Complex interpolation between noncommutative martingale BMO spaces and Hardy–Orlicz spaces

Abstract

Let $\mathcal {M}$ be a semifinite von Neumann algebra and $(\mathcal {M}_n)_{n\ge 0}$ a nondecreasing filtration of von Neumann subalgebras of $\mathcal {M}$. Suppose that $\Phi$ is a p-convex and q-concave Orlicz function with $1< p\le q <\infty$. In this paper, we establish the complex interpolation between the column martingale little BMO space $\textrm{bmo}^c(\mathcal {M})$ and the noncommutative column conditioned martingale Hardy–Orlicz space $h_{\Phi }^c(\mathcal {M})$ associated with the filtration $(\mathcal {M}_n)_{n\ge 0}$.

1 Introduction

The theory of noncommutative martingales has been extensively studied in the past few decades since the pioneer work of Pisier and Xu [19], both for their intrinsic interest and for their applications to noncommutative analysis and operator algebras; see [7, 20, 23] for a detailed description of this history and further references. In recent years, with the development of noncommutative martingales, different noncommutative martingale Hardy spaces naturally appeared; see, for instance, [4, 7, 13, 20, 24]. Moreover, the theory of noncommutative martingale Hardy–Orlicz spaces and symmetric Hardy spaces was deeply investigated in [3, 4, 12, 25] and the references therein.

The goal of this paper is to establish the complex interpolation between noncommutative martingale little BMO spaces and Hardy–Orlicz spaces. Recall that the study of complex interpolations of Orlicz spaces has enjoyed considerable progress; see, for instance, [8, 16, 26]. On the other hand, the interpolations between martingale BMO spaces and martingale Hardy (Orlicz) spaces were studied in [10, 15, 27]. Furthermore, the interpolations between noncommutative martingale BMO spaces and Hardy spaces were initiated by Musat [17] and further studied by Bekjan et al. [2, 4, 5]. Very recently, Randrianantoanina established the interpolation between noncommutative martingale Hardy and BMO spaces for the full range of the index in [21] and proved P. Jones’ interpolation theorem for noncommutative martingale Hardy spaces in [22].

In order to state our main result, we first make the notation a little more precise. Suppose that $\mathcal {M}$ is a semifinite von Neumann algebra equipped with a faithful normal semifinite trace $\tau$. Let $\Phi$ be an Orlicz function. The symbols $\textrm{bmo}^c({\mathcal {M}})$ and $h_{\Phi }^c({\mathcal {M}})$ denote, respectively, the column “little” martingale BMO space and the noncommutative column conditioned martingale Hardy–Orlicz space associated to the fixed filtration $({\mathcal {M}}_n)_{n\ge 0}$; see Sect. 2 below for detailed information. The principal result of this paper is to interpolate the spaces $\textrm{bmo}^c({\mathcal {M}})$ and $h_{\Phi }^c(\mathcal {M})$, which extends the corresponding one in [4].

Theorem 1.1

Assume that $\Phi$ is a p-convex and q-concave Orlicz function for some $1< p\le q<\infty$. If $0<\theta <1$, then

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\theta =h_{\Phi _0}^c({\mathcal {M}}) \end{aligned}$$

with equivalent norms, where $\Phi _0^{-1}:=(\Phi ^{-1})^\theta$.

The main step in the proof of Theorem 1.1 is establishing an equivalent quasinorm for noncommutative martingale Hardy–Orlicz spaces $h_{\Phi }^c(\mathcal {M})$. Using this equivalent characterization, we show that $(h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}}))_\theta =h_{\Phi _1}^c({\mathcal {M}})$, in which $\Phi$ and $\Phi _1$ are two Orlicz functions satisfying $\Phi _1^{-1}(t)=t^{1-\theta }\big (\Phi ^{-1}(t)\big )^\theta$ for every $t>0$. Then, by applying Wolff’s interpolation theorem, we can deduce Theorem 1.1.

The remainder of this paper is divided into two sections. In Sect. 2, we review some backgrounds of Orlicz functions, Orlicz spaces and recall some basic properties of Orlicz functions that will be used in our later considerations. We then introduce the notion of the complex interpolation. At the end of this section, we set up some definitions concerning noncommutative martingale BMO spaces and Hardy–Orlicz spaces. In Sect. 3, we first determine an equivalent quasinorm of the noncommutative martingale Hardy–Orlicz spaces. Then we study the complex interpolation results between different noncommutative martingale Hardy–Orlicz spaces. Finally, we prove Theorem 1.1.

2 Preliminaries

Throughout this paper, for a universal constant $\alpha$, $A\lesssim _\alpha B$ means that $A\le C_\alpha B$ and $C_\alpha$ is a constant only depending on $\alpha$ and $C_\alpha$ varies from line to line. $A \simeq _\alpha B$ means $A\lesssim _{\alpha } B$ and $B\lesssim _{\alpha } A$. Moreover, ${\mathcal {M}}$ will always denote a semifinite von Neumann algebra equipped with a normal semifinite faithful trace $\tau$.

