Boundedness of strong maximal functions with respect to non-doubling measures
- Wei Ding^{1},
- LiXin Jiang^{2} and
- YuePing Zhu^{2}Email author
https://doi.org/10.1186/s13660-016-1229-3
© Ding et al. 2016
Received: 4 March 2016
Accepted: 8 November 2016
Published: 22 November 2016
Abstract
Moreover, we also establish some boundedness result for the Cordoba maximal functions (Córdoba A. in Harmonic Analysis in Euclidean Spaces, pp. 29-50, 1978) associated with the Córdoba-Zygmund dilation in \(\mathbb{R}^{3}\) with respect to some non-doubling measures. This generalizes the result of Fefferman-Pipher (Am. J. Math. 119:337-369, 1997).
Keywords
1 Introduction
The classical theory of one-parameter harmonic analysis for maximal functions and singular integrals on \((\mathbb{R}^{n}; \mu)\) has been developed under the assumption that the underlying measure μ satisfies the doubling property, i.e., there exists a constant \(C > 0\) such that \(\mu(B(x; 2r)) \leq C\mu(B(x; r))\) for every \(x \in \mathbb{R}^{n}\) and \(r > 0\). However, some recent results [4–7] show that it should be possible to dispense with the doubling condition for most of the classical theory. It is well known that the use of doubling measure has two main advantages. One is that we can work with nested property. Another one is that the faces of the cubes have measure zero. As in [4, 6], we will only maintain the last property. If μ is a nonnegative Radon measure without mass-points, one can choose an orthonormal system in \(\mathbb{R}^{n}\) so that any cube Q with sides parallel to the coordinate axes satisfies the property \(\mu(\partial Q)=0\) (Theorem 2 of [4]). The advantage of assuming this property is the continuity of the measure μ on cubes which can ensure that there is a Calderón-Zygmund decomposition [4, 6], which is one of the basic and most frequently used tools in the classical theory.
We first recall some well-known results on one-parameter \(A_{p}(\mu)\) weights with respect to the possibly non-doubling measure μ. We also refer to [8] for the general theory of classical weights. A μ-measurable function ω is said to be a weight if it is nonnegative and μ-locally integrable. A weight ω is said to be an \(A_{p}(\mu)\) weight if ω satisfies the following definition.
Definition 1.1
We use the notation \(A_{\infty}(\mu)=\bigcup_{p>1} A_{p}(\mu)\) to denote the class of weight functions \(\omega\in A_{p}(\mu)\) for some \(p>1\).
When μ is a nonnegative Radon measure without mass-points, Lemma 2.3 in [6] tells us that some classical results for \(\omega\in A_{p}(\mu)\) also hold. We state these results as follows.
Proposition A
- (a)
\(\omega\in A_{\infty}(\mu)\);
- (b)ω satisfies a reverse Hölder inequality; namely, there are positive constants c and δ such that for every cube Qand c may be taken as close to 1 as \(\delta\rightarrow0\);$$\biggl(\frac{1}{\mu(Q)} \int_{Q}\omega^{1+\delta}\, d \mu \biggr)^{1/(1+\delta)} \leq \frac{c}{\mu(Q)} \int_{Q}\omega\, d \mu, $$
- (c)there are positive constants c and ρ such that, for any cube Q and any μ-measurable set F contained in Q,where \(\omega(E)=\int_{E}\omega\, d \mu\);$$\frac{\omega(F)}{\omega(Q)}\leq c \biggl(\frac{\mu(F)}{\mu(Q)} \biggr)^{\rho}, $$
- (d)there are positive constants \(\alpha,\beta<1\) such that whenever F is a measurable set of a cube Q,$$\frac{\mu(F)}{\mu(Q)}\leq\alpha\quad\textit{implies}\quad \frac {\omega(F)}{\omega(Q)}\leq \beta. $$
Remark 1.1
The behavior of the constant c in (b) is not explicitly obtained in [6]. But by a careful examination of its proof, we can find that c may be chosen as \((1-\frac{\delta C_{0}}{C_{1}^{1+\delta}})^{-1/(1+\delta)}\) for two fixed constants \(C_{0}\), \(C_{1}\).
