In this paper, we extend some classical results of martingale transforms from the strong-type spaces (normed space) to the setting of weak-type spaces (quasi-normed space). More precisely, we are interested in the characterization about the interchanging between weak Orlicz-Hardy space ${\rm{w}}\mathcal {H}_{\Phi_{1}}$ and ${\rm{w}}\mathcal {H}_{\Phi_{2}}$ in terms of Burkholder's martingale transforms.

The first motivation in this paper comes from the classical results of Chao and Long [2], as well as the similar results of Garsia [3] and Weisz [10]. The concept of martingale transforms was first introduced by Burkholder [1]. It is shown that the martingale transforms are especially useful to study the relations between the “predictable” Hardy spaces of martingales, such as $\mathcal {H}_{p}$, which is associated with the conditional quadratic variation of martingales. The “characterization” of such spaces via martingale transforms were provided in [2]: the elements in the space $\mathcal {H}_{p_1}$ are none other than the martingale transforms of those in $\mathcal {H}_{p_2}$ for $0 < p_1 < p_2 < \infty$. All of those results can be found also in the monographs of Long [7] and Weisz [11].

Generally, the similar conclusions were obtained also in the case of Orlicz-Hardy spaces for martingales by Ishak and Mogyoródi [4], Meng and Yu [8] and Yu [14-15], according to different situations, respectively.

On the other hand, we also note that in recent years, the weak spaces, including their applications to harmonic analysis and martingale theory, have been got more and more attention. See for example Jiao [5], Nakai [9], Weisz [12-13]. Particularly, Liu, Hou and Wang [6] firstly introduced the weak Orlicz-Hardy spaces of martingales and discussed its basic properties and some martingale inequalities. Jiao [5] investigated the embedding relations between weak Orlicz martingale spaces.

This article will focus its attention on the relationship between the weak Orlicz-Hardy spaces ${\rm{w}}\mathcal {H}_{\Phi_{1}}$ and ${\rm{w}}\mathcal {H}_{\Phi_{2}}$, where $\Phi_{1}$ and $\Phi_{2}$ are two generalized Young functions (not need to be convex) and $\Phi_{1}\preceq\Phi_{2}$ in some sense (see Definition 2.1). It will be shown that the elements in weak Orlicz-Hardy space ${\rm{w}}\mathcal {H}_{\Phi_{1}}$ are none other than the martingale transforms of those in ${\rm{w}}\mathcal {H}_{\Phi_{2}}$, which extend the corresponding results in Chao and Long [2] from strong-type spaces to the setting of weak-type spaces. In this paper, we are interested in the case $\Phi_{1}$ is not convex.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability measure space, let $(\mathcal{F}_{n}, n\in\mathbb{N})$ be a sequence of nondecreasing sub-$\sigma$-algebras of $\mathcal{F}$ such that $\mathcal{F}=\bigvee \mathcal{F}_{n}$, and let $f=(f_n, n\in \mathbb{N})$ be a martingale adapted to $(\mathcal{F}_n, n\in \mathbb{N})$. Denote by $df=(df_n, n\in \mathbb{N})$ the sequence of martingale differences with $df_n=f_n-f_{n-1}$, $n\geq 1$, and set $f_{0}\equiv 0$, $\mathcal{F}_{0}=\{\emptyset, \Omega\}$. The conditional quadratic variation of a martingale $f$ is defined by

$ {{s}_{n}}(f):={{(\sum\limits_{i=1}^{n}{E}(|d{{f}_{i}}{{|}^{2}}|{{\mathcal{F}}_{i-1}}))}^{\frac{1}{2}}},~~s(f):={{(\sum\limits_{i=1}^{\infty }{E}(|d{{f}_{i}}{{|}^{2}}|{{\mathcal{F}}_{i-1}}))}^{\frac{1}{2}}}, $

Then for $0 < p\leq\infty$, we define martingale Hardy space as below

$ \mathcal {H}_p:=\{ f=(f_n, n\in\mathbb{N}): s(f)\in L_p, ~\mbox{and}~\|f\|_{\mathcal {H}_p} :=\|s(f)\|_{p} < \infty \}. $

A non-decreasing function $\Phi(x)$ is called a generalized Young function (convex or concave), if $\Phi(x)=\displaystyle\int_{0}^{x}\varphi(t)\mbox{d}t, x\geq0$, where $\varphi(x)$ is a left-continuous, non-negative function on $[0, +\infty)$. When $\Phi(x)$ is a convex Young function, we can define the inverse of $\varphi(t)$ by $\psi(s):=\inf\{t:\varphi(t)\geq s\}$. It is well known that its integral $\Psi(x)=\displaystyle\int_{0}^{x}\psi(t)\mbox{d}t$ is a convex function and $\Psi(x)$ is called the Young's complementary function of $\Phi$. The upper index and lower index are defined by

$ % MathType!Translator!2!1!LaTeX.tdl!LaTeX 2.09 and later! {p_\Phi } = \mathop {\sup }\limits_{0 < x < \infty } \frac{{x\varphi (x)}}{{\Phi (x)}},\;{q_\Phi } = \mathop {\inf }\limits_{0 < x < \infty } \frac{{x\varphi (x)}}{{\Phi (x)}} $

