Equivalent characterizations of martingale Hardy–Lorentz spaces with variable exponents

Abstract

We prove that under the log-Hölder continuity condition of the variable exponent $p(\cdot )$, a new type of maximal operators, $U_{\gamma ,s}$ is bounded from the variable martingale Hardy–Lorentz space $H_{p(\cdot ),q}$ to $L_{p(\cdot ),q}$, whenever $0<p_-\le p_+ <\infty$, $0<q \le \infty$, $0<\gamma ,s<\infty$ and $1/p_- - 1/p_+ < \gamma +s$. Moreover, the operator $U_{\gamma ,s}$ generates equivalent quasi-norms on the Hardy–Lorentz spaces $H_{p(\cdot ),q}$.

1 Introduction

A measurable function $p(\cdot ): [0,1)\rightarrow (0,\infty ]$ is called a variable exponent. In this paper we suppose that

$$\begin{aligned} 0< p_-:=\text {ess} \inf \limits _{x\in [0,1)} p(x) \le p_+:= \text {ess} \sup \limits _{x\in [0,1)}p(x) < \infty . \end{aligned}$$

Variable Lebesgue spaces $L_{p(\cdot )}$ are investigated very intensively in the literature nowadays (see e.g. Cruz-Uribe and Fiorenza [5], Diening et al. [6], Kokilashvili et al. [15, 16], Nakai and Sawano [19, 25], Kempka and Vybíral [14], Jiao et al. [11,12,13], Yan et al. [36], Liu et al. [17, 18]). Interest in the variable Lebesgue spaces has increased since the 1990 s because of their use in a variety of applications (see the references in Jiao et al. [11]).

As usual in this theory, we also suppose that $p(\cdot )$ satisfy the log-Hölder continuity condition, namely $p(\cdot ) \in C^{\log }$. One of the most important results states that the classical Hardy-Littlewood maximal operator is bounded on the variable $L_{p(\cdot )}$ spaces under this condition (see for example Cruz-Uribe et al. [2], Nekvinda [20], Cruz-Uribe and Fiorenza [5] and Diening et al. [6]).

Nakai and Sawano [19] first introduced the variable Hardy spaces $H_{p(\cdot )}({\mathbb {R}})$. Independently, Cruz-Uribe and Wang [4] also investigated the spaces $H_{p(\cdot )}({\mathbb {R}})$. Cruz-Uribe et al. [3] proved the boundedness of fractional and singular integral operators on weighted and variable Hardy spaces. Sawano [25] improved the results in [19]. Ho [10] studied weighted Hardy spaces with variable exponents. Yan et al. [36] introduced the variable weak Hardy space $H_{p(\cdot ),\infty }({\mathbb {R}})$ and characterized these spaces via radial maximal functions. The Hardy–Lorentz spaces $H_{p(\cdot ),q}({\mathbb {R}})$ were investigated by Jiao et al. in [13]. Similar results for the anisotropic Hardy spaces $H_{p(\cdot )}({\mathbb {R}})$ and $H_{p(\cdot ),q}({\mathbb {R}})$ can be found in Liu et al. [17, 18]. Martingale Musielak–Orlicz Hardy spaces were investigated in Xie et al. [33,34,35]. Recently, we [11] generalized these results for martingale Hardy spaces with variable exponent.

In this paper, we investigate the so called Vilenkin martingales defined as follows. Let $(p_n, n \in {{\mathbb {N}}})$ be a bounded sequence of natural numbers with entries at least 2. Introduce the notations $P_0=1$ and

$$\begin{aligned} P_{n+1}:= \prod _{k=0}^n p_k \qquad (n \in {{\mathbb {N}}}). \end{aligned}$$

We denote the set of natural numbers $\{0,1,\ldots ,\}$ by ${\mathbb {N}}$. By a Vilenkin interval, we mean one of the form $[kP_n^{-1},(k+1)P_n^{-1})$ for some $k,n \in {{\mathbb {N}}}$, $0\le k <P_n$, $k \in {\mathbb {N}}$. Let ${\mathcal {F}}_n$ be the $\sigma$-algebra

$$\begin{aligned} {\mathcal {F}}_{n} = \sigma \{ [kP_n^{-1},(k+1)P_n^{-1}) \, :0 \le k <P_n, k \in {\mathbb {N}} \} \end{aligned}$$

(1)

generated by the Vilenkin intervals. Martingales with respect to $(\mathcal {F}_n, n \in {{\mathbb {N}}})$ are called Vilenkin martingales. Vilenkin martingales were studied in a great number of papers, such as Gát and Goginava [7,8,9], Persson and Tephnadze [21,22,23,24] and Simon [26, 27].

For a fixed $x \in [0,1)$ and $n \in {\mathbb {N}}$, let us denote the unique Vilenkin interval $[kP_n^{-1},(k+1)P_n^{-1})$ which contains x by $I_n(x)$. Then the Doob maximal operator for Vilenkin martingales $f=(f_n, n \in {\mathbb {N}})$ can be rewritten as

$$\begin{aligned} M(f)(x)= \sup _{n \in {\mathbb {N}}} \frac{1}{\lambda (I_n(x))}\left| \int _{I_n(x)} f_n d \lambda \right| . \end{aligned}$$

The boundedness of the Doob martingale maximal operator on the $L_{p(\cdot )}$ spaces was proved in Jiao et al. [11, 12]:

Theorem 1

Suppose that $p(\cdot ) \in C^{\log }$ and $f\in L_{p(\cdot )}$. If $1< p_- \le p_+ < \infty$, then

$$\begin{aligned} \left\| M(f)\right\| _{p(\cdot )} \lesssim \Vert f\Vert _{{p(\cdot )}}. \end{aligned}$$

(2)

If $1 \le p_-\le p_+ < \infty$, then

$$\begin{aligned} \sup _{\rho>0} \left\| \rho \chi _{\{M(f)>\rho \}}\right\| _{p(\cdot )} \lesssim \left\| f\right\| _{p(\cdot )}. \end{aligned}$$

(3)

Later we extended this result to $p_+ =\infty$ in [30]. In this paper the constants C are absolute constants and the constants $C_{p(\cdot )}$ are depending only on $p(\cdot )$ and may denote different constants in different contexts. For two positive numbers A and B, we use also the notation $A \lesssim B$, which means that there exists a constant C such that $A \le CB$.

