# Calderón-Zygmund theory with noncommuting kernels via $H_{1}^{c}$

Antonio Ismael Cano-Mármol  Department of Mathematics
Baylor University
1301 S University Parks Dr
Waco, TX 76798, USA<EMAIL_ADDRESS>and  Éric Ricard
Laboratoire de Mathématiques Nicolas Oresme
UNICAEN, CRNS
1400 Caen, France<EMAIL_ADDRESS>
###### Abstract.

We study an alternative definition of the $H_{1}$-space associated to a
semicommutative von Neumann algebra
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$, first studied by Mei.
We identify a “new” description for atoms in $H_{1}$. We then explain how they
can be used to study $H_{1}^{c}$-$L_{1}$ endpoint estimates for Calderón-
Zygmund operators with noncommuting kernels.

###### Key words and phrases:

Hardy space, BMO space, von Neumann algebra, non-commutative $L_{p}$ space,
Calderón-Zygmund theory, atoms

###### 2020 Mathematics Subject Classification:

42B20, 42B35, 46L51, 46L52

## Introduction

This paper is related to the theory of semicommutative Calderón-Zygmund
operators. This is a research line that takes advantage of the hybrid nature
of certain vector valued $L_{p}$-spaces. Let $(\mathcal{M},\tau)$ be a von
Neumann algebra of operators on a separable Hilbert space, equipped with a
normal semifinite faithful trace $\tau$. Denote by
$\mathcal{A}=\mathcal{M}\overline{\otimes}L_{\infty}(\mathbb{R})$ the weak
operator closure of the space of essentially bounded (strongly measurable)
functions $f:\mathbb{R}\longrightarrow\mathcal{M}$. For the sake of
exposition, we will restrict ourselves to dimension 1, even though our
arguments extend trivially to any finite dimension namely for
$\mathcal{M}\overline{\otimes}L_{\infty}(\mathbb{R}^{n})$. The von Neumann
algebra $\mathcal{A}$ can be identified with the tensor product
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ equipped with the trace

$\displaystyle\varphi(f)=\int_{\mathbb{R}}\tau(f(x))\ dx.$

The noncommutative $L_{p}$-spaces associated with $\mathcal{A}$ are indeed
vector-valued $L_{p}$-spaces: indeed, [P98, Chapter 3]

$L_{p}(\mathcal{A})=L_{p}(\mathbb{R};L_{p}(\mathcal{M})),$

for $1\leq p<\infty$. We are interested in endpoint estimates for operators
acting on $L_{p}(\mathcal{A})$, and in particular in the boundedness of
operators from the operator-valued version of the Hardy space $H_{1}$ into
$L_{1}$. This question was widely studied in the classical setting for scalar-
valued functions [Mey90, MeyC97] as well as for vector-valued functions [F90,
H06], where the existence of the atomic decomposition plays an essential role.
This technique does not seem to have been exploited as often in the
noncommutative setting except perhaps in [HLMP14]. Mei [M07] was the first to
introduce the so-called _operator-valued Hardy space_
$H_{1}(\mathbb{R},\mathcal{M})$ in this context via noncommutative equivalents
of the Poisson integral, the Lusin area integral and the Littlewood-Paley $g$
function. These techniques allowed Mei to identify the dual space of
$H_{1}(\mathbb{R},\mathcal{M})$, which is denoted by
$\mathsf{BMO}(\mathbb{R},\mathcal{M})$, in the spirit of the classical
argument by Fefferman and Stein [FS72]. Moreover, some maximal inequalities,
and several interpolation results via a martingale approach were established.
Mei’s fundamental contribution has been a key in the development of
noncommutative forms of Calderón-Zygmund theory, both in the semicommutative
context and in fully noncommutative ones via transference techniques. For the
first one, the semicommutative Calderón-Zygmund theory was initiated in [P09]
with the obtaining of weak-$L_{1}$ endpoint inequalities for singular
integrals, with an argument that was simplified in recent years [C18, CCP22].
The second line has many instances, among which are [GJP21, JMPX21].

The initial motivation for the present work was to obtain new interpolation
consequences of endpoint estimates of the type
$L_{\infty}(\mathcal{A})$-$\mathsf{BMO}(\mathbb{R},\mathcal{M})$ which rely,
by duality, on the structure of the Hardy space
$H_{1}(\mathbb{R},\mathcal{M})$. Our goal led to two main tasks to tackle: a
completely explicit description of $\mathsf{BMO}(\mathbb{R},\mathcal{M})$, and
the study of the boundedness of Calderón-Zygmund operators on the Hardy space
via its atomic decomposition. This may seem common but another of our goals is
to provide complete proofs; for instance, in many papers the extension from
atoms to the whole Hardy space is overlooked but requires more delicate
arguments than the main estimates (see [B05]). We also want to point out that
up to now there is no notion of noncommutative distributions, thus one has to
be careful when defining Calderón-Zygmund operators.

The operator-valued $\mathsf{BMO}$-space introduced by Mei,
$\mathsf{BMO}(\mathbb{R},\mathcal{M})$, is defined as the intersection of a
column space and a row space, $\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ and
$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ respectively. The reason for
considering both a column and a row space is a ubiquitous phenomenon in
noncommutative analysis (see [LP86] for an outstanding example), and by
symmetry we shall limit our discussion to the column case.
$\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ is set to be the subspace of the
column Hilbert-valued space ([JLX06])
$L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ on which
the seminorm

(MBMO)
$\displaystyle\|g\|_{\mathsf{BMO}_{c}}=\sup_{\begin{subarray}{c}I\subseteq\mathbb{R}\\\
|I|\
\mathrm{finite}\end{subarray}}\Big{\|}\Big{(}\frac{1}{|I|}\int_{I}\big{|}g-g_{I}\big{|}^{2}\Big{)}^{1/2}\Big{\|}_{\mathcal{M}}$

makes sense, where $g_{I}=\frac{1}{|I|}\int_{I}g$. The $\mathsf{BMO_{r}}$
seminorm is $\|g\|_{\mathsf{BMO}_{r}}=\|g^{*}\|_{\mathsf{BMO}_{c}}$. The space
$L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ denotes the
closure of $\mathcal{M}\otimes L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$ with
respect to the weak∗ topology of the von Neumann algebra
$\mathcal{M}\overline{\otimes}B(L_{2}(\mathbb{R},\frac{dt}{1+t^{2}}))$. There
is no guarantee that $\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ is a space of
$\mathcal{M}-$valued functions (in the Bochner sense), and so the integral in
(MBMO) may not be well-defined. Indeed,

$L_{2}\big{(}\mathbb{R},\frac{dt}{1+t^{2}};\mathcal{M}\big{)}\subset
L_{\infty}\big{(}\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}})\big{)},$

but the reverse inclusion fails in general, _a priori_ preventing us from
defining $\mathsf{BMO}_{c}$ as a space of functions. In section 1 and the
first part of section 2, we propose a general construction of
$\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ which recovers Mei’s description
given by (MBMO). We will also study a predual of
$\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ (resp.
$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$). The novelty here is that this
predual space, which we denote as $H_{1}^{r}(\mathcal{A})$ (resp.
$H_{1}^{c}(\mathcal{A})$), will be a row (resp. column) Hardy space which is
exclusively constructed in terms of “new” atomic decompositions, which extend
the work of the second author PhD Thesis [R01].

The key to our approach is the $H_{1}$-$\mathsf{BMO}$ duality product when
elements in the former space are described in terms of atoms. In the classical
case, it is a well-known fact that the norm of $g\in\mathsf{BMO}(\mathbb{R})$
can be characterized through the expression

(atBMO) $\displaystyle\|g\|_{\mathsf{BMO}}=\sup_{a}\big{|}\int ga\big{|},$

so that the supremum is taken over $L_{2}$-atoms [GR85, Mey90, MeyC97]. An
analogous formula for $\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ may shed
light on the structure of atoms in $H_{1}^{c}(\mathcal{A})$. This is exactly
what we achieve. First, it turns out that it is enough to consider only
elements $g\in\mathcal{A}\cap\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$. The
expression (MBMO) is meaningful for $g$, and so duality yields

$\displaystyle\|g\|_{\mathsf{BMO}_{r}}$
$\displaystyle=\|g^{*}\|_{\mathsf{BMO}_{c}}=\sup_{I}\Big{\|}\Big{(}\frac{1}{|I|}\int_{I}\big{|}g^{*}-(g^{*})_{I}\big{|}^{2}\Big{)}^{1/2}\Big{\|}_{\mathcal{M}}$
$\displaystyle=\sup_{I,\|h\|_{L_{2}(\mathcal{M})}\leq
1}\Big{(}\int_{I}\|\frac{1}{\sqrt{|I|}}h(g-g_{I})\|^{2}_{L_{2}(\mathcal{M})}\Big{)}^{1/2}.$

with the supremum taken over $h$ in the unit ball of $L_{2}(\mathcal{M})$.
Now, recalling that $g-g_{I}$ has zero integral over $I$, it follows that

$\|g\|_{\mathsf{BMO}_{r}}=\sup_{I,h}\big{\|}\frac{1}{\sqrt{|I|}}h(g-g_{I})\mbox{\large$\chi$}_{I}\big{\|}_{L_{2}^{\circ}(\mathbb{R};L_{2}(\mathcal{M})}=\sup_{I,h}\sup_{\|f\|_{L_{2}}\leq
1}\big{|}(\tau\circ\int)(h(g-g_{I})\frac{f\mbox{\large$\chi$}_{I}}{\sqrt{|I|}})\big{|}.$

Comparing the latter expression with (atBMO) suggests that an atom in
$H_{1}^{c}(\mathbb{R},\mathcal{M})$ should be an operator of the form $a=bh$
in $L_{1}(\mathcal{A})$, where $h\in L_{2}(\mathcal{M})$ with
$\|h\|_{L_{2}(\mathcal{M})}\leq 1$ and $b\in L_{2}(\mathcal{A})$ is supported
on some interval $I$ and has additional cancellation over $I$ that we will
make precise later. In what follows, $a$ will be called a _$c$ -atom_. Then,
define the column Hardy space $H_{1}^{c}(\mathcal{A})$ as the Banach subspace
of $L_{1}(\mathcal{A})$ of those operators $f$ such that

$\displaystyle f=\sum_{i=0}^{\infty}\lambda_{i}a_{i}\ \mathrm{in}\
L_{1}(\mathcal{A}),\ \mathrm{for\ some}\ (a_{i})_{i}\ c\textrm{-atoms},\
(\lambda_{i})_{i}\in\ell_{1},$

which becomes a Banach space with respect to the norm

$\displaystyle\|f\|_{H_{1}^{c}}=\inf\big{\\{}\sum_{i=0}^{\infty}|\lambda_{i}|\
:\ f=\sum_{i=0}^{\infty}\lambda_{i}a_{i}\big{\\}}.$

The row space $H_{1}^{r}(\mathcal{A})$ is defined analogously and
$H_{1}(\mathcal{A})=H_{1}^{c}(\mathcal{A})+H_{1}^{r}(\mathcal{A})$. By
symmetry, it suffices to show
$H_{1}^{c}(\mathcal{A})^{*}=\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$. The
proof of this duality result strongly relies on the extension of a well-known
argument by Meyer [Mey90]: the space

$\displaystyle L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt)=\big{\\{}f\in
L_{2}(\mathbb{R},(1+t^{2})dt)\ :\ \int_{\mathbb{R}}f=0\big{\\}}$

is a dense subspace of the classical atomic Hardy space $H_{1}(\mathbb{R})$. A
further characterization of the column Hilbert-valued spaces
$L_{\infty}(\mathcal{M};H^{c})$ will be the key to establish an analogous
result in our context. We will briefly explain how to get that the column
Hardy space $H_{1}^{c}(\mathcal{A})$ coincides with the one introduced by Mei
in [M07] in section 4. This will in particular allow us to indeed use
interpolation as usual.

It is worth noting that Mei’s work already contains a description of
$H_{1}^{c}(\mathbb{R},\mathcal{M})$ in terms of certain atomic decompositions.
However, the one included in the present paper seems more useful to deal with
Calderón-Zygmund operators and to make connections with vector valued harmonic
analysis. Indeed, it allows us to consider singular integrals with
noncommuting kernels, something that looks difficult to reach at the
weak-$L_{1}$ level [CCP22]. Let $\mathcal{M}$ be a von Neumann algebra over a
separable Hilbert space, we denote by $\mathcal{S}$ the set of compactly
supported essentially bounded functions $\mathbb{R}\to L_{\infty}\cap
L_{1}(\mathcal{M})$ ( measurable with values in $L_{1}$); it will play the
role of test functions. Let $T$ be a bounded operator on $L_{2}(\mathcal{A})$
for which there exists a kernel $K:\mathbb{R}\times\mathbb{R}\ \setminus\
\\{x=y\\}\longrightarrow\mathcal{M}$ such that

$\displaystyle\int T(f)(x)g(x)\ dx=\int\int K(x,y)f(y)g(x)\ dx\ dy$

holds for any $f,g\in\mathcal{S}$ satisfying that the distance between the
supports $\mathrm{supp}\|f\|_{L_{2}(\mathcal{M})}$ and
$\mathrm{supp}\|g\|_{L_{2}(\mathcal{M})}$ is strictly positive. In that case,
We say that $T$ is a Calderón-Zygmund operator with kernel $K$. Also, assume
that $T$ fulfills

  * •

a right-modularity condition: $T(fh)=T(f)h$ for any $f\in L_{2}(\mathcal{A})$
with compact support and $h\in\mathcal{M}$; we say that $T$ is a left
Calderón-Zygmund operator,

  * •

the _Hörmander condition_

$\displaystyle\int_{|x-y|\geq
2|y^{\prime}-y|}\|K(x,y)-K(x,y^{\prime})\|_{\mathcal{M}}\ dx<\infty.$

Under these assumptions, $T$ extends to a bounded map
$T:H_{1}^{c}(\mathcal{A})\longrightarrow L_{1}(\mathcal{A})$. The proof of
this statement is inspired by [MeyC97] and is divided in two steps. The first
one consists in obtaining an universal constant $C>0$ such that

$\displaystyle\|T(a)\|_{L_{1}(\mathcal{A})}\leq C\ \textrm{for \ any \
$c$-atom}\ a.$

In order to set up this statement, we first check that the extension of $T$ to
$c$-atoms satisfies

$\displaystyle T(a)=T(b)h\ \textrm{for \ any \ $c$-atom}\ a=bh.$

This allows us to exploit the boundedness of $T$ on $L_{2}(\mathcal{A})$. We
rely on an approximation of $T$ and $K$ by certain uniformly bounded kernels,
which also implies that $T$ extends to the whole $H_{1}^{c}(\mathcal{A})$.
Once we have done that, we obtain that whenever $K$ is scalar and modularity
works on both sides, then $T$ extends to a bounded operator from
$H_{1}(\mathcal{A})$ into $L_{1}(\mathcal{A})$. This is somehow a dual
statement of a result in [JMP14] where $L_{\infty}$-BMO estimates are given
and in spirit close to [HLMP14].

The rest of the paper is organized as follows. In section 1, we shall
introduce the column and row Hilbert-valued noncommutative $L_{p}$-spaces.
This enables us to define $\mathsf{BMO}(\mathbb{R,\mathcal{M}})$, as well as
to identify a predual $H_{1}^{c}(\mathcal{A})$ in section 2. In section 3 we
will see how the atomic decomposition provides a boundedness result for
Calderón-Zygmund operators with noncommuting kernels. We then briefly explain
in section 4 how to connect our construction to Mei’s one. We end up with some
technical but useful lemmas in the appendices.

