Boundedness of dyadic maximal operators on variable Lebesgue spaces

Abstract

We introduce three types of dyadic maximal operators and prove that under some conditions on the variable exponent $p(\cdot )$, they are bounded on $L_{p(\cdot )}$ if $1<p_-\le p_+<\infty$. Here we correct Theorem 4.2 of the paper, Szarvas and Weisz (Banach J Math Anal 13:675–696, 2019).

1 Introduction

For a measurable function $p(\cdot )$, the variable Lebesgue space $L_{p(\cdot )}$ consists of all measurable functions f for which $\int _0^{1}|f(x)|^{p(x)}dx<\infty$. If $p(\cdot )$ is a constant, we get back the usual $L_{p}$ space. This topic needs essentially new ideas and is investigated very intensively in the literature nowadays (see e.g., Cruz-Uribe and Fiorenza [1], Diening et al. [2], Nakai and Sawano [9, 10], Jiao et al. [4,5,6], Liu et al. [7, 8]). Interest in the variable Lebesgue spaces has increased since the 1990s because of their use in a variety of applications (see the references in Jiao et al. [4]).

Usually, we suppose that $p(\cdot )$ satisfies the Hölder continuity condition. In [4, 5, 12] as well as in this paper, we suppose a slightly more general condition. It is known (see [4, 5]) that the usual dyadic (or martingale) maximal operator is bounded on $L_{p(\cdot )}$ with $1<p_-\le p_+<\infty$. For the boundedness of the classical Hardy-Littlewood maximal operator see e.g., Cruz-Uribe and Fiorenza [1] and Diening et al. [2].

In [12], we investigated two more dyadic maximal operators denoted by $U_{\beta ,s}$ and $V_\beta$, where $\beta$ and s are positive parameters. We stated there that $U_{\beta ,s}$ is bounded on $L_{p(\cdot )}$ if $1<p_-\le p_+<\infty$ and $\beta ,s>0$. However, there is a mistake in the proof, as we applied Lemma 2 in a wrong way. In this paper, we correct the proof and prove that, for $1<p_-\le p_+<\infty$, $U_{\beta ,s}$ is bounded on $L_{p(\cdot )}$ under the additional condition $\frac{1}{p_-}-\frac{1}{p_+} < \beta +s$. We show also that without this last additional condition, the boundedness of $U_{\beta ,s}$ does not hold. Next, we verify that $V_{\beta }$ is bounded on $L_{p(\cdot )}$ if $1<p_-\le p_+<\infty$ and $\beta >0$, without any additional condition. This was stated in [12] without proof.

In [12], we used the boundedness of the above dyadic maximal operators to prove that the maximal Cesàro (or $(C,\alpha )$) and Riesz operators of the Walsh-Fourier series are bounded from the variable Hardy space $H_{p(\cdot )}$ to $L_{p(\cdot )}$ if $1/(\alpha +1)<p_-<\infty$. Here we correct the theorem and show that under this last condition, the boundedness holds if and only if $\frac{1}{p_-}-\frac{1}{p_+} <1$.

2 Variable Lebesgue spaces

In this section, we recall some basic notations on variable Lebesgue spaces and give some elementary and necessary facts about these spaces. Our main references are Cruz-Uribe and Fiorenza [1] and Diening et al. [2].

For a constant p, the $L_p$ space is equipped with the quasi-norm

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

with the usual modification for $p=\infty$. Here we integrate with respect to the Lebesgue measure $\lambda$.

We are going to generalize these spaces. A measurable function $p(\cdot ): [0,1)\rightarrow (0,\infty )$ is called a variable exponent. For any variable exponent $p(\cdot )$ and any measurable set $A \subset [0,1)$, we will use the notation

$$\begin{aligned} p_{-}(A) := \mathop {\mathrm{ess\,inf}}_{x\in A} p(x) \quad \text{ and } \qquad p_{+}(A) := \mathop {\mathrm{ess\,sup}}_{x\in A} p(x). \end{aligned}$$

