Weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels
© Zhang and Guo; licensee Springer. 2014
Received: 9 February 2014
Accepted: 29 August 2014
Published: 25 September 2014
In this paper, we prove the weighted endpoint estimates for multilinear commutator of singular integral operators with non-smooth kernels.
Let and T be the Calderón-Zygmund operator, the commutator generated by b and T is defined by . A classical result of Coifman et al. (see ) proved that the commutator is bounded on (). In [2, 3], the boundedness properties of the commutators for the extreme values of p are obtained. In this paper, we will introduce the multilinear commutator of singular integral operators with non-smooth kernels and prove the weighted boundedness properties of the operator for the extreme cases.
for some .
- (1)There exists an ‘approximation to the identity’ such that has associated kernel and there exist so that
- (2)There exists an ‘approximation to the identity’ such that has associated kernel which satisfiesand
for some , .
- (1)The weighted BMO space associated with is defined bywhere
and denotes the side length of Q.
- (2)The weighted central BMO space associated with is defined bywhere
For (), set . Given a positive integer m and , we denote by the family of all finite subsets of of j different elements. For , set . For and , set , and .
2 Theorems and proofs
We begin with some preliminaries lemmas.
Let , , and T be the singular integral operators with non-smooth kernels. Then T is boundedness on .
- (a)for every , , and any cube Q,
- (b)for every , , and any cube Q,
where and denotes the side length of Q.
This completes the proof. □
Theorem 1 Let T be the singular integral operators with non-smooth kernels, and with for . Then is bounded from to .
We fix a cube . We decompose f into with , .
This completes the proof of Theorem 1. □
Theorem 2 Let , and with for . Then is bounded from to .
This completes the proof of Theorem 2. □
Project was supported by the National Natural Science Foundation of China (No. 11061003).
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