With the continuous improvement of martingale theory, the applications of harmonic analysis in weak function space and martingale space have become more and more extensive. Herz [1] gave the atomic decomposition method from the modulation analysis into the study of the theory. The atomic decomposition method has been widely used in the martingale theory study, especially the two-parameter atomic decomposition became particularly important to prove inequality theory. Weisz [2] established real-valued single-parameter martingale, the atomic decomposition theory of two-parameter martingale space and the strong atomic decomposition theory of two-parameter strong martingale space in the literature. In the meantime, Weisz used atomic decomposition methods to prove important inequality and duality theory in martingale space. The literature [3-4] studied the situation of single-parameter B-valued martingale space and weak martingale space. They gave the atomic decomposition theories in B-valued martingale space and weak martingale space when the value with certain geometric properties. Chen [5] obtained the atomic decomposition in the two-parameter B-valued martingale space. Ye [6] defined the strong atom and established a strong atomic decomposition theorem in the two-parameter B-valued strong martingale space of $ s\widetilde{\sum}_{\alpha}^p $. Liu [7] introduced a class of important function space–weak Orlicz space. And in the article [8] they studied some basic properties of weak Orlicz space and some applications in harmonic analysis.

In the paper, we will discuss the atomic decomposition for two-parameter B-valued weak Orlicz strong martingale space.

Let $ (\Omega, \mathcal{F}, P) $ be a complete probability space, (B, $ \|\cdot\|) $ be a Banach space with dimension not less than 2. $ \{\mathcal{F }_{n}, n\in N^{2}\} $ is a non-decreasing partial order sub $ \sigma $ algebraic sequence of $ \mathcal{F} $ in $ N^{2} $ and which satisfies the condition $ F_{4} $. For $ \forall n=(n_{1}, n_{2})\in N^{2} $, we define $ \mathcal{F}_{n}^-=\sigma(\mathcal{F}_{n_{1}-1, n_{2}}\bigcup\mathcal {F}_{n_{1}, n_{2}-1}) $, $ \mathcal{F}^{+}_{n}=\sigma(\mathcal{F}_{n_{1}+ 1, n_{2}}\bigcup\mathcal{F}_{n_{1}, n_{2}+1}) $ and $ \mathcal{F}^{*}=\sigma(\bigcup\limits_{n\in N} \mathcal{F}_{\infty}) $. We denote $ E $, $ E_{n} $, $ E^{-}_{n} $, $ E^{+}_{n} $ and $ E^{*}_{n} $ respectively represent the expectation operator conditional expectation operator, and with respect to $ \mathcal{F}_{n} $, $ \mathcal{F}_{n}^{-} $, $ \mathcal{F} _{n}^{+} $ and $ \mathcal{F}^{\ast} $.

For two sets $ G, $ $ H\subset N^2 $ consisting of incomparable pairs of numbers, we write $ H\not\ll G $(or $ H\not\gg G $) if $ \forall m\in G $ there exists $ v_t\in H $ such that $ v_t\not\ll m $(or $ v_t\not\gg m $). Denote by $ \inf H $: the set $ v_t\in H: $ there does not exist $ m\in H $ such that $ v_t<m $.

We define $ d_nf $ as follows:

$ d_{n}f= \begin{cases} 0, n_{1}n_{2}=0\\ f_{n_{1}, n_{2}}-f_{n_{1}-1, n_{2}}-f_{n_{1}, n_{2}-1}+f_{n_{1}-1, n_{2}-1}, n_{1}n_{2}\neq0. \end{cases} $

Let $ \Phi $ be convex and non-decreasing in $ [0, \infty) $ which satisfies $ \Phi(0)=0 $ and $ \lim\limits_{t\rightarrow \infty} \Phi(t)=\infty $, $ \varphi(t) $ is a right continuous derivative function for $ \Phi $. There are two major parameters for the function $ \Phi $,

$ \begin{equation} q_\Phi=\inf\limits_{t>0}\frac{t\varphi(t)}{\Phi(t)}, \ \ \ \ p_\Phi=\sup\limits_{t>0}\frac{t\varphi(t)}{\Phi(t)}. \end{equation} $

(2.1)

It is called that $ \Phi $ satisfies the condition of $ \Delta_2 $, if for any $ a>1 $, there exists a constant $ C_a> 0 $ such that $ \Phi(at)\leq C_a\Phi(t) $, $ \forall t> 0 $. This condition is equivalent to $ p_\Phi<\infty $. $ \Phi $ is strictly convex, if $ q_\Phi> 1 $. In particular, if $ 0 < q_\Phi \leq p_\Phi<\infty $, the function $ \frac{\Phi(t)}{t^{q_\Phi}} $ is monotonically increasing on $ (0, \infty) $, and $ \frac{\Phi (t)}{t^{p_\Phi}} $ is monotonically decreasing on $ (0, \infty) $.