In [11, 29, 31, 32], we generalized the Doob maximal operator and introduced the operator $U_{\gamma ,s}$ for Vilenkin martingales, where $\gamma$ and s are two positive constants. These operators were the key point in the proof of the boundedness of the maximal Fejér operator of the Walsh- and Vilenkin-Fourier series from the variable Hardy space $H_{p(\cdot )}$ to $L_{p(\cdot )}$ (see [11, 31]). Recall the definition of $U_{\gamma ,s}$. For a Vilenkin interval I with length $P_n^{-1}$, $i,j,n \in {\mathbb {N}}$, $l=0,\ldots ,p_j-1$, let us use the notation

$$\begin{aligned} I^{l,j,i}:= I \dot{+} [0,P_{i}^{-1}) \dot{+} l P_{j+1}^{-1} \end{aligned}$$

for the translation of I, where $\dot{+}$ denotes the Vilenkin addition (see Sect. 3). Parallel, we denote $I_n(x)^{l,j,i}:= (I_n(x))^{l,j,i}$. For a Vilenkin martingale $f=(f_n, n \in {\mathbb {N}})$ and $0<\gamma ,s<\infty$, let

$$\begin{aligned} U_{\gamma ,s}(f)(x) := \sup _{n \in {\mathbb {N}}} \sum _{m=0}^{n} \sum _{j=0}^{m} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=j}^{m} \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1}\frac{1}{\lambda (I_n(x)^{l,j,i})}\left| \int _{I_n(x)^{l,j,i}} f_n d \lambda \right| . \end{aligned}$$

(4)

We will see later that $M(f) \le U_{\gamma ,s}(f)$ for all $0<\gamma ,s<\infty$. So the next theorem proved in [31, 32], generalizes (2).

Theorem 2

Let $p(\cdot ) \in C^{\log }$, $1<p_-\le p_+<\infty$ and $0<\gamma ,s<\infty$. If

$$\begin{aligned} \frac{1}{p_-}-\frac{1}{p_+} < \gamma +s, \end{aligned}$$

(5)

then

$$\begin{aligned} \left\| U_{\gamma ,s}(f) \right\| _{p(\cdot )} \lesssim \Vert f\Vert _{{p(\cdot )}} \qquad (f\in L_{p(\cdot )}). \end{aligned}$$

Obviously, inequality (5) and Theorem 2 hold if $p_->\max (1/(\gamma +s),1)$. We proved in [31] that condition (5) is also necessary, the results are not true without this condition.

In [29, 32], we generalized Theorem 2 and, under the same conditions, we obtained also the boundedness of $U_{\gamma ,s}$ from the martingale Hardy space $H_{p(\cdot )}$ to $L_{p(\cdot )}$ for $0<p_-\le p_+<\infty$. In this paper, we generalize these results to variable Lorentz and Hardy–Lorentz spaces. We will prove that $U_{\gamma ,s}$ is bounded from the martingale Hardy–Lorentz space $H_{p(\cdot ),q}$ to $L_{p(\cdot ),q}$, where $0<q \le \infty$. More exactly, we have

Theorem 3

Let $p(\cdot ) \in C^{\log }$, $0<p_-\le p_+<\infty$, $0<q \le \infty$ and $0<\gamma ,s<\infty$. If (5) holds, then

$$\begin{aligned} \Vert U_{\gamma ,s}(f)\Vert _{p(\cdot ),q} \lesssim \left\| f\right\| _{H_{p(\cdot ),q}} \qquad (f\in H_{p(\cdot ),q}). \end{aligned}$$

As a corollary, we get $U_{\gamma ,s}$ is bounded from the Lorentz space $L_{p(\cdot ),q}$ to $L_{p(\cdot ),q}$ and we generalize (3).

Corollary 1

Let $p(\cdot ) \in C^{\log }$ satisfy (5), $0<\gamma ,s<\infty$. If $1<p_-\le p_+<\infty$, $0<q \le \infty$ and $f\in L_{p(\cdot ),q}$, then

$$\begin{aligned} \left\| U_{\gamma ,s}(f) \right\| _{p(\cdot ),q} \lesssim \Vert f\Vert _{{p(\cdot ),q}}. \end{aligned}$$

If $1 \le p_-\le p_+<\infty$ and $f\in L_{p(\cdot )}$, then

$$\begin{aligned} \sup _{\rho>0} \left\| \rho \chi _{\{U_{\gamma ,s}(f)>\rho \}}\right\| _{p(\cdot )} \lesssim \Vert f\Vert _{{p(\cdot )}}. \end{aligned}$$

Moreover, we obtain an equivalent characterization of the martingale Hardy–Lorentz space $H_{p(\cdot ),q}$, namely, we show that $\Vert U_{\gamma ,s}(f)\Vert _{L_{p(\cdot ),q}}$ is equivalent to $\Vert f\Vert _{H_{p(\cdot ),q}}$:

Corollary 2

Let $p(\cdot ) \in C^{\log }$, $0<p_-\le p_+<\infty$, $0<q \le \infty$ and $0<\gamma ,s<\infty$. If (5) holds and $f\in H_{p(\cdot ),q}$, then

$$\begin{aligned} \Vert f\Vert _{H_{p(\cdot ),q}} \le \Vert U_{\gamma ,s}(f)\Vert _{p(\cdot ),q} \le C_{p(\cdot )} \Vert f\Vert _{H_{p(\cdot ),q}}. \end{aligned}$$

Finally, we note again that condition (5) is also necessary.

I would like to thank the referees for reading the paper carefully and for their useful comments and suggestions.

2 Variable Lebesgue and Lorentz spaces

Let $\lambda$ denote the Lebesgue measure on the unit interval [0, 1). For a constant p, the $L_p$ space is equipped with the quasi-norm

$$\begin{aligned} \Vert f\Vert _p:=\left( \int _{0}^{1}|f(x)|^p \, d \lambda (x)\right) ^{1/p} \qquad (0<p<\infty ), \end{aligned}$$

with the usual modification for $p=\infty$.

