\section{introduction}

This paper is devoted to the study of maximal singular integral theory in noncommutative analysis. The starting point of this problem is based on the boundedness theory of noncommutative Calder\'on-Zygmund (abbrieviated as CZ) operators (cf. \cite{M, JP1}), and this theory immediately showed great generality in solving problems in other fields (see e.g. \cite{GJP,JMP1,JMP2}).

Let us first set some notation. Recall that a function $k:\;\mathbb{R}^d \times \mathbb{R}^d \setminus \{(x,x):\;x\in\mathbb R^d\}\rightarrow\mathbb C$ is called a {\emph{standard}} kernel if it satisfies for some constant $C>0$ the size condition
\begin{align}\label{1}
	|k(x,y)| \ \leq \
	\frac{C}{|x-y|^d}
\end{align}
and for some $\gamma\in(0,1)$ the Lipschitz regularity condition 
\begin{align}\label{232}
	\begin{array}{rcl} \big| k(x,y) - k(x,z) \big|+\big| k(y,x) - k(z,x) \big|& \leq &
		\displaystyle \frac{C|y-z|^\gamma}{|x-y|^{d+\gamma}}   \hskip1pt
	\end{array}
\end{align}
whenever $|x-y|\geq2|y-z|$. An operator $T$ is said to be a {\it standard} CZ operator if it is associated with a {\emph{standard}} kernel $k$, that is for any compactly supported bounded function $f$,
\begin{equation}\label{30}
	Tf(x) = \int_{\mathbb{R}^d} k(x,y) f(y) \, dy,\ \  \ x \notin{\mathrm{supp}}  \hskip1pt f
\end{equation}
and $T$ admits a bounded extension on $L_2(\mathbb R^d)$.



Let $({\mathcal M},\tau)$ be a semifinite von Neumann algebra equipped with a normal semifinite faithful (abbrieviated as \emph{n.s.f}) trace. The noncommutative $L_p$-spaces associated with pair $({\mathcal M},\tau)$ are denoted by $L_p({\mathcal M})$. It is trivial that a {\it standard} CZ operator $T$ admits a bounded extension to noncommutative space $L_2(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$, which is still denoted by $T$ without causing any confusion. The extension to $L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$ for $1<p\neq2<\infty$ was first justified by vector-valued theory (see e.g. \cite{HNVW16}) since  $L_p(\mathcal{M})$ are UMD spaces. After the appearance of noncommutative harmonic analysis,
it was shown remarkable that any {\emph{standard}} CZ operator $T$ is of noncommutative weak type $(1,1)$ (cf. \cite{JP1})  and is bounded from $L_\infty$ to noncommutative BMO space (cf. \cite{M}), which also deduce the strong $L_p$ results by interpolation. However, the study on boundedness theory of maximal CZ operators acting on operator-valued functions does not make any progress until ten years later (see \cite{HLX}). An immediate difficulty lies in the fact that the classical maximal function of the  form $M=\sup_{i\in I}|f_i|$ no longer exists in general when 
$(f_i)_{i\in I}$ is a family of operators.  Although, the definition of noncommutative weak type $(1,1)$ maximal inequalities exists in the early stage of noncommutative ergodic theory, the formulation of $L_p$ maximal inequalities was not proposed until the vector-valued noncommutative $L_p$-spaces (see Section 2) appeared two decades ago introduced by Pisier \cite{pisier} and Junge \cite{J1} and has been widely used since then.


For any $\varepsilon>0$, the associated truncated singular integral operator $T_{\varepsilon}$ is defined by
\begin{equation}\label{maximal12}
T_{\varepsilon}f(x) =\int_{|x-y|>\varepsilon} k(x,y) f(y)dy.
\end{equation}
In particular, it is well-known from classical theory (see e.g. \cite[Proposition 4.1.11]{Gra2014}) that there exists a CZ operator $T_0$ and $b\in L_\infty(\mathbb R^d)$ such that
\begin{align}\label{po}
Tf=T_0f+bf
\end{align}
for all $f\in L_2(\mathbb{R}^{d})$.
Here $T_0$ is the weak operator limit in $\mathcal B(L_2(\mathbb{R}^{d}))$ of truncated operators $T_{\varepsilon_j}$ for some subsequence $(\varepsilon_j)_{j\in{\mathbb N}}\subset(0,\infty)$ with $\varepsilon_j\rightarrow0$ as $j\rightarrow\infty$, whose distributional kernel coincides with $k$ on $\mathbb{R}^d \times \mathbb{R}^d \setminus \{(x,x):\;x\in\mathbb R^d\}$. 



When the kernel satisfies the $L_2$-H\"ormander condition (see e.g. \cite{CCP,HLX}) which is quite weaker than the Lipchitz regularity, we obtain a weak type $(1,1)$ criterion  under the assumption of a strong type $(p_0,p_0)$ of $(T_\varepsilon)_{\varepsilon>0}$ with $1<p_0<\infty$ (see \cite[Theorem 1.1]{HLX}). As an important application, we obtained the boundedness theory of {\emph{standard}} maximal
CZ operators of convolution type (see \cite[Theorem 1.3]{HLX}) and left the strong type $(p,p)$ estimates of non-convolution case as an open question.


In the present paper, we give the answer of above question.
\begin{thm}\label{thm:maximal}
Let $T$ be a Calder\'on-Zygmund operator with a {\emph{standard}} kernel $k$. Let $T_{\varepsilon}$ be defined as (\ref{maximal12}). Then the following conclusions hold.
\begin{itemize}
\item[(i)] For $1<p<\infty$, $(T_{\varepsilon})_{\varepsilon>0}$ is of strong type $(p,p)$. More precisely, there exists a constant $C_p$ such that for all $f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$,
\begin{align}\label{maximal}
\big\|{\sup_{\varepsilon>0}}^+T_{\varepsilon}f\big\|_p\leq C_p\|f\|_{p}
\end{align}
with $C_p\leq O({p^4}/{(p-1)^3})$.