If $p_{\Phi} < +\infty$, then the inverse function $\Phi^{-1}$ of $\Phi$ exists and has the form

$ \Phi^{-1}(x)=\displaystyle\int_{0}^{x}m_{\Phi}(t)\mbox{d}t. $

If $\Phi$ is convex then $m_{\Phi}(t)$ is a decreasing function and we can easily see that (see Ishak and Mogyoródi [4])

$ m_{\Phi}(t)=\frac{1}{\varphi(\Phi^{-1}(t))}, ~~t > 0. $

A function $\Phi(x)$ is said to satisfy the $\Delta_{2}$ condition (denote $\Phi\in\Delta_{2}$) if there is a constant $C$ such that $\Phi(2t)\leq C\Phi(t)$ for all $t > 0$. It is well known that if $\Phi(x)$ is a convex function with $p_{\Phi} < +\infty$ then $\Phi\in\Delta_{2}$ and if $\Phi(x)$ is a concave function with $q_{\Phi} > 0$ then $\Phi\in\Delta_{2}$.

Let $\Phi(x)$ be a generalized Young function. We say that the random variable $f$ belongs to the weak Orlicz space ${\rm{w}}L_{\Phi}={\rm{w}}L_{\Phi}(\Omega, \mathcal{F}, \mathbb{P})$ if there exists an $c > 0$ such that $\Phi(\frac{t}{c})\mathbb{P}(|f| > t) < +\infty$ for all $t > 0$. In this case we put

$ \|f\|_{{\rm{w}}L_{\Phi}}:=\inf\Big\{c > 0:~\Phi\Big(\frac{t}{c}\Big)\mathbb{P}(|f| > t)\leq1, ~\forall t > 0\Big\}. $

The class ${\rm{w}}L_{\Phi}$ is said to be a weak Orlicz space. Some basic facts on weak Orlicz spaces were discussed in Liu, Hou and Wang [6]. For example, $\|\cdot\|_{{\rm{w}}L_{\Phi}}$ is a quasi-norm, ${\rm{w}}L_{\Phi}$ is a quasi-Banach space, and $L_{\Phi}\hookrightarrow {\rm{w}}L_{\Phi}$. If $\|f\|_{{\rm{w}}L_{\Phi}} < +\infty$, then

$ \mathop {\sup }\limits_{t > 0} \Phi (\frac{t}{{\parallel f{\parallel _{{\rm{w}}{L_\Phi }}}}})\mathbb{P}(|f| > t) \le 1. $

We define the weak Orlicz-Hardy spaces of martingales as below

$ {\rm{w}}\mathcal {H}_{\Phi}:=\{ f=(f_n, n\in\mathbb{N}): s(f)\in {\rm{w}}L_{\Phi}, ~\mbox{and}~\|f\|_{{\rm{w}}\mathcal {H}_{\Phi}} :=\|s(f)\|_{{\rm{w}}L_{\Phi}} < \infty \}. $

A new type of partial ordering between pairs of Young functions was introduced by [14-15] as below.

Definition 2.1[14-15]  Let $\Phi_{1}$, $\Phi_{2}$ be two generalized Young functions. We call that $\Phi_{2}$ is more convex than $\Phi_{1}$, $\Phi_{2}\succeq \Phi_{1}$ or $\Phi_{1}\preceq \Phi_{2}$ in symbols, if the composition $\Phi^{-1}_{1}\circ\Phi_{2}$ is a convex function.

Lemma 2.1(see [16])  Let $\Phi_{1}\curlyeqprec\Phi_{2}$ be two generalized Young functions having lower index $q_{\Phi_{1}} > 0$ and upper index $p_{\Phi_{2}} < \infty$. Then $q_{\Phi_{1, 2}}>0$ and $p_{\Phi_{1, 2}} < \infty$. More exactly, we have that

(ⅰ)$\frac{{{q}_{{{\Phi }_{2}}}}}{{{p}_{{{\Phi }_{1}}}}}\le {{q}_{{{\Phi }_{1, 2}}}}\le \frac{{{q}_{{{\Phi }_{2}}}}}{{{q}_{{{\Phi }_{1}}}}};~~~ $

(ⅱ)$\frac{{{p}_{{{\Phi }_{2}}}}}{{{p}_{{{\Phi }_{1}}}}}\le {{p}_{{{\Phi }_{1, 2}}}}\le \frac{{{p}_{{{\Phi }_{2}}}}}{{{q}_{{{\Phi }_{1}}}}}. $

Remark 2.1  Since $\Phi_{1, 2}(x)$ is a convex Young function, we denote by $\varphi_{1, 2}(x)$ and $\psi_{1, 2}(x)$ the density functions such that $\Phi_{1, 2}(x)=\displaystyle\int^{x}_{0}\varphi_{1, 2}(t)\mbox{d}t$ and its Young's complementary function $\Psi_{1, 2}(x)=\displaystyle\int^{x}_{0}\psi_{1, 2}(t)\mbox{d}t$, respectively.

