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 \(\alpha =\gamma =1\).