As well known, the space $ wL_{\Phi} $ is the collection of all measurable functions $ f $ for which

$ \|f\|_{wL_{\Phi}}=\inf\{c>0:\sup\limits_{t>0}\Phi(\frac{t}{c})P(\|f\|>t)\leq1\}. $

Definition 2.1    The B-valued martingale $ f=(f_{n}, n\in N^{2}) $ is a strong martingale, when $ E^{*}_{n-1}d_{n}f=0 $, $ \forall n\in N^{2} $.

This article uses $ \|f\|_{p} $, $ f^{*} $, $ S^{(p)}(f) $, $ \sigma^{(p)}(f) $ to indicate its $ L_{p}- $modulus, maximum function, $ p $-mean square function and conditional $ p $-mean square function related to $ \mathcal{F}_{n-1} $:

$ \|f\|_{p}=\sup\limits_{n\in N^{2}}\|f_{n}\|_{p}, \ \ \ f^{*}_{n}=\sup\limits_{i\leq n}\|f_{i}\|, \ \ \ f^{*}=\sup\limits_{n\in N^{2}}\|f_{n}\|; $

$ S^{(p)}_{n}(f)=(\sum\limits_{0\leq i\leq n}\|d_{i}f\|^{p})^{\frac{1}{p}}, \ \ \ \ \ \ S^{(p)}(f)=\sup\limits_{n\in N^{2}}S^{(p)}_{n}(f), 1<p<\infty; $

$ \sigma^{(p)}_{n}(f)=(\sum\limits_{0\leq i\leq n}E_{i-1}\|d_{i}f\|^{p})^{\frac{1}{p}}, \ \ \ \ \sigma^{(p)}(f)=\sup\limits_{n\in N^{2}}\sigma^{(p)}_{n}(f), 1<p<\infty. $

Moreover, for the two-parameter B-valued strong martingale $ f=(f_{n}, n\in N^{2}) $, we introduce a conditional $ p $-mean square function about $ \mathcal{F}^{-}_{n} $ :

$ \widetilde{\sigma}^{(p)}(f):=(\sum\limits_{n\in N^{2}}E_{n}^{-}\parallel d_{n}f\parallel^{p})^{\frac{1}{p}}. $

We define the weak Orlicz spaces as follows:

$ \begin{eqnarray*} wL_{\Phi}&=&\{f=(f_{n}, n\in N^{2}):\|f\|_{wL_{\Phi}}=\sup\limits_{n\in N^{2}}\|f_{n}\|_{wL_{\Phi}}<\infty\}, \\ wH^{*}_{\Phi}&=&\{f=(f_{n}, n\in N^{2}):\|f\|_{wH_{\Phi}}=\|f^{*}\|_{wL_{\Phi}}<\infty\}, \\ wH_{\Phi}^{S}&=&\{f=(f_{n}, n\in N^{2}):\|f\|_{wH_{\Phi}^{S}}=\|S^{(p)}(f)\|_{wL_{\Phi}}<\infty\}, \\ wH_{\Phi}^{\sigma}&=&\{f=(f_{n}, n\in N^{2}):\|f\|_{wH_{\Phi}^{s}}=\|\sigma^{(p)}(f)\|_{wL_{\Phi}}<\infty\}, \\ w\widetilde{H}_{\Phi}^{\sigma}&=&\{f=(f_{n}, n\in N^{2}):\|f\|_{wH_{\Phi}^{s}}=\|\widetilde{\sigma}^{(p)}(f)\|_{wL_{\Phi}}<\infty\}. \end{eqnarray*} $

Definition 2.2    A strong martingale $ g\in L_p $ is said to be a strong $ (\alpha, p) $ atom $ (0<\alpha\leq p, 1<p\leq\infty) $, if there is a stopping time $ v\in T(\mathcal{F}^{+}_n) $ such that:

(1) if $ \{v\not\ll n\} $, $ g_{n}:=E_{n}g=0 $;

(2) $ \|g^{*}\|_{p}\leq P(v\neq\infty)^{\frac{1}{p}-\frac{1}{\alpha}} $.