If $A=[0,1)$, then the numbers $p_{-}(A)$ and $p_{+}(A)$ are denoted simply by $p_{-}$ and $p_{+}$. Denote by ${\mathcal {P}}$ the collection of all variable exponents $p(\cdot )$ satisfying

$$\begin{aligned} 0<p_-\le p_+<\infty . \end{aligned}$$

The variable Lebesgue space $L_{p(\cdot )}$ contains all measurable functions f, for which

$$\begin{aligned} \Vert f\Vert _{L_{p(\cdot )}}:=\inf \left\{ \rho \in (0,\infty ): \int _0^{1} \left( \frac{|f(x)}{\rho }\right) ^{p(x)}\,dx \le 1\right\} <\infty . \end{aligned}$$

Instead of the log-Hölder continuity condition, in [4, 12], we introduced the slightly more general condition

$$\begin{aligned} \lambda (I)^{p_-(I)-p_+(I)} \le C \end{aligned}$$

(1)

for all dyadic intervals $I \subset [0,1)$. By a dyadic interval, we mean one of the form $[k2^{-n},(k+1)2^{-n})$ for some $k,n \in {{\mathbb {N}}}$, $0\le k <2^{n}$.

Remark 1

There exist a lot of functions $p(\cdot )$ satisfying (1). For concrete examples we mention the function $a+cx$ for parameters a and c such that the function is positive ($x \in [0,1)$). All positive Lipschitz functions with order $0< \beta \le 1$ also satisfy (1).

The following lemma was proved in Cruze-Uribe and Fiorenza [1] and Hao and Jiao [3].

Lemma 1

Let $p(\cdot )\in {\mathcal {P}}$satisfy (1). Then, for any dyadic interval $I \subset [0,1)$,

$$\begin{aligned} \lambda (I)^{1/{p_-(I)}} \sim \lambda (I)^{1/{p(x)}} \sim \lambda (I)^{1/{p_+(I)}} \sim \Vert \chi _I\Vert _{p(\cdot )} \quad (\forall x\in I), \end{aligned}$$

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

The following lemma can be found in Jiao et al. [4, 5].

Lemma 2

Let $p(\cdot )\in {\mathcal {P}}$, $1\le p_-\le p_+ < \infty$, satisfy (1). Suppose that $f\in L_{p(\cdot )}$with $\left\| f\right\| _{p(\cdot )}\le 1/2$and $f=f \chi _{\{|f| \ge 1\}}$. Then, for any dyadic interval $I \subset [0,1)$and $x\in I$,

$$\begin{aligned} \left( \frac{1}{\lambda (I)} \int _I |f(t)| \, dt\right) ^{p(x)} \le \left( \frac{C}{\lambda (I)} \int _I |f(t)|^{p(t)}\, dt\right) . \end{aligned}$$

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 $A \le CB$.

3 Dyadic maximal operators

In this section, in addition to the well known Doob’s maximal operator, we introduce two new types of maximal operators. The Doob’s maximal operator is given by

$$\begin{aligned} Mf(x) := \sup _{x\in I} \frac{1}{\lambda (I)}\left| \int _{I} f d \lambda \right| , \end{aligned}$$

where the supremum is taken over all dyadic intervals. The following result was proved in Jiao et al. [5]. For the boundedness of the classical Hardy–Littlewood maximal operator see e.g. Cruz-Uribe and Fiorenza [1] and Diening et al. [2].

Theorem 1

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

$$\begin{aligned} \left\| Mf\right\| _{p(\cdot )} \le C_{p(\cdot )} \Vert f\Vert _{{p(\cdot )}} \qquad (f\in L_{p(\cdot )}). \end{aligned}$$

For an integrable function $f \in L_1$, we define the second maximal operator by

$$\begin{aligned} U_{\beta ,s}f(x) := \sup _{x\in I} \sum _{m=1}^{n} \sum _{j=0}^{m-1} 2^{(j-n)\beta } \sum _{i=j}^{m-1} 2^{(j-i)s} \frac{1}{\lambda (I^{j,i})}\left| \int _{I^{j,i}} f d \lambda \right| , \end{aligned}$$

where I is a dyadic interval with length $2^{-n}$, $\beta ,s$ are positive constants and

$$\begin{aligned} I^{j,i}:= I\dot{+} [2^{-j-1},2^{-j-1} \dot{+} 2^{-i}). \end{aligned}$$

