It's well known that variable exponent Lebesgue spaces have been got more and more attention in modern analysis and functional space theory. In particular, Fan and Zhao [1] investigated various properties of variable exponent Lebesgue spaces and Sobolev spaces. Diening [2, 3] and Cruz-Uribe [4] proved the boundedness of Hardy-Littlewood maximal operator on variable exponent Lebesgue function spaces $ L^{p(\cdot)}(R^n) $ under the conditions that the exponent $ p(\cdot) $ satisfies so called log-Hölder continuity and decay restriction. Many other authors studied its applications to harmonic analysis and some other subjects.

The situation of martingale spaces is different from function spaces. For example, the good-$ \lambda $ inequality method used in classical martingale theory can not be used in variable exponent case. However, recently, variable exponent martingale spaces have been paid more attention too. Aoyama [5] proved that, if $ p(\cdot) $ is $ \mathcal{F}_0 $-measurable, then there exists a positive constant $ c $ such that $ \|M(f)\|_{L_{p(\cdot)}}\leq c\|f\|_{ L_{p(\cdot)} } $ for $ f\in L_{p(\cdot)} $. Nakai and Sadasue [6] pointed out that the inverse is not true, namely, there exists a variable exponent $ p(\cdot) $ such that $ p(\cdot) $ is not $ \mathcal{F}_0 $-measurable, and the above inequality holds, under the assumption that every $ \sigma $-algebra $ \mathcal{F}_n $ is generated by countable atoms. Zhiwei Hao [7] established an atomic decomposition of a predictable martingale Hardy space with variable exponents defined on probability spaces. Motivated by them, we research dyadic martingale Hardy space with variable exponents.

In this paper the unit interval $ [0, 1) $ and Lebesgue measure $ P $ are to be considered. Throughout this paper, $ \textbf{Z} $, $ \textbf{N} $ denote the integer set and nonnegative integer set. By a dyadic interval we mean one of the form $ [k2^{-n}, (k+1)2^{ -n}) $ for some $ k\in \textbf{N}, 0\leq k<2^ n $. Given $ n \in \textbf{ N} $ and $ x \in [0, 1) $, let $ I_n(x) $ denote the dyadic interval of length $ 2^{-n} $ which contains $ x $. The $ \sigma $-algebra generated by the dyadic intervals $ \{I_n(x):x \in [0, 1)\} $ will be denoted by $ \mathcal{F}_n, $ more precisely,

$ \mathcal{F}_n = \sigma\{[k2^{-n}, (k+1)2^{-n}) : 0\leq k < 2^n\}. $

Obviously, $ (\mathcal{F}_n) $ is regular. Define $ \mathcal{F} = \sigma(\cup_n\mathcal{F}_n) $ and denote the set of dyadic intervals by $ A(\mathcal{F}_n) $ and write $ A = \cup_nA(\mathcal{F}_n) $. The conditional expectation operators relative to $ \mathcal {F}_n $ are denoted by $ E_n. $ For a complex valued martingale $ f = (f_n)_{n\geq 0} $, denote $ df_i = f_i-f_{i-1} $ (with convention $ df_{-1} = 0 $) and

$ \begin{eqnarray*} M_n(f)& = &\sup\limits_{0\leq i\leq n}|f_i|, \quad M(f) = \sup\limits_{n\geq 0}|f_n|, \\ s_n(f)& = &(\sum\limits_{i = 1}^{n}E_{i-1}|df_i|^2)^{1/2}, s(f) = (\sum\limits_{i = 1}^{\infty}E_{i-1}|df_i|^2)^{1/2}. \end{eqnarray*} $

Remark 2.1 (see [8]) If $ (\mathcal{F}_n) $ is regular, then for all nonnegative adapted processes $ \gamma = (\gamma_n) $ and $ \lambda\geq \|\gamma_0\|_\infty $, there exist a constant $ c > 0 $ and a stopping time $ \tau_\lambda $ such that

$ \begin{eqnarray*} &\{M(\gamma)>\lambda\}\subset \{\tau_\lambda<\infty\}, \; \; P(\tau_\lambda<\infty)\leq c P(M(\gamma)>\lambda), \\ &{\sup\nolimits _{n \le {t_\lambda }}}\gamma_n = M_{\tau_\lambda}(\gamma)\leq \lambda, \; \; \lambda_2\geq \lambda_1\geq \|\gamma_0\|_\infty\Rightarrow \tau_{\lambda_1}\leq \tau_{\lambda_2}. \end{eqnarray*} $

Let $ p(\cdot):[0, 1)\rightarrow(0, \infty) $ be an $ \mathcal{F} $-measurable function, we define $ p_B^- = \mbox{ess}\inf \{p(x):x\in B\}, p_B^+ = \mbox{ess}\sup \{p(x):x\in B\}, \ B\subset [0, 1). $ We use the abbreviations $ p^ + = p_{[0, 1)}^ + $ and $ p^ - = p_{[0, 1)}^- $.