2.1 Orlicz functions and noncommutative Orlicz spaces

Let $\Phi$ be an Orlicz function on $[0,\infty )$, that is, a continuous, increasing and convex function satisfying $\lim _{t\rightarrow \infty }\Phi (t)=\infty$ and $\Phi (0)=0$. $\Phi$ is said to satisfy the $\Delta _2$ -condition if there exists a positive constant C such that $\Phi (2t)\le C \Phi (t)$ for all $t>0$. For $1\le p\le q<\infty$, an Orlicz function $\Phi$ is said to be p -convex if the function $t\mapsto \Phi (t^{1/p})$ is convex, and to be q -concave if the function $t\mapsto \Phi (t^{1/q})$ is concave. The function $\Phi$ satisfies the $\triangle _2$-condition if and only if it is q-concave for some $q<\infty$; see, for instance, [1, Lemma 5].

We now present some essential properties for Orlicz functions. The following lemma shows that a class of convex functions can be expressed as integrals of elementary functions, which was proved in [11, p. 133].

Lemma 2.1

Let $\Phi$ be a p-convex and q-concave Orlicz function with $1\le p\le q<\infty$. Then there exists a positive Borel measure $\nu$ on $(0,\infty )$ such that, for any $t>0,$

$$\begin{aligned} \Phi (t)\simeq _{p,q}\int \limits _0^\infty \min \{(ts)^p,(ts)^q\}d\nu (s). \end{aligned}$$

Let

$$\begin{aligned} \Psi (t):=\int \limits _0^\infty (t^{2-p}s^{-p}+t^{2-q}s^{-q})^{-1}\,d\nu (s),\quad \forall \,t>0, \end{aligned}$$

(2.1)

where $\nu$ is the positive measure in Lemma 2.1. The following essential properties will be used later and the proof is similar to that of [25, Proposition 3.3]. We omit the details.

Lemma 2.2

Let $\Phi$ be a p-convex and q-concave Orlicz function for $1\le p\le q<2$. Then

• (i) the function $t\mapsto \Psi (t)$ is operator monotone decreasing;

• (ii) the function $t\mapsto \widetilde{\Phi }^{-1}(t)$ is operator monotone increasing,

where

$$\begin{aligned} \widetilde{\Phi }:=\int \limits _0^\infty \min \{(ts)^p,(ts)^q\}d\nu (s). \end{aligned}$$

(2.2)

Assume that $\Phi$ is p-convex and q-concave for $1\le p\le q<2$. Consider the Orlicz function $\Theta$ which satisfies the following condition:

$$\begin{aligned} \Theta ^{-1}(u)=u^{-1/2}\Phi ^{-1}(u),\quad \forall \,u>0. \end{aligned}$$

From [16, Theorem 10.1], we deduce that, for any $u,v>0,$

$$\begin{aligned} \Phi (uv)\le u^2+\Theta (v). \end{aligned}$$

(2.3)

The following lemma can be found in [25, Proposition 3.3].

Lemma 2.3

Let $\Phi ,\Psi$ and $\Theta$ be as above. Then

• (i) $\Psi (t)\simeq _{p,q}t^{-2}\Phi (t)$ for all $t>0$;

• (ii) $\Theta (\Psi (t)^{-1/2})\simeq _{p,q}\Phi (t)$ for all $t>0$;

• (iii) the function $t\mapsto \Psi (t^{1/2})$ is operator monotone decreasing.

The lemma below shows that how to construct an intermediate Orlicz function from two given ones; see, for instance, [26, p. 223].

Lemma 2.4

Let $\Phi _0$, $\Phi _1$ be two Orlicz functions satisfying the $\triangle _2$-condition and let $0<\theta <1$. Suppose that $\Phi ^{-1}:=\big (\Phi _0^{-1}\big )^{1-\theta }\big (\Phi _1^{-1}\big )^\theta$. Then $\Phi$ is an Orlicz function and satisfies the $\triangle _2$-condition.

Assume that $\mathcal {M}$ is a subalgebra of the algebra of all bounded operators acting on some Hilbert space $\mathcal {H}$. A closed densely defined operator a on $\mathcal {H}$ is said to be affiliated with $\mathcal {M}$ if $u^*au=a$ for all unitary operators u in the commutant $\mathcal {M}'$ of $\mathcal {M}$. Let a be a densely defined self-adjoint operator on $\mathcal {H}$ and $a=\int _{-\infty }^\infty \lambda de_\lambda$ stand for its spectral decomposition. Then for any Borel subset B of $\mathbb {R}$, the spectral projection of a corresponding to the set B is defined by $\chi _B(a)=\int _{-\infty }^\infty \chi _B(\lambda )de_\lambda$. A closed densely defined operator a affiliated with $\mathcal {M}$ is said to be $\tau$ -measurable if there exists $s>0$ such that $\tau (\chi _{(s,\infty )}(|a|))<\infty$. The set of all $\tau$-measurable operators is denoted by $L_0(\mathcal {M},\tau )$. For $x\in L_0({\mathcal {M}},\tau )$, we define the generalized singular value $\mu (x)$ of x by

$$\begin{aligned} \mu _t(x):=\inf \{s>0:\ \tau (\chi _{(s,\infty )}(|x|))\le t\},\quad \forall \,t>0. \end{aligned}$$

Given an Orlicz function $\Phi$, the Orlicz space $L_\Phi (0,\infty )$ is defined to be the set of all Lebesgue measurable functions f such that, for some constant $c>0$,

$$\begin{aligned} \Vert f\Vert _{L_\Phi }:=\inf \left\{ c>0:\ \int \limits _0^\infty \Phi \big (|f(t)|/c\big )dt\le 1\right\} <\infty . \end{aligned}$$

We now define the noncommutative Orlicz space by setting

$$\begin{aligned} L_\Phi ({\mathcal {M}},\tau )=\{x\in L_0({\mathcal {M}},\tau ):\ \mu (x)\in L_\Phi (0,\infty )\}. \end{aligned}$$

Equipped with the norm $\Vert x\Vert _{L_\Phi ({\mathcal {M}},\tau )}:=\Vert \mu (x)\Vert _{L_\Phi (0,\infty )}$, the linear space $L_\Phi ({\mathcal {M}},\tau )$ becomes a complex Banach space. For simplicity, we use $L_\Phi ({\mathcal {M}})$ to denote $L_\Phi ({\mathcal {M}},\tau )$.