To introduce the variable Lebesgue spaces let

$$\begin{aligned} \rho (f):=\int _0^1 |f(x)|^{p(x)} d \lambda (x), \end{aligned}$$

where $p(\cdot ): [0,1)\rightarrow (0,\infty ]$ is a variable exponent. The variable Lebesgue space $L_{p(\cdot )}$ is the collection of all measurable functions f for which there exists $\nu >0$ such that

$$\begin{aligned} \rho \left( {f}/{\nu }\right) <\infty . \end{aligned}$$

We equip $L_{p(\cdot )}$ with the quasi-norm

$$\begin{aligned} \Vert f\Vert _{p(\cdot )}{:}{=}\inf \{\nu >0:\rho ({f}/{\nu })\le 1\}. \end{aligned}$$

If $p(\cdot )=p$ is a constant, then we get back the definition of the usual $L_p$ spaces. For any $f\in L_{p(\cdot )}$, we have $\rho (f)\le 1$ if and only if $\Vert f\Vert _{p(\cdot )}\le 1$. It is known that $\Vert \nu f\Vert _{p(\cdot )}=|\nu |\Vert f\Vert _{p(\cdot )}$ and

$$\begin{aligned} \left\| |f|^s \right\| _{p(\cdot )}=\Vert f\Vert _{s p(\cdot )}^s, \end{aligned}$$

where $s\in (0,\infty )$ and $\nu \in {{\mathbb {C}}}$. Details can be found in the monographs Cruz-Uribe and Fiorenza [5] and Diening et al. [6]. Moreover, for

$$\begin{aligned} 0<b\le \min \{p_-,1\} =:{\underline{p}}, \end{aligned}$$

we have

$$\begin{aligned} \Vert f+g\Vert _{p(\cdot )}^{b} \le \Vert f\Vert _{p(\cdot )}^{b} +\Vert g\Vert _{p(\cdot )}^{b}. \end{aligned}$$

(6)

The variable exponent $p'(\cdot )$ is defined pointwise by

$$\begin{aligned} \frac{1}{p(x)}+\frac{1}{p'(x)}=1, \quad x\in [0,1). \end{aligned}$$

The next lemma is well known, see Cruz-Uribe and Fiorenza [5] or Diening et al. [6].

Lemma 1

Let $1\le p_- \le p_+ \le \infty$. For all $f \in L_{p(\cdot )}$ and $g \in L_{p'(\cdot )}$,

$$\begin{aligned} \int _0^{1} \left| fg\right| \, d\lambda \le C_{p(\cdot )} \left\| f\right\| _{p(\cdot )} \left\| g\right\| _{p'(\cdot )}. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert f\Vert _{p(\cdot )}\sim \sup _{\Vert g\Vert _{p'(\cdot )}\le 1}\left| \int _{0}^{1} fg \, d \lambda \right| , \end{aligned}$$

where $\sim$ denotes the equivalence of the numbers.

The variable Lorentz spaces were introduced and investigated by Kempka and Vybíral [14]. $L_{p(\cdot ),q}$ is defined to be the space of all measurable functions f such that

$$\begin{aligned} \Vert f\Vert _{p(\cdot ),q}:=\left\{ \begin{array}{ll} \displaystyle \left( \int _0^\infty \rho ^q \left\| \chi _{\{x\in [0,1):\ |f(x)|>\rho \}} \right\| _{p(\cdot )}^q \, \frac{d \rho }{\rho }\right) ^{1/q}, &{} \hbox {if } 0<q<\infty ;\\ \displaystyle \sup _{\rho \in (0,\infty )} \rho \left\| \chi _{\{x\in [0,1):\ |f(x)|>\rho \}} \right\| _{p(\cdot )}, &{} \hbox {if } q=\infty \end{array}\right. \end{aligned}$$

is finite. If $p(\cdot )$ is a constant, we get back the classical Lorentz spaces (see Bergh and Löfström [1]). In contrary to the spaces with constant $p(\cdot )$, the variable Lorentz spaces $L_{p(\cdot ),q}$ do not include the variable Lebesgue spaces $L_{p(\cdot )}$ as a special cases.

3 Maximal operators

We always suppose that the sequence $(p_n)$ of natural numbers is bounded. Let

$$\begin{aligned} {\widehat{p}}:= \sup _{n \in {\mathbb {N}}}p_n<\infty . \end{aligned}$$

(7)

The conditional expectation operators relative to ${{\mathcal {F}}}_n$ are denoted by $E_n$, where ${{\mathcal {F}}}_n$ was defined in (1). An integrable sequence $f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}}$ is said to be a Vilenkin martingale if $f_{n}$ is $\mathcal {F}_{n}$-measurable for all $n \in {{\mathbb {N}}}$ and $E_{n} f_{m} = f_{n}$ in case $n \le m$. It is easy to show (see e.g. Weisz [28]) that the sequence $(\mathcal {F}_n, n \in {{\mathbb {N}}})$ is regular, i.e., there exist a constant $R>0$ such that $f_n\le R \cdot f_{n-1}$ for all non-negative Vilenkin martingales. We can see easily that $R \ge {\widehat{p}}$, where ${\widehat{p}}$ is defined in (7).

For a Vilenkin martingale $f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}}$, the Doob maximal function is defined by

$$\begin{aligned} M(f):= \sup _{n \in {{\mathbb {N}}}} \left| f_{n} \right| . \end{aligned}$$

If $f\in L_1$, then we can replace $f_n$ by f in the integral.

In the literature the log-Hölder continuity condition is usually supposed. Under this condition, the Hardy-Littlewood maximal operator is bounded on $L_{p(\cdot )}$ if $1<p_- \le p_+$. We denote by $C^{\log }$ the set of all variable exponents $p(\cdot )$ satisfying the so-called log-Hölder continuous condition, namely, there exists a positive constant $C_{\log }(p)$ such that, for any $x,y\in [0,1)$,

$$\begin{aligned} |p(x)-p(y)|\le \frac{C_{\log }(p)}{\log (e+1/|x-y|)}. \end{aligned}$$

(8)

In [31, 32], we generalized the Doob martingale maximal operator as follows. Every point $x \in [0,1)$ can be written in the following way:

$$\begin{aligned} x= \sum _{k=0}^{\infty } \frac{x_k}{P_{k+1}} \qquad (0 \le x_k<p_k, \; x_k \in {{{\mathbb {N}}}}). \end{aligned}$$

If there are two different forms, choose the one for which $\lim _{k \rightarrow \infty } x_k =0$. The so called Vilenkin addition is defined by

$$\begin{aligned} x\dot{+} y = \sum _{k=0}^{\infty } \frac{z_k}{P_{k+1}}, \qquad \hbox {where } z_k:= x_k+y_k \ \text{ mod } p_k, (k \in {{{\mathbb {N}}}}). \end{aligned}$$

We defined the maximal operator $U_{\gamma ,s}$ in (4), where $0<\gamma ,s<\infty$. Of course, if $f\in L_1$, then we can write in the definition f instead of $f_n$. Let us define $I_{k,n}:= [kP_n^{-1},(k+1)P_n^{-1})$, where $0\le k<P_n$, $n\in {{\mathbb {N}}}$. The definition of $U_{\gamma ,s}(f)$ can be rewritten to

$$\begin{aligned} U_{\gamma ,s}(f) = \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{P_n-1} \chi _{I_{k,n}}\sum _{m=0}^{n} \sum _{j=0}^{m} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=j}^{m} \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1} \frac{1}{\lambda (I^{l,j,i}_{k,n})}\left| \int _{I^{l,j,i}_{k,n}} f_n d \lambda \right| , \end{aligned}$$

where $I^{l,j,i}_{k,n}:=(I_{k,n})^{l,j,i}$. Now we point out four special cases of this operator.