##  1\. Column/row Hilbert-valued $L_{p}$-spaces

In this section we collect some basic facts on classical vector-valued
$L_{p}$-spaces [HNVW16], as well as its interaction with row and column
Hilbert-valued $L_{p}$-spaces [JLX06].

Before giving definitions, some facts about vector-valued functions and von
Neumann algebras should be tackled (see [DU77, Chapter 2] and [S71, Section
1.22]). Let $\mathbb{X}$ be a Banach space, and let $(\Omega,\mu)$ be a
$\sigma$-finite measure space (or more generally, a localizable measure space
[S73]). Then, a function $f:\Omega\longrightarrow\mathbb{X}$ is said to be
_$\mu$ -measurable_ whenever there exists a sequence of simple functions
$(f_{n})_{n\geq 1}$, $f_{n}=\sum_{i}x_{i}^{n}\
\mbox{\large$\chi$}_{A_{i}^{n}}$ for some $x_{i}^{n}\in\mathbb{X}$ and
$\mu-$measurable sets $A_{i}^{n}$, such that

$\displaystyle\lim_{n\rightarrow\infty}\|f_{n}-f\|_{\mathbb{X}}=0\
\mu\mathrm{-almost\ everywhere}.$

For $1\leq p<\infty$, one can then define $L_{p}(\Omega,\mu;\mathbb{X})$ as
the space of $\mu$-measurable functions $f$ for which
$\|f(\cdot)\|_{\mathbb{X}}\in L_{p}(\Omega,\mu)$ with the norm
$\|f\|_{p}=\|\|f(\cdot)\|_{\mathbb{X}}\|_{L_{p}(\Omega,\mu)}$. In our
situation, we recall the classical identities:

$\displaystyle
L_{1}(\Omega,\mu;L_{1}(\mathcal{M}))=L_{1}(L_{\infty}(\Omega,\mu)\overline{\otimes}\mathcal{M})=L_{1}(\Omega,\mu)\widehat{\otimes}_{\pi}L_{1}(\mathcal{M});$
$\displaystyle
L_{2}(\Omega,\mu;L_{2}(\mathcal{M}))=L_{2}(L_{\infty}(\Omega,\mu)\overline{\otimes}\mathcal{M})=L_{2}(\Omega,\mu)\otimes_{2}L_{2}(\mathcal{M}),$

where $\widehat{\otimes}_{\pi}$ and $\otimes_{2}$ are the Banach space
projective tensor product and the Hilbert space tensor product. On the other
hand, the identification of the von Neumann algebra
$L_{\infty}(\Omega,\mu)\overline{\otimes}\mathcal{M}$ as a space of vector-
valued functions requires a more subtle construction. In general, a function
$f:\Omega\longrightarrow\mathbb{X}^{*}$ is said to be _weak ∗
$\mu$-measurable_ if $J_{x}\circ f$ is measurable for each $x\in\mathbb{X}$,
where $J_{x}$ denotes the continuous functional on $\mathbb{X}^{*}$ given by
$J_{x}(x^{*})=x^{*}(x)$ for every $x^{*}$ in $\mathbb{X}^{*}$. Consider
$\mathbb{X}=L_{1}(\mathcal{M})$ (which is assumed to be separable to avoid
measurability issues) and define $L_{\infty}(\Omega,\mu;\mathcal{M})$ as the
Banach space of all $\mathcal{M}$-valued weak∗ $\mu-$measurable functions
which are essentially bounded, that is,

$\displaystyle{\mathrm{ess}\sup}_{t\in\Omega}\|f(t)\|_{\mathcal{M}}<\infty.$

Then, $L_{\infty}(\Omega,\mu;\mathcal{M})$ is a von Neumann algebra under the
pointwise multiplication and the map

$\displaystyle f\otimes m\mapsto f(t)m,\quad f\in L_{\infty}(\Omega),\
m\in\mathcal{M},$

can be extended to a weak∗ isomorphism of
$L_{\infty}(\Omega,\mu)\overline{\otimes}\mathcal{M}$ onto
$L_{\infty}(\Omega,\mu;\mathcal{M})$.

Let $H$ be a separable Hilbert space. $B(H)$ can be identified as the space of
bounded infinite matrices acting on $H$ (once a basis is fixed). When equipped
with the usual trace for matrices $\mathrm{Tr}$, it gives rise to the Schatten
classes $S_{p}(H)=L_{p}(B(H),\mathrm{Tr})$ for any $0<p\leq\infty$. Along this
work, the inner product in $H$ is assumed to be linear in the second variable
and antilinear in the first one. Moreover, elements of $H$ viewed in the dual
Hilbert space $H^{*}$ will be represented with an overlined letter. We mean
that $\overline{h}$ will denote the continuous functional

$\displaystyle\overline{h}:k\in H\mapsto\langle h,k\rangle\ \mathrm{for\ any\
}k\in H.$

Given two elements $\xi$ and $\eta$ of $H$, we consider the _rank-one
operator_ $\xi\otimes\overline{\eta}$ acting on $H$ as follows

$\displaystyle(\xi\otimes\overline{\eta})(h)=\langle\eta,h\rangle\ \xi\
\mathrm{for\ any\ }h\in H.$

In the following, let $\mathds{1}$ denote a fixed element of $H$ with
$\|\mathds{1}\|_{H}=1$, and let
$p_{\mathds{1}}=\mathds{1}\otimes\overline{\mathds{1}}$ denote the rank one
projection onto $\mathrm{span}\\{\mathds{1}\\}$. Assume that $\mathcal{M}$ is
an arbitrary semifinite von Neumann algebra equipped with a normal semifinite
faithful trace $\tau$. Then, we define the _column Hilbert-valued
$L_{p}$-space_

$\displaystyle
L_{p}(\mathcal{M};H^{c})=L_{p}(\mathcal{M}\overline{\otimes}B(H))(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}}).$

for any $0<p\leq\infty$. Identify $L_{p}(\mathcal{M})$ as a subspace of
$L_{p}(\mathcal{M}\overline{\otimes}B(H))$ via the map $m\mapsto m\otimes
p_{\mathds{1}}=(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})(m\otimes\mathbf{1}_{B(H)})(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})$. This is equivalent to the identity

$\displaystyle L_{p}(\mathcal{M})=(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})L_{p}(\mathcal{M}\overline{\otimes}B(H))(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}}).$

Then, given an element $f$ in $L_{p}(\mathcal{M};H^{c})$,

$f^{*}f\in(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})L_{p/2}(\mathcal{M}\overline{\otimes}B(H))(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})=L_{p/2}(\mathcal{M}).$

This justifies defining

$\displaystyle\|f\|_{L_{p}(\mathcal{M};H^{c})}=\|f\|_{\
L_{p}(\mathcal{M}\overline{\otimes}B(H))}=\|(f^{*}f)^{1/2}\|_{L_{p}(\mathcal{M})}$

on $L_{p}(\mathcal{M};H^{c})$. Analogously, the _row Hilbert-valued
$L_{p}$-space_ is

$\displaystyle L_{p}(\mathcal{M};H^{*r})=(\mathbf{1}_{\mathcal{M}}\otimes
p_{\mathds{1}})L_{p}(\mathcal{M}\overline{\otimes}B(H)).$

equipped with the norm

$\displaystyle\|f\|_{L_{p}(\mathcal{M};H^{*r})}=\|(ff^{*})^{1/2}\|_{L_{p}(\mathcal{M})}=\|f^{*}\|_{L_{p}(\mathcal{M};H^{c})}.$

Given elements $h\in H$ and $m\in L_{p}(\mathcal{M})$, we will write $m\otimes
h\in L_{p}(\mathcal{M})\otimes H$ for the element
$m\otimes(h\otimes\mathds{1})\in L_{p}(\mathcal{M};H^{c})$ and similarly for
$m\otimes\overline{h}$ for $m\otimes(\mathds{1}\otimes\overline{h})\in
L_{p}(\mathcal{M};H^{*r})$. With these notation, for
$f=\sum_{i=1}^{n}m_{i}\otimes h_{i}$ and
$g=\sum_{i=1}^{n}m_{i}\otimes\overline{h}_{i}$:

$\displaystyle\|f\|_{L_{p}(\mathcal{M};H^{c})}$ $\displaystyle=$
$\displaystyle\Big{\|}\Big{(}\sum_{i,j=1}^{n}\langle h_{i},h_{j}\rangle_{H}\
m_{i}^{*}m_{j}\Big{)}^{1/2}\Big{\|}_{L_{p}(\mathcal{M})},$
$\displaystyle\|g\|_{L_{p}(\mathcal{M};H^{*r})}$ $\displaystyle=$
$\displaystyle\Big{\|}\Big{(}\sum_{i,j=1}^{n}\langle h_{i},h_{j}\rangle_{H}\
m_{i}m_{j}^{*}\Big{)}^{1/2}\Big{\|}_{L_{p}(\mathcal{M})}.$

In this way, we do not need to refer to the function $\mathds{1}$. We will use
without reference that $L_{p}(\mathcal{M})\otimes H$ is dense (resp. weak∗
dense) in $L_{p}(\mathcal{M};H^{c})$ for $1\leq p<\infty$ (resp. $p=\infty$)
and similarly for rows. Moreover, we have that $(m\otimes
h)^{*}=m^{*}\otimes\overline{h}$ as operators.

In fact, column and row Hilbert-valued $L_{p}$-spaces satisfy the expected
duality relations expressed via the natural duality bracket

(1) $\displaystyle\langle f,g\rangle_{r,c}=\mathrm{Tr\otimes\tau}(fg),$
$\displaystyle\langle m\otimes\overline{h},m^{\prime}\otimes
h^{\prime}\rangle_{r,c}=\tau(mm^{\prime})\langle h,h^{\prime}\rangle.$

In particular, it holds linearly isometrically

$\displaystyle
L_{p}(\mathcal{M};H^{c})^{*}=L_{p^{\prime}}(\mathcal{M};H^{*r})\ \mathrm{and}\
L_{p}(\mathcal{M};H^{*r})^{*}=L_{p^{\prime}}(\mathcal{M};H^{c}).$

for any $1\leq p<\infty$ whenever $1/p+1/p^{\prime}=1$.

We will use without reference that $L_{p}(\mathcal{M})\otimes H$ is dense
(resp. weak∗ dense) in $L_{p}(\mathcal{M};H^{c})$ for $1\leq p<\infty$ (resp.
$p=\infty$) and similarly for rows. We also recall the following fact about
homogeneity of column and row Hilbert spaces, see [JLX06, Lemma 2.4]. We state
it only for columns.

######  Corollary 1.1.

Let $H$ and $K$ be two Hilbert spaces, and let $T:H\longrightarrow K$ be a
bounded linear operator. Then $\mathrm{Id}_{\mathcal{M}}\otimes T$ admits a
unique continuous extension (resp. weak∗ continuous) $\widetilde{T}$ from
$L_{p}(\mathcal{M};H^{c})$ into $L_{p}(\mathcal{M};K^{c})$ for $1\leq
p<\infty$ (resp. $p=\infty)$ with the same norm.

Some obvious properties of these extension maps will be crucial in the
following sections.

######  Lemma 1.2.

Let $H$ and $K$ be two Hilbert spaces, let $S$, $T$, and
$(T_{j})_{j=1}^{\infty}$ be some bounded operators from $H$ to $K$ and let
$\widetilde{S},\widetilde{T},(\widetilde{T_{j}})_{j=1}^{\infty}$ be the
corresponding extensions from $L_{p}(\mathcal{M};H^{c})$ to
$L_{p}(\mathcal{M};K^{c})$. Then the following holds.

  1. (1)

$\widetilde{ST}=\widetilde{S}\widetilde{T}$,

  2. (2)

If $S$ and $T$ commute, then $\widetilde{S}$ and $\widetilde{T}$ also commute,

  3. (3)

whenever $\sum_{j=1}^{\infty}T_{j}$ converges in the norm of $B(H)$, there
holds

$\displaystyle\widetilde{\sum_{j=1}^{\infty}T_{j}}=\sum_{j=1}^{\infty}\widetilde{T_{j}}.$

  4. (4)

$({\widetilde{S}})^{*}=\widetilde{\overline{S}^{*}}$ for $p<\infty$.

In the last statement, one needs a conjugation because the ∗ on the left is
for the Banach space adjoint whereas the right one is the Hilbert space sense.
Of course similarly statements hold for row spaces considering bounded maps
$T:H^{*}\to K^{*}$.

###  1.1. Noncommutative spaces $L_{p}(\mathcal{M};L^{c}_{2}(\Omega))$

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space. A remarkable setting
for noncommutative Hilbert-valued column/row $L_{p}$-spaces is the case
$H=L_{2}(\Omega):=L_{2}(\Omega,\mu)$. Notice that under these conditions, the
duality bracket (1) is given by the expression

$\displaystyle\langle m_{1}\otimes\overline{f}_{1},m_{2}\otimes
f_{2}\rangle_{r,c}$ $\displaystyle=\tau_{\mathcal{M}}(m_{1}m_{2})\
\int_{\Omega}\overline{f_{1}}f_{2}\ d\mu.$

In particular, identifying $L_{2}(\Omega,\mu)^{*}$ and $L_{2}(\Omega,\mu)$ and
using the bilinear pairing $(f,g)\mapsto\int_{\Omega}fg\ d\mu$, then

$\displaystyle
L_{p^{\prime}}(\mathcal{M};L_{2}^{r}(\Omega,\mu))=L_{p}(\mathcal{M};L_{2}^{c}(\Omega,\mu))^{*}\
\mathrm{for\ }1\leq p<\infty.$

This duality identity will allow us to drop the conjugation or the involution
$*$ from now on. Moreover, for $F=\sum_{i=1}^{n}m_{i}\otimes f_{i}\in
L_{p}(\mathcal{M})\otimes L_{2}(\Omega)$ with $p<\infty$:

$\|F\|^{p}_{L_{p}(\mathcal{M};L_{2}^{c}(\Omega,\mu))}=\tau\Big{(}\int_{\Omega}\Big{|}\sum_{i=1}^{n}f_{i}(t)m_{i}\Big{|}^{2}\Big{)}^{p/2}$

Since $\sum_{i=1}^{n}f_{i}(t)m_{i}$ can be interpreted as $F(t)$, it is very
tempting to consider elements $L_{p}(\mathcal{M};L_{2}^{c}(\Omega))$ as
functions. Indeed for $p\leq 2$ (see [JLX06, Proposition 2.5]):

######  Proposition 1.3.

Let $1\leq p\leq 2$, the identity on $L_{p}(\mathcal{M})\otimes L_{2}(\Omega)$
extends to an injective contraction $L_{p}(\mathcal{M};L_{2}^{c}(\Omega))\to
L_{2}(\Omega;L_{p}(\mathcal{M}))$.

###### Proof.

First note that by definition

$L_{2}(\mathcal{M})\otimes_{2}L_{2}(\Omega)=L_{2}(\mathcal{M};L_{2}^{c}(\Omega))=L_{2}(\Omega;L_{2}(\mathcal{M})).$

Let $p<2$ and set $q$ be so that $\frac{1}{p}=\frac{1}{2}+\frac{1}{q}$. Take
$F=\sum_{i=1}^{n}m_{i}\otimes f_{i}\in L_{p}(\mathcal{M})\otimes
L_{2}(\Omega)\subset L_{p}(\mathcal{M};L_{2}^{c}(\Omega))$, it corresponds to
an element of $L_{p}(\mathcal{M}\overline{\otimes}B(L_{2}(\Omega)))$. Its
modulus $|F|^{2}=\int_{\Omega}|F(t)|^{2}d\mu$ falls into
$L_{p/2}(\mathcal{M})$, thus using its polar decomposition it can be written
as $F=ab$ with $b\in L_{q}(\mathcal{M})$ and $a\in
L_{2}(\mathcal{M};L_{2}^{c}(\Omega))$ with
$\|a\|_{2}\cdot\|b\|_{q}=\|F\|_{p}$. Actually $a$ is also a simple function
and

$\int_{\Omega}\|F(t)\|_{p}^{2}d\mu\leq\|b\|_{q}^{2}\int_{\Omega}\|a(t)\|_{2}^{2}d\mu=\|F\|_{p}^{2}.$

This shows that the identity is indeed a contraction and thus extends to a
contraction $\iota$ by density.