Lemma 2.1([6])    Let B be a Banach space, $ 1<p\leq2 $, then the following statements are equivalent :

(1) B is $ p $ smooth;

(2) For any two-parameter B-valued strong martingale $ f=(f_n, n\in N^2) $, the $ f_n $ converges in probability when $ \sum\limits_{n\in N^{2}}E\|d_nf\|^p<\infty $.

Lemma 2.2([6])    Let B be a Banach space, $ 1<p\leq2 $, $ 0<\alpha\leq p $, then the following statements are equivalent :

(1) B is $ p $ smooth;

(2) There is a constant $ C $ such that for any two-parameter B-valued strong martingale $ f $: $ \|S^{(p)}(f)\|_{\alpha}\leq C\|\widetilde{\sigma}^{(p)}(f)\|_{\alpha} $;

(3) There is a constant $ C $ such that for any two-parameter B-valued strong martingale $ f $: $ \|f^*\|_{\alpha}\leq C\|\widetilde{\sigma}^{(p)}(f)\|_{\alpha} $.

Theorem 3.1    Let B be a Banach space, $ 1<p\leq2 $, $ \Phi $ satisfies the condition of $ \Delta_2 $. If B is $ p $ smooth, $ 0<\alpha\leq p $, then for any two-parameter B-valued strong martingale $ f=(f_{n}, n\in N^{2})\in w\widetilde{H}_{\Phi}^{\sigma} $, there is a strong $ (\alpha, p) $ atom $ (g^{(k)}, k\in Z) $ and a column of non-negative real number $ \mu=\mu_k, k\in Z\in l_\alpha $, so that for all $ n\in N^2 $:

$ \begin{equation} f_n=\sum\limits_{k=-\infty}^\infty\mu_{k}E_ng^{(k)}, a.e., \end{equation} $

(3.1)

$ \begin{equation} (\sum\limits_{k\in Z}\mu_{k}^\alpha)^{\frac{1}{\alpha}}\leq C\Vert f\Vert_{w\widetilde{H}_\Phi^\sigma}, \end{equation} $

(3.2)

where $ C $ is only related to $ p $ and $ \alpha $.

Proof    Let $ f\in w\widetilde{H}_{\Phi}^{\sigma} $, for every $ k\in Z $, we denote $ F_{k}=\{\widetilde{\sigma}^{(p)}(f)>2^{k}\} $ and stopping time $ v_{k}=\inf\{n\in N^{2}:E_{n}^{+}\chi_{(F_{k})}>\frac{1}{2}\} $. Obviously, $ v_{k} $ is non-decreasing stopping time, and $ v_{k}\rightarrow \infty $ when $ k\rightarrow \infty $.

Let $ f_{n}^{(v_{k})}=\sum\limits_{m\leq n}\chi(v_{k}\not\gg m)d_{m}f $, so $ (f_{n}^{(v_{k})}, n\in N^{2}) $ is stopping martingale and

$ \begin{equation} f_{n}^{(v_{k+1})}-f_{n}^{(v_k)}=\sum\limits_{m\leq n}\chi(v_{k}\ll m \not\gg v_{k+1})d_{m}f. \end{equation} $

(3.3)

For $ \forall n\in N^2 $, we have

$ \begin{equation} f_n=\sum\limits_{k\in Z}\{f_{n}^{(v_{k+1})}-f_{n}^{(v_k)}\}. \end{equation} $

(3.4)

The detail of $ (3.4) $ proving process can be seen in Reference [6].

Suppose $ \mu_{k}=(\frac{p}{p-1})^{2}(2C)^{\frac{1}{p}}2^{k+1}P(v_{k}\neq \infty)^{\frac{1}{\alpha}} $, $ g^{(k)}_{n}=(\frac{1}{\mu_{k}})(f_{n}^{(v_{k+1})}-f_{n}^{(v_{k})}) $, where $ C $ is the smooth coefficient of $ p $. It is easy to verify for all $ k\in Z $, $ g^{(k)}=(g^{(k)}_{n}, n\in N^{2}) $ which is a strong martingale. When $ v_{k}\not\gg n $, $ f_{n}^{(v_{k+1})}=f_{n}^{(v_{k})} $ then $ g_{n}^{(k)}=0 $. Since B is $ p $ smooth, there is $ C>0 $ such that