Remark 2.2  It is shown in Lemma 2.1 that $\Phi_{1, 2}(x)=\Phi^{-1}_{1}\circ\Phi_{2}(x)$ has finite upper index, then the inverse function $\Phi^{-1}_{1, 2}(x)=\Phi^{-1}_{2}\circ\Phi_{1}(x)$ of $\Phi_{1, 2}(x)$ exists and it has the form

$ \Phi^{-1}_{1, 2}(x)=\displaystyle\int^x_0m_{\Phi_{1, 2}}(t)\mbox{d}t, ~~x\geq0. $

Since $\Phi_{1, 2}(x)$ is convex, then its inverse function $\Phi^{-1}_{1, 2}(x)$ is concave, therefore $m_{\Phi_{1, 2}}(x)$ is a decreasing function and we also have that

$ m_{\Phi_{1, 2}}(x)=\frac{1}{\varphi_{1, 2}\circ\Phi^{-1}_{1, 2}(x)}. $

Lemma 2.2(see [6])  Let $\Phi\in\Delta_{2}$, then there exists a constant $K_{\Phi}\geq1$ depending only on $\Phi$, such that

$ \|f+g\|_{{\rm{w}}L_{\Phi}}\leq K_{\Phi}(\|f\|_{{\rm{w}}L_{\Phi}}+\|g\|_{{\rm{w}}L_{\Phi}}), ~~\forall f, g\in{{\rm{w}}L_{\Phi}}. $

Let $v=(v_n, n\in\mathbb{N})$ be a process adapted to $(\mathcal{F}_n, n\in\mathbb{N})$, the martingale transform $T_v$ for a given martingale $f$ is defined by $T_vf=(T_vf_n, n\in\mathbb{N})$ where $T_vf_n:=\sum\limits^{n}_{i=1}v_{i-1}\cdot df_i$. It can easily be seen that $T_vf$ is still a martingale.

The Lemma below is well known and can be found in Long [7] and Weisz [11].

Lemma 2.3(see [7, 13])  Let $f=(f_n, n\in \mathbb{N})$ be a martingale. Then $f_n$ converges a.s. on the set of $\{\omega:~s(f) < \infty\}$.

At first, we prove a necessary lemma, which can be seen as a weak version of the generalized Hölder's inequality and has an independent existence value.

Lemma 3.1  Let $\Phi_{1}$ be a concave Young function with $q_{\Phi_1} > 0$, $\Phi_{2}$ a concave Young function with $q_{\Phi_2} > 0$ or a convex Young function with $p_{\Phi_2} < +\infty$, and let $\Phi_{1}\preceq \Phi_{2}$, $\Phi_{1, 2}(x)=\Phi^{-1}_{1}\circ\Phi_{2}(x)$ with Young's complementary function $\Psi_{1, 2}(x)$. If $f\in {\rm{w}}L_{\Phi_2}$, $g\in {\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}$, then $f\cdot g\in {\rm{w}}L_{\Phi_1}$ and we have

$ \|f\cdot g{{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}\le 2{{K}_{{{\Phi }_{1}}}}\|f{{\|}_{\text{w}{{L}_{{{\Phi }_{2}}}}}}\cdot \|g{{\|}_{\text{w}{{L}_{{{\Phi }_{1}}{}^\circ {{\Psi }_{1, 2}}}}}}. $

(3.1)

Proof  For any $f\in {\rm{w}}L_{\Phi_2}$ and $g\in {\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}$, if $\|f\|_{{\rm{w}}L_{\Phi_2}}\cdot\|g\|_{{\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}}=0$, then (3.1) is obvious. Now we assume that $\|f\|_{{\rm{w}}L_{\Phi_2}}\cdot\|g\|_{{\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}} > 0$. For the sake of convenience, denote $\|f\|_{{\rm{w}}L_{\Phi_2}}=A$ and $\|g\|_{{\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}}=B$. Because $(\Phi_{1, 2}, \Psi_{1, 2})$ is a pair of conjugate Young functions, by Young's inequality, we have that

$ \frac{|f\cdot g|}{A\cdot B}\leq \Phi^{-1}_{1}\circ\Phi_{2}\Big(\frac{|f|}{A}\Big)+\Psi_{1, 2}\Big(\frac{|g|}{B}\Big). $

Since $q_{\Phi_1} > 0$ and $0 < q_{\Phi_2}\leq p_{\Phi_2} < +\infty$, $\Phi_1, \Phi_2\in\Delta_2$. Applying Lemma 2.2, we obtain

$ \frac{\|f\cdot g{{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}}{A\cdot B}\le {{K}_{{{\Phi }_{1}}}}(\|\Phi _{1}^{-1}{}\circ{{\Phi }_{2}}(\frac{|f|}{A}){{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}+\|{{\Psi }_{1, 2}}(\frac{|g|}{B}){{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}). $

(3.2)