To show the injectivity, consider $m\in L_{q}(\mathcal{M})$ and $h\in
L_{2}(\Omega)$. The linear form associated to $m\otimes h$ satisfies $\langle
F,m\otimes h\rangle_{c,r}=\int_{\Omega}\tau(\iota(F)(t)m)h(t)d\mu$. Indeed,
this is clear for $F\in L_{p}(\mathcal{M})\otimes_{2}L_{2}(\Omega)$, and for
all $F$ by density. Since $L_{q}(\mathcal{M})\otimes L_{2}(\Omega)$ is norm-
dense in $L_{q}(\mathcal{M};L_{2}^{r}(\Omega))$, we can conclude. ∎

As a consequence, we can identify elements in
$L_{p}(\mathcal{M};L_{2}^{c}(\Omega))$ with a.e. Bochner measurable functions
from $\Omega$ to $L_{p}(\mathcal{M})$ when $1\leq p\leq 2$. This will be
convenient for some identifications. Unfortunately, when $p>2$ we have no way
to consider elements in $L_{p}(\mathcal{M};L_{2}^{c}(\Omega))$ as functions.
Indeed, in the following sections, the case $p=\infty$ will be specially
relevant. For that reason, the extension of some useful operators on
$L_{2}(\Omega)$ to $L_{p}(\mathcal{M};L_{2}^{c}(\Omega))$ will be carefully
studied.

######  Lemma 1.4.

Let $A,B$ be two measurable sets, and let $w$ and $w^{\prime}$ be strictly
positive functions belonging to $L_{\infty}(\Omega,\mu)$. Consider the
following operators on $L_{2}(\Omega,\mu)$:

$\displaystyle T_{w}$ $\displaystyle:f\mapsto w^{1/2}f,$ $\displaystyle P_{A}$
$\displaystyle:f\mapsto\mbox{\large$\chi$}_{A}f.$

These maps extend to bounded operators on
$L_{p}(\mathcal{M};L^{t}_{2}(\Omega,\mu))$, for $t=c,r$ and any $1\leq
p\leq\infty$ such that $\|\widetilde{T}_{w}\|=\|w\|_{L_{\infty}(\Omega,\mu)}$,
$\|\widetilde{P}_{A}\|=\|\mbox{\large$\chi$}_{A}\|_{L_{\infty}(\Omega,\mu)}$.
Moreover, they satisfy the following relations:

  1. (1)

$\widetilde{T}_{w}\widetilde{T}_{w^{\prime}}=\widetilde{T}_{w^{\prime}}\widetilde{T}_{w}$,

  2. (2)

$\widetilde{P}_{A}\widetilde{T}_{w}=\widetilde{T}_{w}\widetilde{P}_{A}$,

  3. (3)

whenever $w^{-1}$ is bounded,
$\widetilde{T}_{w}\widetilde{T}_{w^{-1}}=\mathrm{Id}=\widetilde{T}_{w^{-1}}\widetilde{T}_{w}$.

  4. (4)

$\widetilde{P}_{A}=\widetilde{P}_{A}\widetilde{P}_{B}=\widetilde{P}_{B}\widetilde{P}_{A}$
whenever $A\subseteq B$,

  5. (5)

$\widetilde{P}_{B\setminus A}=\widetilde{P}_{B}-\widetilde{P}_{A}$ whenever
$A\subseteq B$.

###### Proof.

By Corollary 1.1, the extension operators $\widetilde{T}_{w}$,
$\widetilde{P}_{A}$ are bounded as long as the original ones are bounded on
$L_{2}(\Omega,\mu)$. The maps $T_{w}$ and $P_{A}$ are bounded with norm
$\|w^{1/2}\|_{L_{\infty}}$ and $1$ respectively, since they are pointwise
multiplication operators. Claims (1)-(4) follow from Lemma 1.2, while
linearity of the map $T\mapsto\widetilde{T}$ implies (5). ∎

##  2\. Duality between Hardy spaces and $\mathsf{BMO}$-spaces

Consider the measure space $(\mathbb{R},\frac{dt}{1+t^{2}})$. Set
$\omega(t)=1+t^{2}$. Then, $L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$ is a Hilbert
space with the inner product

$\langle g,f\rangle_{1/\omega}:=\int_{\mathbb{R}}\overline{g(t)}\ f(t)\
\frac{dt}{1+t^{2}}.$

We will consider the associated column space
$L_{p}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ for
$0<p\leq\infty$. We will choose $\mathds{1}$ to be the constant function
$1/\sqrt{\pi}$, which satisfies the condition
$\|\mathds{1}\|_{L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})}=1$. For the sake of
exposition, we define some operators on
$L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ which can
be described in terms of the maps appearing in Lemma 1.4. Let $A$ be a
measurable set with nonzero measure and define the map

$R_{A}=\frac{1}{|A|}\widetilde{T}_{\omega}\widetilde{P}_{A}$

so that $R_{A}$ is the extension of the operator acting on
$L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$ as follows:

$f\mapsto\frac{(1+t^{2})^{1/2}}{\sqrt{|A|}}\mbox{\large$\chi$}_{A}f.$

On the other hand, denote by $a_{A}$ the extension to
$L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ of the map

$a_{A}:f\mapsto f_{A}\ 1=\Big{(}\frac{1}{|A|}\int_{A}f\Big{)}\ 1.$

Similarly the linear form $f\mapsto f_{A}$, has a weak∗ extension from
$L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ to
$\mathcal{M}=L_{\infty}(\mathcal{M};\mathbb{C}^{c})$. Thus we may also use the
notation $f_{A}$ for $f\in
L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$.

Some other Hilbert spaces over the real line will be considered through this
work. Since the distinguished vectors are not relevant, we will always write
them $\mathds{1}$ in all spaces.

######  Lemma 2.1.

Let $A$ be a measurable set. Then, the pre-adjoint maps for $R_{A}$ and
$a_{A}$ on $L_{1}(\mathcal{M};L_{2}^{r}(\mathbb{R},dt/(1+t^{2})))$ act as
follows on any operator $m\otimes f\in L_{1}(\mathcal{M})\otimes
L_{2}(\mathbb{R},dt/(1+t^{2}))$

$\displaystyle(R_{A})_{*}$ $\displaystyle:m\otimes f\longmapsto
m\otimes\frac{\sqrt{1+t^{2}}}{\sqrt{|A|}}\mbox{\large$\chi$}_{I}f$
$\displaystyle(a_{A})_{*}$ $\displaystyle:m\otimes f\longmapsto
m\otimes\langle
1,f\rangle_{1/\omega}\frac{1+t^{2}}{|A|}\mbox{\large$\chi$}_{A}.$

Moreover, the identity operator

$\begin{array}[]{cccc}V&:L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},(1+t^{2})dt))&\longmapsto&L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}}))\\\
&m\otimes f&\longmapsto&m\otimes f,\end{array}$

is contractive and admits a pre-adjoint

$\begin{array}[]{cccc}V_{*}&:L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))&\longmapsto&L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},(1+t^{2})dt))\\\
&m\otimes f&\longmapsto&m\otimes\frac{f}{1+t^{2}}.\end{array}$

###### Proof.

These are consequences of the same results on Hilbert spaces using Proposition
1.2. One just need to care of conjugations which actually play no role here. ∎

Now, we are ready to define the column and row $\mathsf{BMO}$-spaces.

######  Definition 2.2.

Given a von Neumann algebra $\mathcal{M}$ with n.s.f. trace $\tau$, set the
_column $\mathsf{BMO}$-space_, denoted as
$\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$, to be the subspace of operators
$f$ in $L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$
satisfying

(2)
$\displaystyle\|f\|_{\mathsf{BMO}_{c}}:=\sup_{I\subseteq\mathbb{R}}\big{\|}R_{I}(\mathrm{Id}-a_{I})f\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R};\frac{dt}{1+t^{2}}))}<\infty,$

where the supremum is considered over finite intervals $I$ of $\mathbb{R}$.
Likewise, the row $\mathsf{BMO}$-space,
$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$, is the subspace of elements in
$L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}}))$ for which
the norm $\|f\|_{\mathsf{BMO}_{r}}:=\|f^{*}\|_{\mathsf{BMO}_{c}}$ is finite.

It is clear that $\|\cdot\|_{\mathsf{BMO}_{c}}$ is a norm modulo
$\mathcal{M}$. This expression will be convenient for abstract questions. We
point out that it admits a much more tractable form. The multiplication by
$\sqrt{1+t^{2}}$ is just an isometry from $L_{2}(\mathbb{R})$ to
$L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$. For a finite interval $I$, let
$\widetilde{\iota}_{I}$ denote the extension of the map
$f\mapsto\mbox{\large$\chi$}_{I}f$ from $L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$
to $L_{2}(\mathbb{R})$, then clearly:

(3)
$\displaystyle\Big{\|}R_{I}(\mathrm{Id}-a_{I})f\Big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}=\Big{\|}\frac{1}{\sqrt{|I|}}\big{(}\widetilde{\iota}_{I}\big{(}f-(f_{I}\otimes
1)\big{)}\big{)}\Big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}$

In particular, for operators in $\mathcal{M}\otimes
L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$, we recover the expression which
determines the definition for the $\mathsf{BMO}_{c}$-norm introduced in [M07]:

######  Lemma 2.3.

For any operator $f$ in $\mathcal{M}\otimes
L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$ it holds

$\|f\|_{\mathsf{BMO}_{c}}=\sup_{I\subseteq\mathbb{R}}\Bigg{\|}\Bigg{(}\frac{1}{|I|}\int_{I}|f-f_{I}|^{2}\Bigg{)}^{\frac{1}{2}}\Bigg{\|}_{\mathcal{M}}.$

Along the next section, the study of the boundedness of Calderón-Zygmund
operators on operator-valued Hardy spaces will require a concrete formulation
in terms of atomic decompositions. In order to justify introducing these
spaces, we will check that the dual of this new description of the column
(resp. row) Hardy space coincides with
$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ (resp.
$\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$).

######  Definition 2.4.

Let $\mathcal{M}$ be a von Neumann algebra with n.s.f. trace. A _$c$ -atom_ is
a function $a\in L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$
which admits a factorization of the form $a=bh$ for some function
$b:\mathbb{R}\rightarrow L_{2}(\mathcal{M})$ and an norm-one operator $h\in
L_{2}(\mathcal{M})$, satisfying

  1. (1)

$\mathrm{supp}_{\mathbb{R}}(b)\subseteq I$ for some interval $I$,

  2. (2)

$\int_{I}b=0$,

  3. (3)

$\|b\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\leq\frac{1}{\sqrt{|I|}}$.

Then, the _column Hardy space_ $H_{1}^{c}(\mathcal{A})$ is defined to be the
subspace of elements in
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ of the form

(4) $\displaystyle\sum\limits_{i=0}^{\infty}\lambda_{i}a_{i}\ \mathrm{where\
}(\lambda_{i})_{i}\in\ell_{1}\ \mathrm{and}\ (a_{i})_{i}\
c\mbox{-}\mathrm{atoms}$

with respect to the norm

$\displaystyle\|f\|=\inf\\{\sum_{i=0}^{\infty}|\lambda_{i}|\ :\
f=\sum_{i=0}^{\infty}\lambda_{i}a_{i}\ \mathrm{in\
}L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}),\
(\lambda_{i})_{i}\in\ell_{1},\ (a_{i})_{i}\ c\mbox{-}\mathrm{atoms}\\}.$

Under the above definition, any $c$-atom satisfies

$\|a\|_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}\leq 1$

since, by the Hölder inequality,

$\displaystyle\|a\|_{L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})}\leq\|b\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\
\|h\mbox{\large$\chi$}_{I}\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\leq|I|^{-1/2}\
|I|^{1/2}=1.$

Therefore, $H_{1}^{c}(\mathcal{A})$ is contractively contained into
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$.

We choose to give the explicit decomposition of atoms as $a=bh$ rather than
just saying $a\in L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$ with norm less
than $\frac{1}{\sqrt{|I|}}$ and mean zero. We do so as this makes explicit the
connection with vector valued harmonic analysis and will make all the proofs
transparent. We leave the following to the reader:

######  Proposition 2.5.

The column Hardy space $(H_{1}^{c}(\mathcal{A}),\|\cdot\|_{H_{1}^{c}})$ is a
Banach space.

Given a Banach space $\mathbb{X}$, let $L_{2}^{\circ}(\mathbb{R},(1+t^{2})\
dt;\mathbb{X})$ denote the subspace of functions $f$ in
$L_{2}(\mathbb{R},(1+t^{2})\ dt;\mathbb{X})$ satisfying

$\displaystyle\int_{\mathbb{R}}f(t)\ dt=0.$

Then, the classical argument by Meyer [Mey90, Chapter 5, Proposition 1]
extends to the Banach-valued setting yielding the inclusion as a subspace of
$L_{2}^{\circ}(\mathbb{R},(1+t^{2})\ dt;\mathbb{X})$ into the vector-valued
Hardy space $H_{1}(\mathbb{R};\mathbb{X})$ [H06]. More clearly, given $f\in
L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt;L_{2}(\mathcal{M}))$, there exists a
sequence of atoms $(b_{i})_{i}\subseteq H_{1}(\mathbb{R};L_{2}(\mathcal{M}))$
and $(\lambda_{i})_{i}\in\ell_{1}$ such that

(5) $\displaystyle f=\sum_{i=0}^{\infty}\lambda_{i}b_{i}\ \mathrm{in}\
L_{1}(\mathbb{R};L_{2}(\mathcal{M}))\ \mathrm{and}\
\sum_{i=0}^{\infty}|\lambda_{i}|\lesssim\|f\|_{L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt;L_{2}(\mathcal{M}))}.$

Since any $c$-atom $a=bh$ is the product of an $L_{2}$-atom $b$ in
$H_{1}(\mathbb{R};L_{2}(\mathcal{M}))$ and an element $h$ in
$L_{2}(\mathcal{M})$, the argument by Meyer still works in the semicommutative
case.

######  Proposition 2.6.

The formal identity map

$L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt)\otimes L_{1}(\mathcal{M})\to
L_{1}(\mathbb{R})\otimes L_{1}(\mathcal{M})$

extends to an injective and contractive map
$Q:L_{1}(\mathcal{M};L^{\circ,c}_{2}(\mathbb{R},(1+t^{2})dt))\to
H_{1}^{c}(\mathcal{A})$.

###### Proof.

First note that simple tensors $f\otimes m\in
L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt)\otimes L_{1}(\mathcal{M})$ are both in
$H_{1}^{c}(\mathcal{A})$ and in $L_{1}(\mathbb{R})\otimes L_{1}(\mathcal{M})$.
Indeed by Meyer’s result in the scalar case, $f=\sum_{i}\lambda_{i}a_{i}$
where $a_{i}$ are scalar valued atoms. Choose $\alpha,\beta\in
L_{2}(\mathcal{M})$ with $m=\alpha\beta$ and $\|\alpha\|_{2}=1$. Then
$a_{i}\otimes\alpha$ are atoms in $H_{1}(\mathbb{R};L_{2}(\mathcal{M}))$, so
$a_{i}\otimes m$ is a multiple of a $c$-atom and $f\otimes m$ is in
$H_{1}^{c}(\mathcal{A})$.