Here $\dot{+}$ denotes the dyadic addition (see e.g., Schipp, Wade, Simon and Pál [11]). Let us define $I_{k,n}:= [k2^{-n},(k+1)2^{-n})$ with $0\le k<2^{n}$, $n\in {{\mathbb {N}}}$. The preceding definition can be rewritten to

$$\begin{aligned} U_{\beta ,s}f := \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^{n}-1} \chi _{I_{k,n}}\sum _{m=1}^{n} \sum _{j=0}^{m-1} 2^{(j-n)\beta } \sum _{i=j}^{m-1} 2^{(j-i)s} \frac{1}{\lambda (I^{j,i}_{k,n})}\left| \int _{I^{j,i}_{k,n}} f d \lambda \right| . \end{aligned}$$

The following theorem was proved in [12].

Theorem 2

For all $1<p<\infty$and all $0<\beta ,s<\infty$, we have

$$\begin{aligned} \Vert U_{\beta ,s} f\Vert _{p} \le C_{p} \Vert f\Vert _{p} \qquad (f\in L_{p}). \end{aligned}$$

Now we generalize this theorem to variable Lebesgue spaces.

Theorem 3

Let $p(\cdot ) \in {\mathcal {P}}$satisfy (1), $1<p_-\le p_+<\infty$and $0<\beta ,s<\infty$. If

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

(2)

then

$$\begin{aligned} \Vert U_{\beta ,s}f\Vert _{p(\cdot )} \le C_{p(\cdot )} \Vert f\Vert _{{p(\cdot )}} \qquad (f\in L_{p(\cdot )}). \end{aligned}$$

Proof

It is easy to see that we may suppose the conditions $\Vert f\Vert _{p(\cdot )}\le 1/2$, $|f| \ge 1$ or $f=0$ and

$$\begin{aligned} \frac{1}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}} |f(t)| \, dt >1. \end{aligned}$$

We denote by $I_{k,n,j,i,1}$ (resp. $I_{k,n,j,i,2}$) those points $x \in I_{k,n}$ for which $p(x) \le p_+(I_{k,n}^{j,i})$ (resp. $p(x) > p_+(I_{k,n}^{j,i})$). Then

$$\begin{aligned} \int _0^{1}|U_{\beta ,s}f(x)|^{p(x)} \,dx \; &\lesssim \sum _{l=1}^{2}\int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \\&\qquad \left. \frac{\chi _{I_{k,n,j,i,l}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{p(x)} \, dx \\&=: (A)+(B). \end{aligned}$$

Let $q(x):=p(x)/p_0>1$ for some $1<p_0<p_-$. Using convexity and the fact that $q(x) \le q_+(I_{k,n}^{j})$ on $I_{k,n,j,l,1}$, we get that

$$\begin{aligned} (A)&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \left( \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \right. \\&\qquad \left. \left. \frac{\chi _{I_{k,n,j,i,1}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx \\&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \\&\qquad \left. \left( \frac{\chi _{I_{k,n,j,i,1}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx \\&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \\&\qquad \left. \left( \frac{\chi _{I_{k,n,j,i,1}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{q_+(I_{k,n}^{j,i})}\right) ^{p_0} \, dx . \end{aligned}$$

Lemma 2 and Theorem 2 imply

$$\begin{aligned} (A)&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \\&\qquad \left. \frac{\chi _{I_{k,n,j,i,1}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)|^{q(t)} \, dt\right) ^{p_0} \, dx \\&\lesssim \left\| U_{\beta ,s}(|f|^{q(\cdot )}) \right\| _{p_0}^{p_0} \lesssim \left\| |f|^{q(\cdot )} \right\| _{p_0}^{p_0} \le C. \end{aligned}$$

Choosing $0<\beta _0 < \beta$ and $0<r<s+\beta _0$, we obtain

$$\begin{aligned} (B)&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x)\left( \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)\beta } 2^{(j-i)s} \right. \right. \\&\qquad \left. \left. \frac{\chi _{I_{k,n,j,i,2}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx \\&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r)} \right. \\&\qquad \left. \left( 2^{(j-i)r} \frac{\chi _{I_{k,n,j,i,2}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx. \end{aligned}$$