We now introduce the notion of complementary functions for Orlicz functions. In fact, for a given Orlicz function $\Phi$, we have the following integral representation

$$\begin{aligned} \Phi (u)=\int \limits _0^u\phi (s)ds,~u>0, \end{aligned}$$

where $\phi$ is a nondecreasing right-continuous function defined on $[0,\infty )$. Let

$$\begin{aligned}\psi (t)=\sup \{s:\ \phi (s)\le t\}\end{aligned}$$

stand for the right inverse of $\phi$. We define the Orlicz complementary function to $\Phi$ by

$$\begin{aligned} \Phi ^*(v)=\int \limits _0^v\psi (t)dt,~v>0. \end{aligned}$$

It should be mentioned that $\Phi ^*$ is an Orlicz function and there exists a duality between the noncommutative Orlicz spaces $L_\Phi ({\mathcal {M}},\tau )$ and $L_{\Phi ^*}({\mathcal {M}},\tau )$. We refer to [16] for more results on the connections between $\Phi$ and $\Phi ^*$.

Given an operator $x \in L_0(\mathcal {M},\tau )$ and an Orlicz function $\Phi$, we may define $\Phi (|x|)$ through functional calculus, that is, if $|x|=\int _0^\infty s\ de_s^{|x|}$ is the spectral decomposition of |x|, then

$$\begin{aligned} \Phi (|x|)=\int \limits _0^\infty \Phi (s) \ de_s^{|x|}. \end{aligned}$$

The operator $\Phi (|x|)$ is then a positive $\tau$-measurable operator. It is important to observe that

$$\begin{aligned} \tau \big [\Phi (|x|)\big ]=\int \limits _0^\infty \Phi \big (\mu _t(x)\big )dt, \end{aligned}$$

which can be deduced from [9, Corollary 2.8].

The following lemma was proved in [25, Lemma 3.4].

Lemma 2.5

Let $1\le p\le q<2$, $\Phi$ be a p-convex and q-concave Orlicz function and $\Psi$ be as in (2.1). Then

$$\begin{aligned} \sum _{n\ge 1}\tau \big ((a_{n+1}^2-a_n^2)\Psi (a_{n+1})\big )\lesssim _{p,q}\tau \big (\Phi (a)\big ) \end{aligned}$$

for any increasing sequence of positive operators $a_n\uparrow a$.

2.2 Complex interpolation

A couple of quasi-Banach spaces $(X_0, X_1)$ is called compatible if both of them are embedded into a Hausdorff topological vector space X through continuous injective linear maps. We view $X_0$ and $X_1$ as the subspaces of X. The intersection $X_0\cap X_1$ is equipped with the quasinorm

$$\begin{aligned} \Vert x\Vert _{X_0\cap X_1}=\max \{\Vert x||_{X_0},\Vert x\Vert _{X_1}\}. \end{aligned}$$

And the sum $X_0+ X_1$ is defined by

$$\begin{aligned} X_0+ X_1=\{x_0+x_1:\ x_k\in X_k,\ k=0,1\}, \end{aligned}$$

with the quasinorm

$$\begin{aligned} ||x||_{X_0+X_1}=\inf \left\{ ||x||_{X_0}+||x||_{X_1}:\ x=x_0+x_1,\ x_k\in X_k,\ k=0,1\right\} . \end{aligned}$$

It is clear to verify that $X_0\cap X_1$ and $X_0+X_1$ are quasi-Banach spaces (and Banach spaces if $X_0$ and $X_1$ are).

Set

$$\begin{aligned}\mathcal {B}=\{z\in \mathbb {C}:\ 0\le Rez\le 1\}.\end{aligned}$$

The family of all functions $f:\ \mathcal {B}\rightarrow X_0+X_1$ are denoted as $\mathcal {F}(X_0,X_1)$, where $(X_0,X_1)$ is a compatible couple of complex Banach spaces. Then $\mathcal {F}(X_0,X_1)$ satisfies the following conditions:

• (i) On $\mathcal {B}$, f is continuous; In the interior of $\mathcal {B}$, f is analytic;

• (ii) For $k=0,1$, we have $f(k+it)\in X_k$ for all $t\in \mathbb {R}$. Moreover, the function $t\mapsto f(k+it)$ is continuous from $\mathbb {R}$ to $X_k$;

• (iii) For $k=0,1$, $\lim _{|t|\rightarrow \infty }||f(k+it)||_{X_k}=0$.