The end of the argument is as in Proposition 1.3. Any simple
$x=\sum_{i=1}^{n}f_{i}\otimes m_{i}\in
L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt)\otimes L_{1}(\mathcal{M})$ can be
written as $x=F\beta$ , with $\beta\in L_{2}(\mathcal{M})$ and a simple tensor
$F\in
L_{2}(\mathcal{M};L^{\circ,c}_{2}(\mathbb{R},(1+t^{2})dt))=L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt;L_{2}(\mathcal{M}))$
such that
$\|F\|_{2}\cdot\|\beta\|_{2}=\|x\|_{L_{2}(\mathcal{M};L^{\circ,c}_{2}(\mathbb{R},(1+t^{2})dt))}$.
Using Meyers’ decomposition (5) for $F$ gives the norm estimate. One concludes
to the boundedness of the extension by density. The injectivity follows by the
weak∗ density of simple tensors in the dual spaces as in 1.3. ∎

Consequently any linear form on $H_{1}^{c}(\mathcal{A})$ induces another one
on $L_{1}(\mathcal{M};L^{\circ,c}_{2}(\mathbb{R},(1+t^{2})dt))$. Precomposing
it with the map $V_{*}$ of Lemma 2.1 (which obviously has dense range), this
allows us to represent $H_{1}^{c}(\mathcal{A})^{*}$ as a subspace of
$L_{1}(\mathcal{M};L^{\circ,c}_{2}(\mathbb{R},1/(1+t^{2})dt))^{*}\subset
L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},1/(1+t^{2})dt))$.

######  Theorem 2.7.

Given a semifinite von Neumann algebra $\mathcal{M}$, we have a contractive
inclusion

$H_{1}^{c}(\mathcal{A})^{*}\subseteq\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M}).$

###### Proof.

Let $g\in H_{1}^{c}(\mathcal{A})^{*}$, we already explained that $VQ^{*}(g)\in
L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},1/(1+t^{2})dt))$. We estimate its
$\mathsf{BMO}_{r}$ norm,

$\displaystyle\|VQ^{*}g\|_{\mathsf{BMO}_{r}}$
$\displaystyle=\sup_{I\subseteq\mathbb{R}}\|R_{I}(\mathrm{Id}-a_{I})(VQ^{*}g)\|_{L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}}))}$
$\displaystyle=\sup_{I}\sup_{f}|\langle
f,R_{I}(\mathrm{Id}-a_{I})(VQ^{*}g)\rangle|$

where the supremum is taken over $f$ in the unit ball of
$L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$. We can as well
take it on simple tensors by density (we could as well consider it as a
function with values in $L_{1}(\mathcal{M})$). Moreover by the factorization
argument of Proposition 1.3, we can even write it as
$f=\sum_{i=1}^{n}m_{i}h\otimes f_{i}$ with $f_{i}\in
L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$, $m_{i},h\in L_{2}(\mathcal{M})$ such
that $\|h\|_{2}=\|\sum_{i=1}^{n}m_{i}\otimes
f_{i}\|_{L_{2}(\mathbb{R},\frac{dt}{1+t^{2}};L_{2}(\mathcal{M}))}=1$. Using
the formulas of Lemma 2.1,

$QV_{*}(\mathrm{Id}-a_{I})_{*}R_{I*}(f)=\sum_{i=1}^{n}m_{i}h\otimes F_{i}$

where for $i=1,...,n$

$\displaystyle
F_{i}=\frac{\mbox{\large$\chi$}_{I}}{\sqrt{|I|}\sqrt{1+t^{2}}}f_{i}$
$\displaystyle-\frac{\mbox{\large$\chi$}_{I}}{|I|}\langle
1,\frac{\sqrt{1+t^{2}}}{\sqrt{|I|}}\mbox{\large$\chi$}_{I}f_{i}\rangle_{1/\omega}\in
L_{2}(\mathbb{R},(1+t^{2})dt).$

Set $b=\sum_{i=1}^{n}m_{i}\otimes F_{i}\in
L_{2}(\mathcal{M};L_{2}(\mathbb{R},(1+t^{2})dt))$. Then, clearly
$\mbox{supp}_{\mathbb{R}}(b)\subseteq I$, $\int_{I}b=0$ since
$\int_{I}F_{i}=0$. Let
$G=\frac{\mbox{\large$\chi$}_{I}}{\sqrt{|I|}\sqrt{1+t^{2}}}\sum_{i=1}^{n}m_{i}\otimes
f_{i}$. Using that $b$ is the projection of $G$ onto the orthogonal of
constant functions on $I$, one gets

$\Big{(}\int_{I}\big{\|}b(t)\big{\|}_{2}^{2}\
dt\Big{)}^{1/2}\leq\big{\|}G\big{\|}_{L_{2}(\mathbb{R})}\leq\frac{1}{\sqrt{|I|}}\Big{\|}\sum_{i=1}^{n}m_{i}\otimes
f_{i}\Big{\|}_{L_{2}(\mathbb{R},\frac{dt}{1+t^{2}};L_{2}(\mathcal{M}))}\leq\frac{1}{\sqrt{|I|}}.$

In other words, $QV_{*}(\mathrm{Id}-a_{I})_{*}R_{I*}(f)$ is a $c$-atom, hence

$\|VQ^{*}g\|_{\mathsf{BMO}_{r}}\leq\|g\|_{H_{1}^{c}(\mathcal{A})^{*}}.$

∎

The reverse inclusion $\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})\subseteq
H_{1}^{c}(\mathcal{A})^{*}$ is more involved. We need to check that every
operator in $\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ induces a continuous
functional on $H_{1}^{c}(\mathcal{A})$. The starting point of our argument is
that every operator $\varphi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$
induces a functional on the algebraic vector space $\mathcal{H}$ generated by
the $c$-atoms in $H_{1}^{c}(\mathcal{A})\subset
L_{1}(\mathbb{R};L_{1}(\mathcal{M}))$.

Recall that $L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},dt))$ can be also be
interpreted as a space of functions (defined a.e.) with values in
$L_{1}(\mathcal{M})$. By definition, as a function any $c$-atom $a$ is the
pointwise product of a compactly supported function in
$L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$ with a constant element $h\in
L_{2}(\mathcal{M})$, thus it is an element in
$L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},dt))$ and moreover
$\|a\|_{L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},dt))}\leq 1$. To emphasize, we
will denote the inclusion by $\gamma:\mathcal{H}\to
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},dt))$. The arguments we just gave also
say that it is continuous for the norm on $\mathcal{H}$ given by

$\|f\|_{\mathcal{H}}=\inf\\{\sum_{i=1}^{N}|\lambda_{i}|\;:\;f=\sum_{i=1}^{N}\lambda_{i}a_{i}\;,N\geq
1,\;\lambda_{i}\in\mathbb{C}\textrm{ and }a_{i}\;c\textrm{-atoms}\\}.$

We will denote by $A_{g}$ the pointwise multiplication by $g$ on a.e.
functions with values in $L_{1}(\mathcal{M})$ or $L_{2}(\mathcal{M})$. It is
well defined on all elements that have compact support in the spaces we
consider. The same kind arguments justify that $A_{\omega}(a)\in
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},1/(1+t^{2})dt))$ and
$A_{\omega}:\mathcal{H}\to
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},1/(1+t^{2})dt))$ is of course linear.
$A_{\omega}$ is not continuous but if $a$ is an atom supported on $I$ then
$A_{\omega}(a)=\widetilde{M}_{\omega}^{I}(\gamma(a))$, where
$M_{\omega}^{I}:L_{2}(\mathbb{R})\to L_{2}(\mathbb{R},1/(1+t^{2})dt)$ is the
multiplication by $\omega\mbox{\large$\chi$}_{I}$ (which is bounded).

This allows us to define a duality pairing for $\mathcal{H}$ and
$\mathsf{BMO}_{r}$ by $\varphi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$,
$f\in\mathcal{H}$:

$\langle\varphi,f\rangle_{\mathsf{BMO}_{r},\mathcal{H}}=\langle\varphi,A_{\omega}(f)\rangle_{L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}})),L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}.$

This definition is made so that if $\varphi=\sum_{i=1}^{n}m_{i}\otimes
f_{i}\in\mathcal{M}\otimes L_{2}(\mathbb{R},\frac{dt}{1+t^{2}})$ then

$\langle\varphi,f\rangle_{\mathsf{BMO}_{r},\mathcal{H}}=\int_{\mathbb{R}}\tau(f(t)m_{i})f_{i}(t)dt.$

So, we recover the classical duality pairing for functions.

######  Lemma 2.8.

Let $\varphi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$. If $a$ is a $c$-atom
in $H_{1}^{c}(\mathcal{A})$, there holds

$\displaystyle|\langle\varphi,a\rangle_{\mathsf{BMO}_{r},\mathcal{H}}|\leq\
\|\varphi\|_{\mathsf{BMO}_{r}}.$

###### Proof.

Let $I$ be the interval for which $a=bh$ satisfies the definition of a
$c$-atom. Then $A_{\omega}(a)=|I|(R_{I*})^{2}(a)$ and the mean zero condition
exactly means that $a_{I*}(A_{\omega}(a))=0$. Thus, there holds

$\langle\varphi,a\rangle_{\mathsf{BMO}_{r},\mathcal{H}}=\langle
R_{I}(\mathrm{Id}-a_{I})\varphi,|I|R_{I*}(a)\rangle_{L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}})),L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}.$

It remains to show that
$\||I|R_{I*}(a)\|_{L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}\leq
1$, but this comes from the factorization property in terms of operators

$|I|R_{I*}(a)=(\sqrt{|I|}A_{\sqrt{\omega}}(b))\cdot h$

where
$\|\sqrt{|I|}A_{\sqrt{\omega}}(b)\|_{L_{2}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}=\sqrt{|I|}\
\|b\|_{L_{2}(\mathcal{M};L_{2}^{c}(\mathbb{R},{dt}))}\leq 1$ and $h$ is a
norm-one operator. ∎

The second step in the proof is to extend the duality pairing to
$H_{1}^{c}(\mathcal{A})$. This will require some care and approximations.

Given a compactly supported continuous function $\xi:\mathbb{R}\to\mathbb{R}$,
we denote by $R_{\xi}$ the convolution on $L_{2}(\mathbb{R})$ against $\xi$.
We could consider its extension $\widetilde{R}_{\xi}$ to any
$L_{p}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$. On the other hand, using vector-
valued integration, as the convolution against $\xi$ in
$L_{p}(\mathbb{\mathbb{R}};L_{p}(\mathcal{M}))$ is feasible; we denote it by
$\xi*x$. When $p=2$, we have $\widetilde{R}_{\xi}(x)=\xi*x$ using the
identification
$L_{2}(\mathbb{\mathbb{R}};L_{2}(\mathcal{M}))=L_{2}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$.
Moreover, we consider here a third possible extension which will be used along
this section.

######  Lemma 2.9.

Assume $\xi$ is a compactly supported continuous function on $\mathbb{R}$.
Given any $f\in L_{1}(\mathbb{\mathbb{R}};L_{1}(\mathcal{M}))$, then $\xi*f\in
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$ (viewed as a function space).
Moreover, this induces a continuous linear map
$C_{\xi}:L_{1}(\mathbb{\mathbb{R}};L_{1}(\mathcal{M}))\to
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$.

###### Proof.

Recall that $L_{1}(\mathbb{R};L_{1}(\mathcal{M}))$ is the projective tensor
product $L_{1}(\mathbb{R})\widehat{\otimes}_{\pi}L_{1}(\mathcal{M})$. It
suffices to prove the statement for a simple tensor $f=g\otimes m$ with $g\in
L_{1}(\mathbb{R})$ and $m\in L_{1}(\mathcal{M})$. Then the function $\xi*f$
corresponds to the operator $m\otimes(\xi*g)\in
L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))$. Factorizing $m=rs$ with $r,s\in
L_{2}(\mathcal{M})$ and $\|r\|_{2}=\|s\|_{2}=\|m\|_{1}^{1/2}$, we obtain a
factorization in term of operators $m\otimes(\xi*g)=(r\otimes(\xi*g))s$ so
that
$\|r\otimes(\xi*g)\|_{L_{2}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}=\|r\|_{2}\|\xi*g\|_{2}$.
Thus, we can conclude due to the Young inequality that

$\|m\otimes(\xi*g)\|_{L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}\leq\|r\|_{2}\|s\|_{2}\|\xi*g\|_{2}\leq\|m\|_{1}\|g\|_{1}\|\xi\|_{2}.$

We get that the norm of $C_{\xi}$ is controlled by $\|\xi\|_{2}$. ∎

Let fix some non-negative even continuous function
$\phi:\mathbb{R}\to\mathbb{R}$ with support in $(-1,1)$ such that
$\int_{\mathbb{\mathbb{R}}}\phi(t)dt=1$ and denote $\phi_{n}(x)=n\phi(nx)$.

######  Lemma 2.10.

Given a $c$-atom $a=bh$ supported on $I$, the following holds:

  1. (1)

$\frac{1}{2}\phi_{n}*a$ is a finite convex combination of $c$-atoms

  2. (2)

If $n\geq\frac{2}{|I|}$, $\phi_{n}*a-a=\lambda_{n}a_{n}$, where $a_{n}$ is a
$c$-atom supported on $2I$, $\lambda_{n}\in\mathbb{C}$ with
$\lim_{n\to\infty}\lambda_{n}=0$.

  3. (3)

We have $\gamma(\phi_{n}*a)=C_{\phi_{n}}(a)$ for all $n\geq 1$.

###### Proof.

For (1), cut $(-\frac{1}{n},\frac{1}{n})$ into $N$ disjoint intervals
$K_{j}^{n}$ whose lengths are less than $|I|$ (we can have $N=1$) and consider
$b_{j}^{n}=(\phi_{n}\mbox{\large$\chi$}_{K_{j}^{n}})*b$. For all $j$,
$b_{j}^{n}$ is supported on an interval of length $2|I|$, has mean 0 and
$L_{2}$-norm less than
$\frac{2\|\phi_{n}\mbox{\large$\chi$}_{K_{j}^{n}}\|_{1}}{\sqrt{|I|}}$. Thus
$b_{j}^{n}h$ is an atom up to a factor
$2\|\phi_{n}\mbox{\large$\chi$}_{K_{j}^{n}}\|_{1}$ and
$\phi_{n}*a=\sum_{j=1}^{N}b_{j}^{n}h$. But there holds
$\sum_{j=1}^{N}2\|\phi_{n}\mbox{\large$\chi$}_{K_{j}^{n}}\|_{1}=2$, it yields
(1).

Similarly, for $(2)$, $\phi_{n}*b-b$ has support in $2I$ with mean $0$. Its
$L_{2}$-norm goes to $0$ giving the result.