By Hölder’s inequality,

$$\begin{aligned} (B)&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r)} \right. \\&\qquad \left. 2^{(j-i)rq(x)} \left( \frac{\chi _{I_{k,n,j,i,2}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)|^{q_-(I_{k,n}^{j,i})} \, dt\right) ^{q(x)/q_-(I_{k,n}^{j,i})}\right) ^{p_0} \, dx \\&\lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \right. \\&\qquad \left. \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r)} 2^{(j-i)rq(x)} 2^{iq(x)/q_-(I_{k,n}^{j,i})} \right. \\&\qquad \left. \chi _{I_{k,n,j,i,2}}(x)\left( \int _{I_{k,n}^{j,i}}|f(t)|^{q_-(I_{k,n}^{j,i})} \, dt\right) ^{q(x)/q_-(I_{k,n}^{j,i})}\right) ^{p_0} \, dx. \end{aligned}$$

Since $|f| \ge 1$ or $f=0$, $q(x)>q_-(I_{k,n}^{j,i})$ on $I_{k,n,j,l,2}$, $q_-(I_{k,n}^{j,i}) \le q(t)< p(t)$ for all $t \in I_{k,n}^{j,i}$ and

$$\begin{aligned} \int _{I_{k,n}^{j,i}}|f(t)|^{q_-(I_{k,n}^{j,i})} \, dt \le \int _{I_{k,n}^{j,i}}|f(t)|^{p(t)} \, dt \le \frac{1}{2}, \end{aligned}$$

we can see that

$$\begin{aligned} (B)& \; \lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \right. \\ & \qquad \left. \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r)} 2^{(j-i)rq(x)} 2^{iq(x)/q_-(I_{k,n}^{j,i})-i} \right. \\&\qquad \left. \frac{\chi _{I_{k,n,j,i,2}}(x)}{\lambda (I_{k,n}^{j,i})} \int _{I_{k,n}^{j,i}}|f(t)|^{q_-(I_{k,n}^{j,i})} \, dt\right) ^{p_0} \, dx. \end{aligned}$$

For fixed k and n let $J_j$ denote the dyadic interval with length $2^{-j}$ and $I_{k,n} \subset J_j$. Then $I_{k,n}^{j,i} \subset J_j\dot{+} 2^{-j-1} = J_j$. Inequality (1) implies that $2^{-jp(x)} \sim 2^{-jp_-(I_{k,n}^{j,i})}$ for $x \in I_{k,n}$. It is easy to check that for $x \in I_{k,n,j,l,2}$,

$$\begin{aligned} 2^{jrq(x)}&= 2^{jrq(x)} 2^{jq(x)} 2^{-jq(x)} \lesssim 2^{jrq(x)} 2^{jq_-(I_{k,n}^{j,i})} 2^{-jq(x)} \\&< 2^{j\left( rq(x) - \frac{q(x)-q_-(I_{k,n}^{j,i})}{q_-(I_{k,n}^{j,i})}\right) } = 2^{j\left( rq(x) - \frac{q(x)}{q_-(I_{k,n}^{j,i})}+1 \right) }. \end{aligned}$$

Furthermore,

$$\begin{aligned} rq(x) - \frac{q(x)}{q_-(I_{k,n}^{j})}+1&\ge q(x)\left( r - \frac{1}{q_-}\right) +1 \\&\ge \left\{ \begin{array}{ll} 1, &{} \hbox {if } r - \frac{1}{q_-} \ge 0; \\ q_+\left( r - \frac{1}{q_-}\right) +1, &{} \hbox {if } r - \frac{1}{q_-} < 0. \end{array} \right. \end{aligned}$$

Let $r_0:= \min \left( 1,q_+\left( r - \frac{1}{q_-}\right) +1\right)$. Then $r_0>0$ if and only if

$$\begin{aligned} \frac{1}{q_-}-\frac{1}{q_+} <r. \end{aligned}$$

(3)