We say that $ p $ is log-Hölder continuous if

$ \begin{equation} |p(x)-p(y)| \leq \frac{ c}{ -\log d(x, y)} , \end{equation} $

(2.1)

when $ d(x, y) \leq 1/2 $.

The Lebesgue space with variable exponent $ p(\cdot) $ denoted by $ L_{p(\cdot)} $ is defined as the set of all $ \mathcal{F} $-measurable functions $ f $ satisfying

$ \|f\|_{p(\cdot)} = \inf\biggl\{\lambda>0:\rho_{p(\cdot)}(|f(x)|/\lambda)\leq 1\biggl\}<\infty, $

where $ \rho_{p(\cdot)}(f) = \int_0^1 |f(x)|^{p(x)}dP. $

If $ f = (f_n) $ is a martingale, we define $ \|f\|_{p(\cdot)} = \sup\limits_{n\geq 1}\|f_n\|_{p(\cdot)} $.

Remark 2.2 (see [7]) If $ p $ is log-Hölder continuous, then we have

$ P(B)^{1/p^-_{B}}\approx P(B)^{1/p^+_{B}}\approx P(B)^{1/p(x)}\approx \|\chi_{B}\|_{p(\cdot)}, \quad x\in B\subset [0, 1). $

The weak Lebesgue space with variable exponent $ p(\cdot) $ denoted by $ wL_{p(\cdot)} $ is defined as the set of all $ \mathcal{F} $-measurable functions $ f $ satisfying

$ \|f\|_{wL_{p(\cdot)}}: = \sup\limits_{\lambda>0} \lambda\|\chi_{\{|f|>\lambda\}}\|_{p(\cdot)}<\infty. $

Then we define the strong and weak variable exponent dyadic martingale Hardy spaces as follows

$ \begin{eqnarray*} H_{p(\cdot)}& = &\biggl\{f = (f_n):M(f)\in L_{p(\cdot)}\biggl\}, \|f\|_{H_{p(\cdot)}} = \|M(f)\|_{p(\cdot)}, \\ wH^s_{p(\cdot)}& = &\biggl\{f = (f_n):s(f)\in wL_{p(\cdot)}\biggl\}, \|f\|_{wH^{s}_{p(\cdot)}} = \|s(f)\|_{wLp(\cdot)}. \end{eqnarray*} $

We always denote by $ c $ some positive constant, but its value may be different in each appearance.

Definition 3.1 A measurable $ a $ is called a weak- atom if there exists a stopping time $ \nu $ such that

(1) $ E_n(a) = 0, n\geq \nu $,

(2) $ \|s(a)\|_\infty<\infty $.

Theorem 3.2 Suppose that $ p $ is log-Hölder continuous and $ 0<p^-<p^+\leq 1 $. For any $ f = (f_n)\in wH^s_{p(\cdot)} $, there exist $ (a^k) $ of weak atoms with the corresponding stopping times $ \nu_k $ and $ (u_k) $ of nonnegative real numbers such that

$ \begin{eqnarray} &&f_n = \sum\limits_{k\in \textbf{Z}}u_kE_na^k \; \; \; \text{a.e.}, \end{eqnarray} $

(3.1)

$ \begin{eqnarray} &&s(a^k)\leq 3\cdot 2^k, \end{eqnarray} $

(3.2)

$ \begin{eqnarray} &&\|f\|_{wH^s_{p(x)}}\sim \inf \sup\limits_ {k\in\textbf{Z}}2^k\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)}, \end{eqnarray} $

(3.3)

where the infimum is taken over all preceding decompositions of $ f. $

Proof Assume that $ f = (f_n)\in wH^s_{p(\cdot)} $. Let us define the stopping times $ \nu_k: = \inf\{n\in \textbf{N}:s_{n+1}(f)>2^k\}. $ Consequently, $ f_n $ can be written as

$ \begin{equation} f^\nu_n = \sum\limits_{m = 0}^{n}\chi_{\{\nu\geq m\}}(f_m-f_{m-1}). \end{equation} $

(3.4)

