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Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces
Journal of Inequalities and Applications volume 2016, Article number: 4 (2016)
Abstract
The purpose of this paper is to investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main result generalizes the Sawano, Sugano, and Tanaka result to a weighted setting.
1 Introduction
The aim of this paper is to investigate the weighted inequalities of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The main results particularly is related to [1] and [2]. The authors introduced the condition of weights in [1]. Under a certain condition of the weights, we investigate the weighted estimates of commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. The results recover the inequality in [2].
For \(1< p<\infty\), we define \(p':=\frac{p}{p-1}\). In this paper, a symbol C is a positive constant. Whenever we evaluate the operator, the constant C may be change from one constant to another. Let \(|E|\) denote the Lebesgue measure of E. Let \(\mathcal{D}(\mathbb{R}^{n})\) be the collection of all dyadic cubes on \(\mathbb{R}^{n}\). All cubes are assumed to have their sides parallel to the coordinate axes. For a cube \(Q\subset\mathbb{R}^{n}\), we use \(l(Q)\) to denote the side-length \(l(Q)\) and cQ to denote the cube with the same center as Q but with side-length \(cl(Q)\). The integral average of a measurable function f over Q is written
By a ‘weight’ we will mean a non-negative function w that is positive measure a.e. on \(\mathbb{R}^{n}\). Given a weight w and a measurable set E, let
First we define the Morrey spaces.
Definition 1
Let \(1< p\leq p_{0}<\infty\). We define the Morrey space \(\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})\) by
where for all measurable functions f, we define
Remark 1
-
(a)
The ordinary Morrey norm is equivalent to the Morrey norm in this paper (see [1]):
$$\sup_{\substack{Q\subset\mathbb{R}^{n},\\Q\text{: cubes}}} |Q|^{\frac {1}{p_{0}}} \biggl( \fint_{Q} \bigl\vert f(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}\cong \Vert f\Vert _{\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})}. $$ -
(b)
Hölder’s inequality gives us the following inequality: If \(1< p\leq q \leq p_{0}<\infty\), then we have
$$ \Vert f \Vert _{\mathcal{M}_{p}^{p_{0}}} \leq \Vert f \Vert _{\mathcal{M}_{q}^{p_{0}}}. $$
We define the BMO space (see [3, 4]) as follows.
Definition 2
For an \(L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\)-function b, define
where the supremum is taken over all cubes \(Q\subset\mathbb{R}^{n}\). Define
We define the fractional maximal and integral operators.
Definition 3
-
(1)
Let \(0\leq\alpha< n\),
$$M_{\alpha}f(x):=\sup_{Q\ni x} l(Q)^{\alpha} \fint_{Q} \bigl\vert f(y)\bigr\vert \,dy, $$where the supremum is taken over all cubes \(Q\subset\mathbb{R}^{n}\) such that \(x\in Q\).
-
(2)
Let \(0<\alpha<n\),
$$I_{\alpha}f(x):= \int_{\mathbb{R}^{n}} \frac{f(y)}{|x-y|^{n-\alpha}} \,dy. $$
The point-wise inequality holds:
for all positive measurable function f.
It is well known that the following inequality holds (see [5]). The celebrated result is called the Adams inequality.
Theorem A
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}<\infty\). Assume that
Then we have
for all \(f\in\mathcal{M}_{p}^{p_{0}}(\mathbb{R}^{n})\).
Let \(m\in\mathbb{Z}_{+}\). The m-fold commutator \([b, I_{\alpha}]^{(m)}\) is given by the following definition.
Definition 4
Let \(0<\alpha<n\) and \(b\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})\). Then we define
as long as the integral in the right-hand side makes sense.
Remark 2
The following inequality holds:
As shall be verified in the proof of Theorem 1, we virtually consider the operator
and hence we may assume that the integral defining \([b,I_{\alpha}]^{(m)}f(x)\) converges for a.e. \(x\in\mathbb{R}^{n}\).
Di-Fazio and Ragusa [6] obtained the next theorem.