We equip $\mathcal {F}(X_0,X_1)$ with the following norm:

$$\begin{aligned} ||f||_{\mathcal {F}(X_0,X_1)}=\max \left\{ \sup \limits _{t\in \mathbb {R}}||f(it)||_{X_0},\sup \limits _{t\in \mathbb {R}}||f(1+it)||_{X_1}\right\} . \end{aligned}$$

Then $\mathcal {F}(X_0,X_1)$ is a Banach space. Assume $0<\theta <1$. The space of all those $x\in X_0+X_1$ for which there exists $f\in \mathcal {F}(X_0,X_1)$ with $f(\theta )=x$ is denoted as the complex interpolation space $(X_0,X_1)_\theta$. The norm is defined by

$$\begin{aligned} ||x||_\theta :=\{||f||_{\mathcal {F}(X_0,X_1)}:\ f(\theta )=x,\ f\in \mathcal {F}(X_0,X_1)\}. \end{aligned}$$

The map $f\mapsto f(\theta )$ is a contraction from $\mathcal {F}(X_0,X_1)$ to $X_0+X_1$ according to the maximal principle. We refer to [6] for more information about the complex interpolation.

Let us recall the classical result in [8]. Given two Orlicz functions $\Phi _0$, $\Phi _1$, $0<\theta <1$, and $\Phi _2^{-1}=(\Phi _0^{-1})^{1-\theta }\big (\Phi _1^{-1}\big )^\theta$, then it follows from [8, Corollary 4.2] that

$$\begin{aligned} (L_{\Phi _0}(\Omega ),L_{\Phi _1}(\Omega ))_\theta =L_{\Phi _2}(\Omega ) \end{aligned}$$

with equal norms, where $\Omega$ is a measurable space. Combined this with [20, Corollary 2.2], we can derive the following lemma.

Lemma 2.6

Suppose that ${\mathcal {M}}$ is a semifinite von Neumann algebra. Let $\Phi _0$, $\Phi _1$ and $\Phi _2$ be Orlicz functions satisfying $\Phi _2^{-1}=(\Phi _0^{-1})^{1-\theta }\big (\Phi _1^{-1}\big )^\theta$ with $0<\theta <1$. We have

$$\begin{aligned} \left( L_{\Phi _0}({\mathcal {M}}),L_{\Phi _1} ({\mathcal {M}})\right) _\theta =L_{\Phi _2}({\mathcal {M}}) \end{aligned}$$

with equivalent norms.

2.3 Noncommutative martingale Hardy–Orlicz and BMO spaces

Let us recall the general setup for noncommutative martingales. Let $(\mathcal {M}_n)_{n\ge 0}$ be a filtration, that is, a nondecreasing sequence of von Neumann subalgebras of $\mathcal {M}$ whose union is weak$^*$-dense in $\mathcal {M}$. Then for any $n\ge 0$, there exists a normal conditional expectation $\mathcal {E}_n$ from $\mathcal {M}$ onto $\mathcal {M}_n$ such that

• (i) $\mathcal {E}_n(axb)=a\mathcal {E}_n(x)b$ for all $a,\,b\in \mathcal {M}_n$ and $x\in \mathcal {M}$;

• (ii) $\tau \circ \mathcal {E}_n=\tau$.

Note that the conditional expectations satisfy the tower property $\mathcal {E}_m\mathcal {E}_n=\mathcal {E}_n\mathcal {E}_m=\mathcal {E}_{\min (m,n)}$ for all nonnegative integers m and n. Since each $\mathcal {E}_n$ is trace preserving, it can be extended to a contractive projection from $L_\Phi (\mathcal {M},\tau )$ onto $L_\Phi (\mathcal {M}_n,\tau _n)$, where $\tau _n$ is the restriction of $\tau$ to $\mathcal {M}_n$.

A sequence $x=(x_n)_{n\ge 0}$ in $L_1(\mathcal {M})+{\mathcal {M}}$ is called a noncommutative martingale (with respect, or adapted to $(\mathcal {M}_n)_{n\ge 0}$), if for any $n\ge 0$, we have the equality

$$\begin{aligned} \mathcal {E}_n(x_{n+1})=x_n.\end{aligned}$$

The associated difference sequence $(dx_n)_{n\ge 0}$ is defined by setting, $dx_0=x_0$ and $dx_n=x_n-x_{n-1}$ for $n\ge 1$. For any given Orlicz function $\Phi$, if $x=(x_n)_{n\ge 0}\subset L_\Phi (\mathcal {M})$ and

$$\begin{aligned} \Vert x\Vert _\Phi =\sup _{n\ge 0}\Vert x_n\Vert _\Phi <\infty ,\end{aligned}$$

then x is said to be a bounded $L_\Phi$ -martingale. Assume that $x_\infty \in L_\Phi (\mathcal {M})$ and $x_n=\mathcal {E}_n(x_\infty )$. Then $x=(x_n)_{n\ge 0}$ is a bounded $L_\Phi$-martingale and $\Vert x\Vert _\Phi \simeq \Vert x_\infty \Vert _{\Phi }$. Conversely, if $\Phi$ is of p-convex and q-concave for $1<p<q\le \infty$, then any bounded $L_\Phi ({\mathcal {M}})$-martingale $x=(x_n)_{n\ge 0}$ is of the form $(\mathcal {E}_n(x_\infty ))_{n\ge 1}$, where $x_\infty \in L_\Phi ({\mathcal {M}})$ satisfies $\Vert x\Vert _{L_\Phi ({\mathcal {M}})}\approx _{\Phi }\Vert x_\infty \Vert _{L_\Phi ({\mathcal {M}})}$. Consequently, one can identify the space of bounded $L_\Phi$-martingales with the space $L_\Phi (\mathcal {M})$ in the case when $\Phi$ is strictly convex, with the identification given by $x=(x_n)_{n\ge 0}\mapsto x_\infty$.