Now let $ a^k_n: = f_n^{\nu_{k+1}}-f^{\nu_k}_n. $ Thus $ s(a^k)\leq s(f^{\nu_{k+1}})+s(f^{\nu_k})\leq 3\cdot 2^k<\infty. $ Then there exists $ a^k $ such that $ a_n^k = E_n(a^k) $. It is clear that $ a^k $ is really a weak atoms for $ k\in \textbf{Z}. $

Since $ \{\nu_k\neq \infty\} = \{s(f)>2^k\} $, by the definition we have

$ \begin{equation} 2^k\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)} = 2^k\|\chi_{\{s(f)>2^k\}}\|_{p(\cdot)}\leq \|s(f)\|_{wL_{p(\cdot)}}. \end{equation} $

(3.5)

Thus

$ \begin{equation} \sup\limits_{k\in \textbf{Z}}2^k\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)}\leq \|s(f)\|_{wL_{p(\cdot)}}. \end{equation} $

(3.6)

Conversely, let $ D: = \sup\limits_{k\in\textbf{Z}}2^k\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)} $. For a fixed $ y>0 $ choose $ j\in \textbf{Z} $ such that $ 2^j\leq y<2^{j+1} $. Then

$ f_n = \sum\limits_{k = -\infty}^{j-1} a_n^k + \sum\limits_{k = j}^{\infty}: = g_n+h_n $

and $ p(\cdot)/p^-\geq 1 $ implies that $ s(f)\leq s(g)+s(h) $ and

$ \begin{eqnarray} \|\chi_{\{s(f)>6y\}}\|_{p(\cdot)} &\leq &c(\|\chi_{\{s(g)>3y\}}\|_{p(\cdot)/p^-}+\|\chi_{\{s(h)>3y\}}\|_{p(\cdot)/p^-})^{1/p^-}. \end{eqnarray} $

(3.7)

Since $ s(a^k)\leq 3\cdot 2^k $, thus we get

$ \begin{equation} s(g)\leq \sum\limits_{k = -\infty}^{j-1}s(a^k)\leq 3\sum\limits_{k = -\infty}^{j-1}2^k\leq 3\cdot2^j \end{equation} $

(3.8)

and so $ \{s(g)>3y\}\subset \{s(g)>3\cdot 2^j\} = \emptyset $.

The inequality $ s(h)\leq \sum\limits_{k = j}^{\infty}s(a^k) $ implies that $ \{s(h)\neq 0\}\subset \cup_{k = j}^\infty\{\nu_k\neq \infty\} $. Consequently,

$ \begin{eqnarray} \|\chi_{\{s(f)>6y\}}\|_{p(\cdot)}&\leq& c(\|\chi_{\{s(g)>3y\}}\|_{p(\cdot)/p^-}+\|\chi_{\{s(h)>3y\}}\|_{p(\cdot)/p^-})^{1/p^-}\\ &\leq& c\|\chi_{\{s(h)>0\}}\|_{p(\cdot)/p^-}^{p^-}\ \leq c\sum\limits_{k = j}^\infty\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)}\\ & \leq& c\sum\limits_{k = j}^\infty D2^{-k} = 2^{-j}D\leq cD/y, \end{eqnarray} $

(3.9)

which implies

$ \begin{equation} \|f\|_{wH^s_{p(\cdot)}} = \sup\limits_{y>0}6y \|\chi_{\{s(f)>6y\}}\|_{p(\cdot)}\leq cD = c\sup\limits_{k\in\textbf{Z}}2^k\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)}. \end{equation} $

(3.10)

Thus we complete the proof.

Theorem 3.3 Suppose that $ p $ is log-Hölder continuous, $ 1/2<p^-\leq p^+\leq 1 $ and suppose that sublinear $ T $ is bounded from $ L_2 $ to $ L_2. $ If

$ \begin{equation} P(Ta>0)\leq c P( I) \end{equation} $

(3.11)

for all weak atom $ a $ supported on the interval $ I $, then

$ \begin{equation} \|Tf\|_{wL_{p(\cdot)}}\leq c\|f\|_{wH^s_{p(\cdot)}} \quad(f\in wH^s_{p(\cdot)}). \end{equation} $

(3.12)

Proof We may suppose $ \|f\|_{wH^s_{p(\cdot)}} = 1. $ Taking the atomic decomposition and the martingales $ g $ and $ h $ given in the proof of Theorem 3.2 we get that for any given $ y>0 $,

$ Tf\leq Tg+Th\quad \text{ and}\quad \chi_{\{Tf>2y\}}\leq \chi_{\{Tg>y\}}+\chi_{\{ Th>y\}}. $