If \(\mathcal{B}_{x}\) is the collection of all the cubes containing \(x\in\mathbb{R}^{n}\) (centered at x) whose sides are parallel to the coordinate axes, then we obtain the usual Hardy-Littlewood maximal function \(M_{d\nu}f(x)\) with respect to the measure dν (centered maximal function \(\bar{M}_{d\nu}f(x)\)). By means of the Besicovitch covering lemma, it is easy to prove that \(\bar{M}_{d\nu}\) maps \(L^{1}(d\nu)\) into weak \(L^{1}(d\nu)\), and \(L^{p}(d\nu)\) into \(L^{p}(d\nu)\) for \(p>1\). In dimension one, the non-centered maximal operator \(M_{d\nu}\) is also shown to be bounded on \(L^{p}(d\nu)\) for \(p>1\) (see [9] and [10]). However, it is in general not true that \(M_{d\nu}f(x)\) has these boundedness properties. We refer to [10] for counterexamples.
When \(\mathcal{B}_{x}\) denotes the collection of all rectangles R containing \(x\in\mathbb{R}^{n}\) whose sides parallel to the coordinate axes, \(\mathcal{M}\triangleq M_{d\nu}^{n}\) is the strong maximal operator with respect to measure ν in dimension n. When \(d\nu=dx\) is the Lebesgue measure on \(\mathbb{R}^{n}\), Jessen, Marcinkiewcz and Zygmund [11] and Fava [12] showed that \(M_{dx}^{n}\) is bounded on \(L^{p}\) for all \(p>1\). However, Fefferman [1] showed that it is generally not true for the boundedness properties for an arbitrary measure dν. It is thus natural to ask when is \(M_{d\nu}^{n}\) bounded on \(L^{p}(d\nu)\). Obviously, if ν on \(\mathbb{R}^{n}\) is a product measure of n one-dimensional nonnegative Radon measures, then the method of iteration works perfectly to show that \(M_{d\nu}^{n}\) is bounded on \(L^{p}(d\nu)\), \(1< p<\infty\). For a general measure, the iteration method no longer works. In [1], Fefferman constructed a measure ν for which \(M_{d\nu}^{n}\) is unbounded on \(L^{p}(d\nu)\) for all \(p<+\infty\), and gave a sufficient condition on ω for the \(L^{p}(\omega \,dx)\) boundedness of \(M_{\omega \,dx}^{n}\). Fefferman’s result [1] can be stated as follows.
Theorem 1.1
Suppose that \(d\nu(x)=\omega(x)\,dx\) on \(\mathbb{R}^{n}\) where ω is a function which has the property of being uniformly in the class \(A_{\infty}^{1}\) in each variable separately. Then \(M^{n}_{d\nu}\) is a bounded operator on \(L^{p}(d\nu)\) for all \(1< p\leq\infty\).
In fact, the proof given in [1] also established a stronger result. This is given in Fefferman and Pipher [3].
Theorem 1.2
Suppose that \(d\nu(x)=\omega(x)\,dx\) is a positive absolutely continuous measure on \(\mathbb{R}^{n}\). Assume that dν is doubling with respect to the family of all rectangles with sides parallel to the axes, and that ω is a function which has the property of being uniformly in the class \(A_{\infty}^{1}\) in each variable separately except one. Then \(M^{n}_{d\nu}\) is a bounded operator on \(L^{p}(d\nu)\) for all \(1< p\leq\infty\).
If we replace the Lebesgue measure dx by a more general measure dμ, which is not necessarily doubling, and use the notation \(d\nu(x)=\omega(x)\,d\mu\), then it is not known whether the strong maximal function \(M^{n}_{d\nu}\) is bounded on \(L^{p}(d\nu)\) for all \(1< p<\infty\). This is exactly one of the motivations of this paper. To this end, we first define the notion of \(A_{p}\) weights with respect to the possibly non-doubling measure dμ.
Definition 1.2
Remark 1.2
If we replace the Lebesgue measure dx by a product measure \(d\mu(x)=d\mu_{1}(x_{1})\times d\mu_{2}(x_{2})\times\cdots\times d\mu_{n}(x_{n})\), inspired by the work of Fefferman [1], it is natural to ask the following.
Question 1
What conditions on \(\omega(x)\) and \(d\mu(x)\) can ensure the boundedness of the strong maximal function \(M_{\omega \,d\mu}^{n}\) with respect to the measure \(\omega \,d\mu\) on \(L^{p}(\omega \,d\mu)\) for \(1< p<\infty\)?