Because $0 < A=\|f\|_{{\rm{w}}L_{\Phi_{2}}} < +\infty$, so $\Phi_{2}\Big(\frac{t}{A}\Big)\mathbb{P}(|f| > t)\leq1$, for all $t > 0$. Since both $\Phi_{1}$ and $\Phi_{2}$ are continuous and bijective from $[0, +\infty)$ to itself, then for any $s > 0$, there exists a $t > 0$ such that $\Phi_{1}(s)=\Phi_{2}(t/A)$. Moreover, for any $s > 0$, we have

$ \begin{align} &\ \ \ \ {{\Phi }_{1}}(s)\mathbb{P}(\Phi _{1}^{-1}\circ {{\Phi }_{2}}(|f|/A) > s)={{\Phi }_{1}}(s)\mathbb{P}({{\Phi }_{2}}(|f|/A) > {{\Phi }_{1}}(s)) \\ & ={{\Phi }_{1}}(s)\mathbb{P}({{\Phi }_{2}}(|f|/A) > {{\Phi }_{2}}(t/A))={{\Phi }_{2}}(t/A)\mathbb{P}(|f| > t)\le 1. \\ \end{align} $

This implies that $\big\|\Phi^{-1}_{1}\circ\Phi_{2}\big(\frac{|f|}{A}\big)\big\|_{{\rm{w}}L_{\Phi_{1}}}\leq1$. Similarly, we can prove that $\big\|\Psi_{1, 2}\big(\frac{|g|}{B}\big)\big\|_{{\rm{w}}L_{\Phi_{1}}}\leq1$. Substituting these to (3.2), then (3.1) is proved.

Theorem 3.1  Let $\Phi_{1}$ be a concave Young function with $q_{\Phi_1} > 0$, $\Phi_{2}$ a concave Young function with $q_{\Phi_2} > 0$ or a convex Young function with $p_{\Phi_2} < +\infty$, and $\Phi_{1}\preceq \Phi_{2}$. Let $f=(f_{n}, n\in\mathbb{N})\in {\rm{w}}\mathcal {H}_{\Phi_{1}}$, and define the martingale transform $T(f)$ by

$ T{{f}_{0}}=0,~a.s.,~~T{{f}_{n}}=\sum\limits_{i=1}^{n}{{{m}_{{{\Phi }_{1,2}}}}}({{s}_{i}}(f))\cdot d{{f}_{i}},~~n\ge 1. $

Then, the martingale $T(f)=(Tf_{n}, n\in\mathbb{N})$ belongs to ${\rm{w}}\mathcal {H}_{\Phi_{2}}$ and

$ \|T(f){{\|}_{\text{w}{{\mathcal{H}}_{{{\Phi }_{2}}}}}}\le \|\Phi _{2}^{-1}{}\circ {{\Phi }_{1}}(s(f)){{\|}_{\text{w}{{L}_{{{\Phi }_{2}}}}}}\le \|f{{\|}_{\text{w}{{\mathcal{H}}_{{{\Phi }_{1}}}}}}. $

(3.3)

Additionally, $\{Tf_{n}\}_{n\geq1}$ converges a.s. to a limit $Tf_{\infty}$.

Proof  Setting $s_{0}(f)=0$, for all $i\geq1$, we have $E(|df_{i}|^{2}|\mathcal{F}_{i-1})=s^{2}_{i}(f)-s^{2}_{i-1}(f)$, and

$ E(|d(T{f_i}){|^2}|{{\cal F}_{i - 1}}) = E(m_{{\Phi _{1,2}}}^2({s_i}(f))|d{f_i}{|^2}|{{\cal F}_{i - 1}}) = m_{{\Phi _{1,2}}}^2({s_i}(f)) \cdot E(|d{f_i}{|^2}|{{\cal F}_{i - 1}}). $

Then, for all $n\geq1$, we have

$ s_{n}^{2}(T(f))=\sum\limits_{i=1}^{n}{E}(|d(T{{f}_{i}}){{|}^{2}}|{{\mathcal{F}}_{i-1}})=\sum\limits_{i=1}^{n}{m_{{{\Phi }_{1,2}}}^{2}}({{s}_{i}}(f))(s_{i}^{2}(f)-s_{i-1}^{2}(f)). $

The sequence $\{s_{n}(f)\}_{n\geq1}$ is non-negative and non-decreasing, the function $m_{\Phi_{1, 2}}(x)$ is non-negative and decreasing, so, for all $i\geq1$, we have