(Ⅰ) If $ 0<y\leq 1 $, we choose integer $ j $ such that $ 2^j\leq y< 2^{j+1}. $ We know that

$ \begin{eqnarray} \|g\|_2&\leq& \sum\limits_{k = -\infty}^{j-1}\|a^k\|_2\leq \sum\limits_{k = -\infty}^{j-1}\|M(a^k)\|_2 = \sum\limits_{k = -\infty}^{j-1}\|M(a^k)\chi_{\{\nu_k\neq \infty\}}\|_2\\ &\leq& \sum\limits_{k = -\infty}^{j-1}2^kP(\nu_k\neq \infty)^{1/2}. \end{eqnarray} $

(3.13)

Since

$ \|\chi_B\|_{p(\cdot)}\approx P(B)^{1/p^+_B}\approx P(B)^{1/p^-_B}\approx P(B)^{1/p(\omega)}, \omega\in B $

for $ B = \{\nu_k\neq \infty\} $, we have

$ \begin{eqnarray} \|g\|_2 &\leq&\sum\limits_{k = -\infty}^{j-1}2^kP(B)^{1/2} = \sum\limits_{k = -\infty}^{j-1}2^k(P(\nu_k\neq \infty)^{1/p^+_{B}})^{p^+_{B}/2}\\ &\leq&c\sum\limits_{k = -\infty}^{j-1}2^k(\|\chi_{B}\|_{p(\cdot)})^{p^+_{B}/2} \leq c\sum\limits_{k = -\infty}^{j-1}2^{k(1-p^+_B/2)}\|f\|_{wH_{p(\cdot)}}^{p^+_{B}/2}\\ &\leq&cy^{(1-p^+_B/2)}\|f\|_{wH_{p(\cdot)}}^{p^+_{B}/2} \leq cy^{1/2}. \end{eqnarray} $

(3.14)

Define $ B_1 = \{Tg>y\} $, thus we have

$ \begin{eqnarray} \|\chi_{\{Tg>y\}}\|_{p(\cdot)}&\leq& c P(Tg>y)^{1/p^-_{B_1}}\leq c\bigl(y^{-1}E(Tg)\bigl)^{1/p^-_{B_1}}\leq c\bigl(y^{-1}\|Tg\|_2\bigl)^{1/p^-_{B_1}}\\ &\leq& c\bigl(y^{-1}\|g\|_2\bigl)^{1/p^-_{B_1}} \leq c\bigl(y^{-1/2}\bigl)^{1/p^-_{B_1}} \leq cy^{-1}. \end{eqnarray} $

(Ⅱ) If $ y>1. $ Thus

$ \begin{eqnarray} \|g\|_2 &\leq&\sum\limits_{k = -\infty}^{j-1}2^kP(B)^{1/2} = \sum\limits_{k = -\infty}^{j-1}2^k(P(\nu_k\neq \infty)^{1/p^-_{B}})^{p^-_{B}/2}\\ &\leq&c \sum\limits_{k = -\infty}^{j-1}2^k(\|\chi_{B}\|_{p(\cdot)})^{p^-_{B}/2} \leq c\sum\limits_{k = -\infty}^{j-1}2^{k(1-p^-_B/2)}\|f\|_{wH_{p(\cdot)}}^{p^-_{B}/2}\\ &\leq&cy^{(1-p^-_B/2)}\|f\|_{wH_{p(\cdot)}}^{p^-_{B}/2} \leq cy^{3/4}. \end{eqnarray} $

(3.15)

We also have

$ \begin{eqnarray} \|\chi_{\{Tg>y\}}\|_{p(\cdot)}&\leq&c P(Tg>y)^{1/p^-_{B_1}}\leq c\bigl(y^{-2}E(Tg)^2\bigl)^{1/p^-_{B_1}}\leq c\bigl(y^{-2}\|g\|_2^2\bigl)^{1/p^-_{B_1}}\\ &\leq& c\bigl(y^{-2}y^{3/2}\bigl)^{1/p^-_{B_1}} \leq c\bigl(y^{-1/2}\bigl)^{1/p^-_{B_1}} \leq cy^{-1}. \end{eqnarray} $

Combining (3.15) and (3.17) we get

$ \begin{equation} \|Tg\|_{wL_{p(\cdot)}} = \sup\limits_{y>0}y\|\chi_{\{Tg>y\}}\|_{p(\cdot)}\leq c\|f\|_{wH^s_{p(\cdot)}}. \end{equation} $

(3.16)

On the other hand, let $ B_2 = \{Ta^{k}>0\} $, $ I_k $ is support of $ a^k $, we have

