P. Jones’ interpolation theorem for noncommutative martingale Hardy spaces
Authors: Randrianantoanina Narcisse 
Published: October 2022

Let

M

\mathcal {M}

be a semifinite von Nemann algebra equipped with an increasing filtration

(

M

n

)

n
≥
1

(\mathcal {M}_n)_{n\geq 1}

of (semifinite) von Neumann subalgebras of

M

\mathcal {M}

. For

0
>
p
≤
∞

0>p \leq \infty

, let

h

p
c

(

M

)

\mathsf {h}_p^c(\mathcal {M})

denote the noncommutative column conditioned martingale Hardy space associated with the filtration

(

M

n

)

n
≥
1

(\mathcal {M}_n)_{n\geq 1}

and the index

p
p

. We prove that for

0
>
p
>
∞

0>p>\infty

, the compatible couple
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sans-serif h Subscript p Superscript c Baseline left-parenthesis script upper M right-parenthesis comma sans-serif h Subscript normal infinity Superscript c Baseline left-parenthesis script upper M right-parenthesis right-parenthesis">

(

h

p
c

(

M

)
,

h

∞
c

(

M

)

)

\big (\mathsf {h}_p^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )

is

K
K

-closed in the couple

(

L
p

(

N

)
,

L
∞

(

N

)

)

\big (L_p(\mathcal {N}), L_\infty (\mathcal {N}) \big )

for an appropriate amplified semifinite von Neumann algebra

N

⊃

M

\mathcal {N}\supset \mathcal {M}

. This may be viewed as a noncommutative analogue of P. Jones interpolation of the couple

(

H
1

,

H
∞

)

(H_1, H_\infty )

.

As an application, we prove a general automatic transfer of real interpolation results from couples of symmetric quasi-Banach function spaces to the corresponding couples of noncommutative conditioned martingale Hardy spaces. More precisely, assume that

E
E

is a symmetric quasi-Banach function space on

(
0
,
∞
)

(0, \infty )

satisfying some natural conditions,

0
>
θ
>
1

0>\theta >1

, and

0
>
r
≤
∞

0>r\leq \infty

. If

(
E
,

L
∞

)

θ
,
r

=
F

(E,L_\infty )_{\theta ,r}=F

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sans-serif h Subscript upper E Superscript c Baseline left-parenthesis script upper M right-parenthesis comma sans-serif h Subscript normal infinity Superscript c Baseline left-parenthesis script upper M right-parenthesis right-parenthesis Subscript theta comma r Baseline equals sans-serif h Subscript upper F Superscript c Baseline left-parenthesis script upper M right-parenthesis period">

(

h

E
c

(

M

)
,

h

∞
c

(

M

)

)

θ
,
r

=

h

F

c

(

M

)
.

\big (\mathsf {h}_E^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{F}^c(\mathcal {M}).

\]
As an illustration, we obtain that if

Φ
\Phi

is an Orlicz function that is

p
p

-convex and

q
q

-concave for some

0
>
p
≤
q
>
∞

0>p\leq q>\infty

, then the following interpolation on the noncommutative column Orlicz-Hardy space holds: for

0
>
θ
>
1

0>\theta >1

,

0
>
r
≤
∞

0>r\leq \infty

, and

Φ
0

−
1

(
t
)
=
[

Φ

−
1

(
t
)

]

1
−
θ

\Phi _0^{-1}(t)=[\Phi ^{-1}(t)]^{1-\theta }

for

t
>
0

t>0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sans-serif h Subscript normal upper Phi Superscript c Baseline left-parenthesis script upper M right-parenthesis comma sans-serif h Subscript normal infinity Superscript c Baseline left-parenthesis script upper M right-parenthesis right-parenthesis Subscript theta comma r Baseline equals sans-serif h Subscript normal upper Phi 0 comma r Superscript c Baseline left-parenthesis script upper M right-parenthesis comma">

(

h

Φ
c

(

M

)
,

h

∞
c

(

M

)

)

θ
,
r

=

h

Φ
0

,
r

c

(

M

)
,

\big (\mathsf {h}_\Phi ^c(\mathcal {M}), \mathsf {h}_\infty ^c(\mathcal {M})\big )_{\theta , r}=\mathsf {h}_{\Phi _0, r}^c(\mathcal {M}),

\]
where

h

Φ
0

,
r

c

(

M

)

\mathsf {h}_{\Phi _0,r}^c(\mathcal {M})

is the noncommutative column Hardy space associated with the Orlicz-Lorentz space

L

Φ
0

,
r

L_{\Phi _0,r}

.