\item[(ii)]$(T_{\varepsilon})_{\varepsilon>0}$ is of weak type $(1,1)$. More precisely, there exists a constant $C$ such that for all $f\in L_1(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$ and any $\lambda>0$, there is a projection $e\in L_\infty(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}$ satisfying
		$$\forall \varepsilon>0\ \ \|e(T_{\varepsilon}f)e\|_{\infty}\leq\lambda\ \ \mbox{and}\ \ \tau\otimes\int(e^{\perp})\leq C \frac{\|f\|_1}{\lambda}.$$
		
		\item[(iii)]
	
		For any $f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$ with $1\leq p<\infty$,
		$$T_{\varepsilon_j}f\xrightarrow{b.a.u.}T_0f$$
		
	
	where $(\varepsilon_j)_{j\in{\mathbb N}}$ and $T_0$ are introduced in (\ref{po}).
\end{itemize}
\end{thm}
\begin{remark}\label{constant}
	From the corresponding classical result, it is well-known that $C_p\approx p$ as $p\rightarrow\infty$. When $T_\varepsilon$ in \eqref{maximal} is replaced by $M_\varepsilon$, it is known that $C_p\leq C p^2/(p-1)^2$ from the noncommutative Hardy-Littlewood or Doob's maximal inequalities, see e.g. \cite{M, JX}. But at the moment of writing, we are unable to obtain such an order in the present setting. The difficulty can be explained as the lack of a Marcinkiewicz interpolation theorem for a family of general self-adjoint linear operators (see \cite{CaRi}).
\end{remark}

Let us now describe the strategy of the proof of Theorem \ref{thm:maximal}. The deduction of (ii) from (i) is the main result of our previous work \cite[Theorem 1.1]{HLX}.
So the novel part of Theorem \ref{thm:maximal} lies in (i) for CZ operators of non-convolution type, which follows from the following noncommutative Cotlar inequality.

\begin{prop}\label{prop:nc cotlar}
Let $T$ be a Calder\'on-Zygmund operator with a {\emph{standard}} kernel $k$ and $T_{\varepsilon}$ be defined as (\ref{maximal12}). Then for any $1<s<\infty$, $1<p<\infty$ and $f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})_{+}$
\begin{equation}\label{nc cotlar}
\begin{split}
\big\|{\sup_{\varepsilon>0}}^{+}T_{\varepsilon} f\big\|_p &\leq  C_{d,\gamma,s}\Big(\|{\sup_{\varepsilon>0}}^{+}M_{\varepsilon} (Tf)\big\|_p+\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p\Big),
\end{split}
\end{equation}
where $M_\varepsilon$ is the Hardy-Littlewood averaging operator
\begin{equation}\label{e:20Mr}
M_\varepsilon g(x)=\frac{1}{c_d\varepsilon^{d}}\int_{|x-y|\leq \varepsilon}g(y)dy
\end{equation}
defined for any reasonable operator-valued function $g$.
\end{prop}

As the experts might be aware of that the Cotlar inequality \eqref{nc cotlar} is much weaker than the classical one (see e.g. \cite[Theorem 4.2.4]{{Gra2014}}):
for all $0<s\leq1$,
\begin{align}\label{weak cotlar}
\sup_{\varepsilon>0}|T_\varepsilon f(x)|\leq C_{d,\gamma,s}\Big(\sup_{\varepsilon>0}(M_\varepsilon(|Tf|^s)(x))^{\frac1s}+\sup_{\varepsilon>0}M_\varepsilon (|f|)(x)\Big),
\end{align}
which yields not only the strong $L_p$ estimates but also the weak $L_1$. As it has been explained briefly in \cite{HLX}, a noncommutative version of \eqref{weak cotlar} in the case $s<1$ seems unavailable. And for operators with kernels of convolution type, the inequality \eqref{nc cotlar} in the case $s=1$ has been obtained in \cite{HLX}. Our approach to \eqref{nc cotlar} and thus Theorem \ref{thm:maximal}(i) seems new even in the classical setting.  Due to the noncommutativity and nonconvolution, at the moment of writing, we are unable to decrease $s$ to 1 or below in \eqref{nc cotlar}.



\section{Preliminaries}

\subsection{Noncommutative $L_{p}$-spaces}
Throughout this paper,  ${\mathcal M}$ will denote a semifinite von Neumann algebra equipped with a \emph{n.s.f} trace $\tau$. Let $\mathcal{S_{{\mathcal M}+}}$ be the collection of $x\in{\mathcal M}_{+}$ such that $\tau(\mathrm{supp}x)<\infty$, where ${\mathcal M}_{+}$ denotes the positive part of ${\mathcal M}$ and $\mathrm{supp}x$ is the support of $x$. Let $\mathcal{S}_{{\mathcal M}}$ be the linear span of $\mathcal{S_{{\mathcal M}+}}$. Then $\mathcal{S}_{{\mathcal M}}$ is a $w^{*}$-dense $\ast$-subalgebra of ${\mathcal M}$. Given $1\leq p<\infty$, we set
$$\|x\|_{p}=[\tau(|x|^p)]^{1/p}\ \ x\in\mathcal{S}_{{\mathcal M}},$$
where $|x|=(x^{\ast}x)^{\frac{1}{2}}$. The completion of the norm space $(\mathcal{S}_{{\mathcal M}},\|\cdot\|_{p})$ is denoted by  $L_{p}({\mathcal M})$. As usual, we set $L_{\infty}({\mathcal M}) = {\mathcal M}$ equipped with the operator norm $\|\cdot\|_{{\mathcal M}}$, and the positive part of $L_{p}({\mathcal M})$ is written as $L_{p}({\mathcal M})_{+}$. We refer the reader to \cite{FK,PX2} for further details on noncommutative $L_p$-spaces.