Assume that $x=(x_n)_{n\ge 0}$ is a martingale in $L_2({\mathcal {M}})+{\mathcal {M}}$. The column and row conditioned square function $s_c(x)$ and their truncated versions are defined as follows:

$$\begin{aligned} s_{c,n}(x)=\left( \sum _{k=1}^n\mathcal {E}_{k-1}|dx_k|^2\right) ^{1/2},\quad s_{c}(x)=\left( \sum _{k=1}^\infty \mathcal {E}_{k-1}|dx_k|^2\right) ^{1/2}, \end{aligned}$$

and

$$\begin{aligned} s_{r,n}(x)=\left( \sum _{k=1}^n\mathcal {E}_{k-1}|dx_k^*|^2\right) ^{1/2},\quad s_{r}(x)=\left( \sum _{k=1}^\infty \mathcal {E}_{k-1}|dx_k^*|^2\right) ^{1/2}. \end{aligned}$$

The noncommutative column martingale Hardy–Orlicz space $h_{\Phi }^c({\mathcal {M}})$ associated to conditioned square functions is defined to be the completion of the set of all finite martingales in $L_1({\mathcal {M}})\cap {\mathcal {M}}$ with $\Vert x\Vert _{h_\Phi ^c}<\infty$, where the norm is defined by

$$\begin{aligned} \big \Vert x\big \Vert _{h_\Phi ^c}=\big \Vert s_c(x)\big \Vert _{L_\Phi ({\mathcal {M}})}. \end{aligned}$$

The noncommutative row conditioned martingale Hardy–Orlicz space $h_{\Phi }^r({\mathcal {M}})$ can be defined similarly.

The column little BMO space $\textrm{bmo}^c({\mathcal {M}})$ is defined as follows:

$$\begin{aligned} \textrm{bmo}^c({\mathcal {M}}):=\big \{x\in L_2({\mathcal {M}}):\ \Vert x\Vert _{\textrm{bmo}^c({\mathcal {M}})}<\infty \big \} \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert x\Vert _{\textrm{bmo}^c}=\max \left( \Vert \mathcal {E}_{1}(x)\Vert _\infty ,\sup \limits _{n\ge 1}\Vert \mathcal {E}_{n}|x-x_n|^2\Vert _\infty ^{1/2}\right) . \end{aligned}$$

The row little BMO space $\textrm{bmo}^r({\mathcal {M}})$ is defined similarly. Note that the dual space of $h_1^c({\mathcal {M}})$ (resp., $h_1^r({\mathcal {M}})$) is the space $\textrm{bmo}^c({\mathcal {M}})$ (resp., $\textrm{bmo}^r({\mathcal {M}})$). We refer to [14, 18] for details.

3 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1. To accomplish this, we firstly provide an equivalent characterization for $\Vert \cdot \Vert _{h_\Phi ^c}$ under certain assumption on $\Phi$. Let W be the set of all sequences $\{w_n\}_{n\ge 0}$ satisfying that $\{w_n\}_{n\ge 0}$ is non-decreasing, and each $w_n\in L_1^+({\mathcal {M}}_n)$ is invertible with bounded inverse and $\Vert w_n\Vert _1\le 1$. For any $x\in h_\Phi ^c({\mathcal {M}})$, let

$$\begin{aligned} N_\Phi ^c(x):=\inf _W\left\{ \tau \left( \sum _{n\ge 0}\Psi \big (\Phi ^{-1}(w_n)\big )\left[ s_{c,n+1}^2(x)-s_{c,n}^2(x)\right] \right) \right\} ^{1/2}. \end{aligned}$$

The lemma below describes the relation between $N_\Phi ^c(x)$ and $\Vert x\Vert _{h_\Phi ^c}$, which extends [4, Proposition 3.2] to the context of noncommutative Hardy–Orlicz spaces.

Lemma 3.1

Let $1< p\le q<2$ and $\Phi$ be a p-convex and q-concave Orlicz function. Then for any $x\in h_\Phi ^c({\mathcal {M}})$,

$$\begin{aligned} C_{p,q}^{-1/2}N_\Phi ^c(x)\le \Vert x\Vert _{h_\Phi ^c}\le \widetilde{C}_{p,q}^{1/p}N_\Phi ^c(x). \end{aligned}$$

(3.1)

Proof

Let $x\in L_2({\mathcal {M}})+{\mathcal {M}}$ with $\Vert x\Vert _{h_\Phi ^c}\le 1$. By a standard approximation, we may assume that $s_{c,n}(x)$ is invertible with bounded inverse for every $n\ge 1$. By (2.2), we obtain $\{\widetilde{\Phi }(s_{c,n+1}(x))\}_{n\ge 0}\in W$. From this and the definition of $N_\Phi ^c(x)$, we deduce that

$$\begin{aligned} N_\Phi ^c(x)\le \left[ \tau \left( \sum _{n\ge 0}\Psi \left( \Phi ^{-1}(\widetilde{\Phi }(s_{c,n+1}(x)))\right) \left( s_{c,n+1}^2(x)-s_{c,n}^2(x)\right) \right) \right] ^{1/2}. \end{aligned}$$

