cond}(\mathcal{M};\ell_{2}^{c})}\lesssim_{p,q}\big{\|}\xi\|_{L_{\Phi}(\mathcal{M};\ell_{2}^{c})}.$

###### Proof.

Let $\Theta$ be an Orlicz function such that $L_{\Phi}=L_{2}\odot L_{\Theta}$.
Consider the factorization $\xi=\alpha\,.\,\beta$ where $\alpha$ is a lower
triangular matrix in
$L_{2}(\mathcal{M}\overline{\otimes}B(\ell_{2}(\mathbb{N})))$ and $\beta\in
L_{\Theta}^{\rm ad}(\mathcal{M};\ell_{2}^{c})$ according to Proposition 2.12.
Define the strictly lower triangular matrix $\alpha^{-}$ by setting
$\alpha^{-}_{n,j}=\alpha_{n,j}$ for $1\leq j<n$ and the diagonal matrix
$d=\sum_{n\geq 1}\alpha_{n,n}\otimes e_{n,n}$. Clearly, $\alpha=\alpha^{-}+d$.
Set:

$y=d\,.\,\beta\quad\text{and}\quad z=\alpha^{-}\,.\,\beta.$

Then $\xi=y+z$ and from Proposition 2.12(ii), $y$ and $z$ are adapted. Since
$\alpha^{-}$ is strictly lower triangular and $\beta$ is adapted, $z$ is a
scalar multiple of an algebraic $L_{\Phi}^{c,\rm cond}$-atom. The verification
of the norm estimates are straightforward. ∎

We now turn our attention to Davis type inequalities involving other classes
of symmetric spaces of measurable operators using interpolation results from
the previous section. The next result deals with the case of Lorentz spaces.

######  Proposition 4.9.

Let $\xi=(\xi_{n})_{n\geq 1}$ be an adapted sequence that belongs to
$L_{2/3}(\mathcal{M};\ell_{2}^{c})\cap L_{2}(\mathcal{M};\ell_{2}^{c})$. Then
there exist two adapted sequences $y=(y_{n})_{n\geq 1}$ and $z=(z_{n})_{n\geq
1}$ in $L_{2/3}(\mathcal{M})\cap L_{2}(\mathcal{M})$ such that:

  1. (i)

$\xi=y+z$;

  2. (ii)

for every $2/3<p<2$ and $0<q\leq\infty$, the following holds:

$\big{\|}y\big{\|}_{L_{p,q}(\mathcal{M}\overline{\otimes}\ell_{\infty})}+\big{\|}z\big{\|}_{L_{p,q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}\lesssim_{p,q}\big{\|}\xi\big{\|}_{L_{p,q}(\mathcal{M};\ell_{2}^{c})}.$

###### Proof.

The key ingredients for the proof are the decomposition from [43, Theorem 3.1]
and the two interpolations results from Proposition 3.12 and Theorem 3.13.

Fix $\xi\in L_{2/3}^{\rm ad}(\mathcal{M};\ell_{2}^{c})\cap L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$. Choose a discrete representation of $\xi$ with
respect to the couple $(L_{2/3}^{\rm ad}(\mathcal{M};\ell_{2}^{c}),L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c}))$,

$\xi=\sum_{\nu\in\mathbb{Z}}\xi^{(\nu)}$

with $\xi^{(\nu)}\in L_{2/3}^{\rm ad}(\mathcal{M};\ell_{2}^{c})\cap L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$ for every $\nu\in\mathbb{Z}$, the series is
convergent in $L_{2/3}^{\rm ad}(\mathcal{M};\ell_{2}^{c})+L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$, and so that

$J\big{(}\xi^{(\nu)},2^{\nu},L_{2/3}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}\leq 4K\big{(}\xi,2^{\nu},L_{2/3}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}.$

We refer to [3, Lemma 3.3.2] for the existence of a representation satisfying
the properties discribed above.

