightly new noncommutative Calder\'on-Zygmund decomposition achieved by further splitting of the bad function into a convolution part and a remainder part.”

In establishing the main theorem, the proof then proceeds via the following stated structure: (1) present the refined decomposition. We first prove the following theorem.

\begin{theorem} \label{nc CZ decomposition with convolution}
	Fix $f\in\mN_{c,+}$, $\lambda>0$, and $s\in\N$.
	Let $(q_n)_{n\in\Z}$ and $(p_n)_{n\in\Z}$ be the sequences of projections in Lemma \ref{cuculescu's construction}, where $q_n = \sum\limits_{Q\in\mQ_n} q_Q \chi_Q$, $p_n = \sum\limits_{Q\in\mQ_n} p_Q \chi_Q$.
	Suppose $h\in C_{c}^{\infty} (\R^d)$ satisfies
    \begin{align*}
    \supp h \subset B_1,\quad \int_{\R^d} h =1, \quad h\geq 0.
    \end{align*}
    Define $h_r(x)= \frac{1}{r^d} h(\frac{x}{r})$ and $r_n= 2^{-n-1}\sqrt{d}$.
	Then, $f$ admits the decomposition:
	\begin{align*}
	f = g+ b
	  = g + \widetilde{b}_{\text{d}} + b_{\text{d}}^0 + \widetilde{b}_{\text{off}} + b_{\text{off}}^0,
	\end{align*}
	where $g$ is positive and $b$ is self-adjoint in $\mN$.
	More precisely, these components are defined as follows:
	    \[g=qfq + \sum\limits_{n\in\Z}p_n f_n p_n,\]
        \begin{gather*}
        \widetilde{b}_{\text{d}} = \sum\limits_{n\in\Z} \widetilde{b}_{\text{d},n} = \sum\limits_{n\in\Z}\sum\limits_{Q\in\mQ_n}\widetilde{b}_{\text{d},n}^Q ,\quad
        b_{\text{d}}^0 = \sum\limits_{n\in\Z} b_{\text{d},n}^0=\sum\limits_{n\in\Z} \sum\limits_{Q\in\mQ_n}(b_{\text{d},n}^Q-\widetilde{b}_{\text{d},n}^Q),\\
        \widetilde{b}_{\text{off}} = \sum\limits_{n\in\Z} \widetilde{b}_{\text{off},n} = \sum\limits_{n\in\Z} \sum\limits_{Q\in\mQ_n} \widetilde{b}_{\text{off},n}^Q ,\quad
        b_{\text{off}}^0 = \sum\limits_{n\in\Z} b_{\text{off},n}^0 = \sum\limits_{n\in\Z}\sum\limits_{Q\in\mQ_n}(b_{\text{off},n}^Q-\widetilde{b}_{\text{off},n}^Q),
        \end{gather*}
     where for each $Q\in\mQ_n$,
        \begin{gather*}
     	b_{\text{d},n}^Q = p_Q(f-f_n)p_Q\chi_Q,
     	\quad b_{\text{off},n}^Q = p_Q(f-f_n)q_Q\chi_Q + q_Q(f-f_n)p_Q\chi_Q, \\
     	\widetilde{b}_{\text{d},n}^Q = b_{\text{d},n}^Q * h_{r_n},
     	\quad \widetilde{b}_{\text{off},n}^Q = b_{\text{off},n}^Q * h_{r_n}.
        \end{gather*}
    \end{theorem}

(2) next, “start the proof of the main result, reducing it to proving the weak type (1,1) estimates for the lacunary sequences with real kernels (Theorem~\ref{Second theorem});” and (3) “with this reduction established,” prove Theorem~\ref{Second theorem} later. \label{Proof of the main theorem}
Note that every operator in a von Neumann algebra can be decomposed into a linear combination of four positive elements.
By this fact and the density of $\mN_{c,+}$ i