$ \begin{eqnarray*} \|\chi_{\{Th>y\}}\|_{p(\cdot)}&\leq& \|\chi_{\{Th>0\}}\|_{p(\cdot)} = \|\chi_{\{Th>0\}}\|^{p^-}_{p(\cdot)/p^-}\\ &\leq& c (\sum\limits_{k = j}^\infty\|\chi_{\{Ta^{k}>0\}}\|_{p(\cdot)/p^-})^{p^-} \leq c \sum\limits_{k = j}^\infty\|\chi_{\{Ta^{k}>0\}}\|_{p(\cdot)}\\ &\leq& c \sum\limits_{k = j}^\infty P(Ta^{k}>0)^{1/p^-_{B_2}} \leq c \sum\limits_{k = j}^\infty P(I_k)^{1/p^-_{B_2}} \leq c \sum\limits_{k = j}^\infty (P(I_k)^{1/p_I^+})^{p^+_I/p^-_{B_2}}\\ &\leq& c \sum\limits_{k = j}^\infty (P(I_k)^{1/p_I^+})^{2} \leq c \sum\limits_{k = j}^\infty P(I_k)^{1/p_I^+} \leq c\sum\limits_{k = j}^\infty \|\chi_{I_k }\|_{p(\cdot)}\\ &\leq& \sum\limits_{k = j}^\infty\|\chi_{\{\nu_k\neq \infty\}}\|_{p(\cdot)} \leq \sum\limits_{k = j}^\infty D2^{-k} = 2^{-j}D\\ &\leq& cD/y, \end{eqnarray*} $

which implies

$ \begin{equation} \|Tg\|_{wL_{p(\cdot)}}\leq cD\leq c\|f\|_{wH^s_{p(\cdot)}}. \end{equation} $

(3.17)

By (3.18) and (3.19), we have

$ \|Tf\|_{wL_{p(\cdot)}}\leq c(\|Tg\|_{wL_{p(\cdot)}}+\|Th\|_{wL_{p(\cdot)}})\leq c\|f\|_{wH^s_{p(\cdot)}}. $

Thus we complete the proof of Theorem 3.3.

First we introduce the Walsh system. Every point $ x \in [0, 1) $ can be written in the following way

$ x = \sum\limits_{k = 0}^\infty \frac{x_k}{2^{k+1}} , 0\leq x_k <2, x_k \in \bf N. $

In case there are two different forms, we choose the one for which $ \lim\limits_{k\rightarrow \infty} x_k = 0. $

For $ x, y\in [0, 1) $ we define$ x\oplus y = \sum\limits_{k = 0}^\infty \frac{|x_k-y_k|}{2^{k+1}}: = d(x, y), $ which is also called dyadic distance.

The functions $ r_n(x) : = \exp(\pi x_n \sqrt{-1}) \; (n \in \textbf{N}) $ are called Rademacher functions.

The product system generated by these functions is the Walsh system: $ \omega_n(x) : = \prod\limits_{k = 0}^\infty r_k(x)^{x_R}, $ where $ n = \sum\limits_{k = 0}^\infty n_k2^ k, 0\leq n_k < 2 $ and $ n_k\in \textbf{ N}. $

If $ f \in L_1[0, 1) $, then the number $ \hat{f}(n): = E(f\omega_n) $ is said to be the $ n $-th Walsh-Fourier coefficient of $ f $.

Denote by $ s_nf $ the n-th partial sum of the Walsh-Fourier series of a martingale $ f $, namely,

$ s_nf: = \sum\limits_{k = 0}^{n-1}\hat{f}(k)\omega_k. $

Recall that the Walsh-Dirichlet kernels $ D_n: = \sum\limits_{k = 0}^{n-1}\omega_k $ satisfy

$ \begin{equation} D_{2^n}(x) = \begin{cases} 0& \text{if $x\in [2^{-n}, 1)$ }, \\ 2^n& \text{if $x\in [0, 2^{-n})$}. \end{cases} \end{equation} $

(4.1)

Moreover, for any measurable function $ f $, the sequence $ \{{f*D_{2^n} = s_{2^n}f = f_n}\} $ is a martingale sequence.

The Walsh-Fejér kernels are defined with $ K_n: = \frac{1}{n}\sum\limits_{k = 1}^{n}D_k $ and can be estimated by

$ \begin{equation} |K_n(x)|\leq \sum\limits_{j = 0}^{N-1}2^{j-N}\sum\limits_{i = j}^{N-1}(D_{2^i}(x)+D_{2^{i}}(x\oplus2^{-j-1})), \end{equation} $

(4.2)

where $ x \in [0, 1), n, N\in\textbf{ N}, 2^ {N-1} \leq n < 2^ N $(see [9]).