$ \begin{align} E\|f_n^{(v_{k+1})}-f_n^{(v_k)}\|^p &\leq CE(\sum\limits_{m\leq n}\| d_{n}f\|^{p}\chi(v_{k}\ll m\not\gg v_{k+1}))\\ &\leq C\times2\times2^{p(k+1)}P(v_k\neq\infty). \end{align} $

(3.5)

Since the $ {v_{k}\ll m} $ is $ \mathcal{F}_m^- $ measurable, obtained by (3.5) and Lemma 2.1:

$ \begin{align} E(g^{(k)^*})^p &\leq (\frac{p}{(p-1)})^{2p}\sup\limits_{n\in N^{2}}\|g^{(k)}\|_p^p\\ &\leq\sup\limits_{n\in N^{2}}(\frac{p}{(p-1)})^{2p}\frac{C\times2\times2^{p(k+1)}}{\mu^p} P(v_k\neq\infty)\\ &=P(v_k\neq\infty)^{1-\frac{p}{\alpha}}, \end{align} $

thus

$ \begin{equation} \|g^{(k)^*}\|_p\leq P(v_k\neq\infty)^{\frac{1}{p}-\frac{1}{\alpha}}<\infty.\nonumber\ \end{equation} $

Because B has RN property, there exists $ g^{(k)}\in L_p $ such that $ E_ng^{(k)}=g_n^{(k)} $. From the definition of strong atom we know that $ g^{(k)} $ is a strong $ (\alpha, p) $ atom, which proved (3.1).

For any $ k\in Z $, we have

$ \begin{align} (\sum\limits_{k\in Z}\mu_k^\alpha)^{\frac{1}{\alpha}}&\leq c \bigg(\sum\limits_{k\in Z}2^{k\alpha}P(\nu_k\neq\infty)\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(\sum\limits_{kı Z}2^{k\alpha}P(\sup\limits_{n\in N^{2}}{E_n(\chi_{(F_k)})}>\frac{1}{2})\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(\sum\limits_{kı Z}2^{k\alpha}E(\sup\limits_{n\in N^{2}}{E_n(\chi_{(F_k)})^p})\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(\sum\limits_{kı Z}2^{k\alpha}P(F_k)\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(\sum\limits_{k\in Z}2^{k\alpha}P\big(\widetilde{\sigma}^{(p)}(f)>2^k\big)\bigg)^{\frac{1}{\alpha}} \\ &\leq c\bigg(\sum\limits_{k\in Z}\int_{2^{k-1}}^{2^k}y^{\alpha-1}P(\widetilde{\sigma}^{(p)}(f)>y)dy\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(\int_0^\infty y^{\alpha-1}P\big(\widetilde{\sigma}^{(p)}(f)>y\big)dy\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(1+\int_1^\infty y^{\alpha-1}P\big(\widetilde{\sigma}^{(p)}(f)>y\big)dy\bigg)^{\frac{1}{\alpha}}. \end{align} $

(3.6)

Since the function $ \frac{\Phi(t)}{t_{q_\Phi}} $ is monotonically increasing in $ (0, \infty) $, and $ \sup\limits_{t>0}\Phi(t)P\big(\widetilde{\sigma}^{(p)}(f)>t\big)\leq\|f\|_{w\widetilde{H}_\Phi^\sigma}=1 $, then

$ \begin{align} (\sum\limits_{k\in Z}\mu_k^\alpha)^{\frac{1}{\alpha}}&\leq c\bigg(1+\int_1^\infty\frac{y^{q_\Phi}y^{\alpha-q_\Phi-1}}{\Phi(y)}\Phi(y)P\big(\widetilde{\sigma}^{(p)}(f)>y\big)dy\bigg)^{\frac{1}{\alpha}}\\ &\leq c\bigg(1+\frac{1}{\Phi(1)}\int_1^\infty y^{\alpha-q_\Phi-1}dy\bigg)^{\frac{1}{\alpha}}\\ &=c\bigg(1+\frac{1}{q_\Phi-\alpha}\Phi(1)\bigg)^{\frac{1}{\alpha}}. \end{align} $

(3.7)

Denote by $ C $ the last item, then

$ \begin{align} (\sum\limits_{k\in Z}\mu_{k}^\alpha)^{\frac{1}{\alpha}}\leq C\Vert f\Vert_{w\widetilde{H}_\Phi^\sigma}, \end{align} $

where $ C $ is a constant independent of $ f $. Thus Theorem 3.1 is proved.