\label{Main theorem}
	Suppose $T$ is a Calder\'on-Zygmund operator defined as in
\begin{align}\label{def of Tf}
	T f(x) = \int_{\R^d} K(x,y) f(y) \dif y, \quad x\notin \overrightarrow{\supp}\ f,
    \end{align}
associated with a kernel satisfying
\begin{align} \label{Size condition}
       |K(x,y)| \leq \frac{C_1}{|x-y|^d} ;
       \end{align}
and the $L_2$-mean H\"ormander condition
\begin{align}\label{L2 mean Hormander condition}
	\sum_{m\geq 1} H_2 (m) < \infty,
    \end{align}
	Let $T_{\epsilon}$ be defined as in
\begin{align}\label{def of T epsilon f}
	T_{\epsilon}f(x) = \int_{|x-y|>\epsilon} K(x,y) f(y) \dif y.
    \end{align}
	Suppose $(T_{\epsilon})_{\epsilon>0}$ is of strong type $(p_0,p_0)$ for some $1< p_0 <\infty$, that is, for any $f\in L_{p_0}(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{L_{p_0}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_{p_0}(\mN)}.
	\]
	Then, $(T_{\epsilon})_{\epsilon>0}$ is of weak type $(1,1)$, that is, for any $f\in L_1(\mN)$,
	\[
	\norm{(T_{\epsilon}f)_{\epsilon>0}}_{\Lambda_{1,\infty}(\mN;\ell_{\infty})} \lesssim  \left\| f \right\| _{L_1(\mN)}.
	\]
We refer the reader to
\label{Appendix: Noncommutative Lp spaces and maximal norms}
This part consists of three subsections: noncommutative $L_p$ spaces, noncommutative maximal norms, and noncommutative square functions.
Al
for the definitions of the noncommutative maximal norm $ \left\| \cdot \right\| _{L_p(\mN; \ell_{\infty})}$ and the noncommutative weak maximal norm $ \left\| \cdot \right\| _{\Lambda_{p,\infty}(\mN;\ell_{\infty})}$.,

Question:
Let \(\mathcal M\) be a von Neumann algebra with a normal semifinite faithful trace \(\tau\), and set \(\mathcal N=L_\infty(\mathbb R^d)\bar\otimes \mathcal M\) with tensor trace \(\phi=\int\otimes \tau\). Suppose \(T\) is a Calder\'on-Zygmund operator acting on \(\mathcal N\)-valued functions by
\[
Tf(x)=\int_{\mathbb R^d} K(x,y)f(y)\,dy
\]
for \(x\) outside the support of \(f\), where the kernel satisfies the size estimate
\[
|K(x,y)|\le \frac{C_1}{|x-y|^d}.
\]
Assume also that \(K\) satisfies the \(L_2\)-mean H\"ormander condition
\[
\sum_{m\ge 1} H_2(m)<\infty,
\]
where
\[
H_2(m)=\sup_{y\in\mathbb R^d,\,R>0}\left(\frac{(2^mR)^d}{|B_R|}\int_{|v|\le R}\int_{2^mR\le |x-y|\le 2^{m+1}R}|K(x,y+v)-K(x,y)|^2\,dx\,dv\right)^{1/2},
\]
and \(B_R\) is the Euclidean ball of radius \(R\) centered at the origin. For \(\varepsilon>0\), define the truncated operators
\[
T_\varepsilon f(x)=\int_{|x-y|>\varepsilon} K(x,y)f(y)\,dy.
\]
Suppose that for some \(1<p_0<\infty\), the family \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((p_0,p_0)\), i.e.
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{L_{p_0}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_{p_0}(\mathcal N)}
\quad\text{for all }f\in L_{p_0}(\mathcal N).
\]
Which quantitative estimate is valid under these assumptions?

The family \((T_\varepsilon)_{\varepsilon>0}\) is of weak type \((1,1)\): for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]

Choices:
B.
The family \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((1,1)\): for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{L_1(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]
C.
For every \(f\in L_1(\mathcal N)\) and every fixed \(\varepsilon>0\), the truncated operator \(T_\varepsilon\) is of weak type \((1,1)\):
\[
\|T_\varepsilon f\|_{\Lambda_{1,\infty}(\mathcal N)}\lesssim \|f\|_{L_1(\mathcal N)}.
\]
D.
There exists \(p_0\in(1,\infty)\) such that whenever \((T_\varepsilon)_{\varepsilon>0}\) is of strong type \((p_0,p_0)\), one has for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \Big(\sum_{m\ge 1} H_2(m)^2\Big)^{1/2}\,\|f\|_{L_1(\mathcal N)}.
\]
E.
The family \((T_\varepsilon)_{\varepsilon>0}\) is of weak type \((1,1)\) provided the kernel satisfies only the \(L_2\)-mean H\"ormander condition and the size estimate; equivalently, for every \(f\in L_1(\mathcal N)\),
\[
\|(T_\varepsilon f)_{\varepsilon>0}\|_{\Lambda_{1,\infty}(\mathcal N;\ell_\infty)}\lesssim \|f\|_{L_1(\mathcal N)},
\]
without assuming any strong type \((p_0,p_0)\) bound for \((T_\varepsilon)_{\varepsilon>0}\).