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In this paper, the authors study the boundedness of multilinear Calderón-Zygmund singular integral operators and their commutators in generalized Morrey spaces.
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 , it is said that a function K belongs to the class m-CZK(A, ε) if
The operator T is said to be an m-linear Calderón-Zygmund operator if there exists a function K ∈ m-CZK(A, ε) defined away from the diagonal y0 = y1 = y2 ⋯ = ym in (ℝn)m+1 such that
for , and that T extends to a bounded multilinear operator from Lq1 × ⋯ × Lqm to Lq for some 1 ⩽ qj < ∞ with .
It was shown in  that if , then an m-linear Calderón-Zygmund operator satisfies
when 1 < rj < ∞ for j = 1, …, m and
when 1 ≤ rj < ∞ for j = 1, …, m and at least one rj = 1. In particular,
The theory of multiple weight associated with m-linear Calderón-Zygmund operators was developed by Lerner et al. . Let 1 < pj < ∞ for j = 1, …, m, and , we say if
where B is the ball in ℝn and . They showed that if then
If 1 ⩽ pj < ∞ for j = 1, …, m and at least one of the pj = 1, they also proved
Let be a vector-valued locally integrable function. If in (BMO)m, we denote (see ). The commutator generated by an m-linear Calderón-Zygmund operator T and a (BMO)m function is defined by
where each term is the commutator of bj and T in the jth entry of T, that is,
Pérez and Torres  proved that if then
for 1 < pj < ∞ and 1 < p < ∞ with , where j = 1, …, m. In , the authors proved that if and , then
for 1 < pj < ∞ with , where j = 1, …, m.
Feuto  introduced the generalized weighted Morrey space (Lp(ω),Lq)α. Let 1 ⩽ p ⩽ α ⩽ q ⩽ ∞ and ω be a weight. The space (Lp(ω),Lq)α was defined to be the set of all measurable functions f satisfying ∥f∥(Lp(ω),Lq)α < ∞, where
Similarly, the weak space (Lp,∞(ω),Lq)α is defined with
When p = 1, the space (L1,∞(ω),Lq)α was introduced in .
Feuto proved in  that Calderón-Zygmund singular integral operators, Marcinkiewicz operators, the maximal operators associated to Bochner-Riesz operators and their commutators are bounded on (Lp(ω),Lq)α.
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.
Let T be an m-linear Calderón-Zygmund operator, and .
Let be a multilinear commutator, , with 1 < pj < ∞ and . If p ⩽ α < q ⩽ ∞, then
We first recall the definition of Ap weight. A nonnegative locally integrable function ω belongs to Ap (p > 1) if
where p′ is the conjugate index of p, i.e., 1/p + 1/p′ = 1. We say that ω ∈ A1 if there is a constant C > 0 such that
for any ball B. If ω ∈ Ap, then there exists δ > 0 such that
for any measurable subset E of a ball B. Since the Ap classes are increasing with respect to p, we use the following notation A∞ = ⋃p>1Ap. A ≲ B means A ≤ CB, where C is a positive constant independent of the main parameters. For λ > 0 and a ball B ⊂ ℝn, we write λB for the ball with same center as B and radius λ times radius of B.
Now we give the definition of condition.
Let 1 ⩽ pj < ∞ for j = 1, …, m, and . Given , set
We say that if
p′ is the conjugate index of p. When pj = 1, denote , is understood as (infBωj)−1
Obviously, if m = 1, is the classical Ap class. has the following characterization.
Let . Then if and only if
where the condition is understood as in the case pj = 1.
Assume that satisfies condition. Then there exists a finite constant r > 1 such that .
In order to prove the results for commutators, we need the following properties of BMO. For b ∈ BMO, 1 < p < ∞ and ω ∈ A∞, we get
and for all balls B,
For all nonnegative integers k, we obtain
where ω(B) = ∫Bω(x) dx, (see ).
(1) Let B = B(y, r) be a ball of ℝn, fi = fiχ2B + fiχ(2B)c and denote fiχ2B by and fiχ(2B)c by (i = 1, …, m), χE denotes the characteristic function of set E. For x ∈ B(y, r), we have
where α1, …, αm are not all equal to 0 or ∞ at the same time. We first estimate III. Since 2k−1r ⩽ |x − yi| ⩽ 2k+2r, we have
the Hölder inequality gives us that
By the definition of condition, we obtain
For II, we just consider this case: αi = ∞ for i = 1, …, l and αj = 0 for j = l + 1, …, m,
By (3.2) and the definition of condition, we have
Combining all the cases together, we obtain
For , we have
by the Hölder inequality. Since , we obtain the expected result
It suffices to prove . For B = B(y, r), x ∈ B
where α1, …, αm are not all equal to 0 or ∞ at the same time. We first deal with III′.
So we have
For II′, we just consider this case: αi = ∞ for i = 1, …, l and αj = 0 for j = l + 1, …, m. There are two cases:
We just consider the following case:
The estimate for
is similar to (3.11). We get
so we have
Next the proof is similar to Theorem 1.1 of (1), we get
This work is supported by NNSF-China (Grant Nos. 11171345 and 51234005).
The authors declare that they have no competing interests.
PW put forward the ideas of the paper, and the authors completed the paper together. They also read and approved the final manuscript.
Panwang Wang, Email: moc.liamg@wgnawnap.
Zongguang Liu, Email: nc.ude.btmuc@gzuil.