Theorem B
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}<\infty\). Assume that
If \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), then we have
Conversely if \(n-\alpha\) is an even integer and
then \(b\in\operatorname{BMO}(\mathbb{R}^{n})\).
Komori and Mizuhara [7] removed the restriction ‘\(n-\alpha\) is an even integer’.
Theorem C
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(0< q\leq q_{0}<\infty\). Assume that
If \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), then we have
Conversely if
then \(b\in\operatorname{BMO}(\mathbb{R}^{n})\).
Sawano et al. [2] proved the following inequality.
Theorem D
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\), \(1< q\leq q_{0}<\infty\) and \(1< r\leq r_{0}<\infty\). Assume that
Suppose that \(v\in\mathcal{M}_{r}^{r_{0}}(\mathbb{R}^{n})\). Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
In the case of \(m=0\), we refer to [1, 8, 9]. In this paper, we generalize Theorem D to a weighted setting. On the other hand, in [1], the following theorem is proved.
Theorem E
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}< r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that the weights v and w satisfy the following condition:
Then we have
In this paper, we investigate the boundedness of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces corresponding to Theorem E.
2 Main results and their corollaries
In this paper, we obtain two main theorems.
2.1 One of the main results
Theorem 1
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}< r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that the weights v and w satisfy the condition (2). Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
Remark 3
The condition of Theorem 1 corresponds with the condition of Theorem E. This implies that Theorem 1 gives us the same type of corollaries as in Theorem E.
Taking \(w(x)=M_{\frac{aq}{r_{0}}n} ( v^{aq} )(x)^{\frac{1}{aq}}\), we have the following corollary.
Corollary 1
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}< r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Let v be a weight. Suppose that \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), then we have
Taking \(w(x)\equiv1\), we obtain the following corollary.
Corollary 2
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}< r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that \(v\in\mathcal{M}_{aq}^{r_{0}}(\mathbb{R}^{n})\). Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
On the other hand, letting \(r_{0}\to\infty\), we obtain the weighted Adams type inequality for the m-fold commutator \([b, I_{\alpha}]^{(m)}\).
Corollary 3
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that the weights v and w satisfy the following condition:
Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
Corollary 3 gives us the following inequality in letting \(p=p_{0}\), \(q=q_{0}\) and \(v=w\).
Corollary 4
Let \(0<\alpha<n\), \(1< p<\frac{n}{\alpha}\) and \(1< q<\infty\). Assume that
Suppose that \(w\in A_{p,q}(\mathbb{R}^{n})\), i.e.
Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
Corollary 3 and Theorem C give us the following corollary.
Corollary 5
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}<\infty\). Assume that
and \(a>1\). Suppose that the weights v and w satisfy the condition (3). If
holds, then we have for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\),
According to Theorem 1.8 in [2], we can pass our result to the operator given by
where \(\vec{b}=(b_{1},\ldots, b_{m})\). By a similar argument to [2], as a consequence of Theorem 1 in this paper, we can obtain the following estimate.
Corollary 6
Let \(0<\alpha<n\), \(1< p\leq p_{0}<\infty\) and \(1< q\leq q_{0}< r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that the weights v and w satisfy the condition (2). Then, for \(\vec{b}=(b_{1},\ldots, b_{m})\in\operatorname{BMO}(\mathbb {R}^{n})\times\cdots\times\operatorname{BMO}(\mathbb{R}^{n})\), we have
2.2 Fractional integral operators having rough kernel
We define the following operators (see [10–12] and [4]).
Definition 5
Let \(0<\alpha<n\), a measurable function Ω on \(\mathbb{R}^{n}\backslash\{ 0\}\) and a measurable function b. Then we define
and
Remark 4
The following inequality holds:
As shall be verified in the proof of Theorem 2, we consider the operator
and hence we may assume that the integral defining \([b,I_{\Omega,\alpha }]^{(m)}f(x)\) converges for a.e. \(x\in\mathbb{R}^{n}\).
By a similar argument to the proof of Theorem 1, we have the following estimate.