\subsection{Noncommutative $\ell_{\infty}$-valued $L_{p}$-spaces}
Let $I$ be an index set. 
Given $1\le p\leq\infty$, $L_p(\mathcal {M};\ell_\infty(I))$ is defined as all sequence $x=(x_n)_{n\in I}$ in $L_p(\mathcal {M})$ for which  there are $a,b\in L_{2p}( \mathcal {M})$  and $y=(y_n)_{n\in I}\subset L_\infty(\mathcal {M})$ such that
$$x_n=ay_nb,\ \ \forall\ n\in I.$$
The norm of $x$ in $L_p(\mathcal {M};\ell_\infty(I))$ is defined as
$$\|x\|_{L_p(\mathcal {M};\ell_\infty(I))}=\inf\left\{\big\|a\big\|_{{2p}}\sup_{n\in I}\big\|y_n\big\|_\infty\big\|b\big\|_{{2p}}\right\},$$
where the infimum runs over all possible factorizations as above.
Usually, the norm of $x$ in $L_p(\mathcal {M};\ell_\infty(I))$ is denoted by $\|{\sup_{n\in I}}^+x_n\|_p$, that is $\|{\sup_{n\in I}}^+x_n\|_p\triangleq\|x\|_{L_p(\mathcal {M};\ell_\infty(I))}$. We refer to \cite{J1} for more details.


The following useful characterization can be found in e.g. \cite[Remark 4.1]{CXY13}.
\begin{remark}\label{rk:MaxFunct}
	Let $x=(x_n)_{n\in I}$ be a sequence of selfadjoint operators in $L_p(\mathcal {M})$. Then $x\in L_p(\mathcal {M};\ell_\infty)$ if and only if there exists $a\in L_{p}({\mathcal M})_{+}$
	such that $-a\leq x_n\leq a$ for all $n\in I$. Moreover, in this case,
	$$\|x\|_{L_p(\mathcal {M}; \ell_\infty(I))}=\inf\Big\{\|a\|_p: a\in L_{p}({\mathcal M})_{+}\ \mbox{such that}\ -a\leq x_n\leq a
	,\ \forall n\in I\Big\}.$$
\end{remark}

\noindent In practice, we will omit the index set $I$ if there will be no ambiguity.

\subsection{Almost uniform convergence}
The following definition of (bilaterally) almost uniform convergence considered in Lance's work \cite{Lan76}, which presents a suitable replacement of almost everywhere convergence in the noncommutative setting.
\begin{definition}\label{b.a.u.}\rm
	Let $\mathcal {M}$ be a von Neumann algebra equipped with a \emph{n.s.f} trace $\tau$.
	\begin{itemize}
			\item[(i)]  The sequence $(x_n)$ is said to converge bilaterally almost uniformly (b.a.u. in short) to $x$ if for any $\varepsilon>0$, there is a projection $e\in{\mathcal M}$
			such that
			$$\tau(e^\bot)<\varepsilon \ \ \  \mbox{and} \ \ \ \lim _{n\rightarrow\infty}\|e(x_n-x)e\|_\infty=0.$$
			
			\item[(ii)] The sequence $(x_n)$ is said to converge almost uniformly (a.u. in short) to $x$ if for any $\varepsilon>0$, there is a projection $e\in{\mathcal M}$
			such that
			$$\tau(e^\bot)<\varepsilon \ \ \  \mbox{and} \ \ \ \lim _{n\rightarrow\infty}\|(x_n-x)e\|_\infty=0.$$
		\end{itemize}
\end{definition}

\subsection{Some auxiliary lemmas}
At the end of this section, we present two lemmas.
The first lemma should be well-known to experts, see e.g. \cite[Theorem 4.3]{CXY13}.
\begin{lemma}\label{result of cxy}
Let $\psi$ be an integrable function on $\mathbb{R}^d$  such that $|\psi|$ is radial and radially decreasing.
Let $\psi_\varepsilon(x)=\frac1{\varepsilon^d}\, \psi(\frac x\varepsilon)$ for $x\in\mathbb{R}^d$ and $\varepsilon>0$. Then for $1<p\leq\infty$
 $$\big\|{\sup_{\varepsilon>0}}^+\psi_\varepsilon\ast f\big\|_p\leq C_{d}\big\|{\sup_{\varepsilon>0}}^+M_\varepsilon f\big\|_p,\;\forall f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})_+.$$
Here $\|\cdot\|_p$ denotes $\|\cdot\|_{L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})}$.
\end{lemma}