If $j=i=n=m$, we obtain the first spacial case,

$$\begin{aligned} U_{\gamma ,s}^{(1)}(f)(x)&:= \sup _{n \in {\mathbb {N}}} \sum _{l=0}^{p_n-1} \frac{1}{\lambda (I_n(x)^{l,n,n})}\left| \int _{I_n(x)^{l,n,n}} f_n d \lambda \right| \\&= \sup _{n \in {\mathbb {N}}} \frac{p_n}{\lambda (I_n(x))}\left| \int _{I_n(x)} f_n d \lambda \right| , \end{aligned}$$

which is basically M(f). Note that $I_n(x)^{l,n,n}= I_n(x)$ ($n \in {\mathbb {N}}$, $l=0,\ldots ,p_n-1$).

If $j=i=m$, we have

$$\begin{aligned} U_{\gamma ,s}^{(2)}(f)(x)&:= \sup _{n \in {\mathbb {N}}} \sum _{m=0}^{n} \left( \frac{P_m}{P_n} \right) ^{\gamma } \sum _{l=0}^{p_m-1} \frac{1}{\lambda (I_n(x)^{l,m,m})}\left| \int _{I_n(x)^{l,m,m}} f_n d \lambda \right| \\&= \sup _{n \in {\mathbb {N}}} \sum _{m=0}^{n} \left( \frac{P_m}{P_n} \right) ^{\gamma } \frac{p_m}{\lambda (I_m(x))}\left| \int _{I_m(x)} f_n d \lambda \right| . \end{aligned}$$

Here $I_n(x)^{l,m,m} = I_n(x)\dot{+}[0,P_m^{-1}) \dot{+} l P_{m+1}^{-1}=x\dot{+}[0,P_m^{-1})=I_m(x)$. It is easy to see that

$$\begin{aligned} M(f) \le U_{\gamma ,s}^{(1)}(f) \le U_{\gamma ,s}^{(2)}(f) \le C M(f) \end{aligned}$$

for all $0<\gamma ,s<\infty$ and so Theorem 1 holds also for these two operators.

If $m=n$ and $i=n$, we get that

$$\begin{aligned} U_{\gamma ,s}^{(3)}(f)(x) := \sup _{n \in {\mathbb {N}}} \sum _{j=0}^{n} \left( \frac{P_j}{P_n} \right) ^{\gamma +s} \sum _{l=0}^{p_j-1} \frac{1}{\lambda (I_n(x)^{l,j,n})}\left| \int _{I_n(x)^{l,j,n}} f_n d \lambda \right| . \end{aligned}$$

Note that $I_n(x)^{l,j,n}= I_n(x) \dot{+} [0,P_{n}^{-1}) \dot{+} l P_{j+1}^{-1}= I_n(x) \dot{+} l P_{j+1}^{-1}$.

If $m=n$, we obtain the last special case,

$$\begin{aligned} U_{\gamma ,s}^{(4)}(f)(x) := \sup _{n \in {\mathbb {N}}} \sum _{j=0}^{n} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=j}^{n} \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1}\frac{1}{\lambda (I_n(x)^{l,j,i})}\left| \int _{I_n(x)^{l,j,i}} f_n d \lambda \right| . \end{aligned}$$

The maximal operators $U_{\gamma ,s}^{(3)}(f)$ and $U_{\gamma ,s}^{(4)}(f)$ as well as $U_{\gamma ,s}(f)$ cannot be estimated by M(f) from above pointwise. In [31], we investigated the operators $U_{\gamma ,s}^{(3)}$ and $U_{\gamma ,s}^{(4)}$. Their boundedness on $L_{p(\cdot )}$ was the key point in the proof of boundedness and convergence results for the Fejér means of the Vilenkin-Fourier series (see [31]).

It is easy to see that, for all $0<\gamma ,s<\infty$,

$$\begin{aligned} M(f) \le U_{\gamma ,s}^{(j)}(f) \le U_{\gamma ,s}(f) \qquad (j=1,\ldots ,4). \end{aligned}$$

(9)

4 Martingale Hardy–Lorentz spaces

Now we introduce the variable martingale Hardy–Lorentz spaces by

$$\begin{aligned} H_{p(\cdot ),q}:= & {} \left\{ f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}}: \left\| f\right\| _{H_{p(\cdot ),q}}:= \left\| M(f)\right\| _{p(\cdot ),q} < \infty \right\} . \end{aligned}$$

These spaces have several equivalent characterizations, for example an equivalent quasi-norm can be defined by the quadratic variation and by the conditional quadratic variation (see [11]). In this paper, we will give more equivalent characterizations of these Hardy–Lorentz spaces using the above maximal functions.

The atomic decomposition is a useful characterization of the Hardy–Lorentz spaces. First, we introduce the concept of stopping times (see e.g. [28]). A map $\tau : [0,1) \longrightarrow {\mathbb {N}} \cup \{\infty \}$ is called a stopping time relative to $({{\mathcal {F}}}_{n}, n \in {\mathbb {N}})$ if

$$\begin{aligned} \left\{ x \in [0,1): \tau (x)=n \right\} =: \{ \tau =n \} \in {{\mathcal {F}}}_n. \end{aligned}$$

It is well known that the last condition is equivalent to the conditions

$$\begin{aligned} \{ \tau \le n \} \in {{\mathcal {F}}}_{n} \qquad (n \in {{{\mathbb {N}}}}) \end{aligned}$$

and

$$\begin{aligned} \{ \tau \ge n \} \in {{\mathcal {F}}}_{n-1} \qquad (n \in {{{\mathbb {N}}}}). \end{aligned}$$

This implies that the sequence $(f_n^\tau ,n \in {\mathbb {N}})$ defined by

$$\begin{aligned} f^{\tau }_n:=\sum _{k=0}^{n} \chi _{\{\tau \ge m\}} \left( f_k-f_{k-1}\right) \end{aligned}$$

is again a martingale, called stopped martingale, whenever $(f_n,n \in {\mathbb {N}})$ is a martingale. This fact is used in the proof of Theorem 4.

