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A note on Hardy-Littlewood maximal operators
Journal of Inequalities and Applications volume 2016, Article number: 21 (2016)
Abstract
In this paper, we will prove that, for \(1< p<\infty\), the \(L^{p}\) norm of the truncated centered Hardy-Littlewood maximal operator \(M^{c}_{\gamma}\) equals the norm of the centered Hardy-Littlewood maximal operator for all \(0<\gamma<\infty\). When \(p=1\), we also find that the weak \((1,1)\) norm of the truncated centered Hardy-Littlewood maximal operator \(M^{c}_{\gamma}\) equals the weak \((1,1)\) norm of the centered Hardy-Littlewood maximal operator for \(0<\gamma<\infty\). Moreover, the same is true for the truncated uncentered Hardy-Littlewood maximal operator. Finally, we investigate the properties of the iterated Hardy-Littlewood maximal function.
1 Introduction
Define the centered Hardy-Littlewood maximal function by
and the uncentered Hardy-Littlewood maximal function by
The basic real-variable construct was introduced by Hardy and Littlewood [1] for \(n=1\), and by Wiener [2] for \(n\ge2\). It is well known that the Hardy-Littlewood maximal function plays an important role in many parts of analysis. It is a classical mean operator, and it is frequently used to majorize other important operators in harmonic analysis.
It is clear that
holds for all \(x\in\Bbb {R}^{n}\). Both M and \(M^{c}\) are sublinear operators. Although the study of the boundedness for M or \(M^{c}\) is fairly completed, it is very hard to calculate the precise norm about M or \(M^{c}\).
As is well known, the truncated operator has some important properties. In fact, in most situations, \(L^{p}\) boundedness of the truncated operator and the corresponding oscillatory operator is equivalent. There are many works in this regard and the reader can refer to [3] and [4].
Now we define the truncated centered Hardy-Littlewood maximal operator and the truncated uncentered Hardy-Littlewood maximal operator.
Define
and
for \(x\in \Bbb {R}^{n}\) and some real positive number γ.
Obviously, like the inequality (1.3), in the pointwise sense, we immediately deduce from the definition (1.4) and (1.5) that
and
for all \(x\in \Bbb {R}^{n}\), as long as \(\gamma\le\rho\). Consequently, referring to the two truncated operators \(M^{c}_{\gamma}\) and \(M_{\gamma}\), as the sublinear operators, we naturally obtain
and
if \(\gamma\le\rho\), for \(1< p\le\infty\). Clearly, when γ is fixed, for example \(\gamma=1\), \(\Vert M^{c}_{1}\Vert _{L^{p}(\Bbb {R}^{n})\rightarrow L^{p}(\Bbb {R}^{n})}\) and \(\|M^{c}\| _{L^{p}(\Bbb {R}^{n})\rightarrow L^{p}(\Bbb {R}^{n})}\) are two fixed numbers. We think that it is very significant to make certain the precise relation of the two numbers. In the paper, we will consider the question. Surprisingly, the two numbers are equal whenever \(\gamma>0\). The same is true for \(p=1\).
Now we formulate our main theorems.
Theorem 1.1
Let \(M^{c}_{\gamma}\) be defined by (1.4) and \(\gamma>0\). Then
holds for \(1< p\le\infty\).
Theorem 1.2
Let \(M^{c}_{\gamma}\) be defined by (1.4) and \(\gamma>0\). Then
holds.
For the truncated uncentered Hardy-Littlewood Maximal operator, we have similar conclusions.
Theorem 1.3
Let \(M_{\gamma}\) be defined by (1.5) and \(\gamma>0\). Then
holds for \(1< p\le\infty\).
Theorem 1.4
Let \(M_{\gamma}\) be defined by (1.5) and \(\gamma>0\). Then
holds.
In Section 4, we will investigate the properties of the iterated Hardy-Littlewood maximal function.
2 Auxiliary and some lemmas
To prove our main theorems, we first provide some definitions and lemmas which will be used in the follows. Some lemmas can be found in the classic literature and here we omit their proofs.
Definition 2.1
Let f be a measurable function on \(\Bbb {R}^{n}\). The distribution function of f is the function \(d_{f}\) defined on \([0,+\infty)\) as follows:
where \(|A|\) is the Lebesgue measure of the measurable set A.
Lemma 2.1
For \(f\in L^{p}(\Bbb {R}^{n})\) with \(0< p<\infty\), we have
It is easy for us to verify the lemma by Fubini’s theorem. For more details as regards this lemma, one can refer to [5].
Lemma 2.2
Suppose that μ is a positive measure on a σ-algebra \(\mathbb{M}\). If \(A_{1}\subset A_{2}\subset A_{3}\cdots\), \(A_{n}\in\mathbb {M}\), and \(A=\bigcup_{n=1}^{\infty}A_{n}\), then
Lemma 2.2 can be found in the book [6]. Using Lemma 2.2, we can formulate the following conclusions.
