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Weighted norm inequalities for multilinear Calderón-Zygmund operators in generalized Morrey spaces
Journal of Inequalities and Applications volume 2017, Article number: 48 (2017)
Abstract
In this paper, the authors study the boundedness of multilinear Calderón-Zygmund singular integral operators and their commutators in generalized Morrey spaces.
1 Introduction
Let T be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions, i.e.,
In [1], it is said that a function K belongs to the class m-\(\operatorname{CZK}(A,\varepsilon)\) if
-
(1)
\(|K(y_{0},y_{1},\ldots,y_{m})|\leqslant\frac{A}{(\sum_{k,l=0}^{m}|y_{k}-y_{l}|)^{mn}}\),
-
(2)
if \(|y_{j}-y_{j}'|\leqslant\frac{1}{2}\max_{0\leqslant k\leqslant m}|y_{j}-y_{k}|\),
$$\bigl\vert K(y_{0},\ldots,y_{j},\ldots,y_{m})-K \bigl(y_{0},\ldots,y_{j}',\ldots,y_{m} \bigr)\bigr\vert \leqslant \frac{A|y_{j}-y_{j}'|^{\varepsilon}}{(\sum_{k,l=0}^{m}|y_{k}-y_{l}|)^{mn+\varepsilon}} $$for some \(\varepsilon>0\) and \(j=0,1,2\ldots, m\).
The operator T is said to be an m-linear Calderón-Zygmund operator if there exists a function \(K\in m\mbox{-}\operatorname{CZK}(A,\varepsilon)\) defined away from the diagonal \(y_{0}=y_{1}=y_{2}\cdots=y_{m}\) in \((\mathbb{R}^{n})^{m+1}\) such that
for \(x\notin\bigcap_{j=1}^{m} \operatorname{supp} f_{j}\), and that T extends to a bounded multilinear operator from \(L^{q_{1}}\times\cdots\times L^{q_{m}}\) to \(L^{q}\) for some \(1\leqslant q_{j}<\infty\) with \(\frac{1}{q}=\frac{1}{q_{1}}+\cdots +\frac{1}{q_{m}}\).
It was shown in [1] that if \(\frac{1}{r_{1}}+\cdots+\frac {1}{r_{m}}=\frac{1}{r}\), then an m-linear Calderón-Zygmund operator satisfies
when \(1< r_{j}<\infty\) for \(j=1,\ldots,m\) and
when \(1\leq r_{j}<\infty\) for \(j=1,\ldots,m\) and at least one \(r_{j}=1\). In particular,
The theory of multiple weight associated with m-linear Calderón-Zygmund operators was developed by Lerner et al. [2]. Let \(1< p_{j}<\infty\) for \(j=1,\ldots,m\), \(\frac{1}{p}=\frac {1}{p_{1}}+\cdots+\frac{1}{p_{m}}\) and \(\vec{p}=(p_{1},\ldots,p_{m})\), we say \(\vec{\omega}\in A_{\vec{p}}\) if
where B is the ball in \(\mathbb{R}^{n}\) and \(v_{\vec{\omega}}=\prod_{j=1}^{m}\omega_{j}^{p/p_{j}}\). They showed that if \(\vec{\omega}\in A_{\vec{p}}\) then
If \(1\leqslant p_{j}<\infty\) for \(j=1,\ldots,m\) and at least one of the \(p_{j}=1\), they also proved
Let \(\vec{b}=(b_{1},\ldots,b_{m})\) be a vector-valued locally integrable function. If \(\vec{b}=(b_{1},\ldots,b_{m})\) in \((\mathit{BMO})^{m}\), we denote \(\|\vec{b}\|_{(\mathit{BMO})^{m}}=\sup_{j=1,\ldots,m}\|b_{j}\|_{\mathit{BMO}}\) (see [2]). The commutator generated by an m-linear Calderón-Zygmund operator T and a \((\mathit{BMO})^{m}\) function b⃗ is defined by
where each term is the commutator of \(b_{j}\) and T in the jth entry of T, that is,
Pérez and Torres [3] proved that if \(\vec{b}\in(\mathit{BMO})^{m}\) then
for \(1< p_{j}<\infty\) and \(1< p<\infty\) with \(\frac{1}{p}=\frac {1}{p_{1}}+\cdots+\frac{1}{p_{m}}\), where \(j=1,\ldots,m\). In [2], the authors proved that if \(\vec{\omega}\in A_{\vec{p}}\) and \(\vec{b}\in(\mathit{BMO})^{m}\), then
for \(1< p_{j}<\infty\) with \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac {1}{p_{m}}\), where \(j=1,\ldots,m\).
Feuto [4] introduced the generalized weighted Morrey space \((L^{p}(\omega),L^{q} )^{\alpha}\). Let \(1\leqslant p\leqslant \alpha\leqslant q\leqslant\infty\) and ω be a weight. The space \((L^{p}(\omega),L^{q} )^{\alpha}\) was defined to be the set of all measurable functions f satisfying \(\|f\|_{ (L^{p}(\omega ),L^{q} )^{\alpha}}<\infty\), where
with
When \(\omega\equiv1\), the space \((L^{p},L^{q} )^{\alpha }\) was introduced in [5]. If \(p<\alpha\) and \(q=\infty\), the space \((L^{p}(\omega ),L^{\infty} )^{\alpha}\) is just the weighted Morrey space \(L^{p,\kappa}(\omega)\) with \(\kappa=1-p/\alpha\) defined by Komori and Shirai [6].