A measurable function a is called a $p(\cdot )$-atom if there exists a stopping time $\tau$ such that

• (i) $E_n(a)(\cdot ) = 0$ for all $n \le \tau (\cdot )$,

• (ii) $\Vert M(a)\Vert _{\infty } \le \left\| \chi _{ \left\{ \tau < \infty \right\} } \right\| _{p(\cdot )}^{-1}$.

This form of the atoms was used first in [28] for a constant p. The atomic decomposition of the spaces $H_{p(\cdot ),q}$ were proved in Jiao et al. [11]. The classical case can be found in [28].

Theorem 4

Let $p(\cdot ) \in C^{\log }$, $0< p_-\le p_+ < \infty$ and $0 < q \le \infty$. Then the martingale $f = \left( f_{n}\right) _{n \in {{\mathbb {N}}}} \in H_{p(\cdot ),q}$ if and only if there exists a sequence $(a^{k})_{k \in {{\mathbb {Z}}}}$ of $p(\cdot )$-atoms such that for every $n \in {{\mathbb {N}}}$,

$$\begin{aligned} f_{n} = \sum _{k \in {{\mathbb {Z}}}} \mu _{k} E_{n}a^{k} \qquad \text{ almost } \text{ everywhere }, \end{aligned}$$

where $\mu _{k} = 3 \cdot 2^k \left\| \chi _{ \left\{ \tau _{k} < \infty \right\} } \right\| _{p(\cdot )}$ and $\tau _k$ is the stopping time associated with the $p(\cdot )$-atom $a^k$. Moreover,

$$\begin{aligned} \left\| f \right\| _{H_{p(\cdot ),q}}&\sim \inf \left( \sum _{k \in {{\mathbb {Z}}}} 2^{kq} \left\| \chi _{ \left\{ \tau _{k} < \infty \right\} } \right\| _{p(\cdot )}^{q} \right) ^{1/q}, \end{aligned}$$

respectively, where the infimum is taken over all decompositions of f as above.

5 Proofs

Proof of Theorem 3

According to Theorem 4, we can write f as

$$\begin{aligned} f = \sum _{k \in {{\mathbb {Z}}}} \mu _{k} a^{k} = f_1+f_2, \end{aligned}$$

where $k_0 \in {\mathbb {Z}}$,

$$\begin{aligned} f_1 = \sum _{k=- \infty }^{k_0-1} \mu _{k} a^{k}, \qquad f_2 = \sum _{k=k_0}^{\infty } \mu _{k} a^{k}, \qquad \mu _{k} = 3 \cdot 2^k \left\| \chi _{ \left\{ \tau _{k} < \infty \right\} } \right\| _{p(\cdot )} \end{aligned}$$

and $\tau _k$ is the stopping time associated with the $p(\cdot )$-atom $a^k$. Moreover,

$$\begin{aligned} \left( \sum _{k \in {{\mathbb {Z}}}} 2^{kq} \left\| \chi _{ \left\{ \tau _{k} < \infty \right\} } \right\| _{p(\cdot )}^{q} \right) ^{1/q} \lesssim \left\| f \right\| _{H_{p(\cdot ),q}}. \end{aligned}$$

Since

$$\begin{aligned} U_{\gamma ,s}(f)&\le \sum _{m=0}^{n} \sum _{j=0}^{m} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=j}^{m} \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1} \Vert f\Vert _\infty \end{aligned}$$

(10)

$$\begin{aligned}&\le \sum _{m=0}^{n} \sum _{j=0}^{m} 2^{(j-n)\gamma } \sum _{i=j}^{m} 2^{(j-i)s} p_j \Vert f\Vert _\infty \lesssim \Vert f\Vert _\infty , \end{aligned}$$

(11)

$U_{\gamma ,s}$ is bounded on $L_\infty$. This implies that

$$\begin{aligned} \left\| U_{\gamma ,s}(f_1) \right\| _\infty&\le \sum _{k=-\infty }^{k_0-1} \mu _k \left\| U_{\gamma ,s}(a^k) \right\| _\infty \le \sum _{k=-\infty }^{k_0-1} \mu _k \left\| a^k \right\| _\infty \\&\le \sum _{k=-\infty }^{k_0-1} \mu _k \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1}\le 3 \cdot 2^{k_0}. \end{aligned}$$

Thus

$$\begin{aligned} 2^{k_0}\Vert \chi _{\{U_{\gamma ,s}(f)>6 \cdot 2^{k_0}\}}\Vert _{p(\cdot )} \le 2^{k_0}\Vert \chi _{\{U_{\gamma ,s}(f_2)> 3\cdot 2^{k_0}\}}\Vert _{p(\cdot )}, \end{aligned}$$

so we have to consider

$$\begin{aligned} U_{\gamma ,s}(f_2) \le \sum _{k=k_0}^{\infty } \mu _k U_{\gamma ,s}(a^k) \chi _{\{\tau _k<\infty \}} + \sum _{k=k_0}^{\infty } \mu _k U_{\gamma ,s}(a^k) \chi _{\{\tau _k=\infty \}} =: A_1+A_2.\nonumber \\ \end{aligned}$$

(12)

Obviously,

$$\begin{aligned} \{A_1>3\cdot 2^{k_0-1}\} \subset \{A_1>0\}\subset \bigcup \limits _{k=k_0}^\infty \{\tau _k<\infty \}. \end{aligned}$$

Suppose that $0<q<\infty$ and let us choose $0<\varepsilon <\min ({\underline{p}},q)$ and $0<\delta <1$. Applying (6), we have

$$\begin{aligned} \left\| \chi _{\{A_1>3\cdot 2^{k_0-1}\}} \right\| _{p(\cdot )}&\le \left\| \sum _{k=k_0}^\infty \chi _{\{\tau _k<\infty \}}\right\| _{p(\cdot )} \le \left( \sum _{k=k_0}^\infty \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{{\varepsilon }}\right) ^{1/{\varepsilon }} \\&= \left( \sum _{k=k_0}^\infty 2^{-k\delta {\varepsilon }}2^{k\delta {\varepsilon }} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{{\varepsilon }}\right) ^{1/{\varepsilon }}. \end{aligned}$$