Question 2
What conditions on \(\omega(x)\) and \(d\mu(x)\) can ensure the boundedness of the strong maximal function \(M_{d\mu}^{n}\) with respect to the measure dμ on \(L^{p}(\omega\, d\mu)\) for \(1< p<\infty\)?
If we work in \(\mathbb{R}^{3}\) with the dilation group \(\{\rho_{s,t}\}_{s,t>0}\) given by \(\rho_{s,t}(x,y,z)=(sx,tz,stz)\), that is \(\mathcal{B}_{x}\)= the family of all rectangles containing \(x\in\mathbb{R}^{3}\) whose sides are parallel to the coordinate axes in \(\mathbb{R}^{3}\), and whose side lengths in the x, y, and z directions are given by s, t, and \(s\cdot t\) respectively (these rectangles are called Córdoba-Zygmund rectangles), then we get the Córdoba maximal function \(\mathcal{M}(f)(x)\triangleq\mathbb{M}_{d\nu}(f)(x)\) with respect to the measure dν whose sharp estimates have been obtained by Córdoba [2].
With the Córdoba-Zygmund rectangles, we can define Córdoba’s weights \(\mathcal{A}_{p}(\mu)\triangleq\mathbb{A}_{p}(\mu)\). By the Lebesgue differential theorem, if \(\omega\in\mathbb{A}_{p}(\mu)\), then \(\omega(\cdot, y, z) \in A_{p}^{1}(\mu_{1})\) uniformly in y, z, and \(\omega(x, \cdot, z) \in A_{p}^{1}(\mu_{2})\) uniformly in x, z.
When \(d\nu=dx\), Fefferman [13] proved the following theorem.
Theorem 1.3
Moreover, the following is proved by Fefferman and Pipher in [3].
Theorem 1.4
Suppose \(d\nu=\omega(x, y, z)\,dx\,dy\,dz\) is a positive measure on \(\mathbb{R}^{3}\) which is doubling with respect to all the Zygmund rectangles in \(\mathbb{R}^{3}\) and uniformly in \(A^{1}_{\infty}\) in the x and y variables. Then the Córdoba maximal function \(\mathbb{M}_{d\nu}\) with respect to the measure dν is bounded on \(L^{p}(d\nu)\) for all \(1< p <\infty\).
When we replace the Lebesgue measure \(dx\,dy\,dz\) by \(d\mu(x, y, z)\) which is not necessarily doubling with respect to all the Córdoba-Zygmund rectangles in \(\mathbb{R}^{3}\), it is then interesting to ask the following.
Question 3
Let \(d\mu(x, y, z)\) be a nonnegative Radon measure on \(\mathbb{R}^{3}\). What conditions on ω and dμ can guarantee the boundedness of the Córdoba strong maximal function \(\mathbb{M}_{d\mu}(f)(x, y, z)\) with respect to the measure dμ on \(L^{p}(\omega \,d\mu)\) for \(1< p<\infty\)?
Question 4
Let \(d\nu(x, y, z)=\omega(x, y, z)\,d\mu(x, y, z)\) be a measure on \(\mathbb{R}^{3}\). What conditions on ω and dμ can guarantee the boundedness of the Córdoba strong maximal function \(\mathbb{M}_{d\nu}(f)(x, y, z)\) with respect to the measure dν on \(L^{p}(d\nu)\) for \(1< p<\infty\)?
In this paper, we always assume that \(\mu=\mu_{1}\times\mu_{2}\times\cdots\times\mu_{n}\) is a product measure, where \(\mu_{i}\), \(i=1,\ldots,n\) are all nonnegative Radon measures without mass-points and complete. The assumption that \(\mu_{i}\) are complete is just a technical requirement to allow us change the order of integration. For a rectangle \(R\subseteq \mathbb{R}^{n}\), we mean a rectangle whose sides parallel to the coordinate axes.
The main theorems of this paper are as follows.
Theorem 1.5
Let \(\mu(x)=\mu_{1}(x_{1})\cdot\mu_{2}(x_{2})\cdots\mu_{n}(x_{n})\) be a product measure where \(\mu_{i}\), \(i=1,\ldots, n\) are all nonnegative Radon measures in \(\mathbb{R}\) without mass-points and complete. Moreover, we assume that each \(\mu_{i}\) for \(2\leq i\leq n\) is doubling on \(\mathbb{R}\). If \(\omega_{i}(x_{i})=\omega(x_{1},\ldots,x_{i-1},\cdot, x_{i+1},\ldots,x_{n}) \in A_{\infty}^{1}(\mu_{i})\) uniformly with respect to a.e. \((x_{1},\ldots,x_{i-1}, x_{i+1},\ldots,x_{n})\in \mathbb{R}^{n-1}\) for \(i=1,\ldots,n-1\), then the operator \(M_{\omega \,d\mu}^{n}\) is bounded on \(L^{p}(\omega \,d\mu)\) for all \(1 < p <\infty\).
