In this article, the authors consider the commutators of strongly singular Calderón-Zygmund operator with Lipschitz functions. A sufficient condition is given for the boundedness of the commutators from Lebesgue spaces to certain Campanato spaces .
1 Introduction and result
Let be the classical singular integral operator, and the commutator generated by and a locally integrable function is given by
In 1978, Janson  studied the boundedness of the commutator when , the homogeneous Lipschitz space of order , which is the space of all functions , such that
Janson proved that is bounded from to for if and only if with . In 1995, Paluszyński  made a further study of the problem and proved that is bounded from to some Triebel-Lizorkin spaces if and only if , for and .
In 2015, Zhang et al.  gave another kind of interesting results for when belongs to Lipschitz spaces. They proved that is bounded from to if and only if , for , and , where is Campanato space (see Definition 1.2).
On the other hand, motivated by the study of multiplier operator with symbol given by away from the origin , Alvarez and Milman  introduced the following strongly singular Calderón-Zygmund operator.
 Let be a bounded linear operator. is called a strongly singular Calderón-Zygmund operator if the following conditions are fulfilled.
can be extended to a continuous operator from into itself.
There exists a continuous function on such that
if for some and , and,
for with disjoint supports.
For some , both and its conjugate operator can be extended to continuous operators from into , where .
In 1986, Alvarez and Milman studied the boundedness of strongly singular Calderón-Zygmund operator on Lebesgue spaces and Hardy spaces in [6,7]. Later on, there are many authors discussed the mapping properties of strongly singular Calderón-Zygmund operators in various spaces. See, for instance, [8,9, 10,11]. We would like to note that, as stated in [6,7], the strongly singular Calderón-Zygmund operators include pseudo-differential operator with a symbol in the Hörmander class , where , .
Now, we define the commutator generated by strongly singular Calderón-Zygmund operator and a locally integrable function as follows:
The main result of Alvarez et al. in  yields the boundedness of on , , when . Afterward, the mapping properties of , when belongs to BMO space or Lipschitz space, on Lebesgue spaces, Morrey spaces, Herz type spaces, and Hardy spaces have been studied by several authors. See [10,11,12, 13,14,15] for example.
In this article, we will continue the study of the commutator of strongly singular Calderón-Zygmund operator when the symbol belongs to Lipschitz space. The aim is to extend some of the results in  to a strongly singular Calderón-Zygmund operator.
As usual, let denote the ball centered at with radius . For , stands for the ball concentric with having times its radius, that is, . Denote by the Lebesgue measure of and by its characteristic function. For , we write
Let , , the Campanato space is given by
and the supremum is taken over all balls in .
Our result can be stated as follows.
Let be a strongly singular Calderón-Zygmund operator, and, , and be as in Definition 1.1. Suppose that , , and . If , then is bounded from to , that is, there exists a constant such that for all ,
2 Proof of Theorem 1.1
Let and , then for all ,
 Let , , and and be balls in . If , then
Now we recall the boundedness of strongly singular Calderón-Zygmund operator on Lebesgue spaces. Let us observe that is bounded from to (, Theorem 2.1), from to (, Theorem 4.1), and from to (, Lemma 2), and note the assumption (3) in Definition 1.1, and by interpolation between these estimates, we achieve the following -boundedness of . We refer to  (page 1052) and  (pages 42 and 43), for details.
Let be a strongly singular Calderón-Zygmund operator, and, , and be the same as in Definition 1.1.
If , then is bounded from to itself.
If , then there is a positive number satisfying , such that is bounded from to .
Now, let us prove Theorem 1.1.
Proof of Theorem 1.1
For any , it suffices to prove
for all balls in .
For any ball centered at with radius , we divide the proof into two cases.
Case 1. The case when . Denote by the ball with the same center as and 8 times the radius. Let and . For any real number , by Minkowski’s inequality and Hölder’s inequality, we have
Let and notice that , one has
Now, let us consider . Since for any and any one has , it follows from Definition 1.1 that
Observe that the last term of (2.3) is always independent of and , for any . Then we can write
Applying Hölder’s inequality, Lemma 2.1, and noting that , we obtain
where in the last step we made use of the fact that since and and the fact that the series is convergent since .
For , noting that and applying Lemma 2.2 and Hölder’s inequality, we have
where in the last step we also made use of the fact that and is convergent. This, together with the estimates for , yields
Combining the estimates for , , and leads to (2.1) for the case .
Case 2. The case . Set and denote . Let and . For the same reason as that in (2.2), we deduce that
Similar to , we have
where the last two steps follow from the condition and the fact that since and .
Finally, let us consider . Since and for any and , similar to (2.3), we have
Reasoning as in (2.4), we can also write
To estimate , we first observe the fact that since and and the series is convergent since . Then by Hölder’s inequality and Lemma 2.1 and noticing that , we obtain
For , applying Lemma 2.2 and Hölder’s inequality, and observing again the fact that , the series is convergent and , and we have
The estimate for