$ \begin{array}{*{35}{l}} \ \ \ \ \ m_{{{\Phi }_{1,2}}}^{2}({{s}_{i}}(f))(s_{i}^{2}(f)-s_{i-1}^{2}(f)) \\ =[{{m}_{{{\Phi }_{1,2}}}}({{s}_{i}}(f))({{s}_{i}}(f)-{{s}_{i-1}}(f))]\cdot [{{m}_{{{\Phi }_{1,2}}}}({{s}_{i}}(f))({{s}_{i}}(f)+{{s}_{i-1}}(f))] \\ \le [{{m}_{{{\Phi }_{1,2}}}}({{s}_{i}}(f))({{s}_{i}}(f)-{{s}_{i-1}}(f))]\cdot [{{m}_{{{\Phi }_{1,2}}}}({{s}_{i}}(f)){{s}_{i}}(f)+{{m}_{{{\Phi }_{1,2}}}}({{s}_{i-1}}(f)){{s}_{i-1}}(f)] \\ \le \int_{{{s}_{i-1}}(f)}^{{{s}_{i}}(f)}{{{m}_{{{\Phi }_{1,2}}}}(t)\text{d}t}\cdot \left( \int_{0}^{{{s}_{i}}(f)}{{{m}_{{{\Phi }_{1,2}}}}(t)\text{d}t}+\int_{0}^{{{s}_{i-1}}(f)}{{{m}_{{{\Phi }_{1,2}}}}(t)\text{d}t} \right) \\ =[\Phi _{1,2}^{-1}({{s}_{i}}(f))-\Phi _{1,2}^{-1}({{s}_{i-1}}(f))]\cdot [\Phi _{1,2}^{-1}({{s}_{i}}(f))+\Phi _{1,2}^{-1}({{s}_{i-1}}(f))] \\ ={{[\Phi _{1,2}^{-1}({{s}_{i}}(f))]}^{2}}-{{[\Phi _{1,2}^{-1}({{s}_{i-1}}(f))]}^{2}}. \\ \end{array} $

Consequently, for any $n\geq1$, we get

$ s_n^2(T(f)) \le \sum\limits_{i = 1}^n ( {[\Phi _{1,2}^{ - 1}({s_i}(f))]^2} - {[\Phi _{1,2}^{ - 1}({s_{i - 1}}(f))]^2}) = {[\Phi _{1,2}^{ - 1}({s_n}(f))]^2}. $

In other words, we have that $s(T(f))\leq\Phi^{-1}_{1, 2}(s(f))$ a.s.. Given $f\in{\rm{w}}\mathcal {H}_{\Phi_{1}}$, then $\|s(f)\|_{{\rm{w}}L_{\Phi_{1}}}=\|f\|_{{\rm{w}}\mathcal {H}_{\Phi_{1}}} < +\infty$. By the homogeneity of quasi-norm, we may assume that $\|s(f)\|_{{\rm{w}}L_{\Phi_{1}}}=1$, for simplicity. Then

$ \mathop {\sup }\limits_{t > 0} {\Phi _1}(t)\mathbb{P}(s(f) > t) = \mathop {\sup }\limits_{t > 0} {\Phi _1}(\frac{t}{{\parallel s(f){\parallel _{wL{\Phi _1}}}}})\mathbb{P}(s(f) > t) \le 1. $

Since both $\Phi_{1}$ and $\Phi_{2}$ are bijective from $[0, +\infty)$ to itself, for any $s\in(0, +\infty)$, there exists a $t\in(0, +\infty)$, such that $\Phi_{1}(t)=\Phi_{2}(s)$. For any $s > 0$, we have that

$ \begin{align} & {{\Phi }_{2}}(s)\mathbb{P}(\Phi _{1,2}^{-1}(s(f)) > s)={{\Phi }_{2}}(s)\mathbb{P}(\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f)) > s)={{\Phi }_{2}}(s)\mathbb{P}({{\Phi }_{1}}(s(f)) > {{\Phi }_{2}}(s)) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ={{\Phi }_{1}}(t)\mathbb{P}({{\Phi }_{1}}(s(f)) > {{\Phi }_{1}}(t))={{\Phi }_{1}}(t)\mathbb{P}(s(f) > t)\le 1. \\ \end{align} $

This means that $\Phi^{-1}_{1, 2}(s(f))\in {\rm{w}}L_{\Phi_{2}}$ and $\|\Phi^{-1}_{1, 2}(s(f))\|_{{\rm{w}}L_{\Phi_{2}}}\leq\|s(f)\|_{{\rm{w}}L_{\Phi_{1}}}$. Since

$ {{\Phi }_{2}}(\frac{t}{\|\Phi _{1, 2}^{-1}(s(f)){{\|}_{\text{w}{{L}_{{{\Phi }_{2}}}}}}})\cdot \mathbb{P}(s(T(f)) > t)\le {{\Phi }_{2}}(\frac{t}{\|\Phi _{1, 2}^{-1}(s(f)){{\|}_{\text{w}{{L}_{{{\Phi }_{2}}}}}}})\cdot \mathbb{P}(\Phi _{1, 2}^{-1}(s(f)) > t)\le 1, $

then $\|s(T(f))\|_{{\rm{w}}L_{\Phi_{2}}} \leq\|\Phi^{-1}_{1, 2}(s(f))\|_{{\rm{w}}L_{\Phi_{2}}} \leq\|s(f)\|_{{\rm{w}}L_{\Phi_{1}}}$. This means that $T(f)\in{\rm{w}}\mathcal {H}_{\Phi_{2}}$ and

$ \|T(f)\|_{{\rm{w}}\mathcal {H}_{\Phi_{2}}} \leq \|\Phi^{-1}_{2}\circ\Phi_{1}(s(f))\|_{{\rm{w}}L_{\Phi_{2}}} \leq\|f\|_{{\rm{w}}\mathcal {H}_{\Phi_{1}}}. $

The inequality (3.3) is proved.