Theorem 3.2    Let B be a Banach space, $ 1<p\leq \min\{ q_\Phi, 2\} $, $ \Phi $ satisfies the condition of $ \Delta_2 $, the statements are equivalent:

(1) B is $ p $-uniform smooth;

(2) If $ 0<\alpha\leq1 $, then for any two-parameter B-valued strong martingale $ f=(f_n, n\in N^2)\in{w\widetilde{H}_{\Phi}^{\sigma}} $, there is a strong $ (\alpha, p) $ atom $ (g^{(k)}, k\in Z) $ and a non-negative real number sequence $ \mu=(\mu_{k}, k\in Z)\in l_\alpha $, for all $ n\in N^2, $ $ f_n=\sum\limits_{k=-\infty}\mu_{k}E_ng^{(k)} $, a.e..

Proof   (1)$ \Rightarrow $(2) from Theorem 3.1.

(2) $ \Rightarrow $(1). Without loss of generality, we assume $ \|f\|_{w\widetilde{H}_\Phi^\sigma}=1. $ As the function $ \frac{\Phi(t)}{t_{q_\Phi}} $ is monotonically increasing in $ (0, \infty) $, and $ \sup\limits_{t>0}\Phi(t)P\big(\widetilde{\sigma}^{(p)}(f)>t\big)\leq\|f\|_{w\widetilde{H}_\Phi^\sigma}=1 $. We get from the assumption that

$ \begin{align} \|\widetilde{\sigma}^{(p)}(f)\|_p^p&\leq p\int_0^\infty t^{p-1}P(\|\widetilde{\sigma}^{(p)} f\|>t)dt\\ &\leq c\bigg(1+\int_1^\infty \frac{y^{q_\Phi}y^{p-q_\Phi-1}}{\Phi(y)}\Phi(y)P\big(\widetilde{\sigma}^{(p)}(f)>y\big)dy\bigg)\\ &\leq c\bigg(1+\frac{1}{\Phi(1)}\int_1^\infty y^{p-q_\Phi-1}dy\bigg) \|f\Vert_{w\widetilde{H}_\Phi^\sigma}\\ &=c\bigg(1+\frac{1}{q_\Phi-p}\Phi(1)\bigg) \|f\Vert_{w\widetilde{H}_\Phi^\sigma}<\infty. \end{align} $

(3.8)

If $ g $ is a strong $ (\alpha, p) $ atom, we can get

$ \begin{equation} \|g^*\|^\alpha_\alpha=E(g^{*\alpha}\chi_{\tau\neq\infty}) \leq E(g^{*p})^{\frac{\alpha}{p}}P(\tau\neq\infty)^{1-\frac{\alpha}{p}}\leq 1. \end{equation} $

(3.9)

From the hypothesis, there is a strong $ (\alpha, p) $ atom $ (g^{(k)}, k\in Z) $ and a non-negative real number sequence $ \mu=(\mu_{k}, k\in Z)\in l_\alpha $, for all $ n\in N^2 $, $ f_n=\sum\limits_{k=-\infty}^\infty\mu_{k}E_ng^{(k)} $, a.e..

Therefore, from $ 0<\alpha\leq1 $, such that

$ \begin{align} E\|f_m-f_n\|^\alpha &=\sum\limits_{k=-\infty}^{\infty}\mu_{k}^{\alpha}E\|g_m^{(k)}-g_n^{(k)}\|^{\alpha}\\ &\leq C\sum\limits_{\mid k\mid>k_{0}}\mu_{k}^{\alpha}+\sum\limits_{\mid k\mid>k_{0}}\mu_{k}^{\alpha}(E\|g_m^{(k)}-g_n^{(k)}\|)^{\alpha}. \end{align} $

(3.10)

Set $ m\rightarrow \infty, n\rightarrow \infty $ and $ k_0\rightarrow \infty $, therefore $ (f_n, n\in N^2) $ is $ L_{\alpha} $ Cauchy convergence. Thus $ f_n $ converges according in probability. From Lemma 2.1, B is $ p $-smooth.

Theorem 3.2 is proved.