Similarly, the weak space \((L^{p,\infty}(\omega),L^{q} )^{\alpha}\) is defined with
When \(p=1\), the space \((L^{1,\infty}(\omega),L^{q} )^{\alpha}\) was introduced in [4].
Feuto proved in [4] that Calderón-Zygmund singular integral operators, Marcinkiewicz operators, the maximal operators associated to Bochner-Riesz operators and their commutators are bounded on \((L^{p}(\omega),L^{q} )^{\alpha}\).
In this paper, we aim to study the boundedness of multilinear singular integral operators on the product of generalized Morrey spaces. Inspired by the above mentioned works, we state our main results as follows.
Theorem 1.1
Let T be an m-linear Calderón-Zygmund operator, \(\frac{1}{p}=\frac {1}{p_{1}}+\cdots+\frac{1}{p_{m}}\) and \(\vec{\omega}\in A_{\vec{p}}\).
-
(1)
If \(1< p_{j}<\infty\), \(j=1,\ldots,m\) and \(p\leqslant \alpha< q\leqslant\infty\), then
$$\bigl\Vert T\vec{(f)}\bigr\Vert _{(L^{p}(v_{\vec{\omega}}),L^{q})^{\alpha}}\lesssim \prod _{i=1}^{m}\Vert f_{i}\Vert _{({L^{p_{i}}(\omega_{i}),L^{qp_{i}/p}})^{\alpha p_{i}/p}}. $$ -
(2)
If \(1\leqslant p_{j}<\infty\), \(j=1,\ldots,m\) and at least one of \(p_{j}=1\), \(p\leqslant\alpha< q\leqslant\infty\), then
$$\bigl\Vert T\vec{(f)}\bigr\Vert _{(L^{p,\infty}(v_{\vec{\omega}}),L^{q})^{\alpha }}\lesssim\prod _{i=1}^{m}\Vert f_{i}\Vert _{({L^{p_{i}}(\omega _{i}),L^{qp_{i}/p}})^{\alpha p_{i}/p}}. $$
Theorem 1.2
Let \(T_{\vec{b}}\) be a multilinear commutator, \(\vec{b}\in (\mathit{BMO})^{m}\), \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\) with \(1< p_{j}<\infty\) and \(\vec{\omega}\in A_{\vec{p}}\). If \(p\leqslant \alpha< q\leqslant\infty\), then
Remark 1.3
When \(m=1\), Theorem 1.1 is just Theorem 2.1 in [4] and Theorem 1.2 is just Theorem 2.5 in [4].
2 Notations and preliminaries
We first recall the definition of \(A_{p}\) weight. A nonnegative locally integrable function ω belongs to \(A_{p}\) (\(p>1\)) if
where \(p'\) is the conjugate index of p, i.e., \(1/p+1/p'=1\). We say that \(\omega\in A_{1}\) if there is a constant \(C>0\) such that
for any ball B. If \(\omega\in A_{p}\), then there exists \(\delta>0\) such that
for any measurable subset E of a ball B. Since the \(A_{p}\) classes are increasing with respect to p, we use the following notation \(A_{\infty}=\bigcup_{p>1}A_{p}\). \(A\lesssim B\) means \(A\leq CB\), where C is a positive constant independent of the main parameters. For \(\lambda>0\) and a ball \(B\subset\mathbb{R}^{n}\), we write λB for the ball with same center as B and radius λ times radius of B.
Now we give the definition of \(A_{\vec{p}}\) condition.
Definition 2.1
[2]
Let \(1\leqslant p_{j}<\infty\) for \(j=1,\ldots,m\), \(\vec{p}=(p_{1},\ldots ,p_{m})\) and \(\frac{1}{p}=\frac{1}{p_{1}}+\cdots+\frac{1}{p_{m}}\). Given \(\vec{\omega}=(\omega_{1},\ldots,\omega_{m})\), set
We say that \(\vec{\omega}\in A_{\vec{p}}\) if
\(p'\) is the conjugate index of p. When \(p_{j}=1\), denote \(p'_{j}=\infty\), \((\frac{1}{|B|}\int_{B}\omega _{j}^{1-p{'}_{j}})^{1/p{'}_{j}}\) is understood as \((\inf_{B}\omega_{j})^{-1}\)
Obviously, if \(m=1\), \(A_{\vec{p}}\) is the classical \(A_{p}\) class. \(A_{\vec{p}}\) has the following characterization.
Lemma 2.2
[2]
Let \(\vec{\omega}=(\omega_{1},\ldots,\omega_{m})\). Then \(\vec {\omega}\in A_{\vec{p}}\) if and only if
where the condition \(\omega_{j}^{1-p{'}_{j}}\in A_{mp_{j}{'}}\) is understood as \(\omega_{j}^{1/m}\in A_{1}\) in the case \(p_{j}=1\).