There are many ways to check (3). One can identify them as a.e. functions from
$\mathbb{R}$ to $L_{1}(\mathcal{M})$. Another one is to note that point (1)
implies that the convolution by $\phi_{n}$ is continuous for
$\|\cdot\|_{\mathcal{H}}$. Moreover, atoms of the form $a=bh$ with $b$ a
$L_{2}$-atom which is a simple tensor in $L_{2}(\mathbb{R})\otimes
L_{2}(\mathcal{M})$ and $h\in L_{2}(\mathcal{M})$ are dense in $\mathcal{H}$
for $\|\cdot\|_{\mathcal{H}}$. On such atoms, the formula is obvious and then
extends by continuity. ∎

######  Proposition 2.11.

Let $\psi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ satisfying
$\widetilde{P}_{J}\psi=\psi$ for some finite interval $J$. Then for any family
of atoms $(a_{i})$ and $(\lambda_{i})\in\ell_{1}$ such that
$\sum_{i=1}^{\infty}\lambda_{i}a_{i}=0$ in
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ there holds

$\sum_{i=1}^{\infty}\lambda_{i}\
\langle\psi,a_{i}\rangle_{\mathsf{BMO}_{r},\mathcal{H}}=0.$

###### Proof.

First, by Lemma 2.10, $\phi_{n}*a_{i}\in\mathcal{H}$ for all $n,i\geq 1$. We
come back to the definition

$\displaystyle\langle\psi,\phi_{n}*a_{i}\rangle_{\mathsf{BMO}_{r},\mathcal{H}}$
$\displaystyle=$
$\displaystyle\langle\widetilde{P}_{J}\psi,\widetilde{P}_{J}A_{\omega}(\phi_{n}*a_{i})\rangle_{L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},\frac{dt}{1+t^{2}})),L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))}.$

Using the projection $\widetilde{P}_{J}$, we get
$\widetilde{P}_{J}A_{\omega}(\phi_{n}*a_{i})=\widetilde{P}_{J}\widetilde{M}_{\omega}^{J}\gamma(\phi_{n}*a_{i})$
and since $\gamma(\phi_{n}*a_{i})=C_{\phi_{n}}(a_{i})$:

$\displaystyle\langle\psi,\phi_{n}*a_{i}\rangle_{\mathsf{BMO}_{r},\mathcal{H}}$
$\displaystyle=$
$\displaystyle\langle(\widetilde{M}_{\omega}^{J})^{*}\psi,C_{\phi_{n}}(a_{i})\rangle_{L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R})),L_{1}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}$
$\displaystyle=$ $\displaystyle\langle
C_{\phi_{n}}^{*}(\widetilde{M}_{\omega}^{J})^{*}\psi,a_{i}\rangle_{\mathcal{M}\overline{\otimes}L_{\infty}(\mathbb{R}),L_{1}(\mathcal{M}\overline{\otimes}L_{\infty}(\mathbb{R}))}.$

We can conclude as $\sum_{i=1}^{\infty}\lambda_{i}a_{i}=0$ in
$L_{1}(\mathbb{R};L_{1}(\mathcal{M}))$ that
$\sum_{i=1}^{\infty}\lambda_{i}\langle\psi,\phi_{n}*a\rangle=0$.

Next, by Lemma 2.8, the duality pairing is continuous for the norm
$\|\cdot\|_{\mathcal{H}}$. Lemma 2.10 (1) and (2) mean that
$\|\phi_{n}*a_{i}\|_{\mathcal{H}}\leq 2\|a_{i}\|_{\mathcal{H}}$ and
$\|\phi_{n}*a_{i}-a_{i}\|_{\mathcal{H}}\to 0$ as $n\to\infty$. We can conclude
to $\sum_{i=1}^{\infty}\lambda_{i}\langle\psi,a_{i}\rangle=0$ by the Lebesgue
Theorem. ∎

In order to conclude the argument, it remains to show that any operator
$\varphi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ induces a well-defined
functional on $H_{1}^{c}(\mathcal{A})$. The estimates obtained for compactly
supported $\psi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ will yield a
general result under another suitable approximation argument from Garnett’s
book [G07, p. 261] as stated in [M07, Lemma 1.5] in the semi-commutative case.
See Appendix B for its proof.

######  Lemma 2.12.

Suppose $\varphi\in\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ and suppose $J$
is an interval such that $\varphi_{J}=0$. Let $3J$ be the interval concentric
with $J$ having length $3|J|$. Then there exist some universal $C>0$,
$\psi\in\mathsf{BMO}_{c}(\mathbb{R},\mathcal{M})$ such that

$\widetilde{P}_{3J}\psi=\psi,\qquad\widetilde{P}_{J}(\psi-\varphi)=0,\qquad\|\psi\|_{\mathsf{BMO}_{c}}\leq
C\ \|\varphi\|_{\mathsf{BMO}_{c}}.$

######  Theorem 2.13.

Given a semifinite von Neumann algebra $\mathcal{M}$, there is a contractive
inclusion

$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})\subseteq
H_{1}^{c}(\mathcal{A})^{*}.$

###### Proof.

Let $\varphi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ and let $f\in
H_{1}^{c}(\mathcal{A})$ admit an atomic decomposition
$\sum_{i=1}^{\infty}\lambda_{i}a_{i}$. We want to define the duality pairing
by

$\langle\varphi,f\rangle=\sum_{i=1}^{\infty}\lambda_{i}\langle\varphi,a_{i}\rangle_{\mathsf{BMO},\mathcal{H}}.$

We start by showing that this well defined algebraically, that is if $f=0$ in
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ then the right
hand side is also 0.

Given $\varepsilon>0$, let $N\geq 1$ such that
$\sum_{i>N}|\lambda_{i}|<\varepsilon$ and let $J$ be a finite interval
satisfying for $i=1,...,N$:

$\displaystyle\mathrm{supp}_{\mathbb{R}}(a_{i})\subseteq J.$

Without loss of generality, we can assume that $\varphi_{J}=0$, so by Lemma
2.12, there exists some $\psi\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$
satisfying $\widetilde{P}_{3J}\psi=\psi$, $\widetilde{P}_{J}(\varphi-\psi)=0$
and

$\|\psi\|_{\mathsf{BMO}_{r}}\leq C\|\varphi\|_{\mathsf{BMO}_{r}}$

for some universal constant $C>0$. Therefore, as a consequence of Proposition
2.11,

$\displaystyle\sum_{i=1}^{\infty}\lambda_{i}\langle\varphi,a_{i}\rangle$
$\displaystyle=\sum_{i=1}^{\infty}\lambda_{i}\langle\varphi,a_{i}\rangle-\sum_{i=1}^{\infty}\lambda_{i}\langle\psi,a_{i}\rangle$
$\displaystyle=\sum_{i=1}^{N}\lambda_{i}\
\langle\varphi-\psi,a_{i}\rangle+\sum_{i=N+1}^{\infty}\lambda_{i}\
\langle\varphi-\psi,a_{i}\rangle$
$\displaystyle=\sum_{i=N+1}^{\infty}\lambda_{i}\,\langle\varphi-\psi,a_{i}\rangle.$

We get to

$\displaystyle|\sum_{i=1}^{\infty}\lambda_{i}\ \langle\varphi,a_{i}\rangle|$
$\displaystyle\leq\sum_{i=N+1}^{\infty}|\lambda_{i}|(1+C)\|\varphi\|_{\mathsf{BMO}_{r}}<(1+C)\varepsilon\|\varphi\|_{\mathsf{BMO}_{r}}.$

Since $\varepsilon$ is arbitrarily small, the duality pairing is well defined.
To conclude the norm estimate, one just need to use Lemma 2.8. ∎

This last theorem completes the proof of the isometric identity

$\displaystyle(H_{1}^{c}(\mathcal{A}))^{*}=\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M}).$

Therefore, a new description for a predual of
$\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$ has been obtained just in terms of
a new atomic decomposition, which will be crucial for the study of the
boundedness of Calderón-Zygmund operators from $H_{1}^{c}(\mathcal{A})$ to
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$.

We end this section by briefly mentioning that there is another approach to
define the duality bracket
$H_{1}^{c}(\mathcal{A})-\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$. One can
easily show that there is a natural isometric embedding

$\pi:L_{\infty}(\mathcal{M};L_{2}^{r}(\mathbb{R},1/(1+t^{2})dt))\subset
B\big{(}L_{2}(\mathbb{R},\frac{dt}{1+t^{2}}\otimes_{2}L_{2}(\mathcal{M})),L_{2}(\mathcal{M})\big{)}.$

Then for an atom $a=bh$ and $f\in\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})$,
one can set $\langle a,f\rangle=\tau(\pi(f)(b)h)$, without regard to the
decomposition of $a$. Then, one can rely on Hilbert valued $H_{1}$ and
$\mathsf{BMO}$ noting that
$\|f\|_{\mathsf{BMO}_{r}(\mathbb{R},\mathcal{M})}=\sup_{\|h\|_{2}\leq
1}\|\pi(f)^{*}(h)\|_{\mathsf{BMO}(\mathbb{R},L_{2}(\mathcal{M}))}$.
Nevertheless, one still need all the above approximation arguments.

##  3\. Calderón-Zygmund operators with operator-valued kernels

Let $\mathcal{M}$ be a von Neumann algebra over a separable Hilbert space.
Through this section we establish the conditions under which a kernel $K$,
defined outside the diagonal with values in $\mathcal{M}$, will induce a
Calderón-Zygmund operator from $H_{1}^{c}(\mathcal{A})$ into
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$. As one could
expect, the resulting operator will be defined up to a pointwise
multiplication operator. Recall the identification $L_{\infty}(I\times
J)\overline{\otimes}\mathcal{M}=L_{\infty}(I\times J;\mathcal{M})$ where we
consider weak∗-measurable $\mathcal{M}$-valued functions.

Recall that we denote by $\mathcal{S}$ the set of compactly supported
essentially bounded functions $\mathbb{R}\to L_{\infty}\cap
L_{1}(\mathcal{M})$ that are measurable with values in $L_{1}$. Note that
$B_{L_{p}(\mathcal{A})}\cap\mathcal{S}$ is norm-dense in
$B_{L_{p}(\mathcal{A})}$ when $1\leq p<\infty$ and weak∗-dense for $p=\infty$.
Indeed if $f\in L_{p}(\mathcal{A})$ with polar decomposition $f=u|f|$, take
$p_{n}$ a sequence of finite projections strongly going to 1 in $\mathcal{M}$
and consider $f_{n}=u\max\\{|f|,n\\}\ p_{n}\
\mbox{\large$\chi$}_{[-n;n]}\in\mathcal{S}$. It is clear that
$\|f_{n}\|_{p}\leq\|f\|_{p}$ and $f_{n}\to f$ in norm in $L_{p}$
(weak∗-convergence if $p=\infty$).

######  Definition 3.1.

Assume that $T$ is a bounded operator on
$L_{2}(L_{\infty}(\mathbb{R}){\overline{\otimes}}\mathcal{M})$. We say that
$T$ is a _left Calderón-Zygmund operator_ if there exists some function

$\displaystyle K:\mathbb{R}\times\mathbb{R}\ \setminus\
\\{x=y\\}\longrightarrow\mathcal{M}$

such that for every pair of intervals $I,J$ satisfying $d(I,J)>0$, there exist
$K_{I,J}\in L_{\infty}(I\times J)\overline{\otimes}\mathcal{M}$ such that

  * •

${K}_{I,J}(t)=K(t)$ for almost every $t\in I\times J$,

  * •

for any $f,g\in\mathcal{S}$ supported respectively on $J$ and $I$:

(6) $\displaystyle\Big{(}\tau\circ\int\Big{)}(g\ Tf)=\langle
f(y)g(x),K_{I,J}\rangle_{L_{1}(L_{\infty}(I\times
J)\overline{\otimes}\mathcal{M}),L_{\infty}(I\times
J)\overline{\otimes}\mathcal{M}}$

We say that $T$ is a _left Calderón-Zygmund operator_ if it also satisfies
that for any compactly supported $f\in
L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ and
$h\in\mathcal{M}$:

$\displaystyle T(fh)=T(f)h.$

Let $\lambda>0$, we say that the kernel $K$ (or $T$) satisfies the _Hörmander
condition_ for $\lambda$ whenever

(7)
$\displaystyle\int_{|x-y^{\prime}|\geq\lambda|y-y^{\prime}|}\|K(x,y^{\prime})-K(x,y)\|_{\mathcal{M}}\
dx\leq C_{\lambda}$

for some constant $C_{\lambda}\geq 0$.

We choose $\mathcal{S}$ in order to have a perfectly defined duality in (6) as
$f(y)g(x)\in L_{1}(\mathbb{R}^{2};L_{1}(\mathcal{M}))$. Moreover, the space
$\mathcal{S}$ is an algebra for the pointwise product. We could also have
chosen to state (6) for $L_{1}(\mathbb{R};L_{2}(\mathcal{M}))$ instead of
$\mathcal{S}$ without significant modifications.

The following argument constitutes our main estimate it can be considered as
an arrangement of the scalar case to the semicommutative setting.

######  Lemma 3.2.

Let $\mathcal{M}$ be a von Neumann algebra. Let $T$ be a left Calderón-Zygmund
operator which is bounded on $L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$ and has
associated kernel

$K:\mathbb{R}\times\mathbb{R}\ \setminus\
\\{x=y\\}\longrightarrow\mathcal{M}.$

If $K$ satisfies the Hörmander condition for some $\lambda\geq 1$ and moreover

$\displaystyle T(a)=T(b)h\ \textrm{\rm for any $c$-atom}\ a=bh,$

then there holds

$\displaystyle\|T(a)\|_{L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})}\leq\max\\{C_{\lambda},\lambda^{1/2}\|T\|\\}.$

###### Proof.

Let $a=bh$ be a $c$-atom in $H_{1}^{c}(\mathcal{A})$ with support contained in
the interval $I=[y_{0}-d,y_{0}+d]$ and let $\lambda I=[y_{0}-\lambda
d,y_{0}+\lambda d]$. Then, the norm

$\displaystyle\|T(a)\|_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}=\int_{\lambda
I}\|T(a)\|_{L_{1}(\mathcal{M})}+\int_{(\lambda
I)^{c}}\|T(a)\|_{L_{1}(\mathcal{M})}$

can be bounded in two steps. First, the continuity of $T$ on
$L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$ implies that

$\displaystyle\int_{\lambda I}\|T(a)$
$\displaystyle\|_{L_{1}(\mathcal{M})}=\int_{\lambda
I}\|T(b)h\|_{L_{1}(\mathcal{M})}\leq\Big{(}\int_{\lambda
I}\|T(b)\|^{2}_{L_{2}(\mathcal{M})}\Big{)}^{1/2}\ \Big{(}\int_{\lambda
I}\|h\|_{L_{2}(\mathcal{M})}^{2}\Big{)}^{1/2}$
$\displaystyle=\|T(b)\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\ |\lambda
I|^{1/2}\|h\|_{L_{2}(\mathcal{M})}$
$\displaystyle\leq\|T:L_{2}(\mathbb{R};L_{2}(\mathcal{M}))\rightarrow
L_{2}(\mathbb{R};L_{2}(\mathcal{M}))\|\
\|b\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\ |\lambda I|^{1/2}\
\|h\|_{L_{2}(\mathcal{M})}$ $\displaystyle\leq\lambda^{1/2}\
\|T:L_{2}(\mathbb{R};L_{2}(\mathcal{M}))\rightarrow
L_{2}(\mathbb{R};L_{2}(\mathcal{M}))\|.$

To deal with the second term, we have

$\|T(a)\mbox{\large$\chi$}_{(\lambda
I)^{c}}\|_{1}=\|T(b)h\mbox{\large$\chi$}_{(\lambda
I)^{c}}\|_{1}=\sup_{\begin{subarray}{c}g\in\mathcal{S},\|g\|_{\infty}\leq 1\\\
\mbox{\large$\chi$}_{(\lambda I)^{c}}g=g\end{subarray}}\big{|}\tau\int
T(b)hg\big{|}.$

