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).
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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
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
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
The variable Lebesgue space \(L_{p(\cdot )}\) contains all measurable functions f, for which
Instead of the log-Hölder continuity condition, in [4, 12], we introduced the slightly more general condition
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 \(I \subset [0,1)\),
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 \(x\in I\),
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
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
For an integrable function \(f \in L_1\), we define the second maximal operator by
where I is a dyadic interval with length \(2^{-n}\), \(\beta ,s\) are positive constants and
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
The following theorem was proved in [12].
Theorem 2
For all \(1<p<\infty \)and all \(0<\beta ,s<\infty \), we have
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
then
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
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
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
Choosing \(0<\beta _0 < \beta \) and \(0<r<s+\beta _0\), we obtain
By Hölder’s inequality,
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
we can see that
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 \(x \in I_{k,n,j,l,2}\),
Furthermore,
Let \(r_0:= \min \left( 1,q_+\left( r - \frac{1}{q_-}\right) +1\right) \). Then \(r_0>0\) if and only if
Hence
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
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
Let
Lemma 1 implies that
Thus, by (1),
which tends to infinity as \(n\rightarrow \infty \) if (4) holds. \(\square \)
The third dyadic maximal operator is introduced by
where \(f \in L_1\), I is a dyadic interval with length \(2^{-n}\), \(\beta \) is a positive constant and
Then
The following theorem can be found in [12].
Theorem 5
For all \(1<p<\infty \)and all \(0<\beta <\infty \), we have
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
Proof
Similarly to the proof of Theorem 3, we may suppose again that \(\Vert f\Vert _{p(\cdot )}\le 1/2\) and
Since \(I_{k,n} \subset I_{k,n}^{m}\)\((m=0,\ldots ,n-1)\), we can apply Lemma 2 and Theorem 5 to obtain
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
If \(0 < \alpha \le 1 \le \gamma \)and \(1/(\alpha +1)< p_{-} < \infty \), then
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 \(\alpha =\gamma =1\).
References
Cruz-Uribe, D.V., Fiorenza, A.: Variable Lebesgue spaces. Foundations and harmonic analysis. Birkhäuser/Springer, New York (2013)
Diening, L., Harjulehto, P., Hästö, P., Ružička, M.: Lebesgue and Sobolev spaces with variable exponents. Springer, Berlin (2011)
Hao, Z., Jiao, Y.: Fractional integral on martingale Hardy spaces with variable exponents. Fract. Calc. Appl. Anal. 18(5), 1128–1145 (2015)
Jiao, Y., Weisz, F., Wu, L., Zhou, D.: Variable martingale Hardy spaces and their applications in Fourier analysis. Dissertationes Math. (to appear)
Jiao, Y., Zhou, D., Hao, Z., Chen, W.: Martingale Hardy spaces with variable exponents. Banach J. Math 10, 750–770 (2016)
Jiao, Y., Zuo, Y., Zhou, D., Wu, L.: Variable Hardy–Lorentz spaces \(H^{p(\cdot ), q}({\mathbb{R}}^n)\). Math. Nachr. 292, 309–349 (2019)
Liu, J., Weisz, F., Yang, D., Yuan, W.: Variable anisotropic Hardy spaces and their applications. Taiwanese J. Math. 22, 1173–1216 (2018)
Liu, J., Weisz, F., Yang, D., Yuan, W.: Littlewood–Paley and finite atomic characterizations of anisotropic variable Hardy-Lorentz spaces and their applications. J. Fourier Anal. Appl. 25, 874–922 (2019)
Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262(9), 3665–3748 (2012)
Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integral Equ. Oper. Theory 77, 123–148 (2013)
Schipp, F., Wade, W.R., Simon, P., Pál, J.: Walsh Series: An Introduction to Dyadic Harmonic Analysis. Adam Hilger, Bristol, New York (1990)
Szarvas, K., Weisz, F.: The boundedness of the Cesaro- and Riesz means in variable dyadic Hardy spaces. Banach J. Math. Anal. 13, 675–696 (2019)
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Open access funding provided by Eötvös Loránd University (ELTE).
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This research was supported by the Hungarian National Research, Development and Innovation Office—NKFIH, K115804 and KH130426.
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Communicated by M. S. Moslehian.
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Weisz, F. Boundedness of dyadic maximal operators on variable Lebesgue spaces. Adv. Oper. Theory 5, 1588–1598 (2020). https://doi.org/10.1007/s43036-020-00071-9
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DOI: https://doi.org/10.1007/s43036-020-00071-9