Moreover,

$ K_{2^n}(x) = \frac{1}{2}\biggl(2^{-n}D_{2^n}(x)+ \sum\limits_{j = 0}^{n-1}2^{j-n}D_{2^{n}}(x\oplus 2^{-j-1})\biggl). $

For $ n \in \textbf{ N} $ and a martingale $ f $, the Cesàro mean of order $ n $ of the Walsh-Fourier series of $ f $ is given by $ \sigma_nf: = \frac{1}{n}\sum\limits_{k = 1}^ns_nf. $

It is simple to show that in case $ f\in L_1[0, 1) $ we have

$ \sigma_nf(x) = f*K_n = \int f(t)K_n(x\oplus t)dt. $

Define the maximal operator $ \sigma^*f = \sup\limits_n|\sigma_nf|. $

Definition 4.1 A pair $ (a, B) $ of measurable function $ a $ and $ B\in A(F_n) $ is called a $ p(\cdot) $- atom if (1) $ E_n(a) = 0 $, (2) $ \|M(a)\|_\infty\leq \|\chi_{B}\|^{-1}_{ ^{p(\cdot)} } $, (3) $ \{a\neq 0\}\subset B $.

Lemma 4.2 (see [10]) Suppose that $ p $ is log-Hölder continuous and $ 0<p^-<p^+\leq 1 $. For any $ f = (f_n)\in H_{p(\cdot)} $, there exist $ (a^B, B)_{B\in A} $ of $ p(\cdot) $-atoms and $ (u_B)_{B\in A } $ of nonnegative real numbers such that

$ f_n = \sum\limits_{B\in A}u_BE_na^B \text { a.e.}, \quad \text{and}\quad \inf \biggl\| \sum\limits_{B\in A} (\frac{ u_B\chi_{B }} {\| \chi_{B} \|_{{p(\cdot)} } } ) \biggl \|_{ p(\cdot) }\leq c \|f\|_{ H_{p(\cdot)} }. $

Lemma 4.3 (see [10]) Suppose that the operator $ T $ is sublinear and for each $ p_0 < p(\cdot) \leq 1 $, there exists a constant $ c> 0 $ such that

$ \begin{equation} \int_{[0, 1)\backslash B}|Ta|^{p(\cdot)}d\mu\leq c \end{equation} $

(4.3)

for every $ p(\cdot) $-atom $ (a, B) $. If $ T $ is bounded from $ L_\infty $ to $ L_\infty $, then $ \|Tf\|_{p(\cdot)}\leq c\|f\|_{H_{p(\cdot)}}\; (f\in H_{p(\cdot)}) $

Theorem 4.4 Suppose that $ p $ is log–Hölder continuous and $ 1/2<p^-\leq p^+\leq 1 $. Then for any $ f\in H_p(\cdot) $, we have $ \|\sigma^*f\|_{p(\cdot)}\leq c\|f\|_{H_{p(\cdot)}}. $

Proof By Lemma 4.3, Theorem 4.4 will be complete if we show that the operator $ \sigma^* $ satisfies (4.3) and is bounded from $ L_\infty $ to $ L_\infty. $

Obviously,

$ \begin{eqnarray} |\sigma_nf(x)|& = &|\int_0^1f(t)K_n(x\oplus t)dt|\leq\int_0^1|f(t)(K_n(x\oplus t))|dt\\ &\leq&\|f\|_\infty\int_0^1|K_n(x\oplus t)|dt. \end{eqnarray} $

(4.4)

Since $ \|D_{2^n}\|_1 = 1 $ and (4.2), we can show that $ \|K_n\|_1\leq c $ for all $ n\in \textbf{N}, $ which verifies that $ \sigma^* $ is bounded on $ L_\infty $.

Let $ a \neq 1 $ be an arbitrary $ p(\cdot) $-atom with support $ B $ and $ P(B) = 2^ {-\tau}. $ Without loss of generality, we may suppose that $ B = [0, 2^{-\tau}). $

For $ k<2^\tau $, $ \omega_k $ is constant on $ B $ and so $ \sigma_na(x) = \int_0^1a(t)d\mu = 0. $ Therefore, we may suppose that $ n>2^\tau $.