For each $\nu\in\mathbb{Z}$, we apply [43, Theorem 3.1] to the adapted
sequence $\xi^{(\nu)}$: there exists two adapted sequences $y^{(\nu)}$ and
$z^{(\nu)}$ such that $\xi^{(\nu)}=y^{(\nu)}+z^{(\nu)}$,

$J\big{(}y^{(\nu)},2^{\nu},L_{2/3}(\mathcal{M}\overline{\otimes}\ell_{\infty};L_{2}(\mathcal{M}\overline{\otimes}\ell_{\infty}))\big{)}\lesssim
J\big{(}\xi^{(\nu)},2^{\nu},L_{2/3}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)},$

and

$J\big{(}z^{(\nu)},2^{\nu},L_{2/3}^{\rm
cond}(\mathcal{M};\ell_{2}^{c});L_{2}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})\big{)}\lesssim
J\big{(}\xi^{(\nu)},2^{\nu},L_{2/3}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{2}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}.$

It follows that for $0<\theta<1$ and $0<q\leq\infty$, we have:

(4.2)
$\big{\|}(J(y^{(\nu)},2^{\nu}))_{\nu}\big{\|}_{\lambda^{\theta,q}}\lesssim_{\theta,q}\big{\|}(K(\xi,2^{\nu}))_{\nu}\big{\|}_{\lambda^{\theta,q}}$

and

(4.3)
$\big{\|}(J(z^{(\nu)},2^{\nu}))_{\nu}\big{\|}_{\lambda^{\theta,q}}\lesssim_{\theta,q}\big{\|}(K(\xi,2^{\nu}))_{\nu}\big{\|}_{\lambda^{\theta,q}}$

where for a given sequence $(a_{\nu})_{\nu}$ of scalars, we use the quasi-norm

$\big{\|}(a_{\nu})_{\nu}\big{\|}_{\lambda^{\theta,q}}:=\Big{(}\sum_{\nu\in\mathbb{Z}}\big{(}2^{-\nu\theta}|a_{\nu}|\big{)}^{q}\Big{)}^{1/q}.$

Let $y=\sum_{\nu\in\mathbb{Z}}y^{(\nu)}$ and
$z=\sum_{\nu\in\mathbb{Z}}z^{(\nu)}$. It is clear $y$ and $z$ are adapted
sequences and $\xi=y+z$. For any given $2/3<p<2$ and $0<q\leq\infty$, choose
$\theta=3/2-1/p$. It clearly follows from (4.2) and Theorem 3.13 that:

$\big{\|}y\big{\|}_{L_{p,q}(\mathcal{M}\overline{\otimes}\ell_{\infty})}\lesssim_{p,q}\|\xi\|_{L_{p,q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})}.$

Similarly, we may deduce from (4.3) and Proposition 3.12 that

$\big{\|}z\big{\|}_{L_{p,q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}\lesssim_{p,q}\|\xi\|_{L_{p,q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c}}.$

The desired inequality follows from combining the last two inequalities. ∎

We isolate the following important example. Motivated by the noncommutative
Khintchine inequality for weak-$L_{1}$ spaces ([4]), we state below a
weak-$L_{1}$-version of the Davis-decomposition for adapted sequences. This
appears to be new even for the classical setting.

######  Example 4.10.

There exists a constant $C>0$ so that for every adapted sequence $\xi\in
L_{1,\infty}(\mathcal{M};\ell_{2}^{c})$, there exist two adapted sequences $y$
and $z$ such that:

  1. (i)

$\xi=x+z$;

  2. (ii)

$\big{\|}y\big{\|}_{L_{1,\infty}(\mathcal{M}\overline{\otimes}\ell_{\infty})}+\big{\|}z\big{\|}_{L_{1,\infty}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}\leq
C\big{\|}\xi\big{\|}_{L_{1,\infty}(\mathcal{M};\ell_{2}^{c})}$.

It is worth pointing out that the preceding example allows us to deduce the
noncommutative weak-type $(1,1)$ version of the Burkholder/Rosenthal ([41,
Theorem 3.1]) from the simpler weak-type inequality involving square functions
given in [40, Theorem 2.1].