Theorem 1.6
Let \(\mu(x)=\mu_{1}(x_{1})\cdot\mu_{2}(x_{2})\cdots\mu_{n}(x_{n})\) be a product measure where \(\mu_{i}\), \(i=1,\ldots, n\) are all nonnegative Radon measures in \(\mathbb{R}\) without mass-points and complete. Moreover, we assume that each \(\mu_{i}\) for \(2\leq i\leq n\) is doubling on \(\mathbb{R}\). Then the strong maximal operator \(M_{d\mu }^{n}\) with respect to the measure dμ is bounded on \(L^{p}(\omega \,d\mu)\) if and only if \(\omega\in A_{p}^{n}(d\mu)\) for all \(1 < p <\infty\).
Concerning the Córdoba maximal function associated with the Córdoba-Zygmund dilations \(\rho_{s, t}\) in \(\mathbb{R}^{3}\), when \(\omega(x, y, z) \,d\mu(x, y, z)\) is not necessarily doubling with respect to all the Córdoba-Zygmund rectangles in \(\mathbb{R}^{3}\), we have the following.
Theorem 1.7
Moreover, using Theorem 1.7 we have the following.
Theorem 1.8
Assume \(\mu=\mu_{1}\times\mu_{2}\times\mu_{3}\), where \(\mu_{2}\), \(\mu_{3}\) are nonnegative Radon measures and satisfy the doubling property for all intervals \(I\subseteq\mathbb{R}\), and \(\mu_{1}\) is a nonnegative Radon measure in \(\mathbb{R}\) without mass-points (which is not necessarily doubling). Then the Córdoba maximal operator \(\mathbb{M}_{\mu}\) is bounded on \(L^{p}(\omega \,d\mu)\) if and only if \(\omega\in\mathbb{A}_{p}(\mu)\) for all \(1< p<\infty\).
The organization of the paper is as follows. In Section 2, we will establish the reverse Hölder inequality for weights ω in the class \(A_{p}^{n}(\mu)\) adapted to our general product measure μ, which is not necessarily doubling with respect to the rectangles with sides parallel to the coordinate axes in \(\mathbb{R}^{n}\). Section 3 gives the proofs of Theorems 1.5 and 1.6 of boundedness of strong maximal functions with respect to the non-doubling measures dμ and \(d\nu=\omega \,d\mu\). In Section 4, we establish Theorems 1.7 and 1.8 for the Córdoba strong maximal functions with respect to the Córdoba-Zygmund dilations.
2 Reverse Hölder inequality of weights \(A_{p}^{n}(\mu)\)
The purpose of this section is to establish reverse Hölder inequality of weights in the class \(A_{p}^{n}(\mu)\) adapted to our general product measure μ.
We use the notation \(w(E)=\int_{E} w(x)\,d\mu(x)\) for every measurable set \(E\subset\mathbb{R}^{n}\) in this section.
Lemma 2.1
Proof
Remark 2.1
By the same proof, Lemma 2.1 holds for \(\omega\in \mathbb{A}_{p}(\mu)\) with respect to the Córdoba-Zygmund rectangles in \(\mathbb{R}^{3}\).
Since \([\omega]_{A_{p}^{n}(\mu)}\geq1\), from the inequality (2.1), one can also easily obtain the following lemma.
Lemma 2.2
Remark 2.2
Equation (2.2) is called the \(A_{\infty}^{n}(\mu)\) condition. It is easy to see that if \(\omega\in A_{p}^{n}(\mu)\), \(\omega_{i}(x_{i})=\omega(x_{1},\ldots,x_{i-1},\cdot, x_{i+1},\ldots,x_{n}) \in A_{\infty}^{1}(\mu_{i})\) uniformly with respect to \(x_{1},\ldots,x_{i-1}, x_{i+1},\ldots,x_{n}\).