Using Hölder’s inequality for $\frac{ q-{\varepsilon }}{q}+\frac{{\varepsilon }}{q}=1$, we get

$$\begin{aligned} \left\| \chi _{\{A_1>3\cdot 2^{k_0-1}\}} \right\| _{p(\cdot )}&\le \left( \sum _{k=k_0}^\infty 2^{-k\delta {\varepsilon }\frac{q}{q-{\varepsilon }}}\right) ^{\frac{q-{\varepsilon }}{{\varepsilon }q}} \left( \sum _{k=k_0}^\infty 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )} ^q\right) ^{1/q} \\&\lesssim 2^{-k_0\delta } \left( \sum _{k=k_0}^\infty 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )} ^q\right) ^{1/q}. \end{aligned}$$

Consequently,

$$\begin{aligned} \sum _{k_0=-\infty }^\infty 2^{k_0q} \left\| \chi _{\{A_1>3\cdot 2^{k_0-1}\}} \right\| _{p(\cdot )}^q&\lesssim \sum _{k_0=-\infty }^\infty 2^{k_0(1-\delta )q} \sum _{k=k_0}^\infty 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^q \\&= \sum _{k=-\infty }^\infty 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^q \sum _{k_0=-\infty }^k 2^{k_0(1-\delta )q} \\&\lesssim \sum _{k=-\infty }^\infty 2^{kq} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^q\\&\lesssim \Vert f\Vert _{H_{p(\cdot ),q}}^q. \end{aligned}$$

Next, let us estimate the term $A_2$. For a fixed $k \in {{\mathbb {Z}}}$, the sets $\left\{ \tau _k=K\right\}$ are disjoint and there exist disjoint Vilenkin intervals $I_{k,K,\mu }\in \mathcal {F}_K$ such that

$$\begin{aligned} \left\{ \tau _k=K\right\} =\bigcup _{\mu } I_{k,K,\mu } \qquad (K \in {{\mathbb {N}}}), \end{aligned}$$

where the union in $\mu$ is finite and $\lambda (I_{k,K,\mu })=P_K^{-1}$. Thus

$$\begin{aligned} \left\{ \tau _k<\infty \right\} =\bigcup _{K\in {{\mathbb {N}}}} \bigcup _\mu I_{k,K,\mu }, \end{aligned}$$

where the Vilenkin intervals $I_{k,K,\mu }$ are disjoint for a fixed $k \in {{\mathbb {Z}}}$. Then

$$\begin{aligned} a^{k}= \sum _{K\in {{\mathbb {N}}}} \sum _{\mu } a^{k} \chi _{I_{k,K,\mu }}. \end{aligned}$$

The operator $U_{\gamma ,s}$ can be written as

$$\begin{aligned} U_{\gamma ,s}(a^k)(x) := \sup _{n \in {\mathbb {N}}} \sup _{x \in I} \sum _{m=0}^{n} \sum _{j=0}^{m} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=j}^{m} \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1}\frac{1}{\lambda (I^{l,j,i})}\left| \int _{I^{l,j,i}} a^k d \lambda \right| , \end{aligned}$$

where $I \in \mathcal {F}_n$ is a Vilenkin interval. Since $\int _{I_{k,K,\mu }}a^{k}\, d\lambda =0$, we have

$$\begin{aligned} \int _{I^{l,j,i}} a^{k} \, d \lambda = 0 \end{aligned}$$

if $i \le K$. Thus we can suppose that $i > K$, and so $n \ge m > K$. If $x \notin I_{k,K,\mu }$, $x \in I$ and $j \ge K$, then $I^{l,j,i} \cap I_{k,K,\mu } = \emptyset$. Therefore we can suppose that $j < K$. Similarly, if

$$\begin{aligned} x \in I_{k,K,\mu } \dot{+} [l P_{j+1}^{-1}, (l+1) P_{j+1}^{-1}) \setminus (I_{k,K,\mu } \dot{+} l P_{j+1}^{-1}), \end{aligned}$$

then $I^{l,j,i} \cap I_{k,K,\mu } = \emptyset$, so we may assume that $x \in I_{k,K,\mu }\dot{+}l P_{j+1}^{-1}=I_{k,K,\mu }^{l,j,K}$. Therefore, for $x \not \in I_{k,K,\mu }$,

$$\begin{aligned} U_{\gamma ,s}(a^{k} \chi _{I_{k,K,\mu }})(x)&\le \sup _{n > K} \chi _{I}(x) \sum _{m=K+1}^{n} \sum _{j=0}^{K-1} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=K+1}^{m} \left( \frac{P_j}{P_i} \right) ^{s}\\&\quad \sum _{l=0}^{p_j-1} \frac{1}{\lambda (I^{l,j,i})} \left| \int _{I^{l,j,i}} a^k \, d \lambda \right| \chi _{I_{k,K,\mu }^{l,j,K}}(x). \end{aligned}$$

It is easy to see that

$$\begin{aligned} \sum _{i=K+1}^{m} \left( \frac{1}{P_i} \right) ^{s} = \sum _{i=K+1}^{m} \left( \frac{1}{P_K p_K \cdots p_{i-1}} \right) ^{s} \le \sum _{i=K+1}^{m} \left( \frac{1}{P_K 2^{i-K}} \right) ^{s} \le C_s \left( \frac{1}{P_K} \right) ^{s}. \end{aligned}$$

Hence

$$\begin{aligned}&U_{\gamma ,s}(a^{k} \chi _{I_{k,K,\mu }})(x) \\&\le \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1} \sup _{n> K} \chi _{I}(x) \sum _{m=K+1}^{n} \sum _{j=0}^{K-1} \left( \frac{P_j}{P_n} \right) ^{\gamma } \sum _{i=K+1}^{\infty } \left( \frac{P_j}{P_i} \right) ^{s} \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}}(x)\\&\lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1} \sup _{n> K} (n-K) \left( \frac{P_K}{P_n} \right) ^{\gamma } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{\gamma +s} \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}}(x)\\&\lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1} \sup _{n > K} (n-K) 2^{(K-n)\gamma } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{\gamma +s} \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}}(x). \end{aligned}$$