Lemma 2.3
[2]
Assume that \(\vec{\omega}=(\omega_{1},\ldots,\omega_{m})\) satisfies \(A_{\vec{p}}\) condition. Then there exists a finite constant \(r>1\) such that \(\vec{\omega}\in A_{\vec{p}/r}\).
In order to prove the results for commutators, we need the following properties of BMO. For \(b\in \mathit{BMO}\), \(1< p<\infty\) and \(\omega\in A_{\infty}\), we get
and for all balls B,
For all nonnegative integers k, we obtain
where \(\omega(B)=\int_{B}\omega(x)\, \mathrm{d}x\), \(b_{B}=\frac {1}{|B|}\int_{B}b(x)\, \mathrm{d}x\) (see [4]).
3 Proof of the main results
Proof of Theorem 1.1
(1) Let \(B=B(y,r)\) be a ball of \(\mathbb{R}^{n}\), \(f_{i}=f_{i}\chi _{2B}+f_{i}\chi_{(2B)^{c}}\) and denote \(f_{i}\chi_{2B}\) by \(f_{i}^{0}\) and \(f_{i}\chi_{(2B)^{c}}\) by \(f_{i}^{\infty}\) (\(i=1,\ldots,m\)), \(\chi_{E}\) denotes the characteristic function of set E. For \(x\in B(y,r)\), we have
where \(\alpha_{1},\ldots,\alpha_{m}\) are not all equal to 0 or ∞ at the same time. We first estimate III. Since \(2^{k-1}r\leqslant|x-y_{i}|\leqslant2^{k+2}r\), we have
the Hölder inequality gives us that
By the definition of \(A_{\vec{p}}\) condition, we obtain
For II, we just consider this case: \(\alpha_{i}=\infty\) for \(i=1,\ldots,l\) and \(\alpha_{j}=0\) for \(j=l+1,\ldots,m\),
By (3.2) and the definition of \(A_{\vec{p}}\) condition, we have
Combining all the cases together, we obtain
Taking \(L^{p}(v_{\vec{\omega}})\) norm on the ball \(B(y,r)\) on both sides of (3.6), by (1.2), we get
Multiplying both sides of (3.7) by \(v_{\vec{\omega }}(B)^{1/\alpha-1/q-1/p}\), by Lemma 2.2 and (2.1), we obtain
For \(\frac{1}{p_{1}/p}+\frac{1}{p_{2}/p}+\cdots+\frac {1}{p_{m}/p}=1\), we have
by the Hölder inequality. Since \(\sum_{k=0}^{\infty}\frac {1}{2^{nk\delta(1/\alpha-1/q)}}<\infty\), we obtain the expected result
(2) For \(\lambda>0\), by (3.6) and (1.3), we get
That is,
Multiplying both sides of (3.9) by \(v_{\vec{\omega }}(B)^{1/\alpha-1/q-1/p}\), we conclude as in the case (1). □
Proof of Theorem 1.2
It suffices to prove \(T_{\vec{b}}^{j}\). For \(B=B(y,r)\), \(x\in B\)
where \(\alpha_{1},\ldots,\alpha_{m}\) are not all equal to 0 or ∞ at the same time. We first deal with \(\mathit{III}'\).
Select suitable \(s>1\) to raise \(\int_{({2^{k+1}{B}})^{m}}{\prod_{i=1,i\neq j}^{m}|f_{i}(y_{i})f_{j}(y_{j}) (b_{j}(y_{j})-b_{2^{k+1}B} )|}\,\mathrm{d}\vec{y}\) by the Hölder inequality and such that \(\vec {\omega}\in A_{\vec{p}/s}\) by Lemma 2.3. Then characterization \(A_{\vec{p}/s}\) and (2.3) yield
So we have
For \(\mathit{II}'\), we just consider this case: \(\alpha_{i}=\infty\) for \(i=1,\ldots,l\) and \(\alpha_{j}=0\) for \(j=l+1,\ldots,m\). There are two cases:
or
We just consider the following case:
The estimate for
is similar to (3.11). We get
so we have
Take \(L^{p}(v_{\vec{\omega}})\) norm on the ball \(B(y,r)\) on both sides of (3.15). By (1.5), (2.3), (2.4), we have
Multiplying both sides of (3.16) by \(v_{\vec{\omega }}(B)^{1/\alpha-1/q-1/p}\), by Lemma 2.2 and (2.1), we obtain
Next the proof is similar to Theorem 1.1 of (1), we get
□
References
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Acknowledgements
This work is supported by NNSF-China (Grant Nos. 11171345 and 51234005).
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PW put forward the ideas of the paper, and the authors completed the paper together. They also read and approved the final manuscript.
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Wang, P., Liu, Z. Weighted norm inequalities for multilinear Calderón-Zygmund operators in generalized Morrey spaces. J Inequal Appl 2017, 48 (2017). https://doi.org/10.1186/s13660-017-1325-z
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DOI: https://doi.org/10.1186/s13660-017-1325-z