Under our assumption on μ being a product measure, we also have the reverse Hölder inequality for \(\omega\in A_{p}^{n}(\mu)\).
Lemma 2.3
Proof
If \(\omega\in A_{p}^{n}(\mu)\), \(p >1\), then \(\omega^{1-p'} \in A_{p'}^{n}(\mu)\), where \(1/p+1/p'=1\). Consequently, by Lemma 2.3, it is easy to deduce the following corollary.
Corollary 2.1
Let \(p>1\), and \(\omega\in A_{p}^{n}(\mu)\), then there is an \(\varepsilon>0\) such that \(\omega\in A_{p-\varepsilon}^{n}(\mu)\).
Proof
3 The strong maximal functions with respect to non-doubling measures
The main purpose of this section is to prove Theorem 1.5. We first need to prove the following geometric covering lemma whose proof is inspired by those in [1, 2, 14], and [3] when \(du=dx\). Weak type estimates for strong maximal functions were first studied by Jessen, Marcinkiewcz and Zygmund [11] who first proved the strong differentiation theorem. Córdoba and Fefferman [14] gave a more geometric proof (see also Jawerth and Torchinsky [15]). Their method in [14] relies on a deep understanding of the geometry of rectangles. Namely, they established a deep and difficult geometric covering lemma. This lemma will lead to the weak type \((p,p)\) of \(M_{\omega \,d\mu}^{n}\) as argued in [14]. Then we can complete the proof of Theorem 1.5 by interpolation (see, e.g., [16] and [17]). The proof of Theorem 1.6 is the same as that of Theorem 1.8 in Section 4, we shall omit it here.
Lemma 3.1
Assume that \(\mu(x)=\mu_{1}(x_{1})\cdot\mu_{2}(x_{2})\cdots\mu_{n}(x_{n})\) is a product measure where \(\mu_{i}\), \(i=1,\ldots, n\) are all nonnegative Radon measures in \(\mathbb{R}\) without mass-points and complete. Assume also that each \(\mu_{i}\) for \(2\leq i\leq n\) is doubling on \(\mathbb{R}\) and that \(\omega_{i}\in A_{\infty}^{1}(\mu_{i})\) uniformly, \(i=1,\ldots,n-1\).
Proof
If we can prove it at \(n=2\), then it is easy to complete the proof by induction. Hence we only give the proof when n=2.
4 Córdoba’s maximal function
Proof of Theorem 1.7
Using the fact that \(\omega\in \mathbb{A}_{p}(\mu)\) (\(A_{p}\) weights with respect to the Córdoba-Zygmund rectangles and the not necessarily doubling measure μ), we see that \(w(\cdot, y, z)\) is in \(A_{p}^{1}(d\mu_{1})\) uniformly in y, z and \(w(x, \cdot, z)\) is in \(A_{p}^{1}(d\mu_{2})\) uniformly in x, z. By the assumptions that the measures \(\mu_{2}\), \(\mu_{3}\) are doubling on \(\mathbb{R}\), Theorem 1.7 is an immediate corollary of Theorem 1.5. □
We will prove Theorem 1.4 by an argument similar to the one given in [13]. We first prove the reverse Hölder’s inequality for \(\omega\in\mathbb{A}_{p}(\mu)\). For a fixed number \(a>0\), let \(\mathcal{U}\) be the family of all rectangles whose sides are parallel to the coordinate axes in \(\mathbb{R}^{2}\), and whose side lengths in the x, y directions are given by s and sa, where s is arbitrary. First of all, by Corollary 9.2.4 of [18], using a linear change of scale we obtain the following proposition.
Proposition 4.1
Proceeding as in [13] or the proof of Lemma 2.3, together with the above proposition, we can establish the following reverse Hölder inequality for Córdoba’s weights with respect to a certain non-doubling measure μ. We omit the proof.
Proposition 4.2
Then with a proof similar to that of Corollary 2.1, one has the following result.
Corollary 4.1
Let \(p>1\), and \(\omega\in\mathbb{A}_{p}(\mu)\). Then there is an \(\varepsilon>0\) such that \(\omega\in A_{p-\varepsilon}(\mu)\).
We are now ready to complete the proof of Theorem 1.8.
Proof of Theorem 1.8
Declarations
Acknowledgements
The authors would like to thank the referees very much for the kind advice and useful suggestions.
Article is supported by NNSF of China grants (11501308, 11271209).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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