Lemma 2.3
Suppose that the operators \(M^{c}\) and \(M_{\gamma}^{c}\) are defined as in (1.1) and (1.4). The equality
holds for all \(f\in L^{p}(\Bbb {R}^{n})\) and \(\lambda>0\).
Proof
For a fixed \(x\in \Bbb {R}^{n}\), by the definition of \(M^{c}\) in (1.1), associate to each ε a ball \(B(x,r_{\varepsilon})\) which satisfies
Now taking \(\gamma>r_{\varepsilon}\), it follows from the definition of \(M^{c}_{\gamma}\) that
Note that \(M^{c}_{\gamma}f(x)\) increases as \(\gamma\rightarrow\infty \). Thus we have
Clearly, we have
Hence combining (2.6) with (2.7) yields
Obviously it implies from (2.8) that
Set
and
We have \(A_{n}\subset A_{n+1}\) for \(n=1,2,\ldots\) , and \(A=\bigcup _{n=1}^{\infty}A_{n}\). It follows from Lemma 2.2 and the definition of the distribution function that
This is our desired result. □
Using the same method as in the proof of Lemma 2.3, we obtain Lemma 2.4.
Lemma 2.4
Suppose that the operators M and \(M_{\gamma}\) are defined as in (1.2) and (1.5). For a given \(\lambda>0\), the equality
holds for all \(f\in L^{p}(\Bbb {R}^{n})\).
Lemma 2.5
Let \(1< p<\infty\). For \(\varepsilon>0\), there exists a function \(g\in C_{c}^{\infty}(\Bbb {R}^{n})\) such that
where
Proof
By the definition of the operator norm of \(M^{c}\), we can find a function \(f\in L^{p}(\Bbb {R}^{n})\) such that
Since \(C_{c}^{\infty}(\Bbb {R}^{n})\) is dense in \(L^{p}(\Bbb {R}^{n})\), for \(\delta>0\), there exists a function \(g\in C_{c}^{\infty}(\Bbb {R}^{n})\) which satisfies
Thus it implies from (2.13) that
where the constant A is a bound of the operator \(M^{c}\).
Combining (2.13) with (2.14) yields
If the number δ is small enough, we can immediately deduce that
It implies from (2.15) and (2.16) that the inequality (2.11) holds. □
3 Proof of main theorems
Now we shall prove our main theorems. We first consider the case \(1< p<\infty\).
Proof of Theorem 1.1
For convenience, we first prove
for all \(0<\gamma<\infty\).
From the definition of the operator \(M^{c}_{\gamma}\) in (1.4), we have
for \(x\in \Bbb {R}^{n}\) and \(0<\gamma<\infty\), where \(v_{n}\) is the volume of the unit ball in \(\Bbb {R}^{n}\).
A simple computation implies that
where the dilation operator \(\tau_{\gamma}\) is defined as follows:
for \(\gamma>0\) and \(x\in \Bbb {R}^{n}\).
It follows from (3.2) that
Taking the supremum over all \(f\in L^{p}(\Bbb {R}^{n})\) with \(\|f\|_{L^{p}(\Bbb {R}^{n})}\neq0\) for the two sides of equation (3.4), we have
Next, we will use equation (3.5) to prove
for all \(\gamma>0\).
Since
we merely need to prove
By Lemma 2.5, for \(\varepsilon>0\), there exists a function \(g\in C_{c}^{\infty}(\Bbb {R}^{n})\) such that
We may assume that the support of g is contained in the ball \(B(0,R)\), where R is a positive number. Since \(g\in C_{c}^{\infty}(\Bbb {R}^{n})\) implies \(g\in L^{p}(\Bbb {R}^{n})\), naturally we have \(M^{c}g\in L^{p}(\Bbb {R}^{n})\) by the \(L^{p}\) boundedness of the operator \(M^{c}\). It is not hard to find a positive number S such that
Now we set \(\gamma_{0}=R+S\). Then it can be deduced from the definition of \(M^{c}_{\gamma}\) that
holds for \(|x|< S\).
It follows from (3.6), (3.7), and (3.8) that
Obviously, (3.9) implies that
Consequently, the inequality (3.10) yields
By (3.5) and (3.11), we can derive from the arbitrariness property of ε that
for all \(\gamma>0\).
This finishes the proof of Theorem 1.1. □
Next we will pay attention to proving the weak \((1,1)\) boundedness for the truncated centered Hardy-Littlewood maximal operator.
Proof of Theorem 1.2
First, we prove that
holds for all \(0<\gamma<\infty\).
From the identity (3.2), we have
For any \(\lambda>0\), we derive from (3.13) that
Thus (3.14) implies that
If \(\|f\|_{ L^{1}(\Bbb {R}^{n})}\neq0\), then it follows from (3.15) that
Now taking the supremum over all \(f\in L^{1}(\Bbb {R}^{n})\) with \(\|f\|_{ L^{1}(\Bbb {R}^{n})}\neq0\) for the two sides of (3.16), we have
Next we will use (3.17) to prove that
holds for all \(\gamma>0\).