By applying the operator convexity of $0\leq t\rightarrow t^{p}$ for $1\leq p\leq2$, we can deduce 
 the operator-valued version of the H\"{o}lder inequality (see e.g. \cite[Lemma 3.5]{M}).
\begin{lemma}\label{holderinequality}
Let $(\Omega,\mu)$ be a measure space. Then for $1< p<\infty$ and let $p^{\prime}$ be the conjugate index of $p$, that is $1/ p^\prime +1/p =1$, we have
\begin{equation}\label{maximal132}
\int_{\Omega}f(x)g(x)d\mu(x)\leq\Big(\int_{\Omega}f(x)^{p}d\mu(x)\Big)^{\frac{1}{p}}
\Big(\int_{\Omega}g(x)^{p^{\prime}}d\mu(x)\Big)^{\frac{1}{p^{\prime}}},
\end{equation}
where $f:\Omega\rightarrow L_{1}(\mathcal{M})+L_{\infty}(\mathcal{M})$ and $g:\Omega\rightarrow \mathbb{C}$ are positive functions such that all members of inequality (\ref{maximal132}) make sense.
\end{lemma}
\begin{proof}
If we set
$$d\nu(x)=\frac{g(x)^{p^\prime}d\mu(x)}{\int_{\Omega}g(x)^{p^{\prime}}d\mu(x)},$$
then $\nu(\Omega)=1$. By applying the operator convexity, we obtain
\begin{align*}
\hskip1pt \int_{\Omega}f(x)g(x)d\mu(x)
=&\ \hskip1pt \int_{\Omega}f(x)g(x)^{1-p^{\prime}}d\nu(x)\int_{\Omega}g(x)^{p^{\prime}}d\mu(x)\\
\leq&\ \Big(\int_{\Omega}f(x)^{p}g(x)^{(1-p^{\prime})p}d\nu(x)\Big)^{\frac{1}{p}}\int_{\Omega}g(x)^{p^{\prime}}d\mu(x)\\
=&\ \hskip1pt \Big(\int_{\Omega}f(x)^{p}d\mu(x)\Big)^{\frac{1}{p}}\Big(\int_{\Omega}g(x)^{p^{\prime}}d\mu(x)\Big)^{\frac{1}{p^{\prime}}},
\end{align*}
which finishes the proof.
\end{proof}














\section{The noncommutative Cotlar inequalities}

In this section, we prove  Proposition \ref{prop:nc cotlar}---the noncommutative Cotlar inequalities.
\begin{lemma}\label{mainlemma1}
Let $T$ be a CZ operator defined as (\ref{30}) with a {\emph{standard}} kernel $k$. If we set
$$I_{\varepsilon,N}(x)=\int_{\varepsilon<|x-y|<N}k(x,y)dy,$$
then for $1< p<\infty$,
$$\int_{|x-x_{0}|<N}|I_{\varepsilon,N}(x)|^{p}dx\leq C_{d,\gamma,p} N^{d}\ \ \mbox{for\ all}\ \varepsilon,\ N\ \mbox{and}\ x_{0}.$$
\end{lemma}
The proof of this lemma is quite similar as presented in \cite[Page 306]{St}. For the sake of self-completeness, we present a proof.
\begin{proof}

	Note that each ball with radius $N$ can be covered by a finite number of balls with radius $N/2$. Hence, it suffices to show
	\begin{equation}\label{maximal2}
		\int_{|x-x_{0}|<N/2}|I_{\varepsilon,N}(x)|^{p}dx\leq C_{d,\gamma,p}N^{d}\ \ \mbox{for\ all}\ \varepsilon,\ N\ \mbox{and}\ x_{0}.
	\end{equation}

From the $L_p$-boundedness of $T$ and the assumptions on $k$, we know that the truncated operators $T_\varepsilon$ are uniformly bounded in the $L_p$ norm (see e.g. \cite[Page 31, Proposition 1]{St}), that is
$$\sup_{\varepsilon>0}\|T_\varepsilon f\|_p\leq C_{d,\gamma,p}\|f\|_p,\ \forall f\in L_p(\mathbb{R}^{d}).$$

Now we exploit the above fact by comparing $I_{\varepsilon,N}(x)$ and $T_\varepsilon(\chi^{N,x_0})(x)$ whenever $|x-x_{0}|<N/2$, where $\chi^{N,x_0}$ is the characteristic function of the ball $\{y:|y-x_0|<N\}$.

Noting that under the condition $|x-x_{0}|<N/2$, we have
$$\{y:|x-y|<N\}\Delta\{y:|y-x_0|<N\}\subset\{y:N/2<|x-y|<3N/2\}.$$
As a consequence, if $|x-x_{0}|<N/2$,
\begin{align*}
	\hskip1pt \big|I_{\varepsilon,N}(x)-T_\varepsilon(\chi^{N,x_0})(x)\big|
	=&\ \hskip1pt \Big|\int_{\varepsilon<|x-y|<N}k(x,y)dy-\int_{\varepsilon<|x-y|\atop |y-x_0|<N}k(x,y)dy\Big|\\
	\leq&\ \int_{N/2<|x-y|<3N/2}|k(x,y)|dy
	\leq C_d,
\end{align*}
where the last inequality follows from the size condition (\ref{1}) of the kernel. Therefore, applying the above estimate, the uniform $L_p$ boundedness of $T_\varepsilon$ and the triangle inequality, we   get
\begin{align*}
	\hskip1pt \int_{|x-x_{0}|<N/2}|I_{\varepsilon,N}(x)|^{p}dx\leq&\
	C_{d,p}N^d+\|T_\varepsilon(\chi^{N,x_0})\|^p_p\\
	\leq&\ \hskip1pt C_{d,p}N^d+C_{d,\gamma,p}\|\chi^{N,x_0}\|^p_p\leq C_{d,\gamma,p}N^d.
\end{align*}
This completes the proof.
\end{proof}

