L p Bounds for the Commutators of Oscillatory Singular Integrals with Rough Kernels

and Applied Analysis 3 2. Lemmas We give some lemmas which will be used in the proof of Theorems 1 and 2. Lemma 8. Let mδ(ξ) ∈ C(R) (0 < δ < ∞) be a family of multipliers such that suppmδ ⊂ {ξ : |ξ| ≤ δ}, ∇ξmδ = (∂mδ/∂ξ1, ..., ∂mδ/∂ξn), and for some constants C, 0 < A ≤ 1/2, and α > 0 󵄩󵄩󵄩󵄩mδ 󵄩󵄩󵄩󵄩∞ ≤ Cmin {Aδ, log −(α+1) (2 + δ)} , 󵄩󵄩󵄩󵄩 ∇ξmδ 󵄩󵄩󵄩󵄩∞ ≤ C. (11) Let Tδ be the multiplier operator defined by Tδf(ξ) = mδ(ξ)f(ξ), ξ = (ξ, ξn+1). For b ∈ BMO(R), denote by [b, Tδ] the commutator of Tδ. Then for any 0 < ] < 1, there exists a positive constant C = C(n, ]) such that 󵄩󵄩󵄩󵄩[b, Tδ] f 󵄩󵄩󵄩󵄩2 ≤ C‖b‖BMO(Rn+1)(Aδ) ] log( 1 A ) 󵄩󵄩󵄩󵄩f 󵄩󵄩󵄩󵄩2,

It has been proved that the boundedness of   on   (R  ) can be obtained from the   (R +1 ) boundedness of  ,Ω (see [5]).
Recently, Chen and Ding [20] established the   boundedness of the commutator of singular integrals with the kernel condition Ω ∈   ( −1 ).
It is natural to ask whether the similar result holds for the commutators of oscillatory singular integrals, which is defined by In this paper, we will give a positive answer to the above question by imposing some conditions on .We first prove the boundedness of the commutator of singular integral along surfaces, which is defined by Theorem 1.Let Ω be a function in  1 ( −1 ) satisfying (2) and (3),  ∈ (R +1 ), radial function  ∈  1 ([0, ∞)) with (0) =   (0) = 0, and   is a convex increasing function.If Ω ∈   ( −1 ) for some  > 1, then [,  ,Ω ] is bounded on  2 (R +1 ).
Combining Theorem 4 with Theorems 1 and 2, respectively, we can get the following two theorems immediately.

Lemmas
We give some lemmas which will be used in the proof of Theorems 1 and 2.
Proof.We assume that ‖‖ BMO(R +1 ) = 1.Let x = (,  +1 ) and let Ψ( x) be a radial function such that supp Ψ ⊂ {x : 1/4 ≤ | x| ≤ 4}, and Let  , be the convolution operator whose kernel is  , ; that is,  ,  =  , * .Recall that supp On the other hand, by the Yong inequality, we have Then, using the same argument of the proof of Lemma 2 in [22] we can prove Lemma 8.
Let the measure   on R +1 be defined by for all  ∈ Z. Define the maximal operator in R +1 by Lemma 9 (see [18]).Suppose  * is bounded on   (R +1 ) for all 1 <  < ∞.Then, for arbitrary functions   , the following vector valued inequality: holds with any 1 <  < ∞.
The maximal function in R 2 is defined by We know that the   (R +1 ) boundedness of  * is deduced from the   (R 2 ) boundedness of   by method of rotations, and if  is as in Theorem 1 or Theorem 2,   is a bounded operator on   (R 2 ) for all 1 <  < ∞ (see [23,24]).Let  ∈ S(R  ) be a radial function satisfying 0 ≤  ≤ 1 with its support in the unit ball and For  ∈ Z, denote by Δ  and   the convolution operators whose symbols are  0 (2 − ) and (2 − ), respectively.
Lemma 10 (see [20]).For the multiplier   ( ∈ Z),  ∈ (R  ), and any fixed 0 <  < 1/2, we have where  is independent of  and . and Let  ∈ (R +1 ), and denote by [, Proof.We prove it by using arguments which are essentially the same as those in the proof of Lemma 3.7 in [20].Two things must be modified: (i) instead of Lemma 3.6 in [20], we use Lemma 9; (ii) In [20], ) is the paraproduct of Bony [25] between two functions  and .In the estimate of  1 , we will use the following formulas:

The proof of Theorems 4 and 7
We begin with a lemma, which plays an important role in proving Theorem 4.
Lemma 12. Let () ∈ (R  ), x = (,  +1 ) ∈ R +1 , and Proof.We know where   = (1/||) ∫  () and  is the square in R  whose edges are parallel to the axis.So where Q is the square in R +1 whose edges are parallel to the axis.Consider where  is the projection on R  of Q and  is the side length of Q.Then ‖‖ BMO(R +1 ) = sup Q⊂R +1