The idea used in the proof of Proposition 4.9 can be extended for the case of
general symmetric spaces for the Banach space range. More precisely, we have
the following result:

######  Proposition 4.11.

Let $E$ be a Banach function space. If $E\in{\rm Int}(L_{1},L_{q})$ for
$1<q<2$, then

$E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})=E(\oplus_{n=1}^{\infty}\mathcal{M}_{n})+E^{\rm
cond,ad}(\mathcal{M};\ell_{2}^{c})$

where $E^{\rm cond,ad}(\mathcal{M};\ell_{2}^{c})$ denotes is the subspace of
$E^{\rm cond}(\mathcal{M};\ell_{2}^{c})$ consisting of adapted sequences.

###### Sketch of the proof.

It is easy to see that under the assumption,
$E(\oplus_{n=1}^{\infty}\mathcal{M}_{n})\subseteq E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$. On the other hand, we have from Theorem 4.5
that $E^{\rm cond,ad}(\mathcal{M};\ell_{2}^{c})\subseteq E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$. Thus, we only need to verify one inequality.

The proof rests upon few facts. The interpolation space $E$ is given by a
$K$-method, the simultaneous nature of Proposition 4.9 above, and Proposition
4.3.

Since $E$ is given by a $K$-method, we may fix a Banach function space
$\mathcal{F}$ such that for any given semifinite von Neumann algebra
$\mathcal{N}$,

$C_{E}^{-1}\big{\|}a\big{\|}_{\mathcal{F};K}\leq\big{\|}a\big{\|}_{E(\mathcal{N})}\leq\big{\|}a\big{\|}_{\mathcal{F};K}\
,\quad a\in E(\mathcal{N}).$

Under the assumption $1<q<2$ and Proposition 4.3, we may also state that for
every sequence $\xi\in E^{\rm ad}(\mathcal{M};\ell_{2}^{c})$,

$\big{\|}\xi\big{\|}_{\big{(}L_{1}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}_{\mathcal{F};K}}\approx_{E}\big{\|}\xi\big{\|}_{E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})}.$

Similarly, for every $\zeta\in E^{\rm cond}(\mathcal{M};\ell_{2}^{c})$,

$\big{\|}\zeta\big{\|}_{\big{(}L_{1}^{\rm
cond}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})\big{)}_{\mathcal{F};K}}\approx_{E}\big{\|}\zeta\big{\|}_{E^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}.$

We now outline the argument. Fix $\xi\in L_{1}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\cap L_{q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}$. Repeating the argument in the proof of
Proposition 4.9 (taking into account the fact that the decomposition in
Proposition 4.9 works simultaneously), we obtain a decomposition $\xi=y+z$
where $y=\sum_{\nu\in\mathbb{Z}}y^{(\nu)}$ and
$z=\sum_{\nu\in\mathbb{Z}}z^{(\nu)}$ are representations with respect to the
couple
$(L_{1}(\mathcal{M}\overline{\otimes}\ell_{\infty}),L_{q}(\mathcal{M}\overline{\otimes}\ell_{\infty}))$
and $(L_{1}^{\rm cond}(\mathcal{M};\ell_{2}^{c}),L_{q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c}))$ respectively and further satisfy that for
every $\nu\in\mathbb{Z}$,

$J\big{(}y^{(\nu)},2^{\nu},L_{1}(\mathcal{M}\overline{\otimes}\ell_{\infty});L_{q}(\mathcal{M}\overline{\otimes}\ell_{\infty})\big{)}\lesssim_{q}K\big{(}\xi,2^{\nu},L_{1}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm ad}(\mathcal{M};\ell_{2}^{c})\big{)}$

and

$J\big{(}z^{(\nu)},2^{\nu},L_{1}^{\rm
cond}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c})\big{)}\lesssim_{q}K\big{(}\xi,2^{\nu},L_{1}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})\big{)}.$

Consider the following functions defined on the semi-axis $(0,\infty)$:

$f(t)=J(y^{(\nu)},2^{\nu})\quad\text{for}\ t\in[2^{\nu},2^{\nu+1}),$
$g(t)=J(z^{(\nu)},2^{\nu})\quad\text{for}\ t\in[2^{\nu},2^{\nu+1}),$

and

$h(t)=K(\xi,2^{\nu})\quad\text{for}\ t\in[2^{\nu},2^{\nu+1}).$

It follows that for every $t>0$,

$\max\big{\\{}f(t);g(t)\big{\\}}\lesssim_{q}h(t).$

Taking the norms on the function space $\mathcal{F}$, we have

$\big{\|}f\big{\|}_{\mathcal{F}}+\big{\|}g\big{\|}_{\mathcal{F}}\lesssim_{q}\big{\|}h\big{\|}_{\mathcal{F}}.$

From the definitions of the three functions, we further get that:

(4.4)
$\|y\|_{(L_{1}(\mathcal{M}\overline{\otimes}\ell_{\infty}),L_{q}(\mathcal{M}\overline{\otimes}\ell_{\infty})_{\mathcal{F};J}}+\|z\|_{(L_{1}^{\rm
cond}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c}))_{\mathcal{F};J}}\lesssim_{q}\|\xi\|_{(L_{1}^{\rm
ad}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c}))_{\mathcal{F};K}}.$

From the equivalence of the $J$-methods and $K$-methods relative the function
space $\mathcal{F}$ (see for instance, [30, Theorem 2.9]) and the equivalence
of norms stated at the beginning of the proof, we may conclude that:

$\|y\|_{E(\mathcal{M}\overline{\otimes}\ell_{\infty})}+\|z\|_{E^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}\lesssim_{E}\|\xi\|_{E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})}$

which is the desired inequality. ∎

######  Remark 4.12.

The argument in the proof of Theorem 4.11 can be carried out for the larger
class $E\in{\rm Int}(L_{p},L_{q})$ with $2/3<p<q<2$ to get that every $\xi\in
L_{p}^{\rm ad}(\mathcal{M};\ell_{2}^{c})\cap L_{q}^{\rm
ad}(\mathcal{M};\ell_{2}^{c})$ admits a decomposition into two adapted
sequences $y$ and $z$ satisfying:

$\|y\|_{(L_{p}(\mathcal{M}\overline{\otimes}\ell_{\infty}),L_{q}(\mathcal{M}\overline{\otimes}\ell_{\infty})_{\mathcal{F};J}}+\|z\|_{(L_{p}^{\rm
cond}(\mathcal{M};\ell_{2}^{c});L_{q}^{\rm
cond}(\mathcal{M};\ell_{2}^{c}))_{\mathcal{F};J}}\lesssim_{E}\|\xi\|_{E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})}.$

However, we do not know if the equivalence of the $J$-method and the
$K$-method relative to the function space $\mathcal{F}$ is valid for the case
of quasi-Banach couples.

As an immediate application of Proposition 4.11, we deduce the next result
which partially answers a problem from [43, Remark 3.11].

######  Corollary 4.13.

Let $E$ be a Banach function space. If $E\in{\rm Int}(L_{1},L_{q})$ for
$1<q<2$, then

$\mathcal{H}_{E}^{c}(\mathcal{M})=\mathsf{h}_{E}^{d}(\mathcal{M})+\mathsf{h}_{E}^{c}(\mathcal{M}).$

###### Proof.

We only need to verify the inclusion
$\mathcal{H}_{E}^{c}(\mathcal{M})\subseteq\mathsf{h}_{E}^{d}(\mathcal{M})+\mathsf{h}_{E}^{d}(\mathcal{M})$.
That is to show the existence of a constant $\alpha_{E}$ such that for every
$x\in\mathcal{H}_{E}^{c}(\mathcal{M})$, the following holds:

$\inf\left\\{\big{\|}x^{d}\big{\|}_{\mathsf{h}_{E}^{d}}+\big{\|}x^{c}\big{\|}_{\mathsf{h}_{E}^{c}}\right\\}\leq\alpha_{E}\big{\|}x\big{\|}_{\mathcal{H}_{E}^{c}}$

where the infimum is taken over all $x^{d}\in\mathsf{h}_{E}^{d}(\mathcal{M})$
and $x^{c}\in\mathsf{h}_{E}^{c}(\mathcal{M})$ such that $x=x^{d}+x^{c}$.