Theorem 2
Let \(1< s\leq\infty\), \(0<\alpha<n\), \(1\leq s'< p\leq p_{0}<\infty\), \(1< q\leq q_{0}<\infty\) and \(1< r\leq r_{0}<\infty\). Assume that
and \(1< a<\frac{r_{0}}{q_{0}}\). Suppose that the weights v and w satisfy \([ v^{s'},w^{s'} ]_{\frac{aq_{0}}{s'},\frac{r_{0}}{s'},\frac {aq}{s'},\frac{p}{s'a}}^{\frac{1}{s'}}<\infty\). Moreover, suppose that \(\Omega\in L^{s}(\mathbb{S}^{n-1})\) is homogeneous of order 0: For any \(\lambda>0\), \(\Omega(\lambda x)=\Omega(x)\). Then, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\), we have
Since \([b,I_{\Omega,\alpha} ]^{(0)}=I_{\Omega,\alpha}\), we refer to [12]. Theorem 2 recovers the following result (see [4, 11]).
Corollary 7
Let \(1< s\leq\infty\), \(0<\alpha<n\), \(1\leq s'< p<\frac{n}{\alpha}\) and \(1< q<\infty\). Assume that
and \(w^{s'}\in A_{\frac{p}{s'},\frac{q}{s'}}(\mathbb{R}^{n})\). Suppose that \(\Omega\in L^{s}(\mathbb{S}^{n-1})\) is homogeneous of order 0: For any \(\lambda>0\), \(\Omega(\lambda x)=\Omega(x)\). Then we have, for \(b\in\operatorname{BMO}(\mathbb{R}^{n})\),
3 Some lemmas
In this section, we prepare some lemmas for proving main results. We recall the following inequalities (see [3, 13] and [4]).
Lemma 1
(The John-Nirenberg inequality)
Let \(1\leq p<\infty\) and let Q be a cube. Then there exists a constant \(C>0\) such that
for all \(b\in\operatorname{BMO}(\mathbb{R}^{n})\).
We invoke the following decomposition which is derived in [14–16]. We omit the details; see [1, 12] for the proof.
Let \(\mathcal{D}(Q_{0})\) be the collection of all dyadic subcubes of \(Q_{0}\), that is, all those cubes obtained by dividing \(Q_{0}\) into \(2^{n}\) congruent cubes of half its length, dividing each of those into \(2^{n}\) congruent cubes. By convention \(Q_{0}\) itself to \(\mathcal{D}(Q_{0})\), and so on.
Lemma 2
Let \(\gamma:=m_{3Q_{0}}(f)\) and \(A>2\cdot18^{n}\). For \(k=1,2,\ldots\) we take
For \(\theta_{1}>1\), let
and \(A'> ( 2\cdot18^{n} )^{\frac{1}{\theta_{1}}}\). For \(k=1,2,\ldots\) we take
Considering the maximality cube, we have
Then we have
Let \(E_{k,j}:=Q_{k,j} \backslash D_{k+1}\) and \(E_{k,j}':=Q_{k,j}'\backslash D_{k+1}'\). Moreover we obtain
Lemma 3
Under the condition of Theorem 1, we can choose auxiliary indices \(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta _{4}\) and \(\theta_{5}\) so that the following conditions hold:
-
1.
\(\theta_{1}\), \(\theta_{2}\), \(\theta_{3}\), \(\theta_{4}\) and \(\theta_{5}\in(1,p)\).
-
2.
\(L>1\) and \(s\in(q,r)\) such that \(s\theta_{2}< Lq\) and \(s'\theta_{2}< q'\).
-
3.
For the index \(\theta_{1}\in(1,p)\), we can choose \(a_{*}>1\) such that \(a_{*}\theta_{1}< p\).