With Lemma \ref{mainlemma1}, we can finish the proof of Proposition \ref{prop:nc cotlar}.
\begin{proof}[Proof of Proposition \ref{prop:nc cotlar}.]
Let $\varphi$ be a positive smooth, radially decreasing
function with integral one  supported in the ball $B(0,\frac{1}{2})$. For any $\varepsilon>0$, we will use the notation $\varphi_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\varphi(\frac{x}{\varepsilon})$. Without loss of generality, we may assume that the kernel $k$ is real-valued. Indeed, by writing the kernel $k$ into its real and imaginary parts, we just need to verify each part separately (see e.g. \cite[Lemma 3.1]{HLX}).   On the other hand, since 
$C_c^\infty({\mathbb R}^d)\otimes\mathcal{S}_{{\mathcal M}}$ is dense in $L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$, it suffices to consider any fixed positive function $f\in C_c^\infty({\mathbb R}^d)\otimes\mathcal{S}_{{\mathcal M}}$. In the following, we fix $1<s<\infty$ and $1<p<\infty$. Decompose $T_{\varepsilon}f$ as
\begin{eqnarray}\label{k1}
T_{\varepsilon}f(x)=\varphi_{\varepsilon}\ast(Tf)(x)+
T_{\varepsilon}f(x)-\varphi_{\varepsilon}\ast(Tf)(x).
\end{eqnarray}
The first term of right-hand side of (\ref{k1}) is easy to handle. Indeed, by Lemma \ref{result of cxy}, 
\begin{eqnarray}\label{k2}
\big\|{\sup_{\varepsilon>0}}^+\varphi_{\varepsilon}\ast(Tf)\big\|_p\leq C_{d} \big\|{\sup_{\varepsilon>0}}^{+}M_{\varepsilon} (Tf)\big\|_p.
\end{eqnarray}

On the other hand, recalled in \eqref{po} that there exists a CZ operator $T_0$ such that $Tf=T_0f+bf$,  where $T_0$ is the weak operator limit in $\mathcal B(L_2(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}))$ of truncated operators of $T_{\varepsilon_{j}}$ for some sequence $\varepsilon_{j}\rightarrow0$ and $b\in {L_{\infty}(\mathbb{R}^{d})}$.
As a consequence,
\begin{eqnarray}\label{k7}
\varphi_{\varepsilon}\ast(Tf)(x)=\varphi_{\varepsilon}\ast(T_0f)(x)+\varphi_{\varepsilon}\ast(bf)(x).
\end{eqnarray}
By Lemma \ref{result of cxy} and Lemma \ref{holderinequality},  the second term on the right-hand side of (\ref{k7}) is handled as 
\begin{eqnarray}\label{k8}
	\big\|{\sup_{\varepsilon>0}}^+\varphi_{\varepsilon}\ast(bf)\big\|_p\leq C_{d}\|b\|_\infty \big\|{\sup_{\varepsilon>0}}^{+}M_{\varepsilon} f\big\|_p\leq C_{d} \big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p.
\end{eqnarray}

According to (\ref{k1})-(\ref{k8}), it remains to estimate $T_{\varepsilon}f(x)-\varphi_{\varepsilon}\ast(T_0f)(x)$. Notice that the kernel associated to $\varphi_{\varepsilon}\ast(T_0f)(x)$ is
$$\int_{\mathbb{R}^{d}}\varphi_{\varepsilon}(x-z)k(z,y)dz,$$
which should be understood as the principle value and equals to
$$\lim_{\varepsilon_{j}\rightarrow0}\int_{|z-y|>\varepsilon_{j}}\varphi_{\varepsilon}(x-z)k(z,y)dz.$$
If we set
$K(x,y,\varepsilon)\triangleq k(x,y)
\chi_{|x-y|>\varepsilon}-\int_{\mathbb{R}^{d}}\varphi_{\varepsilon}(x-z)k(z,y)dz,$
then 
$$T_{\varepsilon}f(x)-\varphi_{\varepsilon}\ast(T_0f)(x)=\int_{\mathbb{R}^{d}}K(x,y,\varepsilon)
f(y)dy.$$

In the following, we decompose
$$
K(x,y,\varepsilon)=K_{1}(x,y,\varepsilon)+K_{2}(x,y,\varepsilon)
$$
where
$$K_{1}(x,y,\varepsilon)=K(x,y,\varepsilon)\chi_{|x-y|>\varepsilon}\ \ \mbox{and}\ \ K_{2}(x,y,\varepsilon)=K(x,y,\varepsilon)\chi_{|x-y|\leq\varepsilon}.$$
To estimate $K_{1}(x,y,\varepsilon)$, by the property of $\varphi$, we get
$$K_{1}(x,y,\varepsilon)=\int_{\mathbb{R}^{d}}\varphi_{\varepsilon}(x-z)(k(x,y)-k(z,y))dz\chi_{|x-y|>\varepsilon}.$$
The support of $\varphi$ gives $|x-z|<\frac{\varepsilon}{2}<\frac{1}{2}|x-y|$. Hence, the Lipschitz smoothness condition and the fact that $K_{1}(x,y,\varepsilon)$ is real-valued imply
\begin{align*}
K_{1}(x,y,\varepsilon)
\leq C\int_{\mathbb{R}^{d}}\varphi_{\varepsilon}(x-z)\frac{|x-z|^{\gamma}}{|x-y|^{d+\gamma}}dz\chi_{|x-y|>\varepsilon}
\leq C \frac{\varepsilon^{\gamma}}{|x-y|^{d+\gamma}}\chi_{|x-y|>\varepsilon}.
\end{align*}
Thus by above observations, 
\begin{align*}
T_{K,\varepsilon,1}f(x)\triangleq\int_{\mathbb{R}^{d}}K_{1}(x,y,\varepsilon)f(y)dy
\leq&\ C\varepsilon^{\gamma}\sum_{k=0}^{\infty}
\int_{2^{k}\varepsilon<|x-y|<2^{k+1}\varepsilon}\frac{f(y)}{|x-y|^{d+\gamma}}dy
\\
\leq&\ C \sum_{k=0}^{\infty}2^{-k\gamma}\frac{1}{(2^{k+1}\varepsilon)^{d}}
\int_{|x-y|<2^{k+1}\varepsilon}f(y)dy.
\end{align*}
On the other hand, it is obvious to see
$$
-C\sum_{k=0}^{\infty}2^{-k\gamma}\frac{1}{(2^{k+1}\varepsilon)^{d}}
\int_{|x-y|<2^{k+1}\varepsilon}f(y)dy\leq T_{K,\varepsilon,1}f(x).$$
Therefore, by Remark \ref{rk:MaxFunct} and Lemma \ref{holderinequality}, we deduce tha
\begin{eqnarray}\label{k3}
\big\|{\sup_{\varepsilon>0}}^+T_{K,\varepsilon,1}f\big\|_p
\leq C\sum_{k=0}^{\infty}2^{-k\gamma}\|{\sup_{\varepsilon>0}}^+M_{\varepsilon}f\|_p
\leq C_\gamma\|{\sup_{\varepsilon>0}}^+\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\|_p.
\end{eqnarray}