Hence

$$\begin{aligned} (B)\;& \lesssim \; \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \right. \\&\qquad \left. \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r)} 2^{(j-i)\left( rq(x) - \frac{q(x)}{q_-(I_{k,n}^{j,i})}+1 \right) } \right. \\&\qquad \left. \frac{1}{\lambda (I_{k,n}^{j,i})}\int _{I_{k,n}^{j,i}}|f(t)|^{q(t)} \, dt \right) ^{p_0} \, dx\\&\lesssim \; \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=1}^{n} \sum _{j=0}^{m-1}\sum _{i=j}^{m-1} 2^{(j-n)(\beta -\beta _0)} 2^{(j-i)(\beta _0+s-r+r_0)} \right. \\&\qquad \left. \frac{1}{\lambda (I_{k,n}^{j,i})}\int _{I_{k,n}^{j,i}}|f(t)|^{q(t)} \, dt \right) ^{p_0} \, dx\\& \lesssim \; \left\| U_{\beta - \beta _0,\beta _0+s-r+r_0}(|f|^{q(\cdot )}) \right\| _{p_0}^{p_0} \lesssim \; \left\| |f|^{q(\cdot )} \right\| _{p_0}^{p_0} \le C, \end{aligned}$$

whenever (3) holds. Since r can be arbitrarily near to $s+ \beta _0$ and $\beta _0$ to $\beta$, this completes the proof. $\square$

Remark 2

Inequality (2) and Theorem 3 hold if $p_->\max (1/(\beta +s),1)$.

In [12], we used the parameters $\beta =(1+\alpha )t-r/(r-t)>0$, $s=r/(r-t)-\alpha t>0$ and stated that Theorem 3 holds without the condition (2). However, this is not the case.

Theorem 4

Let $p(\cdot ) \in {\mathcal {P}}$satisfy (1), $1<p_-\le p_+<\infty$and $0<\beta ,s<\infty$. If

$$\begin{aligned} \frac{1}{p_-(I_{0,n}^{1,n-1})}-\frac{1}{p_+(I_{0,n})} > \beta +s \end{aligned}$$

(4)

for all $n \in {{\mathbb {N}}}$, then $U_{\beta ,s}$is not bounded on $L_{p(\cdot )}$.

Proof

Choosing $m=n$, $j=0$ and $i=n-1$, we can see that

$$\begin{aligned}&\int _0^{1}|U_{\beta ,s}f(x)|^{p(x)} \,dx \\&\quad \ge \int _0^{1} \chi _{I_{0,n}}(x) \left( 2^{-n(\beta +s)} \frac{1}{\lambda (I_{0,n}^{1,n-1})} \left| \int _{I_{0,n}^{1,n-1}}f(t) \, dt \right| \right) ^{p(x)} \, dx. \end{aligned}$$

Let

$$\begin{aligned} f(t):= \chi _{I_{0,n}^{1,n-1}}(t) 2^{n/p_-(I_{0,n}^{1,n-1})}. \end{aligned}$$

Lemma 1 implies that

$$\begin{aligned} \Vert f\Vert _{p(\cdot )} =2^{n/p_-(I_{0,n}^{1,n-1})} \Vert \chi _{I_{0,n}^{1,n-1}}\Vert _{p(\cdot )} \le C. \end{aligned}$$

Thus, by (1),

$$\begin{aligned} \int _0^{1}|U_{\beta ,s}f(x)|^{p(x)} \,dx&\ge \int _{I_{0,n}} 2^{-n(\beta +s)p(x)} 2^{np(x)/p_-(I_{0,n}^{1,n-1})} \, dx \\&\ge C \int _{I_{0,n}} 2^{np_+(I_{0,n})(1/p_-(I_{0,n}^{1,n-1})- \beta -s)} \, dx \\&= C 2^{np_+(I_{0,n})(1/p_-(I_{0,n}^{1,n-1})- \beta -s)} 2^{-n} \end{aligned}$$

which tends to infinity as $n\rightarrow \infty$ if (4) holds. $\square$

The third dyadic maximal operator is introduced by

$$\begin{aligned} V_\beta f(x) := \sup _{x \in I} \sum _{m=0}^{n-1} 2^{(m-n)\beta } \, \frac{1}{\lambda (I^{m})} \left| \int _{I^{m}} f \right| , \end{aligned}$$

where $f \in L_1$, I is a dyadic interval with length $2^{-n}$, $\beta$ is a positive constant and

$$\begin{aligned} I^{m}:= I\dot{+} [0,2^{-m}). \end{aligned}$$

Then

$$\begin{aligned} V_\beta f = \sup _{n \in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}} \sum _{m=0}^{n-1} 2^{(m-n)\beta } \frac{1}{\lambda (I_{k,n}^{m})} \left| \int _{I_{k,n}^{m}} f \right| . \end{aligned}$$

The following theorem can be found in [12].