We assert the following equation:
holds for any \(f\in L^{1}(\Bbb {R}^{n})\) with \(\|f\|_{L^{1}}\neq0\).
Clearly the left side of (3.18) is not smaller than the right side, so it suffices to prove the opposite inequality.
It follows from Lemma 2.3 that
Set
For \(\varepsilon>0\), there must be a positive number \(\lambda_{0}\) such that
We conclude that
This is equivalent to
Consequently, (3.18) holds.
Using equation (3.18), we deduce that
Consequently, we immediately obtain our desired conclusion by the two identities (3.17) and (3.19). □
Proof of Theorem 1.3
We conclude from the definition of the operator \(M_{\gamma}\) in (1.5) that
Thus we have
for all \(\gamma>0\) and \(1< p<\infty\).
Next we will prove that
If \(f\in L^{p}(\Bbb {R}^{n})\), then we have \(Mf\in L^{p}(\Bbb {R}^{n})\). It follows from Lemma 2.1, Lemma 2.4, and equation (3.21) that
Since we have the obvious inequality
we derive from (3.22) that
This is our desired result. □
Proof of Theorem 1.4
Using the almost same methods of proving Theorem 1.2, we can formulate the proof of Theorem 1.4. □
4 Iterated Hardy-Littlewood maximal function
In this section, we will consider the iterated Hardy-Littlewood maximal function.
Let M be the uncentered Hardy-Littlewood maximal function defined by (1.2). Define the iterated Hardy-Littlewood maximal function denoted by \(M^{k+1}\) as follows:
for \(k=1,2,\ldots\) , and \(x\in \Bbb {R}^{n}\). Set \(M^{1}f(x):=Mf(x)\).
In order to study the properties of the iterated Hardy-Littlewood maximal function, we first introduce the following lemma.
Lemma 4.1
Suppose that a sequence \(\{c_{i}\}_{i=1}^{\infty}\) satisfies the following two conditions simultaneously:
-
(i)
\(c_{1}=r\in(0,1)\);
-
(ii)
for any \(k\ge1\), \(c_{k+1}=(1-r)c_{k}+r\).
Then \(\{c_{i}\}_{i=1}^{\infty}\) is strictly monotone increasing and we have
Proof
By the mathematical induction and the two conditions (i) and (ii), we can easily obtain \(0< c_{k}<1\) for each \(k\in\Bbb {N}\). Moreover, the condition (ii) implies
This shows that \(\{c_{i}\}_{i=1}^{\infty}\) is strictly monotone increasing. Since \(\{c_{i}\}_{i=1}^{\infty}\) is monotone increasing and has the upper bound, the limit of \(\{c_{i}\}_{i=1}^{\infty}\) exists, and we can easily get
 □
By Lemma 4.1, we have the following theorem.
Theorem 4.2
For any \(f\in L^{\infty}(\Bbb {R}^{n})\), the equation
holds for any \(x\in \Bbb {R}^{n}\).
Proof
If \(\|f\|_{\infty}=0\), the proof is trivial. If \(\|f\|_{\infty}>0\), for any \(\varepsilon\in(0,\|f\|_{\infty})\), define a set
Then we have \(|E_{\varepsilon}|>0\), where \(|E_{\varepsilon}|\) denotes the Lebesgue measure of \(E_{\varepsilon}\). For any fixed point \(a\in \Bbb {R}^{n}\), there exists a number \(R>0\) such that
Denote \(\tilde{E}_{\varepsilon}=E_{\varepsilon}\cap B(a,R)\).
Define a set as
Actually if \(f\in L^{p}(\Bbb {R}^{n})\) with \(1\le p\le\infty\), then \(|(S_{L}(f))^{c}|=0\), where \((S_{L}(f))^{c}\) denotes the complement set of \(S_{L}(f)\). When \(x\in\tilde{E}_{\varepsilon}\cap S_{L}(f)\), we derive from (4.3) that
and
for all \(k=1,2,\ldots\) .
When \(x\in B(a,R)\), we consider the uncentered Hardy-Littlewood maximal function of f at the point x. It follows that
Set
It implies from (4.5) that
A straightforward computation implies from (4.6) that
Denote
and
for \(k=1,2,\ldots\) .
It implies from (4.7) that
Using the inductive method, we can easily obtain
Thus Lemma 4.1 implies that
When \(\varepsilon\rightarrow0\), we have
By the definition of the Hardy-Littlewood function, we obviously deduce
Combining (4.8) with (4.9) yields
By the arbitrary choice of a, we obtain
for all \(x\in \Bbb {R}^{n}\). □
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Acknowledgements
The research was supported by the National Natural Foundation of China (Grant Nos. 11471039, 11271162, and 11561062).
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Wei, M., Nie, X., Wu, D. et al. A note on Hardy-Littlewood maximal operators. J Inequal Appl 2016, 21 (2016). https://doi.org/10.1186/s13660-016-0963-x
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DOI: https://doi.org/10.1186/s13660-016-0963-x