Now we turn to estimate $K_{2}(x,y,\varepsilon)$. However, in this case
$$K_{2}(x,y,\varepsilon)=-\lim_{\varepsilon_{j}\rightarrow0}\int_{|z-y|>\varepsilon_{j}}\varphi_{\varepsilon}(x-z)k(z,y)dz\chi_{|x-y|\leq\varepsilon}.$$
In the following, we decompose $K_{2}(x,y,\varepsilon)$ as
\begin{align*}
K_{2}(x,y,\varepsilon) & =-\Big[\int_{|z-y|\geq\frac{\varepsilon}{2}}\varphi_{\varepsilon}(x-z)k(z,y)dz\chi_{|x-y|\leq\varepsilon}\\
       & \ \ \ \ \ \ +\lim_{\varepsilon_{j}\rightarrow0}\int_{\varepsilon_{j}<|z-y|<\frac{\varepsilon}{2}}(\varphi_{\varepsilon}(x-z)-
       \varphi_{\varepsilon}(x-y))k(z,y)dz\chi_{|x-y|\leq\varepsilon}\\
       & \ \ \ \ \ \ +\lim_{\varepsilon_{j}\rightarrow0}\int_{\varepsilon_{j}<|z-y|<\frac{\varepsilon}{2}}
       \varphi_{\varepsilon}(x-y)k(z,y)dz\chi_{|x-y|\leq\varepsilon}\Big]\\
       &\triangleq -\Big[A_{\varepsilon}^{1}(x,y)+A_{\varepsilon}^{2}(x,y)+A_{\varepsilon}^{3}(x,y)\Big].
\end{align*}

To handle $A_{\varepsilon}^{1}(x,y)$, by applying the size condition (\ref{1}), we have
$$A_{\varepsilon}^{1}(x,y)\leq C
\int_{|z-y|\geq\frac{\varepsilon}{2}}
\frac{\varphi_{\varepsilon}(x-z)}{|z-y|^{d}}dz\chi_{|x-y|\leq\varepsilon}
\leq C_d\frac{1}{\varepsilon^{d}}\chi_{|x-y|\leq\varepsilon}.$$
Therefore, 
$$-C_d\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}f(y)dy\leq
T_{K,\varepsilon,2}^{1}f(x)\triangleq\int_{\mathbb{R}^{d}}A_{\varepsilon}^{1}(x,y)f(y)dy
\leq C_d\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}f(y)dy,$$
which implies that
\begin{eqnarray}\label{k4}
\big\|{\sup_{\varepsilon>0}}^+T^{1}_{K,\varepsilon,2}f\big\|_p
\leq C_d\|{\sup_{\varepsilon>0}}^+M_{\varepsilon}f\|_p\leq C_d\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p.
\end{eqnarray}

For the second term $A_{\varepsilon}^{2}(x,y)$, using the mean value theorem, the size condition (\ref{1}) of the kernel and the polar coordinates formula, we get
$$A_{\varepsilon}^{2}(x,y)\leq C
\frac{1}{\varepsilon^{d+1}}\int_{|z-y|<\frac{\varepsilon}{2}}
\frac{1}{|z-y|^{d-1}}dz\chi_{|x-y|\leq\varepsilon}
\leq C\frac{1}{\varepsilon^{d}}\chi_{|x-y|\leq\varepsilon}.$$
Hence, we deduce that
$$-C\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}f(y)dy\leq
T_{K,\varepsilon,2}^{2}f(x)\triangleq\int_{\mathbb{R}^{d}}A_{\varepsilon}^{2}(x,y)f(y)dy
\leq C\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}f(y)dy.$$
This gives rise to
\begin{eqnarray}\label{k5}
\big\|{\sup_{\varepsilon>0}}^+T^{2}_{K,\varepsilon,2}f\big\|_p
\leq C\|{\sup_{\varepsilon>0}}^+M_{\varepsilon}f\|_p\leq C\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p.
\end{eqnarray}