Moreover, if we denote $\|\Phi^{-1}_{2}\circ\Phi_{1}(s(f))\|_{{\rm{w}}L_{\Phi_{2}}}=A$, then

$ \mathbb{P}\big(\Phi^{-1}_{2}\circ\Phi_{1}(s(f)) > t\big)\leq \frac{1}{\Phi_{2}(t/A)}, ~~\forall t > 0. $

Note that $\mathop {\lim }\limits_{t \to + \infty } {\Phi _2}(t/A) = + \infty $, so

$ \begin{align} \ & \mathbb{P}(\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f))=+\infty )=\underset{n\to \infty }{\mathop{\lim }}\,\mathbb{P}(\bigcap\limits_{k=1}^{n}{\{\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f)) > k\}}) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ }\ \le \underset{n\to \infty }{\mathop{\lim }}\,\mathbb{P}(\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f)) > n)\le \underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{{{\Phi }_{2}}(n/A)}=0. \\ \end{align} $

On the other hand, since $s(T(f))\leq\Phi^{-1}_{2}\circ\Phi_{1}(s(f))$, then $\{s(T(f)) < +\infty\}\supset\{\Phi^{-1}_{2}\circ\Phi_{1}(s(f)) < +\infty\}$. Hence, we have that

$ \begin{align} & 1\ge \mathbb{P}(s(T(f)) < +\infty )\ge \mathbb{P}(\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f)) < +\infty ) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =1-\mathbb{P}(\Phi {{_{2}^{-1}}{}\circ}{{\Phi }_{1}}(s(f))=+\infty )=1. \\ \end{align} $

This means that $s(T(f)) < +\infty$ a.s.. Consequently, by Lemma 2.3, $\{Tf_n\}_{n\geq1}$ converges a.s. to a limit $Tf_{\infty}$. The proof is completed.

Theorem 3.2  Let the generalized Young functions $\Phi_1$ and $\Phi_2$, the martingales $f$ and $T(f)$ be as in Theorem 3.1. Then

$ \|f\|_{{\rm{w}}\mathcal {H}_{\Phi_{1}}} \leq 2\sqrt{2}K_{\Phi_1}\|\varphi_{1, 2}\circ\Phi^{-1}_{1, 2}(s(f))\|_{{\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}} \cdot\|T(f)\|_{{\rm{w}}\mathcal {H}_{\Phi_{2}}}. $

Proof  With $s_{0}(T(f))=0$, we have

$ E(|dT{f_i}{|^2}|{{\cal F}_{i - 1}}) = s_i^2(T(f)) - s_{i - 1}^2(T(f)) $

for all $i\geq1$. From the representation of $Tf_{n}$ figuring in the statement of Theorem 3.1, we have

$ |df_i|=\frac{|dTf_i|}{m_{\Phi_{1, 2}}(s_{i}(f))}, ~~i\geq1 $

(if $m_{\Phi_{1, 2}}(s_{i}(f))=0$, then we can add an $\varepsilon > 0$ to each $s_{i}(f)$ and at the end let $\varepsilon\rightarrow 0$.) Therefore, by Abel's rearrangement, we have

$ \begin{array}{l} s_n^2(f) = \sum\limits_{i = 1}^n E (|dT{f_i}{|^2}|{{\cal F}_{i - 1}}) = \mathop {\mathop \sum \limits^n }\limits_{i = 1} {\mkern 1mu} E[{(\frac{{|dT{f_i}|}}{{{m_{{\Phi _{1,2}}}}({s_i}(f))}})^2}|{{\cal F}_{i - 1}}]\\ \;\;\;\;\;\;\;\;\; = \sum\limits_{i = 1}^n {\frac{{s_i^2(T(f)) - s_{i - 1}^2(T(f))}}{{m_{{\Phi _{1,2}}}^2({s_i}(f))}}} \\ \;\;\;\;\;\;\;\;\; = \sum\limits_{i = 1}^n [ s_i^2(T(f)) - s_{i - 1}^2(T(f))] \cdot \varphi _{1,2}^2(\Phi _{1,2}^{ - 1}({s_i}(f)))\\ \;\;\;\;\;\;\;\;\; = s_n^2(T(f)) \cdot \varphi _{1,2}^2(\Phi _{1,2}^{ - 1}({s_n}(f)))\\ \;\;\;\;\;\;\;\;\;\; - \sum\limits_{i = 1}^{n - 1} {s_i^2} (T(f))[\varphi _{1,2}^2(\Phi _{1,2}^{ - 1}({s_{i + 1}}(f))) - \varphi _{1,2}^2(\Phi _{1,2}^{ - 1}({s_i}(f)))]. \end{array} $