Assume for the moment that $b\in\mathcal{S}$ and $h\in L_{\infty}\cap
L_{1}(\mathcal{M})$. Let $y_{0}$ be the center of $I$. Since $I$ and $(\lambda
I)^{c}$ are disjoint measurable sets and $b$ has integral zero, it follows
using (6) and the Hörmander condition that

$\displaystyle\|T(a)$ $\displaystyle\mbox{\large$\chi$}_{(\lambda
I)^{c}}\|_{1}\leq\sup_{g}\big{|}\tau\int_{(\lambda I)^{c}}\int_{I}K_{(\lambda
I)^{c},I}(x,y)\ b(y)hg(x)\ dxdy\big{|}$
$\displaystyle\leq\sup_{g}\big{|}\tau\int_{(\lambda
I)^{c}}\int_{I}(K_{(\lambda I)^{c},I}(x,y)-K_{(\lambda I)^{c},I}(x,y_{0}))\
b(y)hg(x)\ dx\ dy\big{|}$ $\displaystyle=\big{\|}\int_{I}(K_{(\lambda
I)^{c},I}(x,y)-K_{(\lambda I)^{c},I}(x,y_{0}))\ b(y)h\
dy\big{\|}_{L_{1}((\lambda I)^{c};L_{1}(\mathcal{M}))}$
$\displaystyle\leq\int_{(\lambda I)^{c}}\int_{I}\|(K_{(\lambda
I)^{c},I}(x,y)-K_{(\lambda I)^{c},I}(x,y_{0}))\ b(y)h\|_{L_{1}(\mathcal{M})}\
dx\ dy$ $\displaystyle\leq\int_{(\lambda I)^{c}}\int_{I}\|(K_{(\lambda
I)^{c},I}(x,y)-K_{(\lambda I)^{c},I}(x,y_{0}))\|_{\mathcal{M}}\
\|b(y)\|_{L_{2}(\mathcal{M})}\ \|h\|_{L_{2}(\mathcal{M})}\ dx\ dy$
$\displaystyle\leq C_{\lambda}\ \int_{I}\|b(y)\|_{L_{2}(\mathcal{M})}\
\|h\|_{L_{2}(\mathcal{M})}\ dy\leq C_{\lambda}\|b\|_{2}\
\|h\mbox{\large$\chi$}_{I}\|_{2}\leq C_{\lambda}.$

The general case follows by approximation. Indeed let $b_{n}\in\mathcal{S}$
going to $b$ with $\|b_{n}\|_{2}\leq\|b\|_{2}$ as explained above and
$h_{n}\in L_{\infty}\cap L_{1}(\mathcal{M})$ going to $h$ in
$L_{2}(\mathcal{M})$ with $\|h_{n}\|_{L_{2}(\mathcal{M})}\leq 1$. Centering
$b_{n}$ on $I$ decreases its $L_{2}$-norm, so we can as well assume it has
mean 0\. Let $J_{k}=[-k,k]\cap(\lambda I)^{c}$. Then for any $k\geq 1$,
$T(b_{n})h_{n}\mbox{\large$\chi$}_{J_{k}}\to T(b)h\mbox{\large$\chi$}_{J_{k}}$
in $L_{1}$ as $n\to\infty$ by the $L_{2}$-continuity of $T$. It follows that
$\|T(b)h\mbox{\large$\chi$}_{J_{k}}\|_{1}\leq\limsup_{n}\|T(b_{n})h_{n}\mbox{\large$\chi$}_{J_{k}}\|_{1}\leq
C_{\lambda}$. Letting $k\to\infty$ gives $\|T(b)h\mbox{\large$\chi$}_{(\lambda
I)^{c}}\|_{1}\leq C_{\lambda}$. ∎

In order to extend a left Calderón-Zygmund operator to the whole Hardy space
$H_{1}^{c}(\mathcal{A})$, we will proceed as in [MeyC97]. A family of bounded
kernels will be constructed, yielding a bounded family of Calderón-Zygmund
operators that will approximate the original operator.

######  Lemma 3.3.

Let $T$ be a left Calderón-Zygmund operator whose associated kernel $K$
satisfies the Hörmander condition (for $\lambda=2$) (7). Then, there exist a
sequence of left Calderón-Zygmund operators $(T_{m})_{m\geq 1}$ with kernels
$(K_{m})_{m\geq 1}\subseteq
L_{\infty}(\mathbb{R}\times\mathbb{R})\overline{\otimes}\mathcal{M}$ and a
constant $C^{\prime}>0$ such that

(8) $\displaystyle\int_{|x-y^{\prime}|\geq
4|y-y^{\prime}|}\|K_{m}(x,y^{\prime})-K_{m}(x,y)\|_{\mathcal{M}}\ dx\leq
C^{\prime}.$

Moreover, there holds

$\displaystyle\lim_{m\rightarrow\infty}\|T_{m}(f)-T(f)\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}=0$

for every $f\in L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$.

###### Proof.

Let $\phi$ be an even smooth function supported on $(-1,1)$ with $\int\phi=1$,
and let $R_{m}$ be the operator defined on
$L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$ as $R_{m}(f)=f*\phi_{m}$ where
$\phi_{m}(x)=m\phi(mx)$. We define the function
$K_{m}:\mathbb{R}^{2}\to\mathcal{M}$ by duality. Assume $z\in
L_{1}(\mathcal{M})$ is decomposed as $z=ab$ with $a,b\in L_{2}(\mathcal{M})$,
and define the quantity

(9) $\displaystyle\langle
K_{m}(x,y),z\rangle_{\mathcal{M},L_{1}(\mathcal{M})}=\int\tau\big{(}T((\tau_{y}\phi_{m})(t)a)\tau_{x}({\phi}_{m})(t)b\big{)}dt,$

where $\tau_{x}\phi(t)=\phi(t-x)$. Indeed assume $z=\alpha\beta$ with
$\alpha,\beta\in L_{2}(\mathcal{M})$ and choose $p$ a finite projection so
that $pa,p\alpha\in\mathcal{M}$, then using the right-modularity condition,

$\int_{\mathbb{R}}\tau\big{(}T(\tau_{y}\phi_{m}pa)\tau_{x}{\phi}_{m}\
b\big{)}=\int_{\mathbb{R}}\tau\big{(}T(\tau_{y}\phi_{m}p)\tau_{x}\phi_{m}\
z\big{)}=\int_{\mathbb{R}}\tau\big{(}T(\tau_{y}\phi_{m}p\alpha)\tau_{x}\phi_{m}\
\beta\big{)}.$

Letting $p\to 1$ and using the $L_{2}$-continuity of $T$ yields

$\langle
T(\tau_{y}\phi_{m}a),\tau_{x}({\phi}_{m})b)\rangle_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}=\langle
T(\tau_{y}\phi_{m}\alpha),\tau_{x}({\phi}_{m})\beta)\rangle_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}.$

Linearity is proved in a similar way. We also get

$\displaystyle\|K_{m}(x,y)\|_{\mathcal{M}}$
$\displaystyle=\sup_{\begin{subarray}{c}{a,b\in L_{2}(\mathcal{M})}\\\
\|a\|_{2}=\|b\|_{2}=1\end{subarray}}|(\int\circ\tau)\big{(}T(\tau_{y}\phi_{m}a)\
\tau_{x}{\phi}_{m}b\big{)}|$
$\displaystyle\leq\|T\|_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}\|\phi_{m}\|_{L_{2}(\mathbb{R})}^{2}.$

Assuming that $f$ and $g$ are simple functions, we get by linearity

$\displaystyle\langle R_{m}TR_{m}f,g\rangle$ $\displaystyle=\tau\int\int
T\Big{(}\int\phi_{m}(\cdot-y)\ f(y)\ dy\Big{)}(t)\ \phi_{m}(t-x)\ dt\ g(x)\
dx$ $\displaystyle=\int\int\tau\big{(}K_{m}(x,y)f(x)g(y)\big{)}\ dx\ dy.$

The formula extends to $f,g\in L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$ by
continuity since $K_{m}$ is bounded on the whole $\mathbb{R}^{2}$. Therefore,
$K_{m}$ is the kernel for $R_{m}TR_{m}$ in the sense of Definition 3.1.
Moreover, $K_{m}$ is a continuous function $\mathbb{R}\times\mathbb{R}\to
L_{\infty}(\mathcal{M})$ by continuity of translations on $L_{2}(\mathbb{R})$
as

$\displaystyle\|K_{m}(x,y)-K_{m}(x,y^{\prime})\|_{\mathcal{M}}$
$\displaystyle\leq\|T\|\ \|\phi_{m}\|_{2}\
\|(\tau_{y}-\tau_{y^{\prime}})\phi_{m}\|_{2}$ $\displaystyle\leq\|T\|\
m^{1/2}\ \|\phi\|_{2}\
\big{(}m^{3/2}{|y-y^{\prime}|}\|\phi^{\prime}\|_{\infty}\big{)}$ (10)
$\displaystyle=m^{2}{|y-y^{\prime}|}\|T\|\|\phi\|_{2}\|\phi^{\prime}\|_{\infty}$

A similarly estimate holds for
$\|K_{m}(x,y)-K_{m}(x^{\prime},y)\|_{\mathcal{M}}$.

On the other hand, the kernel $K_{m}$ satisfies the Hörmander condition for
$\lambda=4$. First, consider $x,y,y^{\prime}$ such that $|x-y|\geq
4|y^{\prime}-y|$ and $|x-y|>4/m$. Then, the support of $\tau_{x}{\phi}_{m}$ is
disjoint from that of $\tau_{y}\phi_{m}$ and $\tau_{y^{\prime}}\phi_{m}$. Let
$I_{y,y^{\prime}}$ and $I_{x}$ denote some disjoint intervals containing the
support of $\tau_{y}\phi_{m},\tau_{y^{\prime}}\phi_{m}$ and
$\tau_{x}{\phi}_{m}$ respectively. Then, the identities (6) and (9) for the
kernel $K_{I_{x},I_{y,y^{\prime}}}$ (extending to 0 outside $I_{x}\times
I_{y,y^{\prime}}$) give

$K_{m}(x,z)=K_{I_{x},I_{y,y^{\prime}}}*({\phi}_{m}\otimes\phi_{m})(x,z)$

for $z=y,y^{\prime}$. Using that for $|u|,|v|\leq\frac{1}{m}$,

$|x-u-(y-{v})|\geq|x-y|-\frac{2}{m}>\frac{1}{2}|x-y|\geq 2|y^{\prime}-y|,$

what implies

$\displaystyle\int_{|x-y|\geq
4|y^{\prime}-y|\vee\frac{4}{m}}\|K_{m}(x,y)-K_{m}(x,y^{\prime})\|_{\mathcal{M}}\
dx$ $\displaystyle\leq\int_{|x-y|\geq
4|y^{\prime}-y|\vee\frac{4}{m}}\int\int\big{\|}{K}(x-u,y-{v})-{K}(x-{u},y^{\prime}-{v})\big{\|}_{\mathcal{M}}\
{\phi_{m}}(u)\phi_{m}(v)\ du\ dv\ dx.$

Now, the Hörmander condition for the kernel $K$ implies

$\int_{|x-y|\geq
4|y^{\prime}-y|\vee\frac{4}{m}}\|K_{m}(x,y)-K_{m}(x,y^{\prime})\|_{\mathcal{M}}\
dx\leq C.$

On the other hand, whenever $|x-y|\leq 4/m$ holds, the estimate (10) yields

$\displaystyle\int_{\frac{4}{m}\geq|x-y|\geq
4|y^{\prime}-y|}\|K_{m}(x,y)-K_{m}(x,y^{\prime})\|_{\mathcal{M}}\ dx$
$\displaystyle\leq
m^{2}\|T\|\|\phi^{\prime}\|_{\infty}\|\phi\|_{2}\int_{\frac{4}{m}\geq|x-y|\geq
4|y^{\prime}-y|}|y^{\prime}-y|\ dx\leq 8\
\|T\|\|\phi^{\prime}\|_{\infty}\|\phi\|_{2}.$

Thus $K_{m}$ satisfies condition (8). Finally, given $f\in
L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ there holds

$\displaystyle\|T_{m}(f)-T(f)\|_{L_{2}}$
$\displaystyle\leq\|R_{m}T(R_{m}(f)-f)\|_{L_{2}}+\|(R_{m}-\mathrm{Id})T(f)\|_{L_{2}}$
$\displaystyle\leq\|T\|\
\|R_{m}(f)-f\|_{L_{2}}+\|(R_{m}-\mathrm{Id})T(f)\|_{L_{2}}\rightarrow 0.$

as $m$ grows to infinity. ∎

Before proving that a left Calderón-Zygmund operator extends to a bounded map
from $H_{1}^{c}(\mathcal{A})$ into $L_{1}(\mathcal{A})$, a fundamental
property is included.

######  Proposition 3.4.

Let $\mathcal{M}$ be a von Neumann algebra and let $T$ be a left Calderón-
Zygmund operator which is bounded on $L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$
whose kernel is zero. Then $T$ corresponds to the left multiplication by some
operator $F\in L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$.

###### Proof.

Let $f\in L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ such that
$\mbox{supp}\|f\|_{2}$ is compact and contained in some interval $J$. Then, we
claim that for any $g,g^{\prime}\in\mathcal{S}$ there holds

(11) $\displaystyle\langle
g^{\prime},T(gf)\rangle_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}=\langle
g^{\prime},T(g)f\rangle_{L_{2}(\mathbb{R};L_{2}(\mathcal{M}))}.$

As a consequence of Definition 3.1, given $m\in\mathcal{M}\cap
L_{1}(\mathcal{M})$ and $I$ an arbitrary interval of positive length and
$\lambda<1$, it holds

$\displaystyle\langle
g^{\prime}(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes
1),T(g(\mbox{\large$\chi$}_{\lambda I}\otimes m))\rangle$
$\displaystyle=\langle g(y)(\mbox{\large$\chi$}_{\lambda I}\otimes
m)(y)g^{\prime}(x)(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes
1)(x),K_{(\overline{I})^{c},\lambda I}\rangle=0,$ $\displaystyle\langle
g^{\prime}(\mbox{\large$\chi$}_{\lambda I}\otimes
1),T(g(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes m))\rangle$
$\displaystyle=\langle g(y)(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes
m)(y)g^{\prime}(x)(\mbox{\large$\chi$}_{\lambda I}\otimes 1)(x),K_{\lambda
I,(\overline{I})^{c}}\rangle=0.$

It follows that

$\langle g^{\prime}(\mbox{\large$\chi$}_{\lambda I}\otimes
1),T(g(\mbox{\large$\chi$}_{I^{c}}\otimes m))\rangle=\langle
g^{\prime}(\mbox{\large$\chi$}_{\lambda I}\otimes
1),(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes 1)T(g(1\otimes
m))\rangle.$

Using modularity, we get

$\langle g^{\prime}(\mbox{\large$\chi$}_{\lambda I}\otimes
1),T(g)(\mbox{\large$\chi$}_{I}\otimes m)\rangle=\langle
g^{\prime}(\mbox{\large$\chi$}_{\lambda I}\otimes
1),T(g(\mbox{\large$\chi$}_{I}\otimes m))\rangle.$

By continuity, this also holds for $\lambda=1$ and similarly

$\displaystyle\langle
g^{\prime}(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes
1),T(g)(\mbox{\large$\chi$}_{I}\otimes m)\rangle$ $\displaystyle=0=\langle
g^{\prime}(\mbox{\large$\chi$}_{(\overline{I})^{c}}\otimes
1),T(g(\mbox{\large$\chi$}_{I}\otimes m))\rangle,$

yielding $\langle g^{\prime},T(g)(\mbox{\large$\chi$}_{I}\otimes
m)\rangle=\langle g^{\prime},T(g(\mbox{\large$\chi$}_{I}\otimes m))\rangle$.
By continuity of $T$, this can be extended to any $g,g^{\prime}\in L_{2}$. And
finally, this identity also remains valid when replacing
$\mbox{\large$\chi$}_{I}\otimes m$ by any $f$ by weak∗-density of elementary
tensors (and Kaplansky’s theorem) and by weak∗ continuity of all involved
maps. Therefore, claim (11) follows.