Assume in addition that, for these indices,
Then we obtain
Proof
We examine the second item; \(s\theta_{2}< Lq\) and \(s'\theta_{2}< q'\). For \(0<\varepsilon<1\), we take \(\delta=\frac{\varepsilon}{q^{2}}<\varepsilon\). If \(s=q+\varepsilon\) and \(\theta_{2}=1+\delta\), then we have the following estimate:
On the other hand, we check \(s'\theta_{2}< q'\):
Next we check \(\frac{p}{ ( \theta_{5} ( \frac{p}{\theta_{5}} )' )'}>1\). Since \(\theta_{5}>1\), we obtain
Therefore we have
This gives us
By a similar argument, we obtain
 □
Remark 5
The index \(\theta_{1}\) in Lemma 2 corresponds with the index \(\theta_{1}\) in Lemma 3.
4 Proof of Theorem 1
Proof of Theorem 1
Fix a dyadic cube \(Q_{0}\in\mathcal{D}(\mathbb{R}^{n})\). Let \(\mathcal{D}_{\nu}\) be the collection of dyadic cubes. The volume of the elements of \(\mathcal{D}_{\nu}\) is \(2^{n\nu}\). For \(x\in Q_{0}\), we have
We evaluate A and B in Sections 4.1 and 4.2, respectively.
4.1 The estimate of A
By \(| b(x)-b(y) |^{m}\leq2^{m-1} ( |b(x)-m_{Q}(b)|^{m} + |m_{Q}(b)-b(y)|^{m} )\), we obtain
We take \(\theta_{1}>1\) as in Lemma 2. By Hölder’s inequality for \(\theta_{1}>1\), we have
By Lemma 1, we have
We evaluate I. Let
and
where \(Q_{k,j}\) is in Lemma 2. Then we have
By the duality argument, we have
Let \(g\geq0\), \(\operatorname{supp}(g)\subset Q_{0}\), \(\Vert g \Vert _{L^{q'}(Q_{0})}=1\). Then we have
We evaluate \(I_{k,j}\). If \(Q\in\mathcal{D}_{k,j}(Q_{0})\), then we have
Hence we obtain
Since
we obtain
By Hölder’s inequality for \(\theta_{2}>1\) as in Lemma 3, we obtain
By Lemma 1, we obtain
where \(v_{k,j}=v\chi_{Q_{k,j}}\) and the symbol M is the ordinary Hardy-Littlewood maximal operator. By Lemma 2, we have
We take \(s\in(q,r)\) and \(L>1\) as in Lemma 3. By Hölder’s inequality for \(s>1\), we have
By Hölder’s inequality for \(Lq>1\), we obtain the following inequality:
Since \(s\theta_{2}< Lq\), the boundedness of \(M: L^{\frac{Lq}{s\theta_{2}}}(\mathbb{R}^{n}) \to L^{\frac{Lq}{s\theta _{2}}}(\mathbb{R}^{n})\) gives us the following inequality:
Since \(a\geq L>1\), by Hölder’s inequality for \(\frac{a}{L}\geq1\),
By Lemma 2, this implies that
where
A similar argument gives us the following estimate:
By summing up \(I_{0}\) and \(I_{k,j}\), we obtain
By Hölder’s inequality for \(q>1\), we have
Since \((Lq)'< q'\), the boundedness of \(M:L^{\frac{q'}{ (Lq)'}}(\mathbb{R}^{n})\to L^{\frac {q'}{(Lq)'}}(\mathbb{R}^{n})\) gives us the following inequality:
Since \(s'\theta_{2}< q'\), the boundedness of \(M:L^{\frac{q'}{s'\theta_{2}}}(\mathbb{R}^{n}) \to L^{\frac{q'}{s'\theta_{2}}}(\mathbb{R}^{n})\) gives us the following inequality:
By Hölder’s inequality for \(\frac{p}{a}>1\), we obtain
By the condition (2), we obtain
This implies that
Since
by Theorem A, we have
We evaluate II. Let
and
where \(Q_{k,j}'\) is found in Lemma 2. Then we have
By the duality argument, we have
Let \(g\geq0\) be such that \(\operatorname{supp}(g)\subset Q_{0}\) and \(\Vert g \Vert _{L^{q'}(Q_{0})}=1\). We have
We evaluate \(\mathit{II}_{k,j}\). If \(Q\in\mathcal{D}_{k,j}'(Q_{0})\), then we have