Note that

$$\begin{aligned} \Psi \big (\Phi ^{-1}(\widetilde{\Phi }(s_{c,n+1}(x)))\big )\lesssim _{p,q}\Psi \big (\Phi ^{-1}(\Phi (s_{c,n+1}(x)))\big )=\Psi \big (s_{c,n+1}(x)\big ). \end{aligned}$$

Combining this with Lemma 2.5, we further obtain

$$\begin{aligned} \begin{aligned} N_\Phi ^c(x)&\lesssim _{p,q} \left[ \tau \left( \sum _{n\ge 0}\Psi \big (s_{c,n+1}(x)\big )\big (s_{c,n+1}^2(x)-s_{c,n}^2(x)\big )\right) \right] ^{1/2}\\&\lesssim _{p,q} \left[ \tau \big (\Phi (s_c(x))\big )\right] ^{1/2}, \end{aligned} \end{aligned}$$

which yields the left hand side inequality.

Now we turn to the right hand side estimate. Let $\{w_n\}_{n\in \mathbb {N}}\in W$ and $w:=\sup _n w_n.$ Since $\{w_n\}_{n\in \mathbb {N}}$ is non-decreasing and the fact that $\Phi (t)\simeq _{p,q} \widetilde{\Phi }(t)$ for $t>0$, it follows that

$$\begin{aligned} \Psi \big (\Phi ^{-1}(w_n)\big )\gtrsim _{p,q}\Psi \big (\widetilde{\Phi }^{-1}(w_n)\big )\ge \Psi \big (\widetilde{\Phi }^{-1}(w)\big )\gtrsim _{p,q}\Psi \big (\Phi ^{-1}(w)\big ), \end{aligned}$$

where the second inequality is due to Lemma 2.2. Then

$$\begin{aligned}&\tau \bigg (\sum _{n\ge 0}\Psi (\Phi ^{-1}(w_n)\big )\big (s_{c,n+1}^2(x)-s_{c,n}^2(x)\big )\bigg )\\&\quad \gtrsim _{p,q} \tau \bigg (\sum _{n\ge 0}\Psi (\Phi ^{-1}(w)\big )\big (s_{c,n+1}^2(x)-s_{c,n}^2(x)\big )\bigg )\\&\quad = \tau \big (\Psi \big (\Phi ^{-1}(w)\big )s_{c}^2(x)\big ). \end{aligned}$$

Assume that $N_\Phi ^c(x)\le 1$. Combining the above inequality, Lemma 2.3 (ii), [9, Theorem 4.2] and (2.3), we deduce that

$$\begin{aligned} \begin{aligned} \tau \big (\Phi (s_c(x))\big )&=\int \limits _0^\infty \Phi \bigg (\mu _t\big (\Psi (\Phi ^{-1}(w))^{-1/2}\Psi (\Phi ^{-1}(w))^{1/2}s_c(x)\big )\bigg )dt\\&\le \int \limits _0^\infty \Phi \big (\mu _t(\Psi (\Phi ^{-1}(w))^{-1/2}\big )\mu _t\big (\Psi (\Phi ^{-1}(w))^{1/2}s_c(x))\big )dt\\&\le \int \limits _0^\infty \Theta \big (\mu _t(\Psi (\Phi ^{-1}(w))^{-1/2})\big )dt+\int \limits _0^\infty \big [\mu _t(\Psi (\Phi ^{-1}(w))^{1/2}s_c(x))\big ]^2dt\\&\simeq _{p,q} \tau (w)+\tau \big (|\Psi (\Phi ^{-1}(w))^{1/2}s_c(x)|^2\big )\le 2, \end{aligned} \end{aligned}$$

which means $\tau \big (\Phi (s_c(x))\big )\le \widetilde{C}_{p,q}.$ Without loss of the generality, we assume that $\widetilde{C}_{p,q}>1$. Thus,

$$\begin{aligned} \tau \left( \Phi \bigg (\frac{s_c(x)}{\widetilde{C}_{p,q}^{1/p}}\bigg )\right) \le \big (\widetilde{C}_{p,q}^{1/p}\big )^{-p}\tau \big (\Phi (s_c(x))\big )\le 1. \end{aligned}$$

This implies $\Big \Vert \frac{s_c(x)}{\widetilde{C}_{p,q}^{1/p}}\Big \Vert _{L_\Phi }\le 1$ and thus $\Vert x\Vert _{h_\Phi ^c}\le \widetilde{C}_{p,q}^{1/p}N_\Phi ^c(x)$. The proof is complete. $\square$

The Now, we give the following complex interpolation between column martingale Hardy spaces.

Lemma 3.2

Let $1< p\le q<2$ and $0<\theta <1$. Assume that $\Phi$ is a p-convex and q-concave Orlicz function. Then

$$\begin{aligned} \big (h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}})\big )_\theta =h_{\Phi _1}^c({\mathcal {M}}) \end{aligned}$$

(3.2)

with equivalent norms, where $\Phi _1^{-1}(t)=t^{1-\theta }\big (\Phi ^{-1}(t)\big )^\theta$.