Noticing that both the sequences $\{s_n(T(f))\}_{n\geq0}$ and $\{\varphi_{1,2}\circ\Phi^{-1}_{1,2}(s_{n}(f))\}_{n\geq0}$ are nonnegative and nondecreasing, then we get that

$ s_n^2(f) \le 2s_n^2(T(f)) \cdot \varphi _{1,2}^2(\Phi _{1,2}^{ - 1}({s_n}(f))),\;\;\;\;n \ge 0. $

Therefore

$ s(f) \leq \sqrt{2}s(T(f))\cdot\varphi_{1,2}(\Phi^{-1}_{1,2}(s(f))). $

Thus, applying Lemma 3.1, we have that

$ \begin{array}{l} \parallel f{\parallel _{{\rm{w}}{{\cal H}_{{\Phi _1}}}}} \le \sqrt 2 \parallel s(T(f)) \cdot {\varphi _{1,2}}(\Phi _{1,2}^{ - 1}(s(f))){\parallel _{{\rm{w}}{L_{{\Phi _1}}}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \le 2\sqrt 2 {K_{{\Phi _1}}}\parallel s(T(f)){\parallel _{{\rm{w}}{L_{{\Phi _2}}}}} \cdot \parallel {\varphi _{1,2}}(\Phi _{1,2}^{ - 1}(s(f))){\parallel _{{\rm{w}}{L_{{\Phi _1}^\circ {\Psi _{1,2}}}}}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = 2\sqrt 2 {K_{{\Phi _1}}}\parallel T(f){\parallel _{{\rm{w}}{{\cal H}_{{\Phi _2}}}}} \cdot \parallel {\varphi _{1,2}}(\Phi _{1,2}^{ - 1}(s(f))){\parallel _{{\rm{w}}{L_{{\Phi _1}^\circ {\Psi _{1,2}}}}}}. \end{array} $

This proves the assertion.

Now, combining Theorem 3.1 and 3.2, we obtain the following corollary, one of the main results of the present article.

Corollary 3.1  Let $\Phi_{1}$ be a concave Young function with $q_{\Phi_1} > 0$, $\Phi_{2}$ a concave Young function with $q_{\Phi_2} > 0$ or a convex Young function with $p_{\Phi_2} < +\infty$, and $\Phi_{1}\preceq \Phi_{2}$. Then for any martingale $f=(f_{n}, n\in\mathbb{N})\in {\rm{w}}\mathcal {H}_{\Phi_{1}}$, there exists a martingale $g=(g_{n}, n\in\mathbb{N})\in {\rm{w}}\mathcal {H}_{\Phi_{2}}$, such that $f$ is the martingale transform of $g$. Namely, we have

$ {f_0} = 0,a.s.,\;\;\;{f_n} = \sum\limits_{i = 1}^n {{v_{i - 1}}} \cdot d{g_i},\;\;\;n \ge 1,\ $

where $v_{i}=\varphi_{1, 2}\circ\Phi^{-1}_{1, 2}(s_{i}(f))$, $(i=0, 1, 2, 3, \cdots)$. We have

$ \|v_{\infty}\|_{{\rm{w}}L_{\Phi_{1}\circ\Psi_{1, 2}}} \leq \max\{1, (p_{\Phi_{1, 2}}-1)\|f\|_{{\rm{w}}\mathcal {H}_{\Phi_{1}}}\} $

(3.4)

and

$ \|g\|_{{\rm{w}}\mathcal {H}_{\Phi_{2}}} \leq\|\Phi^{-1}_{2}\circ\Phi_{1}(s(f))\|_{{\rm{w}}L_{\Phi_2}} \leq \|f\|_{{\rm{w}}\mathcal {H}_{\Phi_1}}. $

Proof  From Theorem 3.1 and 3.2, only the inequality (3.4) needs to be proved. In fact, since $(\Phi_{1, 2}, \Psi_{1, 2})$ is a pair of conjugate Young functions, so

$ u\varphi_{1, 2}(u)=\Phi_{1, 2}(u)+\Psi_{1, 2}(\varphi_{1, 2}(u)), ~~\forall\, u > 0. $

(3.5)

Because ${p_{{\Phi _{1,2}}}} = \mathop {\sup }\limits_{u > 0} \frac{{u{\varphi _{1,2}}(u)}}{{{\Phi _{1,2}}(u)}}$, then

$ p_{\Phi_{1, 2}}\Phi_{1, 2}(u)\geq u\varphi_{1, 2}(u), ~~\forall\, u > 0. $

(3.6)

By (3.5) and (3.6), we get

$ p_{\Phi_{1, 2}}\Phi_{1, 2}(u)\geq\Phi_{1, 2}(u)+\Psi_{1, 2}(\varphi_{1, 2}(u)), ~~\forall\, u > 0, $

and then

$ \Psi_{1, 2}(\varphi_{1, 2}(u))\leq (p_{\Phi_{1, 2}}-1)\Phi_{1, 2}(u), ~~\forall\, u > 0. $

(3.7)