If $ j\geq \tau $ and $ x\notin B $, then $ x\oplus 2^{-j-1}\notin B. $ Consequently, for $ x\notin B $ and $ i\geq j\geq \tau $ we have

$ \begin{equation} \int_B|a(t)| D_{2^i}(x\oplus t)dt = \int_B|a(t)|D_{2^{i}}(x\oplus t\oplus 2^{-j-1})dt = 0. \end{equation} $

(4.5)

Moreover for $ 2^{N-1}\leq n< 2^N $ and $ n>2^\tau $ (which implies $ N-1\geq \tau), $

$ \begin{eqnarray} |\sigma_na(x)|&\leq&\int_B|a(t)K_n(x\oplus t)|dt\leq |a|*|K_n|(x) \end{eqnarray} $

(4.6)

$ \begin{eqnarray} &\leq& \sum\limits_{j = 0}^{\tau-1}2^{j-N}\sum\limits_{i = j}^{N-1}\int_0^1|a(t)| (D_{2^i}(x\oplus t)+D_{2^{i}}(x\oplus t\oplus 2^{-j-1})) dt. \end{eqnarray} $

(4.7)

Since $ x\in[0, 1)\backslash B $, we have

$ \begin{eqnarray*} &&\int_{B}D_{2^i}(x\oplus t \oplus2^{-j-1})dt = 2^{i-\tau}\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x)\quad \mbox{if }\quad j\leq i\leq \tau-1, \\ &&\int_{B}D_{2^i}(x\oplus t )dt = 2^{i-\tau}\chi_{[2^{-\tau}, 2^{-i})}(x)\quad \mbox{if }\quad i\in \textbf{N}, \\ &&\int_{B}D_{2^i}(x\oplus t \oplus2^{-j-1})dt = \chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-\tau})}(x)\quad \mbox{if} \quad i\geq \tau. \end{eqnarray*} $

By the definition of an atom, for $ n\geq 2^\tau, 2^{N-1}\leq n<2^N, $ we have

$ \begin{eqnarray*} \sup\limits_{}|a|*|K_n|(x)&\leq & \sum\limits_{j = 0}^{\tau-1}2^{j-N}\sum\limits_{i = j}^{N-1}\int_B|a(t)| (D_{2^i}(x\oplus t)+D_{2^{i}}(x\oplus t\oplus 2^{-j-1})) dt\\ &\leq&\|a\|_\infty\sum\limits_{j = 0}^{\tau-1}2^{j-\tau}\sum\limits_{i = j}^{\tau-1}(2^{i-\tau}\chi_{[2^{-\tau}, 2^{-i})}(x)+2^{i-\tau}\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x))\\ &&+\|a\|_\infty \sum\limits_{j = 0}^{\tau-1}2^{j-N}\sum\limits_{i = \tau}^{N-1}2^{i-\tau}\chi_{[2^{-\tau}, 2^{-i})}(x)+\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x) . \end{eqnarray*} $

To verify (4.3) we have to investigate the integral of $ (\sup_n|a|*|K_n|(x))^{p(\cdot)} $ over $ [0, 1)\backslash B $. Integrating over $ [0, 1)\backslash B, $ we obtain