Proof

Consider a larger von Neumann algebra $({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2)),\tau \otimes Tr)$. Then $h_1^c({\mathcal {M}})$ and $h_{\Phi }^c({\mathcal {M}})$ can be identified with a subspace of $L_1({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2))$ and $L_\Phi ({\mathcal {M}}\bar{\otimes }{} {\textbf {B}} (\ell _2(\mathbb {N}^2))$, respectively. From Lemma 2.6, we deduce that

$$\begin{aligned} L_{\Phi _1}({\mathcal {M}}\bar{\otimes }{} {\textbf {B}} (\ell _2(\mathbb {N}^2))=\big (L_1({\mathcal {M}}\bar{\otimes }{} {\textbf {B}} (\ell _2(\mathbb {N}^2)),L_\Phi ({\mathcal {M}}\bar{\otimes }{} {\textbf {B}} (\ell _2(\mathbb {N}^2))\big )_\theta \end{aligned}$$

with equal norms, which gives the inclusion $(h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}}))_\theta \subset h_{\Phi _1}^c({\mathcal {M}})$.

We now show the inverse inclusion. Let x be a finite martingale in $L_2({\mathcal {M}})+{\mathcal {M}}$ with $\Vert x\Vert _{h_{\Phi _1}^c} < 1$. Let $N_\Phi ^c$ be defined previously. Using (3.1), Lemma 2.3 (i) and Lemma 2.4, we have

$$\begin{aligned} \inf _W\left[ \tau \left( \sum _{n\ge 0}\big (\Phi ^{-1}(w_n)\big )^{-2\theta }w_n^{2\theta -1}\big (s_{c,n+1}^2(x)-s_{c,n}^2(x)\big )\right) \!\right] ^{1/2}\!\simeq _{p,q}N_{\Phi _1}^c(x) < C_{p,q}^{1/2}. \end{aligned}$$

Let $\{w_n\}\in W$ satisfies

$$\begin{aligned} \tau \bigg (\sum _{n\ge 0}\big (\Phi ^{-1}(w_n)\big )^{-2\theta }w_n^{2\theta -1}\big |dx_{n+1}|^2\big )\bigg ) < C_{p,q}. \end{aligned}$$

(3.3)

For $\varepsilon >0$ and $z\in S:=\{z\in \mathbb {C}:\ 0\le Rez\le 1\}$, we define

$$\begin{aligned} f_\varepsilon (z)= \exp (\varepsilon (z^2-\theta ^2))\sum _ndx_{n+1}w_n^{\theta -z}\left[ \Phi ^{-1}(w_n)\right] ^{z-\theta }. \end{aligned}$$

Observe that $f_\varepsilon (\theta )=x$ and $f_\varepsilon$ is continuous on S and analytic on the interior of S. It follows immediately that $f_\varepsilon (it)$ and $f_\varepsilon (1+it)$ tends to 0 as $t\rightarrow \infty$. A direct computation gives that for all $t\in \mathbb {R}$,

$$\begin{aligned} \tau \left( \sum _nw_n^{-1}|d(f_\varepsilon )_{n+1}(it)|^2\right) \exp \big (-2\varepsilon (t^2+\theta ^2)\big )\tau \left( \sum _n\big (\Phi ^{-1}(w_n)\big )^{-2\theta }w_n^{2\theta -1}\big |dx_{n+1}|^2\right) , \end{aligned}$$

and

$$\begin{aligned} \tau \left( \sum _nw_n^{-1}|d(f_\varepsilon )_{n+1}(1+it)|^2\right)&=\exp (2\varepsilon ((1+it)^2-\theta ^2))\tau \\&\qquad \left( \sum _n\big (\Phi ^{-1}(w_n)\big )^{-2\theta }w_n^{2\theta -1}\big |dx_{n+1}|^2\right) . \end{aligned}$$

By (3.1) and (3.3), we obtain

$$\begin{aligned} \Vert f_\varepsilon (it)\Vert _{h_{1}^c}\le \exp (\varepsilon )C_{p,q}^{1/2} \end{aligned}$$

and

$$\begin{aligned} \Vert f_\varepsilon (1+it)\Vert _{h_{1}^c}\le \exp (\varepsilon )C_{p,q}^{1/2}. \end{aligned}$$

Similarly, we have

$$\begin{aligned} \Vert f_\varepsilon (it)\Vert _{h_{\Phi }^c}\le \exp (\varepsilon )C_{p,q}^{1/2}\quad \textrm{and} \quad \Vert f_\varepsilon (1+it)\Vert _{h_{\Phi }^c}\le \exp (\varepsilon )C_{p,q}^{1/2}. \end{aligned}$$

Thus $x=f_\varepsilon (\theta )\in \big (h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}})\big )_\theta$ and

$$\begin{aligned} \Vert x\Vert _{(h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}}))_\theta }\le \exp (\varepsilon )C_{p,q}^{1/2}. \end{aligned}$$

Hence

$$\begin{aligned} \Vert x\Vert _{(h_1^c({\mathcal {M}}),h_\Phi ^c({\mathcal {M}}))_\theta }\le C_{p,q}^{1/2}\Vert x\Vert _{h_{\Phi _1}^c}. \end{aligned}$$

We finish the proof. $\square$

Lemma 3.3

Let $1< p_i\le q_i<\infty$, $i=0,1,2$ and $0<\theta <1$. $\Phi _i$ is a $p_i$-convex and $q_i$-concave Orlicz function. Then, the following holds with equivalent norms,

$$\begin{aligned} \big (h_{\Phi _0}^c({\mathcal {M}}),h_{\Phi _1}^c({\mathcal {M}})\big )_\theta =h_{\Phi _2}^c({\mathcal {M}}), \end{aligned}$$

(3.4)

where $\Phi _2^{-1}(t)=(\Phi _0^{-1}(t))^{1-\theta }\big (\Phi _1^{-1}(t)\big )^\theta$.