Theorem 5

For all $1<p<\infty$and all $0<\beta <\infty$, we have

$$\begin{aligned} \Vert V_{\beta } f\Vert _{p} \le C_{p} \Vert f\Vert _{p} \qquad (f\in L_{p}). \end{aligned}$$

The generalization of this result reads as follows.

Theorem 6

If $p(\cdot ) \in {\mathcal {P}}$satisfies (1), $1<p_-\le p_+<\infty$and $0<\beta <\infty$, then

$$\begin{aligned} \Vert V_{\beta }f\Vert _{p(\cdot )} \le C_{p(\cdot )} \Vert f\Vert _{{p(\cdot )}} \qquad (f\in L_{p(\cdot )}). \end{aligned}$$

Proof

Similarly to the proof of Theorem 3, we may suppose again that $\Vert f\Vert _{p(\cdot )}\le 1/2$ and

$$\begin{aligned} \frac{1}{\lambda (I_{k,n}^{m})} \int _{I_{k,n}^{m}} |f(t)| \, dt >1. \end{aligned}$$

Since $I_{k,n} \subset I_{k,n}^{m}$$(m=0,\ldots ,n-1)$, we can apply Lemma 2 and Theorem 5 to obtain

$$\begin{aligned}&\int _0^{1}|V_{\beta }f(x)|^{p(x)} \,dx\\&\quad \lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \left( \sum _{m=0}^{n-1} 2^{(m-n)\beta } \frac{1}{\lambda (I_{k,n}^{m})} \int _{I_{k,n}^{m}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx \\&\quad \lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=0}^{n-1} 2^{(m-n)\beta } \left( \frac{1}{\lambda (I_{k,n}^{m})} \int _{I_{k,n}^{m}}|f(t)| \, dt\right) ^{q(x)}\right) ^{p_0} \, dx \\&\quad \lesssim \int _0^{1} \left( \sup _{n\in {{\mathbb {N}}}} \sum _{k=0}^{2^n-1} \chi _{I_{k,n}}(x) \sum _{m=0}^{n-1} 2^{(m-n)\beta } \frac{1}{\lambda (I_{k,n}^{m})} \int _{I_{k,n}^{m}}|f(t)|^{q(t)} \, dt\right) ^{p_0} \, dx \\&\quad \lesssim \left\| V_{\beta }(|f|^{q(\cdot )}) \right\| _{p_0}^{p_0} \lesssim \left\| |f|^{q(\cdot )} \right\| _{p_0}^{p_0} \le C, \end{aligned}$$

which proves the theorem. $\square$

4 The maximal Cesàro and Riesz operator on $H_{p(\cdot )}$

Using Theorems 3 and 6, we can prove the boundedness of the maximal Cesàro and Riesz operators of Walsh-Fourier series as in [12].

Theorem 7

Let $p(\cdot ) \in {\mathcal {P}}$satisfy (1) and

$$\begin{aligned} \frac{1}{p_-}-\frac{1}{p_+} <1. \end{aligned}$$

(5)

If $0 < \alpha \le 1 \le \gamma$and $1/(\alpha +1)< p_{-} < \infty$, then

$$\begin{aligned} \left\| \sigma _*^{\alpha }f\right\| _{{p(\cdot )}} + \left\| \sigma _*^{\alpha ,\gamma }f\right\| _{{p(\cdot )}}\lesssim \left\| f\right\| _{H_{p(\cdot )}} \qquad (f\in H_{p(\cdot )}). \end{aligned}$$

The same holds for the space $H_{p(\cdot ),q}$$(0<q \le \infty )$.

For the definitions, details and proof see [12]. We stated there that Theorem 7 holds without the condition (5). However, as we have seen in Jiao et al. [4], this is not true for $\alpha =\gamma =1$.