We have shown that $T$ commutes with right multiplications by $f\in
L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$, thus it has to be a left
multiplication by an element in
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ as the algebra
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ acts in standard form on
$L_{2}(\mathbb{R};L_{2}(\mathcal{M}))$. ∎

######  Theorem 3.5.

Let $\mathcal{M}$ be a von Neumann algebra. Let $T$ be a left Calderón-Zygmund
operator with associated kernel
$K:\mathbb{R}\times\mathbb{R}\setminus\\{x=y\\}\rightarrow\mathcal{M}$. If $K$
satisfies Hörmander condition (7), then $T$ extends to a bounded operator from
$H_{1}^{c}(\mathcal{A})$ into $L_{1}(\mathbb{R};L_{1}(\mathcal{M}))$.

###### Proof.

First the operator $T$ can be defined on $c$-atoms. This is where we use the
modularity. Indeed, given a $c$-atom with decomposition $a=bh$ supported on
$I$, $T(b)h$ is well defined in
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ but it may depend
on the decomposition. Let us justify that it does not. First, for any $m\geq
1$, since $K_{m}(x,\cdot)$ belongs to
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ (actually it is
continuous), we have the formula

$\displaystyle T_{m}(a)(x)=\int K_{m}(x,y)\ b(y)h\ dy=\int K_{m}(x,y)\ b(y)\
dt\cdot h=T_{m}(b)(x)\cdot h.$

Consider another decomposition $a=b^{\prime}h^{\prime}$ supported on $J$.
Then, we have $T_{m}(b)h=T_{m}(b^{\prime})h^{\prime}$ in
$L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$. Thus, $T(a)$ can
indeed be defined as $T(a)=T(b)h=T(b^{\prime})h^{\prime}$ since the norm of
the difference

$\displaystyle\|T(b)h-$ $\displaystyle
T(b^{\prime})h^{\prime}\|_{L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})}\leq\|T(b)h-T_{m}(b)h\|_{L_{1}}+\|T_{m}(b^{\prime})h^{\prime}-T(b^{\prime})h^{\prime}\|_{L_{1}}$
$\displaystyle\leq\sqrt{|I|}\ \|h\|_{L_{2}(\mathcal{M})}\
\|T(b)-T_{m}(b)\|_{L_{2}}+\sqrt{|J|}\ \|h^{\prime}\|_{L_{2}(\mathcal{M})}\
\|T(b^{\prime})-T_{m}(b^{\prime})\|_{L_{2}}$

goes to zero as $m\to\infty$ by Lemma 3.3.

In particular $T$ and $T_{m}$ satisfy the assumptions of Lemma 3.2 and we can
conclude that for some universal $C\geq 0$ (due to (8)):

(12)
$\displaystyle\|T(a)-T_{m}(a)\|_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}\leq C.$

Moreover, $R_{m}a=\phi_{m}*a$ and $\phi_{m}*a-a$ are multiples of $c$-atoms by
Lemma 2.10 and $\|\phi_{m}*a-a\|_{H_{1}^{c}(\mathcal{A})}\to 0$ as
$m\to\infty$. Thus

$\displaystyle\|T(a)-T_{m}(a)\|_{L_{1}}$
$\displaystyle\leq\|R_{m}TR_{m}(a)-R_{m}T(a)\|_{L_{1}}+\|R_{m}T(a)-T(a)\|_{L_{1}}$
(13)
$\displaystyle\leq\|T(R_{m}a-a)\|_{L_{1}}+\|(R_{m}-\mathrm{Id})Ta\|_{L_{1}},$

which goes to $0$ with $m\to\infty$.

In order to define $T$ on $H_{1}^{c}(\mathcal{A})$, assume $f$ is decomposed
as $f=\sum_{i=1}^{\infty}\lambda_{i}a_{i}$ where $a_{i}$’s are $c$-atoms and
$(\lambda_{i})\in\ell_{1}$. Then, we check that

$T(f)=\sum_{i=1}^{\infty}\lambda_{i}T(a_{i})\in
L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$

does not depend on the decomposition. Indeed, assume that $f=0$. Then, the
extension above is well defined and

$\displaystyle\sum_{i=1}^{\infty}\lambda_{i}T_{m}(a_{i})(x)=\int K_{m}(x,y)\
\sum_{i=1}^{\infty}\lambda_{i}a_{i}(y)\ dy=0$

holds since $K_{m}(x,\cdot)$ belongs to
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$. Therefore, we obtain

$\displaystyle\Big{\|}\sum_{i=1}^{\infty}\lambda_{i}\
T(a_{i})\Big{\|}_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}$
$\displaystyle\leq\Big{\|}\sum_{i=1}^{\infty}\lambda_{i}\
(T(a_{i})-T_{m}(a_{i}))\ \Big{\|}_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}$
$\displaystyle\leq\sum_{i=1}^{\infty}|\lambda_{i}|\
\|T(a_{i})-T_{m}(a_{i})\|_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}$
$\displaystyle\leq\sum_{i=1}^{\infty}|\lambda_{i}|\
\|T(a_{i})-T_{m}(a_{i})\|_{L_{1}(\mathbb{R};L_{1}(\mathcal{M}))}.$

The former series tends to $0$ as $m\to\infty$ as consequence of (12) and
(13). The boundedness of $T$ is then clear as a consequence of Lemma 3.2. ∎

######  Remark 3.6.

We chose to write the results for Calderón-Zygmund defined from
$L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ into itself.
Actually, they extend without any modification to maps from
$L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})$ to
$L_{2}(L_{\infty}(\mathbb{R})\overline{\otimes}\widehat{\mathcal{M}})$ with
$\widehat{\mathcal{M}}$-valued kernels as long as
$(\mathcal{M},\tau)\subset(\widehat{\mathcal{M}},\widehat{\tau})$ is an
inclusion of von Neumann algebras with $\widehat{\tau}_{|\mathcal{M}}=\tau$
and $T$ has the right modular property with respect to $\mathcal{M}$.

######  Remark 3.7.

As we already mentioned in the Introduction, all of our results hold for
functions in $\mathbb{R}^{n}$. They also extend to more general measure metric
spaces so long as the underlying measure is doubling, that is, if the
condition

$\mu(B(x,2r))\leq C_{\mu}\mu(B(x,r),$

holds for all $x\in\mathrm{supp}(\mu)$ and all $r>0$ and that one has suitable
approximations (such as in Lemma 2.12 or 3.3). When $\mu$ is a measure on
$\mathbb{R}^{n}$ which fails the doubling condition the definition of the
appropriate $\mathsf{BMO}$-type space and a predual is more involved and due
to Tolsa in the classical case [T01]. A semicommutative definition in that
context can be found in [CP19]. We let the interested reader to try to carry
on all our arguments to that setting.

##  4\. An example

We sketch how to use Theorem 3.5 to recover some results in [M07]. A similar
approach was used in [XXX16] to obtain characterizations of some Hardy spaces
relying on column-Hilbert spaces Calderón-Zygmund kernels, and indeed
constitutes a particular case of our work.

Let $(\mathcal{N},\tau)$ be a semifinite von Neumann algebra and consider

$\mathcal{M}=\mathcal{N}\overline{\otimes}B(L_{2}(\mathbb{R}^{+},t\ dt)^{2})$

with its natural tensor product trace. We fix $\mathds{1}$ a function of norm
1 in $L_{2}(\mathbb{R}^{+},t\ dt)^{2}$.

Let $P(x,y)=\frac{1}{\pi}\frac{y}{x^{2}+y^{2}}$ be the Poisson kernel for the
upper half plane. We can define a continuous convolution kernel
$K:\mathbb{R}\times\mathbb{R}\ \setminus\
\\{x=y\\}\to\mathcal{M}=\mathcal{N}\overline{\otimes}B(L_{2}(\mathbb{R}^{+},t\
dt)^{2})$ by $K(x,v)=\mathcal{K}(x-v)$ where $\mathcal{K}$ is defined as

$\mathcal{K}(x)=1_{\mathcal{N}}\otimes\Big{(}\big{(}\frac{\partial}{\partial
x}P(x,\cdot),\frac{\partial}{\partial
y}P(x,\cdot)\big{)}\otimes\mathds{1}\Big{)}$

for $x\neq 0$. The map $\mathcal{K}$ is continuous and bounded on each closed
interval not containing 0. Since $\mathcal{N}$ does not play any role,
$\mathcal{K}$ actually takes values in a column Hilbert space. It is a
standard fact that it satisfies the Mikhlin condition. All this implies that
$K$ satisfies the Hörmander condition.

The Calderón-Zygmund operator $T$ associated to $K$ can be defined. At the
$L_{2}$-level, it acts only on a column subspace of the
$S_{2}(L_{2}(\mathbb{R}^{+},t\ dt)^{2})$ component. Thus, by homogeneity of
hilbertian operator spaces, we deduce that it is bounded from the classical
result with values in Hilbert spaces (i.e. going from $L_{2}(\mathbb{R})$ to
$L_{2}(\mathbb{R};L_{2}(\mathbb{R}^{+},t\ dt))$). Since $T$ is right-modular
with respect to $\mathcal{N}$ inside the tensor algebra
$\mathcal{N}\overline{\otimes}B(L_{2}(\mathbb{R}^{+},t\ dt)^{2})$ (with
$n\mapsto n\otimes p_{\mathds{1}}$ which preserves the traces), we indeed have
a left Calderón-Zygmund operator from
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{N}$ to
$L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M}$ by Remark 3.6.

Let $f$ be a compactly supported simple function on $\mathbb{R}$ with values
in $\mathcal{N}\cap L_{1}(\mathcal{N})$. If $F$ denotes the harmonic extension
of $f$ to the upper half plane, the column Littlewood-Paley function
associated to $f$ is

$G_{c}(f)(x)=\Big{(}\int_{\mathbb{R}^{+}}|\nabla F(x,y)|^{2}y\
dy\Big{)}^{1/2}.$

We can see $f$ as a function with values in $\mathcal{M}$ by considering
$f\otimes(\mathds{1}\otimes\mathds{1})$ as we already did. Some easy
computations yield

$\big{|}T(f\otimes p_{\mathds{1}})\big{|}^{2}=G_{c}(f)^{2}\otimes
p_{\mathds{1}}.$

After giving suitable definitions for all the objects, Theorem 3.5 yields that
there is a constant $C$ so that there holds

(14)
$\|G_{c}(f)\|_{L_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{M})}\leq
C\|f\|_{H^{c}_{1}(L_{\infty}(\mathbb{R})\overline{\otimes}\mathcal{N})}.$

for every $f\in H^{c}_{1}(\mathcal{A})$. In the same way, one can also get
(14) with the Lusin square function $S_{c}(f)$ instead of the Littlewood-Paley
one.

Consequently our definition for $H_{1}^{c}(\mathcal{A})$ matches that of
[M07]: we have just proved an inclusion and the dual spaces are the same (see
Theorem 2.4 (i) and Corollary 2.7 there).

As another direct application, we recover a dual result of [JMP14] and close
to [HLMP14]: for scalar kernels satisfying the Hörmander condition, the
associated Calderón-Zygmund operators are bounded on
$L_{p}(\mathbb{R};L_{p}(\mathcal{M}))$ when $1<p<\infty$.

##  Appendix A Vector-valued Hardy spaces

This appendix contains an explicit argument for the construction of the map

$\displaystyle
Q:L_{2}^{\circ}(\mathbb{R},(1+t^{2})dt;L_{2}(\mathcal{M}))\longrightarrow
H_{1}(\mathbb{R};L_{2}(\mathcal{M}))$

as stated along the discussion preceding Proposition 2.6. Actually, the map
$Q$ can be considered whenever $L_{2}(\mathcal{M})$ is replaced by any other
Banach space. Moreover, a brief study of _molecules_ , which give an
additional description of the vector-valued Hardy space, is included.

Given a Banach space $\mathbb{X}$ we say that a function $a$ belonging to
$L_{1}(\mathbb{R};\mathbb{X})$ is a $L_{2}(\mathbb{R};\mathbb{X})$-atom in
$H_{1}(\mathbb{R};\mathbb{X})$ whenever it satisfies the following conditions:

  * •

$\mathrm{supp}(a)\subseteq I$ for some interval $I$,

  * •

$\int_{I}a=0$.

  * •

$\|a\|_{L_{2}(\mathbb{R};\mathbb{X})}\leq\frac{1}{\sqrt{|I|}}$,

Then, $H_{1}(\mathbb{R};\mathbb{X})$ is defined as the subspace of those
functions $f$ in $L_{1}$ admitting a decomposition

(15) $\displaystyle
f=\sum\limits_{i=1}^{\infty}\lambda_{i}a_{i}\quad\mathrm{in\
}L_{1}(\mathbb{R};\mathbb{X})$

for some absolutely summable sequence $(\lambda_{i})_{i=1}^{\infty}$ and some
family of $L_{2}(\mathbb{R};\mathbb{X})$-atoms $(a_{i})_{i=1}^{\infty}$.
According to [H06], the definition of $H_{1}(\mathbb{R};\mathbb{X})$ can be
given via maximal functions the same way it is done in the scalar-valued case.
This justifies which sort of convergence is considered at (15). It can be
checked that the norm

$\displaystyle\|f\|_{H_{1}(\mathbb{R};\mathbb{X})}=\inf\\{\sum_{i=1}^{\infty}|\lambda_{i}|\
:\ f=\sum_{i=1}^{\infty}\lambda_{i}a_{i}\ \mathrm{for\ some\
}(\lambda_{i})_{i}\in\ell_{1}\ and\ L_{2}\textrm{-atoms}\ (a_{i})_{i}\\}$

is equivalent to the one defined via maximal functions with values in
arbitrary Banach spaces. On the other hand, we include below the
characterization via _molecules_ of $H_{1}(\mathbb{R};\mathbb{X})$ which
streamlines the proof for the classical case [Mey90, Ch. 5 Sec. 5].

Let $\mathbb{\mathbb{X}}$ be a Banach space and consider as before the
function $\omega(x)=1+x^{2}$. Then, the space of Bochner measurable functions

$\displaystyle L_{2}(\mathbb{R},\omega\ dx;\mathbb{\mathbb{X}})=\big{\\{}f\in
L_{0}(\mathbb{R};\mathbb{\mathbb{X}})\ :\
\int_{\mathbb{R}}\|f(x)\|^{2}_{\mathbb{\mathbb{X}}}\ \omega(x)\ dx\
<\infty\big{\\}}$

is contained in $L_{1}(\mathbb{R};\mathbb{\mathbb{X}})$. We will consider the
subspace

$\displaystyle M(\mathbb{\mathbb{X}})=\big{\\{}f\in L_{2}(\mathbb{R},\omega\
dx;\mathbb{\mathbb{X}})\ :\ \int_{\mathbb{R}}f=0\big{\\}}.$

######  Lemma A.1.

$M(\mathbb{\mathbb{X}})$ is a dense linear subspace of
$H_{1}(\mathbb{R};\mathbb{\mathbb{X}})$.