$ \begin{eqnarray*} &&\int_{[0, 1)\backslash B} (\sup\limits_{n\geq 2^\tau}|a|*K_n(x))^{p(\cdot)}dx\\ &\leq&c\int_{[0, 1)\backslash B} ( \|a\|_\infty\sum\limits_{j = 0}^{\tau-1}2^{j-\tau}\sum\limits_{i = j}^{\tau-1}(2^{i-\tau}\chi_{[2^{-\tau}, 2^{-i})}(x)+2^{i-\tau}\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x)))^{p(\cdot)}dx\\ &&+c \int_{[0, 1)\backslash B} ( \|a\|_\infty \sum\limits_{j = 0}^{\tau-1}2^{j-N}\sum\limits_{i = \tau}^{N-1}2^{i-\tau}\chi_{[2^{-\tau}, 2^{-i})}(x)+\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x)) )^{p(\cdot)}dx\\ &\leq&c (1\vee \|a\|_\infty )^{p^+}(\sum\limits_{j = 0}^{\tau-1}2^{(j-\tau)p^-}\sum\limits_{i = j}^{\tau-1}2^{(i-\tau)p^-} \biggl( \|\chi_{[2^{-\tau}, 2^{-i})}(x)\|_{p(\cdot)}+ \|\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x)\|_{p(\cdot)}\biggl) \\ &&+c (1\vee \|a\|_\infty )^{p^+}\sum\limits_{j = 0}^{\tau-1}2^{(j-N)p^-}\sum\limits_{i = \tau}^{N-1}\biggl( 2^{(i-\tau)p^+} \|\chi_{[2^{-\tau}, 2^{-i})}(x)\|_{p(\cdot)}+ \|\chi_{[2^{-j-1}, 2^{-j-1}\oplus 2^{-i})}(x)\|_{p(\cdot)}\biggl)\\ &\leq&c \|\chi_B\|_{p(\cdot)} ^{-p^+}(\sum\limits_{j = 0}^{\tau-1}2^{(j-\tau)p^-}\sum\limits_{i = j}^{\tau-1}2^{(i-\tau)p^-} 2^{-i} +c P(B)^{-p^+}\sum\limits_{j = 0}^{\tau-1}2^{(j-N)p^-}\sum\limits_{i = \tau}^{N-1}( 2^{(i-\tau)p^+} 2^{-i}+ 2^{-i})\\ &\leq&c \|\chi_B\|_{p(\cdot)} ^{-p^+}(\sum\limits_{j = 0}^{\tau-1}2^{j(2 p^–1)}2^{-2\tau p^-} +c P(B)^{-p^+}\sum\limits_{j = 0}^{\tau-1}2^{(j-N)p^-}\sum\limits_{i = \tau}^{N-1}( 2^{(i-\tau)p^+} 2^{-i}+ 2^{-i})\\ &\leq& c2^{\tau (p^+-2p^-)}\sum\limits_{j = 0}^{\tau}2^{j(2p^–1)}+c\sum\limits_{j = 0}^{\tau-1}2^{(j-N)p^-}\sum\limits_{i = \tau}^{N-1} 2^{i(p^+-1)} +c 2^{\tau p^+}\sum\limits_{j = 0}^{\tau-1}2^{(j-N)p^-}\sum\limits_{i = \tau}^{N-1} 2^{-i}\\ &\leq& c_1+c_2+c_3 = c. \end{eqnarray*} $

By Lemma 4.3, the proof is completed.

Theorem 4.5 Suppose that $ p $ is log-Hölder continuous and $ 1<p^-\leq p^+<\infty $. Then for any $ f\in L_p(\cdot), $ we have $ \|\sigma^*f\|_{p(\cdot)}\leq c\|f\|_{{p(\cdot)}}. $

Proof We assume $ \|f\|_{p(\cdot)} = 1/2. $ If else, we let $ f $ replaced by $ \frac{f}{2\|f\|_{p(\cdot)}}. $ Since

$ \begin{equation*} |K_n(x)|\leq \sum\limits_{j = 0}^{N-1}2^{j-N}\sum\limits_{i = j}^{N-1}(D_{2^i}(x)+D_{2^{i}}(x\oplus2^{-j-1})), \end{equation*} $

where $ x \in [0, 1), n, N\in\textbf{ N}, 2^ {N-1} \leq n < 2^ N $. Thus we have

$ \begin{eqnarray} |\sigma_nf(x)|&\leq &|f|*|K_n|(x)\\ &\leq& \sum\limits_{j = 0}^{N-1}2^{j-N}\sum\limits_{i = j}^{N-1}(|f|*D_{2^i}(x)+|f|*D_{2^{i}}(x\oplus2^{-j-1}))\\ &\leq&\sum\limits_{j = 0}^{N-1}2^{j-N}\sum\limits_{i = j}^{N-1}(|f_n(x)|+|f_n(x\oplus2^{-j-1})|). \end{eqnarray} $

(4.8)

Thus by Doob's inequality of variable exponents martingale spaces, we have

$ \begin{eqnarray*} \|\sigma^*f\|_{p(\cdot)} = \|\sup\limits_{n\geq 1}|\sigma_nf(x)| \|_{p(\cdot)} &\leq& \sup\limits_{N\geq1 }\sum\limits_{j = 0}^{N-1}2^{(j-N)}\sum\limits_{i = j}^{N-1}(\|\sup\limits_{n\geq 1}|f_n|\|_{p(\cdot)}+\|\sup\limits_{n\geq 1}|f_n|\|_{p(\cdot)})\\ &\leq & c\sup\limits_{N\geq1 }\sum\limits_{j = 0}^{N-1}2^{(j-N)}\sum\limits_{i = j}^{N-1}\|f\|_{p(\cdot)} \leq c\sup\limits_{N\geq 1}\sum\limits_{j = 0}^{N-1}2^{(j-N)}(N-j)\\ &\leq& \sum\limits_{k = 0}^\infty2^{-k}k\leq c\|f\|_{p(\cdot)}. \end{eqnarray*} $

Thus we complete the proof.