Suppose $ T: X\rightarrow Y $ is the mapping, where $ X $ is the weak Orlicz strong martingale space on $ (\Omega, \mathcal {F} _n, P) $, and $ Y $ is the space of measurable function on $ (\Omega, \mathcal{F}, P) $. $ T $ is sub-linear, if

$ \begin{eqnarray} |T(f+g)|\leq|Tf|+|Tg|, \ \ \ |T(\alpha f)|=|\alpha||Tf|, a.e.. \end{eqnarray} $

(4.1)

If the $ Tf=0 $ is on the set $ {f=0} $. $ T $ is called bounded, if

$ \begin{eqnarray} \|T\|=\sup\limits_{\|f\|_X\leq1}\|Tf\|_Y<\infty. \end{eqnarray} $

(4.2)

Theorem 4.1   Let $ T:L_p\rightarrow L_p $ be a bounded sublinear operator, B be isomorphic to $ p $-smooth Banach space which satisfies for any $ (\alpha, p) $ atom $ a $,

$ \begin{eqnarray} P(|Ta|>0)\leq C_0P(\nu\neq\infty), \end{eqnarray} $

(4.3)

where $ \nu $ is the stopping time associated with $ a $, and $ C_0 $ is a constant independent of $ a $. If $ 0 <p <q_\Phi\leq p_\Phi<\infty $ and $ p\leq2 $, then there is $ C>0 $ such that

$ \begin{eqnarray} \|Tf\|_{wL_\Phi}\leq C\|f\|_{w\widetilde {H}_{\Phi}^{\sigma}}, \ \ \ \forall f=(f_n)\in{w\widetilde {H}_{\Phi}^{\sigma}(B)}. \end{eqnarray} $

(4.4)

Proof    Let $ f \in w\widetilde{H}_{\Phi}^{\sigma}(B) $ and $ \|f\|_{w\widetilde {H}_{\Phi}^{\sigma}(B)}=\|\widetilde{\sigma}^ {(p)}(f)\|_{wL_\Phi} = 1 $. From Theorem 3.1, there exists atomic decomposition of $ f $ such that satisfies (3.1) and (3.2). For any $ y>0 $, let $ j\in Z $ such that $ 2^j\leq y<2^{j+1} $, we define

$ \begin{eqnarray} g_n=\sum\limits_{k\leq j-1}\mu_ka_n^k, \ \ \ \ h_n=\sum\limits_{k\geq j}\mu_ka_n^k, \ \ \ \ n\geq0. \end{eqnarray} $

(4.5)

Considering $ h = (h_n, n\in N^2) $, we denote that each $ a_k $ satisfies $ \Phi(2^k)P(\nu_k\neq\infty) \leq c\Phi(2^k)\big(P\big(\widetilde{\sigma}^{(p)}(f)>2^k\big)\big) \leq c\sup\limits_{t>0}\Phi(t)\big(P\big(\widetilde{\sigma}^{(p)}(f)>t\big)\big) =c\|f\|_{ w\widetilde {H}_{\Phi}^{\sigma}(B)}=c. $

Condition (4.3) shows

$ \begin{align} \Phi(2^j)P(|Th|>2^j) &\leq\Phi(2^j)P(|Th|>0)\leq\Phi(2^j)P(\sum\limits_{k\geq j}\mu_k|Ta^k|>0)\\ &\leq\sum\limits_{k\geq j}\Phi(2^j)P(|Ta^k|>0)\leq C_0\sum\limits_{k\geq j}\Phi(2^j)P(\nu_k\neq\infty)\\ &=C_0\sum\limits_{k\geq j}\frac{\Phi(2^j)}{\Phi(2^k)}\Phi(2^k)P(\nu_k\neq\infty)\leq C_0\sum\limits_{k\geq j}\frac{\Phi(2^k2^{-(k-j)})}{\Phi(2^k)}\\ &\leq C_0\sum\limits_{k\geq j}\frac{\Phi(2^k)2^{-(k-j)}}{\Phi(2^k)}\leq{C_0}\sum\limits_{k^{\prime}=0}^{\infty}2^{-k^{\prime}}=2C_0. \end{align} $