Therefore we obtain
Since
we obtain
By Hölder’s inequality for \(\theta_{3}>1\) as in Lemma 3, we have
By Lemma 2, we obtain
where
A similar argument gives us the following estimate:
By summing up \(\mathit{II}_{0}\) and \(\mathit{II}_{k,j}\), we obtain
By Hölder’s inequality for \(q>1\), we have
Since \((\theta_{3}q)'< q'\) and \(\operatorname{supp}(g)\subset Q_{0}\), by the boundedness of \(M:L^{\frac{q'}{(\theta_{3}q)'}}(\mathbb{R}^{n})\to L^{\frac{q'}{(\theta _{3}q)'}}(\mathbb{R}^{n})\), we have
Therefore we have
By Hölder’s inequality for \(\frac{p}{a_{*}\theta_{1}}>1\) as in Lemma 3, we have
By Lemma 3, we have \(\theta_{1} ( \frac{p}{\theta_{1}a_{*}} )'\leq ( \frac {p}{a} )' \). By Hölder’s inequality, we have
By the condition (2), we obtain
This implies that
Since
by Theorem A, we have
Therefore we have
4.2 The estimate of B
Since \(|b(x)-b(y)|^{m}\leq2^{m-1} ( | b(x)-m_{Q}(b) |^{m}+| m_{Q}(b)-b(y) |^{m} )\), we have
Therefore we obtain
By Hölder’s inequality and the definition of the Morrey norm we obtain
Since \(\frac{1}{q_{0}}=\frac{1}{p_{0}}+\frac{1}{r_{0}}- \frac{\alpha}{n}\), the integral of \(C_{1}[f,v](x)^{q}\) on \(Q_{0}\) is evaluated as follows:
By Hölder’s inequality for \(\theta_{4}>1\) as in Lemma 3, we have
We evaluate \(\vert b(x)-m_{Q}(b)\vert \). If \(Q\supsetneqq Q_{0}\) and \(Q\in\mathcal{D}(\mathbb{R}^{n})\), then there exists \(k=1,2,\ldots\) , such that \(Q_{k}:=Q\), \(Q_{j}\in\mathcal{D}(\mathbb{R}^{n})\), \(Q_{j}\supsetneqq Q_{j-1}\) and \(|Q_{j}|=2^{n}|Q_{j-1}|\) (\(j=1,2,\ldots,k\)). By the triangle inequality, we obtain
Moreover, we have
where we invoke Definition 2 for the last line. By the inequality \((a+b)^{m}\leq2^{m-1}(a^{m}+b^{m})\):
By the estimates (6), (7), and Hölder’s inequality for \((p/a)'>p'\), we obtain
By the triangle inequality on \(L^{q\theta_{4}'}(\mathbb{R}^{n})\), we obtain
By the estimate (8), we obtain
By Lemma 1, we have
The estimate (9) gives us the following:
As a consequence of (10), we obtain the following inequality:
By the condition (2), we have
Therefore we obtain
Next, we evaluate \(C_{2}[f,v](x)\). By Hölder’s inequality for \(\theta_{5}\in(1,p)\) in Lemma 3, we have
By Hölder’s inequality for \(\frac{p}{\theta_{5}}>1\), we obtain
Taking the Morrey norm, we obtain
Using Lemma 1, we have
Since we have the assumption that \(a\geqq\frac{p}{ ( \theta_{5} ( \frac{p}{\theta_{5}} )' )'}>1\), using Hölder’s inequality, we obtain
The integral of \(C_{2}[f,v](x)^{q}\) on \(Q_{0}\) is evaluated as follows:
By the condition (2), we have
We obtain the desired result.  □
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Acknowledgements
The author wish to thank Professor Y Sawano for a rich lecture of higher order commutators generated by BMO-functions and the fractional integral operator.
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Iida, T. Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces. J Inequal Appl 2016, 4 (2016). https://doi.org/10.1186/s13660-015-0953-4
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DOI: https://doi.org/10.1186/s13660-015-0953-4