Boundedness of dyadic maximal operators on variable Lebesgue spaces

We introduce three types of dyadic maximal operators and prove that under some conditions on the variable exponent p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document}, they are bounded on Lp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{p(\cdot )}$$\end{document} if 1<p-≤p+<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p_-\le p_+<\infty $$\end{document}. Here we correct Theorem 4.2 of the paper, Szarvas and Weisz (Banach J Math Anal 13:675–696, 2019).


Introduction
For a measurable function pðÁÞ, the variable Lebesgue space L pðÁÞ consists of all measurable functions f for which R 1 0 jf ðxÞj pðxÞ dx\1. If pðÁÞ 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 pðÁÞ 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 pðÁÞ with 1\p À p þ \1. 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 b;s and V b , where b and s are positive parameters. We stated there that U b;s is bounded on L pðÁÞ if 1\p À p þ \1 and b; 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 À p þ \1, U b;s is bounded on L pðÁÞ under the additional condition 1 p À À 1 p þ \b þ s. We show also that without this last additional condition, the boundedness of U b;s does not hold. Next, we verify that V b is bounded on L pðÁÞ if 1\p À p þ \1 and b [ 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 ðC; aÞ) and Riesz operators of the Walsh-Fourier series are bounded from the variable Hardy space H pðÁÞ to L pðÁÞ if 1=ða þ 1Þ\p À \1.
Here we correct the theorem and show that under this last condition, the boundedness holds if and only if 1 p À À 1 p þ \1.

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 kf k p :¼ with the usual modification for p ¼ 1.
Here we integrate with respect to the Lebesgue measure k.
We are going to generalize these spaces. A measurable function pðÁÞ : ½0; 1Þ ! ð0; 1Þ is called a variable exponent. For any variable exponent pðÁÞ and any measurable set A & ½0; 1Þ, we will use the notation p À ðAÞ :¼ ess inf x2A pðxÞ and p þ ðAÞ :¼ ess sup x2A pðxÞ: If A ¼ ½0; 1Þ, then the numbers p À ðAÞ and p þ ðAÞ are denoted simply by p À and p þ . Denote by P the collection of all variable exponents pðÁÞ satisfying 0\p À p þ \1: The variable Lebesgue space L pðÁÞ contains all measurable functions f, for which kf k L pðÁÞ :¼ inf q 2 ð0; 1Þ :

\1:
Instead of the log-Hölder continuity condition, in [4,12], we introduced the slightly more general condition for all dyadic intervals I & ½0; 1Þ. By a dyadic interval, we mean one of the form ½k2 Àn ; ðk þ 1Þ2 Àn Þ for some k; n 2 N, 0 k\2 n .
Remark 1 There exist a lot of functions pðÁÞ satisfying (1). For concrete examples we mention the function a þ cx for parameters a and c such that the function is positive (x 2 ½0; 1Þ). All positive Lipschitz functions with order 0\b 1 also satisfy (1).
The following lemma was proved in Cruze-Uribe and Fiorenza [1] and Hao and Jiao [3].
The following lemma can be found in Jiao et al. [4,5]. In this paper the constants C are absolute constants and the constants C pðÁÞ are depending only on pðÁÞ and may denote different constants in different contexts. For two positive numbers A and B, we use also the notation A.B, which means that there exists a constant C such that A CB.

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 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 pðÁÞ 2 P satisfies (1) and 1\p À p þ \1, then Mf k k pðÁÞ C pðÁÞ kf k pðÁÞ ðf 2 L pðÁÞ Þ: For an integrable function f 2 L 1 , we define the second maximal operator by where I is a dyadic interval with length 2 Àn , b; s are positive constants and Here _ þ denotes the dyadic addition (see e.g., Schipp, Wade, Simon and Pál [11]). Let us define I k;n :¼ ½k2 Àn ; ðk þ 1Þ2 Àn Þ with 0 k\2 n , n 2 N. The preceding definition can be rewritten to The following theorem was proved in [12].
Theorem 2 For all 1\p\1 and all 0\b; s\1, we have Now we generalize this theorem to variable Lebesgue spaces.
Proof Choosing m ¼ n, j ¼ 0 and i ¼ n À 1, we can see that where f 2 L 1 , I is a dyadic interval with length 2 Àn , b is a positive constant and I m :¼ I _ þ½0; 2 Àm Þ: Then The following theorem can be found in [12].
Theorem 5 For all 1\p\1 and all 0\b\1, we have kV b f k p C p kf k p ðf 2 L p Þ: The generalization of this result reads as follows.