Substituted $u$ in (3.7) by $\Phi^{-1}_{1, 2}(s(f))$, we have

$ \Psi_{1, 2}(\varphi_{1, 2}\circ\Phi^{-1}_{1, 2}(s(f))) \leq(p_{\Phi_{1, 2}}-1)\Phi_{1, 2}(\Phi^{-1}_{1, 2}(s(f))) \leq(p_{\Phi_{1, 2}}-1)\cdot s(f). $

(3.8)

Employing (3.8), on the one hand, by the convexity of $\Psi_{1, 2}$, for all $t > 0$, we have

$\begin{gathered} {\Phi _1} \circ {\Psi _{1,2}}(\frac{t}{{\max \{ 1,({p_{{\Phi _{1,2}}}} - 1)\parallel s(f){\parallel _{{\text{w}}{L_{{\Phi _1}}}}}\} }}) \leqslant {\Phi _1}(\frac{{{\Psi _{1,2}}(t)}}{{\max \{ 1,({p_{{\Phi _{1,2}}}} - 1)\parallel s(f){\parallel _{{\text{w}}{L_{{\Phi _1}}}}}\} }}) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \leqslant {\Phi _1}(\frac{{{\Psi _{1,2}}(t)}}{{({p_{{\Phi _{1,2}}}} - 1)\parallel s(f){\parallel _{{\text{w}}{L_{{\Phi _1}}}}}}}). \hfill \\ \end{gathered} $

(3.9)

On the other hand, for any $t > 0$, we have

$ \begin{align} &\mathbb{P}({{\varphi }_{1, 2}}{}\circ \Phi _{1, 2}^{-1}(s(f)) > t)=\mathbb{P}({{\Psi }_{1, 2}}({{\varphi }_{1, 2}}{}\circ \Phi _{1, 2}^{-1}(s(f))) > {{\Psi }_{1, 2}}(t)) \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \mathbb{P}(({{p}_{{{\Phi }_{1, 2}}}}-1)s(f)>{{\Psi }_{1, 2}}(t)). \\ \end{align} $

(3.10)

Since $f\in {\rm{w}}\mathcal {H}_{\Phi_{1}}$, we have $s(f)\in {\rm{w}}L_{\Phi_1}$, furthermore, we have $(p_{\Phi_{1, 2}}-1)s(f)\in {\rm{w}}L_{\Phi_1}$ too, and $\|(p_{\Phi_{1, 2}}-1)s(f)\|_{{\rm{w}}L_{\Phi_1}}=(p_{\Phi_{1, 2}}-1)\|s(f)\|_{{\rm{w}}L_{\Phi_1}}=(p_{\Phi_{1, 2}}-1)\|f\|_{{\rm{w}}\mathcal {H}_{\Phi_{1}}}$. Therefore, for any $u > 0$, we have

$ \Phi_{1}\bigg(\frac{u}{\|(p_{\Phi_{1, 2}}-1)s(f)\|_{{\rm{w}}L_{\Phi_1}}}\bigg)\mathbb{P}\big((p_{\Phi_{1, 2}}-1)s(f) > u\big)\leq1. $

(3.11)

From (3.9), (3.10) and (3.11), for any $t > 0$, we have that

$ \begin{align} & \ \ \ \ \ {{\Phi }_{1}}{}\circ {{\Psi }_{1,2}}(\frac{t}{\max \{1,({{p}_{{{\Phi }_{1,2}}}}-1)\|f{{\|}_{\text{w}{{\mathcal{H}}_{{{\Phi }_{1}}}}}}\}})\mathbb{P}({{\varphi }_{1,2}}{}\circ \Phi _{1,2}^{-1}(s(f)) > t) \\ & ={{\Phi }_{1}}{}\circ {{\Psi }_{1,2}}(\frac{t}{\max \{1,({{p}_{{{\Phi }_{1,2}}}}-1)\|s(f){{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}\}})\mathbb{P}({{\varphi }_{1,2}}{}\circ \Phi _{1,2}^{-1}(s(f)) > t) \\ & \le {{\Phi }_{1}}(\frac{{{\Psi }_{1,2}}(t)}{\|({{p}_{{{\Phi }_{1,2}}}}-1)s(f){{\|}_{\text{w}{{L}_{{{\Phi }_{1}}}}}}})\mathbb{P}(({{p}_{{{\Phi }_{1,2}}}}-1)s(f) > {{\Psi }_{1,2}}(t))\le 1. \\ \end{align} $

This implies that

$ \begin{align} &\|{{v}_{\infty }}{{\|}_{\text{w}{{L}_{{{\Phi }_{1}}{}\circ {{\Psi }_{1, 2}}}}}}=\|{{\varphi }_{1, 2}}{}\circ \Phi _{1, 2}^{-1}(s(f)){{\|}_{\text{w}{{L}_{{{\Phi }_{1}}{}\circ {{\Psi }_{1, 2}}}}}} \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le \max \{1, ({{p}_{{{\Phi }_{1, 2}}}}-1)\|f{{\|}_{\text{w}{{\mathcal{H}}_{{{\Phi }_{1}}}}}}\}. \\ \end{align} $