We now deal with the last term $A_{\varepsilon}^{3}(x,y)$. To see this, we rewrite $A_{\varepsilon}^{3}(x,y)$ as
$$A_{\varepsilon}^{3}(x,y)=\lim_{\varepsilon_{j}\rightarrow0}I_{\varepsilon_{j},\frac{\varepsilon}{2}}(y)
       \varphi_{\varepsilon}(x-y)\chi_{|x-y|\leq\varepsilon},$$
where
$$I_{\varepsilon_{j},\frac{\varepsilon}{2}}(y)=\int_{\varepsilon_{j}<|z-y|<\frac{\varepsilon}{2}}k(z,y)dz.$$
By applying the H\"{o}lder inequality stated in Lemma \ref{holderinequality}, we find
\begin{align*}
T_{K,\varepsilon,2}^{3}f(x)\triangleq&\ \int_{|x-y|\leq\varepsilon}A_{\varepsilon}^{3}(x,y)f(y)dy\leq\int_{|x-y|\leq\varepsilon}|A_{\varepsilon}^{3}(x,y)|f(y)dy\\
\leq&\ \Big(\int_{|x-y|\leq\varepsilon}\lim_{\varepsilon_{j}\rightarrow0}|I_{\varepsilon_{j},\frac{\varepsilon}{2}}(y)|^{s^{\prime}}
\varphi_{\varepsilon}(x-y)dy\Big)^{\frac{1}{s^{\prime}}}
\Big(\int_{|x-y|\leq\varepsilon}
\varphi_{\varepsilon}(x-y)f(y)^{s}dy\Big)^{\frac{1}{s}}.
\end{align*}
Using the
Fatou lemma, the support of $\varphi$ and Lemma \ref{mainlemma1}, we can see that the first term above can be controlled by
$$\Big(\lim_{\varepsilon_{j}\rightarrow0}\int_{|x-y|\leq\frac{\varepsilon}{2}}|I_{\varepsilon_{j},\frac{\varepsilon}{2}}(y)|^{s^{\prime}}
\varphi_{\varepsilon}(x-y)dy\Big)^{\frac{1}{s^{\prime}}}\leq C\Big(\frac{1}{\varepsilon^{d}}\lim_{\varepsilon_{j}\rightarrow0}\int_{|x-y|\leq\frac{\varepsilon}{2}}
|I_{\varepsilon_{j},\frac{\varepsilon}{2}}(y)|^{s^{\prime}}
dy\Big)^{\frac{1}{s^{\prime}}}\leq C_{d,\gamma,s}.$$
On the other hand, it is easy to see that
$$\Big(\int_{|x-y|\leq\varepsilon}
\varphi_{\varepsilon}(x-y)f(y)^{s}dy\Big)^{\frac{1}{s}}\leq
C\Big(\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}
f(y)^{s}dy\Big)^{\frac{1}{s}}.$$
Furthermore, it is trivial that
$$-C_{d,\gamma,s}
\Big(\frac{1}{\varepsilon^{d}}\int_{|x-y|\leq\varepsilon}
f(y)^{s}dy\Big)^{\frac{1}{s}}\leq T_{K,\varepsilon,2}^{3}f(x).$$
Therefore, by Remark \ref{rk:MaxFunct},
\begin{eqnarray}\label{k6}
\big\|{\sup_{\varepsilon>0}}^+T^{3}_{K,\varepsilon,2}f\big\|_p\leq C_{d,\gamma,s}\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p
\end{eqnarray}

Finally,
putting (\ref{k1})-(\ref{k6}) together with the Minkowski inequality, we  obtain
$$
\big\|{\sup_{\varepsilon>0}}^{+}T_{\varepsilon} f\big\|_p\leq C_{d}\|{\sup_{\varepsilon>0}}^{+}M_{\varepsilon} (Tf)\big\|_p+C_{d,\gamma,s}\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p.
$$
This completes the argument of Proposition \ref{prop:nc cotlar}.
\end{proof}


\section{Proof of Theorem \ref{thm:maximal}}

Now we are at a position to prove Theorem \ref{thm:maximal}.
\begin{proof}[Proof of Theorem \ref{thm:maximal} (i) and (ii).]
As we pointed out previously, combining \cite[Theorem 1.1]{HLX}, it suffices to show the strong type $(p,p)$ estimates in (i).

Now let $1<p<\infty$ and $f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}).$ By decomposing $f = f_{1}-f_{2} +i(f_{3}-f_{4})$ with positive $f_{j}$ ($j=1,2,3,4$), we assume that $f$ is positive. By Proposition \ref{prop:nc cotlar}, it suffices to estimate the two terms on the right hand of (\ref{nc cotlar}). To this end, we first use the noncommutative Hardy-Littlewood maximal $L_p$ estimate \cite{M} to deduce that
\begin{align}\label{cotlar norm}
\big\|{\sup_{\varepsilon>0}}^{+}M_{\varepsilon} (Tf)\big\|_p\leq C_{d}\frac{p^2}{(p-1)^2}\|Tf\|_p\leq C_{\gamma,d}\frac{p^4}{(p-1)^3}\|f\|_p,
\end{align}
where the second inequality follows from the strong type $(p,p)$ boundedness of $T$ with bound $C_{\gamma,d}\frac{p^2}{p-1}$ (see \cite[Theorem A]{JP1}).

It remains to estimate the second term appeared on the right of (\ref{nc cotlar}).  Choose $s$ such that $1<s<p$. By applying the noncommutative Hardy-Littlewood maximal inequalities to $f^{s}\in L_{p/s}(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$, there exists a positive operator $F\in L_{p/s}(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$ such that for any $\varepsilon>0$,
$$\|F\|_{p/s}\leq C_{d}\frac{p^{2}}{(p-s)^{2}}\|f^{s}\|_{p/s}\quad\mbox{and}\quad
    M_{\varepsilon}(f^{s})\le F.$$
As a consequence, by the monotone increasing property of $0\leq a\rightarrow a^{t}$ for $0<t<1$, we infer that for any $\varepsilon>0$,
$$\big(M_{\varepsilon}(f^{s})\big)^{^{1/s}}\leq F^{1/s}\ \mbox{and}\ \|F^{1/s}\|_{p}=\|F\|^{1/s}_{p/s}\leq C^{1/s}_{d}\frac{p^{2/s}}{(p-s)^{2/s}}\|f\|_{p}.$$
Therefore, we find
\begin{eqnarray}\label{k676}
\big\|{\sup_{\varepsilon>0}}^{+}\big(M_{\varepsilon} f^{s}\big)^{\frac{1}{s}}\big\|_p\leq\|F^{1/s}\|_{p}\leq C^{1/s}_{d}\frac{p^{2/s}}{(p-s)^{2/s}}\|f\|_{p}.
\end{eqnarray}