Let $\xi=(dx_{n})_{n\geq 1}\in E^{\rm ad}(\mathcal{M};\ell_{2}^{c})$. It is
enough to take martingales $x^{c}$ and $x^{d}$ with for every $n\geq 1$,

$dx_{n}^{c}=z_{n}-\mathcal{E}_{n-1}(z_{n})\quad\text{and}\quad
dx_{n}^{d}=y_{n}-\mathcal{E}_{n-1}(y_{n}),$

where $y$ and $z$ are the adapted sequences from Theorem 4.11. Clearly,
$x=x^{d}+x^{c}$. Since the map $(a_{n})_{n\geq
1}\mapsto(\mathcal{E}_{n-1}(a_{n}))_{n\geq 1}$ is a contraction in
$L_{p}(\oplus_{n=1}^{\infty}\mathcal{M}_{n})$ for every $1\leq p<\infty$, it
follows by interpolation that it is also bounded in
$E(\oplus_{n=1}^{\infty}\mathcal{M}_{n})$. In particular,

$\|x^{d}\|_{\mathsf{h}_{E}^{d}}\lesssim_{E}\|y\|_{E(\oplus_{n=1}^{\infty}\mathcal{M}_{n})}.$

On the other hand, since for every $n\geq 1$,
$\mathcal{E}_{n-1}|dx_{n}^{c}|^{2}\leq\mathcal{E}_{n-1}|z_{n}|^{2}$, we
immediately get that

$\|x^{c}\|_{\mathsf{h}_{E}^{c}}\leq\|z\|_{E^{\rm
cond}(\mathcal{M};\ell_{2}^{c})}.$

Combining these two inequalities, we arrive at

$\|x^{d}\|_{\mathsf{h}_{E}^{d}}+\|x^{c}\|_{\mathsf{h}_{E}^{c}}\lesssim_{E}\|\xi\|_{E^{\rm
ad}(\mathcal{M};\ell_{2}^{c})}=\|x\|_{\mathcal{H}_{E}^{c}}.$

The proof is complete. ∎

We recall that the conclusion of Corollary 4.13 also applies to interpolation
space $E\in{\rm Int}(L_{p},L_{2})$ for $1<p<2$ (see [43, Theorem 3.9]). We
suspect that the preceding corollary is valid for any Banach function space in
${\rm Int}(L_{1},L_{2})$ but our method is restricted to $1<q<2$ (see also
[42, Problem 4.2] for a related question).

As examples of spaces that are not covered by previously known results, we
consider the general Lorentz space $\Lambda_{1,w}$ where $w$ is a positive
decreasing function on $(0,\infty)$ with $\int_{0}^{\infty}w(t)\ dt=\infty$.
The Lorentz space $\Lambda_{1,w}$ is the linear space consisting of all $f\in
L_{0}$ such that

$\big{\|}f\big{\|}_{\Lambda_{1,w}}=\int_{0}^{\infty}\mu_{t}(f)w(t)\
dt<\infty.$

The space $(\Lambda_{1,w},\|\cdot\|_{\Lambda_{1,w}})$ is a fully symmetric
Banach function space. According to [29], $\Lambda_{1,w}$ is $r$-concave if
and only if (for $1/r+1/r^{\prime}=1$),

$\Big{(}\frac{1}{t}\int_{0}^{t}w(s)^{r^{\prime}}\
ds\Big{)}^{1/r^{\prime}}\lesssim w(t),\quad t>0.$

With the preceding criterion, one can isolate the family of weights $w$ for
which $\Lambda_{1,w}\in{\rm Int}(L_{1},L_{q})$ for some $1<q<2$ and therefore
the Davis decomposition applies to martingales from the corresponding Hardy
spaces $\mathcal{H}_{\Lambda_{1,w}}^{c}(\mathcal{M})$.

Acknowledgments. I am grateful to the anonymous referee for a careful reading
of the paper and for useful comments which improved the presentation of the
paper.

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