Since the function $x \mapsto x2^{-\gamma x}$ is bounded, we obtain that

$$\begin{aligned} U_{\gamma ,s}(a^{k} \chi _{I_{k,K,\mu }})(x)&\lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1} \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{\gamma +s} \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}}(x). \end{aligned}$$

From this it follows that, for $x \in \{\tau _k=\infty \}$,

$$\begin{aligned} U_{\gamma ,s}(a^{k})(x)&\lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{-1} \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{\gamma +s} \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}}(x). \end{aligned}$$

(13)

Let us choose $0<\beta <1$ and $0<\epsilon <{\underline{p}}$. By (13),

$$\begin{aligned}&\left\| |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}} \right\| _{p(\cdot )/\epsilon } \\&\lesssim \left\| \chi _{\{\tau _k<\infty \}}\right\| _{p(\cdot )}^{-\beta \epsilon } \left\| \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon } \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}} \right\| _{p(\cdot )/\epsilon }. \end{aligned}$$

Choose $\max (1,\beta p_+)<r<\infty$. By Lemma 1, there exists a function $g\in L_{(\frac{p(\cdot )}{\varepsilon })'}$ with $\Vert g\Vert _{{(\frac{p(\cdot )}{\varepsilon })'}}\le 1$ such that

$$\begin{aligned}&\left\| |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}} \right\| _{p(\cdot )/\epsilon } \left\| \chi _{\{\tau _k<\infty \}}\right\| _{p(\cdot )}^{\beta \epsilon }\\&\quad \lesssim \int _0^{1}\sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon } \sum _{l=0}^{p_j-1} \chi _{I_{k,K,\mu }^{l,j,K}} g \, d \lambda \\&\quad \le \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon } \sum _{l=0}^{p_j-1} \left\| \chi _{I_{k,K,\mu }^{l,j,K}} \right\| _{{\frac{r}{\beta \epsilon }}} \left\| \chi _{I_{k,K,\mu }^{l,j,K}} g \right\| _{{(\frac{r}{\beta \epsilon })'}}\\&\quad \lesssim \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \sum _{j=0}^{K-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon }\\&\qquad \sum _{l=0}^{p_j-1} \int _0^{1} \chi _{I_{k,K,\mu }}(x) \left( \frac{1}{\lambda (I_{k,K,\mu }^{l,j,K})}\int _{I_{k,K,\mu }^{l,j,K}} \left| g\right| ^{(\frac{r}{\beta \epsilon })'} \, d \lambda \right) ^{1/(\frac{r}{\beta \epsilon })'} \,dx. \end{aligned}$$

We use Hölder’s inequality to obtain

$$\begin{aligned}&\left\| |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}} \right\| _{p(\cdot )/\epsilon } \left\| \chi _{\{\tau _k<\infty \}}\right\| _{p(\cdot )}^{\beta \epsilon }\\&\lesssim \int _0^{1} \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \chi _{I_{k,K,\mu }}(x) \sum _{j=0}^{K-1} \sum _{l=0}^{p_j-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon (1/(\frac{r}{\beta \epsilon })+1/(\frac{r}{\beta \epsilon })')} \\&\qquad \left( \frac{1}{\lambda (I_{k,K,\mu }^{l,j,K})}\int _{I_{k,K,\mu }^{l,j,K}} \left| g\right| ^{(\frac{r}{\beta \epsilon })'} \, d \lambda \right) ^{1/(\frac{r}{\beta \epsilon })'} \,dx \\&\lesssim \int _0^{1} \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \chi _{I_{k,K,\mu }}(x) \left( \sum _{j=0}^{K-1} \sum _{l=0}^{p_j-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon } \right) ^{1/(\frac{r}{\beta \epsilon })} \\&\qquad \left( \sum _{j=0}^{K-1} \sum _{l=0}^{p_j-1} \left( \frac{P_j}{P_K} \right) ^{(\gamma +s) \beta \epsilon } \frac{1}{\lambda (I_{k,K,\mu }^{j,K})}\int _{I_{k,K,\mu }^{j,K}} \left| g\right| ^{(\frac{r}{\beta \epsilon })'} \, d \lambda \right) ^{1/(\frac{r}{\beta \epsilon })'} \,dx \\&\le \int _0^{1} \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \chi _{I_{k,K,\mu }} \left( U_{\gamma \beta \epsilon ,s\beta \epsilon }^{(3)} \left( |g|^{(\frac{r}{\beta \epsilon })'} \right) \right) ^{1/(\frac{r}{\beta \epsilon })'} \, d \lambda \\&\le \left\| \sum _{K \in {{\mathbb {N}}}} \sum _{\mu } \chi _{I_{k,K,\mu }} \right\| _{{p(\cdot )/\epsilon }} \left\| \left( U_{\gamma \beta \epsilon ,s\beta \epsilon }^{(3)} \left( |g|^{(\frac{r}{\beta \epsilon })'} \right) \right) ^{1/(\frac{r}{\beta \epsilon })'}\right\| _{{(p(\cdot )/\epsilon )'}}. \end{aligned}$$

Inequality (5) is equivalent to

$$\begin{aligned} \frac{p_+- \epsilon }{p_+}-\frac{p_--\epsilon }{p_-} < (\gamma +s)\epsilon . \end{aligned}$$

We can choose $\beta$ near to 1 such that

$$\begin{aligned} \frac{p_+- \epsilon }{p_+}-\frac{p_--\epsilon }{p_-} < (\gamma +s)\beta \epsilon . \end{aligned}$$

Next we can choose r so large that

$$\begin{aligned} \frac{1}{\left( (p(\cdot )/\epsilon )'/(r/\beta \epsilon )'\right) _-}-\frac{1}{\left( (p(\cdot )/\epsilon )'/(r/\beta \epsilon )'\right) _+}&= \frac{r/(r- \beta \epsilon )}{p_+/(p_+-\epsilon )} - \frac{r/(r- \beta \epsilon )}{p_-/(p_--\epsilon )} \\&< (\gamma +s)\beta \epsilon . \end{aligned}$$