Proof

The spaces considered here are compatible. Each $h_{\Phi _i}^c({\mathcal {M}})$, $i=0,1,2$ can be regarded as a complemented subspace of $L_{\Phi _i}({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2)))$. It follows from Lemma 2.6 that

$$\begin{aligned} \left( L_{\Phi _0}({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2)),L_{\Phi _1} ({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2))\right) _\theta =L_{\Phi _2}\left( {\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2))\right) . \end{aligned}$$

Since each $h_{\Phi _i}^c({\mathcal {M}})$, $i=0,1,2$ is identified with a complemented subspace of $L_{\Phi _i}({\mathcal {M}}\bar{\otimes }{} {\textbf {B}}(\ell _2(\mathbb {N}^2)))$, we have (3.4). This completes the proof. $\square$

Now we are ready to prove the main result of this paper which deals with complex interpolation between the spaces $\textrm{bmo}^c({\mathcal {M}})$ and $h_{\Phi }^c({\mathcal {M}})$.

Proof of Theorem 1.1

We divide the proof into three cases.

Case 1: $2<p\le q<\infty$. By [16, Corollary 11.6], we know that $\Phi ^*$ is a $q'$-convex and $p'$-concave Orlicz function with $1<q'\le p'<2$. By Lemma 3.2, we infer that, for every $0<\theta <1$,

$$\begin{aligned} \big (h_1^c({\mathcal {M}}),h_{\Phi ^*}^c({\mathcal {M}})\big )_\theta =h_{\Phi _1}^c({\mathcal {M}}), \end{aligned}$$

where, for every $t>0,$ $\Phi _1^{-1}(t):=t^{1-\theta }\big ({\Phi ^*}^{-1}(t)\big )^\theta$. Applying the duality result [6, Theorem 4.5.1] and the fact that $h_\Phi ^c({\mathcal {M}})$ is reflexive, we obtain

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\theta =h_{\Phi _1^*}^c({\mathcal {M}}). \end{aligned}$$

Using [16, Theorem 8.3], we have $(\Phi _1^*(t))^{-1}\simeq \Phi _0^{-1}(t)$. It follows that

$$\begin{aligned}\big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\theta =h_{\Phi _0}^c({\mathcal {M}}). \end{aligned}$$

Case 2: $p\le 2$ and $p\theta ^{-1}>2$. It follows immediately that $\Phi _0$ is $p\theta ^{-1}$-convex and $q\theta ^{-1}$-concave. Fix $1-\frac{p}{2}<\upsilon <1 - \theta$ and define $\Phi ^{-1}_1(t)=[\Phi ^{-1}(t)]^{1-\upsilon }$, $t>0$. Note that $\Phi _1(t)$ is $p(1-\upsilon )^{-1}$-convex with $p(1-\upsilon )^{-1}>2$ and $\Phi _0^{-1}(t)=[\Phi _1^{-1}(t)]^{\epsilon }$ where $\epsilon =\theta /(1-\upsilon )\in (0,1)$. Then by Case1 we have

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi _1}^c({\mathcal {M}})\big )_\epsilon =h_{\Phi _0}^c({\mathcal {M}}). \end{aligned}$$

By Lemma 3.3, we have

$$\begin{aligned} \big (h_{\Phi _0}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\eta =h_{\Phi _1}^c({\mathcal {M}}), \end{aligned}$$

where $\eta =1-\upsilon /(1-\theta )$. Employing Wolff’s interpolation theorem in [28], we obtain

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\zeta =h_{\Phi _0}^c({\mathcal {M}}), \end{aligned}$$

where $\zeta =\frac{\epsilon \eta }{1-\epsilon +\epsilon \eta }$. A simple calculation shows that $\zeta =\theta$.

Case 3: $p\le 2$ and $p\theta ^{-1}\le 2$. Define $\Phi ^{-1}_3(t)=[\Phi _0^{-1}(t)]^{p/2}$, $t>0$. Using the result obtained in Case 2, we have

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi _0}^c({\mathcal {M}})\big )_\epsilon =h_{\Phi _3}^c({\mathcal {M}}) \quad \textrm{with} \quad \epsilon =p/2, \end{aligned}$$

and

$$\begin{aligned} \big (h_{\Phi _3}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\eta =h_{\Phi _0}^c({\mathcal {M}}) \quad \textrm{with} \quad \eta =\frac{\theta -\theta \epsilon }{1-\theta \epsilon }. \end{aligned}$$

Applying Wolff’s interpolation Theorem again, we deduce

$$\begin{aligned} \big (\textrm{bmo}^c({\mathcal {M}}),h_{\Phi }^c({\mathcal {M}})\big )_\zeta =h_{\Phi _0}^c({\mathcal {M}}), \end{aligned}$$

where $\zeta =\frac{\eta }{1-\epsilon +\epsilon \eta }$. A simple calculation shows that $\zeta =\theta$. This completes the proof. $\square$

Remark 3.4

The row version of Theorem 1.1 also holds true by replacing the equivalent quasinorm $N_\Phi ^c$ of $\Vert \cdot \Vert _{h_{\Phi }^c({\mathcal {M}})}$ with the equivalent quasinorm $N_\Phi ^r$ of $\Vert \cdot \Vert _{h_{\Phi }^r({\mathcal {M}})}$ in Lemma 3.1 and repeating the steps in the proof of Theorem 1.1.