Finally, together with (\ref{cotlar norm}) and (\ref{k676}) and letting $s\rightarrow1$,
we see that 
$$\big\|{\sup_{\varepsilon>0}}^+T_{\varepsilon}f\big\|_p\leq C_{d,\gamma}\frac{p^4}{(p-1)^3}\|f\|_{p}.$$
This finishes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:maximal} (iii).]
Let $f\in L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$ and $(\varepsilon_j)_{j\in{\mathbb N}}$ be the sequence that appeared in (\ref{po}). It is already known from  (\ref{po}) that $T_{\varepsilon_j}f\rightarrow T_0f$ in  the sense of the weak operator topology in  $\mathcal B(L_2(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}))$. Hence, we just need to show the b.a.u convergence of $(T_{\varepsilon_j}f)_j$ whenever $j\rightarrow\infty$, since then the limit must be $T_0f$ (see e.g. \cite{CXY13}). Moreover,
by \cite[Proposition 1.3]{CL1}, it is enough to show  for any $\delta>0$, there exists a projection $e\in L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}$ such that 
\begin{eqnarray}\label{k6761}
\int\otimes\tau(1-e)\leq\delta\ \ \mbox{and}\ \ \|e(T_{\varepsilon_k}f-T_{\varepsilon_\ell}f)e\|_{\infty}\rightarrow0\ \mbox{as}\ k,\ell\rightarrow\infty.
\end{eqnarray}

Fix $\delta>0$. For each $n\geq1$, by the density of $C^\infty_c(\mathbb R^d)\otimes S_{\mathcal M}$ in $L_p(L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M})$, we can find $g_n\in C^\infty_c(\mathbb R^d)\otimes S_{\mathcal M}$ such that $\|f-g_n\|_p^p\leq\frac{\delta}{2^n n^p}$. Then using the fact that $(T_{\varepsilon_{j}})_{j\in{\mathbb N}}$ is of weak type $(p,p)$ ($1\leq p<\infty$), there is a projection $e_{n}\in L_{\infty}(\mathbb{R}^{d})\overline{\otimes}\mathcal{M}$ such that
\begin{equation*}\sup_{j}\|e_{n}T_{\varepsilon_{j}} (f-g_{n})e_{n}\|_{\infty}<\frac1n\ \mbox{and}\
	\int\otimes\tau(1-e_n)<n^p\|f-g_n\|_p^p\leq\frac{\delta}{2^{n}}.
\end{equation*}
Let $e=\wedge_{n}e_{n}$. Then 
\begin{align}\label{err}\int\otimes\tau(e^\perp)<\delta\ \mbox{and}\ \sup_{j}\|eT_{\varepsilon_{j}} (f-g_{n})e\|_{\infty}<\frac{1}{n},\ \forall\ n\geq1.\end{align}

On the other hand, we have
\begin{align}\label{uni}
	\lim_{\ell,k\rightarrow+\infty}\|T_{\varepsilon_{\ell}}g_{n}-T_{\varepsilon_{k}}g_{n}\|_{\infty}=0\ \forall n\geq1.
\end{align}
Indeed, write $g_n=\sum_i\phi_{i}\otimes m_{i}$ with finite sum of $i$ such that $\phi_{i}\in C^\infty_c(\mathbb R^d)$ and $m_{i}\in S_{\mathcal M}$. Then for $\varepsilon_\ell<\varepsilon_k$, one has
\begin{align*}
	T_{\varepsilon_{\ell}}g_{n}(x)-T_{\varepsilon_{k}}g_{n}(x) & = \sum_{i}\int_{\varepsilon_{\ell}<|x-y|\leq\varepsilon_{k}}k(x,y)\phi_{i}(y)dy\otimes m_{i}


\end{align*}
which tends to 0 as $\ell,k\rightarrow+\infty$ (see e.g. \cite[Page 234]{Gra2014}). Therefore, by (\ref{uni})
we know that for any $n\geq1$, there is a positive constant $N_{n}$ such that for any $\ell,k>N_{n}$
$$\|T_{\varepsilon_{\ell}}g_{n}-T_{\varepsilon_{k}}g_{n}\|_{\infty}<\frac{1}{n}.$$
Then together with \eqref{err}, we finally get for any $\ell,k>N_{n}$,
\begin{align*}
	\|e(T_{\varepsilon_{\ell}}f-T_{\varepsilon_{k}}f)e\|_{\infty} &\leq \|T_{\varepsilon_{\ell}}g_{n}-T_{\varepsilon_{k}}g_{n}\|_{\infty}+\|e
	T_{\varepsilon_{\ell}}(f-g_{n})e\|_{\infty}
	+\|eT_{\varepsilon_{k}}(f-g_n)e\|_{\infty}<\frac{3}{n},
\end{align*}
which yields
$$\lim_{\ell,k\rightarrow\infty}\|e(T_{\varepsilon_{\ell}}f-T_{\varepsilon_{k}}f)e\|_{\infty}=0.$$
This completes the proof.
\end{proof}