Since $(r/\beta \epsilon )'<{(p(\cdot )/\epsilon )'}$, we can apply Theorem 2 and conclude

$$\begin{aligned}&\left\| |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}} \right\| _{p(\cdot )/\epsilon } \left\| \chi _{\{\tau _k<\infty \}}\right\| _{p(\cdot )}^{\beta \epsilon }\\&\quad \lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{{p(\cdot )/\epsilon }} \left\| U_{\gamma \beta \epsilon ,s\beta \epsilon }^{(3)} \left( |g|^{(\frac{r}{\beta \epsilon })'} \right) \right\| _{\frac{(p(\cdot )/\epsilon )'}{(r/\beta \epsilon )'}}^{1/(r/\beta \epsilon )'}\\&\quad \lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{{p(\cdot )/\epsilon }} \left\| |g|^{(\frac{r}{\beta \epsilon })'} \right\| _{\frac{(p(\cdot )/\epsilon )'}{(r/\beta \epsilon )'}}^{1/(r/\beta \epsilon )'}\\&\quad \lesssim \left\| \chi _{\{\tau _k<\infty \}} \right\| _{{p(\cdot )/\epsilon }}. \end{aligned}$$

From this it follows that

$$\begin{aligned} \left\| \chi _{\{A_2>3 \cdot 2^{k_0-1} \}} \right\| _{p(\cdot )}&\le \left\| \frac{\sum _{k=k_0}^{\infty } \mu _k^\beta |U_{\gamma ,s}(a^{k})|^\beta \chi _{\{\tau _k=\infty \}}}{3^\beta 2^{\beta (k_0-1)}} \right\| _{{p(\cdot )}} \nonumber \\&\lesssim 2^{-\beta k_0} \left\| \sum _{k=k_0}^{\infty } \mu _k^{\beta \epsilon } |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}} \right\| _{{p(\cdot )/\epsilon }}^{1/\epsilon }\nonumber \\&\lesssim 2^{-\beta k_0} \left( \sum _{k=k_0}^{\infty } \mu _k^{\beta \epsilon } \left\| |U_{\gamma ,s}(a^{k})|^{\beta \epsilon } \chi _{\{\tau _k=\infty \}}\right\| _{{p(\cdot )}/\epsilon } \right) ^{1/\epsilon } \nonumber \\&\lesssim 2^{-\beta k_0} \left( \sum _{k=k_0}^{\infty } 2^{k\beta \epsilon } \left\| \chi _{\{\tau _k<\infty \}} \right\| _{{p(\cdot )}/\epsilon } \right) ^{1/\epsilon } \nonumber \\&\le 2^{-\beta k_0} \left( \sum _{k=k_0}^{\infty } 2^{k(\beta -\delta )\epsilon } 2^{k\delta \epsilon } \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{\epsilon } \right) ^{1/\epsilon }, \end{aligned}$$

(14)

where $\beta<\delta <1$. Let us again use Hölder’s inequality with $\frac{q-\varepsilon }{q}+\frac{\varepsilon }{q}=1$:

$$\begin{aligned} \left\| \chi _{\{A_2>3 \cdot 2^{k_0-1} \}} \right\| _{p(\cdot )}&\lesssim 2^{-\beta k_0} \left( \sum _{k=k_0}^{\infty } 2^{k(\beta -\delta )\epsilon \frac{q}{q-\epsilon }}\right) ^{\frac{q-\epsilon }{\epsilon q}} \left( \sum _{k=k_0}^{\infty } 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{q}\right) ^{1/q}\\&\lesssim 2^{-k_0\delta } \left( \sum _{k=k_0}^{\infty } 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{q}\right) ^{1/q}. \end{aligned}$$

By changing the order of the sums, we obtain

$$\begin{aligned} \sum _{k_0=-\infty }^\infty 2^{k_0q}\left\| \chi _{\{A_2>3 \cdot 2^{k_0-1} \}} \right\| _{p(\cdot )}^{q}&\lesssim \sum _{k_0=-\infty }^\infty 2^{k_0(1-\delta )q} \sum _{k=k_0}^{\infty }2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{q} \\&=\sum _{k=-\infty }^\infty 2^{k\delta q} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{q} \sum _{k_0=-\infty }^{k} 2^{k_0(1-\delta )q} \\&\lesssim \sum _{k=-\infty }^\infty 2^{kq} \left\| \chi _{\{\tau _k<\infty \}} \right\| _{p(\cdot )}^{q}\\&\lesssim \Vert f\Vert _{H_{p(\cdot ),q}}^q. \end{aligned}$$

This finishes the proof of Theorem 3 when $0<q<\infty$. The proof is very similar for $q=\infty$, so we omit it. $\square$

Remark 1

Inequality (5) obviously holds if $1/(\gamma +s) \le p_-\le p_+<\infty$. If $p_-<1/(\gamma +s)$, then (5) is equivalent to

$$\begin{aligned} p_+ < \frac{p_-}{1-(\gamma +s)p_-}. \end{aligned}$$

Proof of Corollary 1

Jiao et al. [11] proved that $H_{p(\cdot ),q}$ is equivalent to $L_{p(\cdot ),q}$, whenever $1<p_-\le p_+<\infty$ and $0<q \le \infty$. Then the first inequality follows from Theorem 3. By Theorem 3 and (3),

$$\begin{aligned} \sup _{\rho>0} \left\| \rho \chi _{\{U_{\gamma ,s}(f)>\rho \}}\right\| _{p(\cdot )}&= \left\| U_{\gamma ,s}(f)\right\| _{{p(\cdot ),\infty }} \lesssim \left\| f\right\| _{H_{p(\cdot ),\infty }} \\&= \left\| M(f)\right\| _{{p(\cdot ),\infty }} \lesssim \left\| f\right\| _{{p(\cdot )}}, \end{aligned}$$

which proves the second inequality. $\square$

Finally, besides Corollary 2, we give equivalent characterizations of the Hardy–Lorentz spaces with the help of the maximal operators defined above.

Corollary 3

Let $p(\cdot ) \in C^{\log }$, $0<p_-\le p_+<\infty$, $0<q \le \infty$ and $0<\gamma ,s<\infty$. If (5) holds, $f\in H_{p(\cdot ),q}$ and $j=1,\ldots ,4$, then

$$\begin{aligned} \Vert f\Vert _{H_{p(\cdot ),q}} = \Vert M(f)\Vert _{p(\cdot ),q} \le \Vert U_{\gamma ,s}^{(j)}(f)\Vert _{p(\cdot ),q} \le \Vert U_{\gamma ,s}(f)\Vert _{p(\cdot ),q} \le C_{p(\cdot )} \Vert f\Vert _{H_{p(\cdot ),q}}. \end{aligned}$$

Proof

The inequalities follow from (9) and Theorem 3. $\square$