The basic thing is $ \Phi(at)\leq a\Phi(t) $, $ \forall t>0 $, $ 0<a\leq1 $, which derive from the convexity of $ \Phi $. For $ g = (g_n, n\in N^2) $, we know $ \|g^*\|_p\leq C\|\widetilde\sigma^{(p)}(g)\|_p $ $ (1\leq p\leq2) $ from Lemma 2.2. Since $ T:L_p\rightarrow L_p $ is a bounded sublinear operator, then

$ \begin{align} \|Tg\|_{L_p}&\leq C\|g\|_{L_p}\leq C\|g^*\|_{L_p}\leq C^{\prime}\|\widetilde{\sigma}^{(p)}(g)\|_{L_p}\\ &\leq C^{\prime}\|\sum\limits_{k\leq j-1}\mu_k\widetilde{\sigma}^{(p)}(a^k)\|_{L_p}\\ &\leq C^{\prime}\sum\limits_{k\leq j-1}\mu_k\|\widetilde{\sigma}^{(p)}(a^k)\|_{L_p}\\ &\leq C^{\prime}\sum\limits_{k\leq j-1}\mu_k(E\widetilde{\sigma}^{(p)}(a^k)^p\chi_{\{\nu_{k\neq\infty}\}})^{\frac{1}{p}}\\ &\leq C^{{\prime}{\prime}}\sum\limits_{k\leq j-1}2^kP(\nu_k\neq \infty)^\frac{1}{p} \end{align} $

and

$ P(|Tg|>2^j)\leq\frac{1}{2^{pj}}\|Tg\|_{L_p}^p\leq C^{\prime p}\bigg(\frac{1}{2^j}\sum\limits_{k\leq j-1}2^kP(\nu_k\neq\infty)^\frac{1}{p}\bigg)^p. $

Since the function $ \frac{\Phi(t)}{t_{p_\Phi}} $ is monotonically decreasing, therefore $ \frac{\Phi{(2^j)}}{\Phi{{(2^k)}}}\leq\frac{2^{jp_\Phi}}{2^{kp_\Phi}}=2^{(j-k)p_\Phi} $, if $ k\leq j $, such that

$ \begin{align} \Phi(2^j)P(|Tg|>2^j) &\leq C^{{\prime}{\prime}p}\bigg(\sum\limits_{k\leq j-1}2^{k-j}\bigg(\frac{\Phi(2^j)}{\Phi(2^k)}\cdot\Phi(2^k)P(\nu_k\neq\infty)\bigg)^{\frac{1}{p}} \bigg)^p\\ &\leq C^{{\prime}{\prime}p}\bigg(\sum\limits_{k\leq j-1}2^{k-j}\bigg(\frac{\Phi(2^j)}{\Phi(2^k)}\bigg)^{\frac{1}{p}} \bigg)^p\\ &\leq C^{{\prime}{\prime}p}\bigg(\sum\limits_{k\leq j-1}2^{k-j}2^{(j-k)\frac{p_\Phi}{p}}\bigg)^p\\ &\leq C^{{\prime}{\prime}p}\bigg(\sum\limits_{k^{\prime}\geq1}2^{-k^{\prime}(1-\frac{p_\Phi}{p})}\bigg)^p\\ &\leq C^{{\prime}{\prime}{\prime}p}2^{p_\Phi-p}, \end{align} $

which $ C^{{\prime}{\prime}{\prime}} $ is a constant independent of $ f $, so

$ \begin{align} \sup\limits_{y>0}\Phi(2y)P(|Tf|>2y) &\leq\sup\limits_{y>0}\Phi(2y)\bigg(P(|Tg|>y)+P(|Th|>y)\bigg)\\ &\leq C\sup\limits_{j\in Z}\Phi(2^j)\bigg(P(|Tg|>2^j)+P(|Th|>2^j)\bigg)\\ &\leq C(C^{{\prime}{\prime}{\prime}p}2^{p_\Phi-p}+2C_0):=\tilde{C}. \end{align} $

We assume $ \tilde{C}\geq 1 $ with loss of generality, then

$ \sup\limits_{y>0}\Phi(\frac{2y}{\tilde{C}})P(|Tf|>2y)\leq\frac{1}{\tilde{C}}\sup\limits_{y>0}\Phi(2y)(P(|Tf|>2y)\leq1. $

Finally,

$ \begin{align} \|Tf\|_{wL_\Phi}\leq \tilde{C}=\tilde{C}\|f\|_{w\widetilde {H}_{\Phi}^{\sigma}}. \end{align} $

Thus, we prove $ (4.1) $.