###### Proof.

Let $f\in M(\mathbb{\mathbb{X}})$ and define

$\displaystyle f_{0}=f\ \mbox{\large$\chi$}_{\\{|x|\leq 1\\}}\ \mathrm{and}\
f_{j}=f\ \mbox{\large$\chi$}_{\\{2^{j-1}<|x|\leq 2^{j}\\}}\ \mathrm{for\ any\
}j>0.$

These functions satisfy

$\displaystyle\|f_{j}\|_{L_{2}(\mathbb{R};\mathbb{\mathbb{X}})}$
$\displaystyle=\Big{(}\int_{2^{j-1}<|x|\leq
2^{j}}\|f(x)\|_{\mathbb{\mathbb{X}}}^{2}\ dx\Big{)}^{1/2}$
$\displaystyle\leq\Big{(}\int_{2^{j-1}<|x|\leq
2^{j}}\|f(x)\|^{2}_{\mathbb{X}}\ \omega(x)\ dx\Big{)}^{1/2}\
2^{-(j-1)}=:R_{j}\ 2^{-j}$

so that the sequence $(R_{j})_{j\geq 0}$ belongs to $\ell_{2}$. Let $I_{j}$ be
the integral $\int_{\mathbb{R}}f_{j}$. Then, by the Cauchy-Schwarz inequality
and a similar computation, it follows that, for $j\geq 0$

$\displaystyle\|I_{j}\|_{\mathbb{X}}$
$\displaystyle\leq\Big{(}\int_{2^{j-1}<|x|\leq
2^{j}}\|f(x)\|^{2}_{\mathbb{X}}\ \omega(x)\ dx\Big{)}^{1/2}\
\Big{(}\int_{2^{j-1}<|x|\leq 2^{j}}\omega(x)^{-1}\ dx\Big{)}^{1/2}$
$\displaystyle\leq R_{j}\ 2^{-j/2}.$

Therefore, this yields some estimates for $S_{j}=\sum_{k\geq j}I_{j}$. Indeed,

$\displaystyle\|S_{j}\|_{\mathbb{X}}\leq c\sum\limits_{k\geq j}R_{k}\
2^{-k/2}.$

Now, let’s replace the functions $f_{j}$ by some _perturbed_ atoms $a_{j}$
given by

$\displaystyle a_{j}(x)=f_{j}(x)+S_{j+1}\ 2^{-(j+2)}\
\mbox{\large$\chi$}_{|x|\leq 2^{j+1}}(x)-S_{j}\ 2^{-(j+1)}\
\mbox{\large$\chi$}_{|x|\leq 2^{j}}(x).$

Then, the sequence $(a_{j})_{j\geq 0}$ satisfies

$\displaystyle\int_{\mathbb{R}}a_{j}(x)\ dx$
$\displaystyle=\int_{2^{j-1}<|x|\leq 2^{j}}f(x)\ dx+S_{j+1}-S_{j}$
$\displaystyle=\int_{2^{j-1}<|x|\leq 2^{j}}f(x)\ dx-I_{j}=0.$

Moreover, it is easy to check that, by hypothesis, the support of $a_{j}$ is
contained in $[-2^{j+1},2^{j+1}]$ and using the triangle inequality,

$\displaystyle\|a_{j}\|_{L_{2}(\mathbb{R};\mathbb{X})}$ $\displaystyle\leq
R_{j}\ 2^{-j}+2\sum\limits_{k\geq j}R_{k}\ 2^{-k/2}2^{-(j+1)/2}=\lambda_{j}.$

Then $(\lambda_{i})_{i\in\mathbb{N}}\in\ell_{1}$ by previous computations and
since

$\displaystyle\sum_{j=0}^{\infty}\sum_{k\geq j}2^{-(j+1)/2}R_{j}\ 2^{-k/2}$
$\displaystyle\lesssim\sum_{j=0}^{\infty}R_{j}2^{-j}\lesssim\|f\|_{L_{2}(\mathbb{R},(1+t^{2})dt;L_{2}(\mathcal{M}))}.$

Moreover, redefining $a_{i}$ as $a_{i}/\lambda_{i}$, we obtain the expression

$\displaystyle f=\sum_{i\in\mathbb{N}}\lambda_{i}\ a_{i}$

where each $a_{i}$ is an atom with support $[-2^{j+1},2^{j+1}]$ and
convergence holds in the sense of $L_{1}(\mathbb{R};\mathbb{X})$. In
conclusion, $f$ belongs to $H_{1}(\mathbb{R};\mathbb{X})$. ∎

######  Definition A.2.

A _molecule_ $f$ with values in $\mathbb{X}$, centered at $x_{0}$ and of width
$d>0$, is defined to be a function belonging to $M(\mathbb{X})$ which is
normalized by

$\displaystyle\Big{(}\int_{\mathbb{R}}\|f(x)\|_{\mathbb{X}}^{2}\
\Big{(}1+\frac{|x-x_{0}|^{2}}{d^{2}}\Big{)}\ dx\Big{)}^{1/2}\leq d^{-1/2}.$

The $H_{1}(\mathbb{R};\mathbb{X})$-norm is invariant by translations and
homogeneous for composition with homothethies. It follows that a molecule $f$
sits in $H_{1}(\mathbb{R};\mathbb{X})-$ with a norm controlled by an absolute
constant. One can check that an atom is a molecule. Thus, one can use
molecules instead of atoms in the definition of
$H_{1}(\mathbb{R};\mathbb{X})$.

##  Appendix B Proof of Lemma 2.12

The justification for Lemma 2.12 is included in this appendix. This result was
stated in the commutative setting by Garnett [G07], and also by Mei without
proof [M07]. For that reason, a general version of the argument is developed
here. Before giving the explicit construction, some preliminary classical
results are showed. We will use the easier form (3) and to lighten the
notation we drop $\otimes\mathds{1}$.

######  Lemma B.1.

Suppose $f\in
L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R},\frac{dt}{1+t^{2}}))$ and let
$I\subset J$ be finite intervals. Then, there holds

$\big{\|}\frac{1}{\sqrt{|I|}}\widetilde{\iota}_{I}(f-f_{I})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}\leq\sqrt{\frac{|J|}{|I|}}\big{\|}\frac{1}{\sqrt{|J|}}\widetilde{\iota}_{J}(f-f_{J})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}.$

###### Proof.

This is clear because one goes from $\iota_{J}(f-f_{J})$ just by restricting
and centering with respect to $I$ and all these operations are contractions at
the $L_{2}$-level. ∎

Let $J=(a,b]$ be a finite interval and let $c_{J}=\frac{a+b}{2}$ denote the
center of $J$. Write

$\displaystyle
J=J_{0}\cup\bigcup_{n=1}^{\infty}J_{n}\cup\bigcup_{n=1}^{\infty}J_{n}^{\prime}$

where $d(J_{n},\partial J)=|J_{n}|$ for $n\geq 0$ and
$d(J_{n}^{\prime},\partial J)=|J_{n}^{\prime}|$ for $n\geq 1$. Then $J_{0}$
coincides with the middle third of $J$, that is,

$\displaystyle
J_{0}=\frac{1}{3}J=\big{(}c_{J}-\frac{1}{2}\frac{1}{3}|J|,c_{J}+\frac{1}{2}\frac{1}{3}|J|\big{]},$

while for any $n\geq 1$,

$\displaystyle J_{n}$
$\displaystyle=\big{(}c_{J}+\frac{|J|}{3}\sum_{k=0}^{n-1}\frac{1}{2^{k}},c_{J}+\frac{|J|}{3}\sum_{k=0}^{n}\frac{1}{2^{k}}\big{]}$
$\displaystyle J_{n}^{\prime}$
$\displaystyle=\big{(}c_{J}-\frac{|J|}{3}\sum_{k=0}^{n}\frac{1}{2^{k}},c_{J}-\frac{|J|}{3}\sum_{k=0}^{n-1}\frac{1}{2^{k}}\big{]}.$

Thus, $|J_{n}|=|J_{n}^{\prime}|=\frac{|J|}{3}\frac{1}{2^{n}}$. Set $K_{n}$
(respectively $K_{n}^{\prime}$) as the reflection of $J_{n}$ (respectively
$J_{n}^{\prime}$) across $b$ (respectively $a$). Then,

$\displaystyle K_{n}$
$\displaystyle=\big{[}b+\frac{|J|}{3}-\frac{|J|}{3}\sum_{k=1}^{n}\frac{1}{2^{k}},b+\frac{|J|}{3}-\frac{|J|}{3}\sum_{k=1}^{n-1}\frac{1}{2^{k}}\big{)},$
$\displaystyle K_{n}^{\prime}$
$\displaystyle=\big{[}a-\frac{|J|}{3}+\frac{|J|}{3}\sum_{k=1}^{n-1}\frac{1}{2^{k}},a-\frac{|J|}{3}+\frac{|J|}{3}\sum_{k=1}^{n}\frac{1}{2^{k}}\big{)},$

so $|K_{n}|=|K_{n}^{\prime}|=|J_{n}|=|J_{n}^{\prime}|$. Finally, define
$L=[b+\frac{|J|}{3},\infty)$ and $L^{\prime}=(-\infty,a-\frac{|J|}{3})$. This
construction yields the desired operator $\psi$, which is given by the
following expression assuming the normalization $\varphi_{J}=0$ (recall that
$\varphi$ is defined up to a constant factor):

$\displaystyle\psi$ $\displaystyle=\widetilde{P}_{J}\varphi+\sum_{n\geq
1}\varphi_{J_{n}}\otimes\mbox{\large$\chi$}_{K_{n}}+\sum_{n\geq
1}\varphi_{J_{n}^{\prime}}\otimes\mbox{\large$\chi$}_{K_{n}^{\prime}}.$

Let $J^{+}=\cap_{n\geq 1}J_{n}$ and $J^{-}=\cap_{n\leq 1}J_{n}$. It is worth
to mention that if $\mathbb{E}^{+}$ denotes the conditional expectation on
$L_{2}(J^{+})$ corresponding to the partition $\\{J_{n};n\geq 1\\}$, then
$\sum_{n\geq 1}\varphi_{J_{n}}\otimes\mbox{\large$\chi$}_{K_{n}}$ is nothing
but
$\widetilde{R}_{+}(\widetilde{\mathbb{E}^{+}})(\widetilde{P}_{J^{+}}\varphi)$
where $R_{+}$ is the reflection with respect to $b$. A similar formula holds
for $\sum_{n\geq
1}\varphi_{J_{n}^{\prime}}\otimes\mbox{\large$\chi$}_{K_{n}^{\prime}}$.

We turn to the proof of Lemma 2.12. We need to bound, for all finite intervals
$I$, the quantity

$\big{\|}\frac{1}{\sqrt{|I|}}\iota_{I}(\psi-\psi_{I})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}.$

We distinguish according to $I$:

Case 1: $I\subset J$; there is nothing to do as
$\iota_{I}(\psi-\psi_{I})=\iota_{I}(\varphi-\varphi_{I})$.

Case 2: $b\in I$.

Using Lemma (B.1), we can assume that $I$ is symmetric around $b$ losing a
factor $\sqrt{2}$.

If $a\notin I$, then $I\subset\cup_{n=n_{0}}^{\infty}(J_{n}\cup K_{n})$ with
$J_{n_{0}-1}\cap I=\emptyset$. Similarly, by enlarging it and losing another
factor $\sqrt{2}$, we can assume it is has the form
$\cup_{n=n_{0}}^{\infty}(J_{n}\cup K_{n})$ because $|J_{n_{0}}|\leq|I|$. Then
$\psi_{I}=\psi_{I^{+}}=\psi_{I^{-}}=\varphi_{I^{-}}$ where
$I^{-}=\cup_{n=n_{0}}^{\infty}J_{n}$ and $I^{+}=\cup_{n=n_{0}}^{\infty}K_{n}$,
$|I|=2|I^{-}|$. Note that $\widetilde{R}_{+}$ yields an isometry and

$\|\frac{1}{\sqrt{|I^{+}|}}\iota_{I^{+}}(\psi-\psi_{I^{+}})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}=\|\frac{1}{\sqrt{|I^{-}|}}\widetilde{\mathbb{E}^{+}}\iota_{I^{-}}(\varphi-\varphi_{I^{-}})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}.$

Thus using that $\widetilde{\mathbb{E}^{+}}$ is a contraction commuting with
$\iota_{I^{-}}$, we get that

$\|\frac{1}{\sqrt{|I|}}\iota_{I}(\psi-\psi_{I})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}^{2}\leq\|\frac{1}{\sqrt{|I^{+}|}}\iota_{I^{+}}(\psi-\psi_{I^{+}})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}^{2}\leq\|\varphi\|_{\mathsf{BMO}_{c}}.$

If $a\in I$, then we can simply replace it with $3J$ which has a comparable
size. then a similarly reasoning as above also gives a control by
$\|\frac{1}{\sqrt{|J|}}\iota_{J}(\varphi-\varphi_{J})\big{\|}_{L_{\infty}(\mathcal{M};L_{2}^{c}(\mathbb{R}))}$.

Case 3: $a\in I$. One can proceed as in case 1.

Case 4: $I\subset(b,\infty)$.

If $I\supset K_{n}$ for some $n$, then $|I|\geq|K_{n}|$ and losing a factor
$\sqrt{2}$ we can replace it by the bigger interval $\cup_{k\geq n}K_{k}\cup
I$ which has size less than $2|I|$. We are then back to case 2.

If $I\subset K_{n}$ for some $n$ there is noting to do as
$\iota_{I}(\psi-\psi_{I})=0$.

If $I\subset K_{n}\cup K_{n+1}$ for some $n\geq 1$. By losing a constant
factor we can assume that $I$ is symmetric with respect to left border of
$K_{n}$ say $b_{n}$. If the size of $I$ is smaller than $2|K_{n+1}|$, then
$\iota_{I}(\psi-\psi_{I})$ is a simple function taking two values
$\pm\frac{1}{2}(\psi_{K_{n}}-\psi_{K_{n+1}})$. Then the norm of
$\frac{1}{\sqrt{|I|}}\iota_{I}(\psi-\psi_{I})$ is
$\|\frac{1}{2}(\psi_{K_{n}}-\psi_{K_{n+1}})\|$. It the same as if we consider
$(b_{n}-|K_{n+1}|,b_{n}+|K_{n+1}|)$ and we are back to situation where
$K_{n+1}\subset I$ (or we can use the argument of case 2).

The only remaining situation is when $I\subset K_{1}\cup L$. Then an easy
calculation gives that the involved norm is controlled by
$\|\sqrt{2}\psi_{K_{1}}\|=\|\sqrt{2}\varphi_{J_{1}}\|$. This is where we use
that $\varphi_{J}=0$ to say that (recall $6|J_{1}|=|J|$)

$\|\varphi_{J_{1}}\|\leq\sqrt{6}\|\frac{1}{\sqrt{|J|}}\widetilde{\mathbb{E}^{+}}i_{J}(\varphi-0)\|\leq\sqrt{6}\|\varphi\|_{\mathsf{BMO}_{c}}.$

Case 5: $I\subset(-\infty,a)$. It can be done similarly.

We have dealt with all possible situations for $I$. This conclude Lemma 2.12
with at most $C=2\sqrt{6}$.

Acknowledgement. The first author was supported by Grant SEV-2015-0554-19-3
funded by MCIN/AEI/10.13039/501100011033 and by ESF Investing in your future.
In addition, he was partially supported by Grant PID 2019-107914GB-I00 (MCIN
PI J. Parcet), and by the Madrid Government Program V under PRICIT Grant
SI1/PJI/2019-00514.

## References