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

Abstract

We establish the 𝐿^𝑝 boundedness for some commutators of oscillatory singular integrals with the kernel condition which was introduced by Grafakos and Stefanov. Our theorems contain various conditions on the phase function.

1. Introduction

The homogeneous singular integral operator 𝑇_Ω is defined by 𝑇_Ω𝑓(𝑥)=p.v._ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛𝑓𝑦𝑑𝑦, (1) where Ω∈𝐿¹(𝑆^𝑛−1) satisfies the following conditions.(a)Ω is homogeneous function of degree zero on ℝ^𝑛⧵{0}; that is, Ω(𝑡𝑥)=Ω(𝑥) (2) for any 𝑡>0 and 𝑥∈ℝ^𝑛⧵{0}.(b)Ω has mean zero on 𝑆^𝑛−1, the unit sphere in ℝ^𝑛; that is, _𝑆𝑛−1Ω𝑥^′𝑑𝜎𝑥^′=0. (3)

The oscillatory singular integral we will consider here is defined by 𝑇_𝜙𝑓(𝑥)=p.v._ℝ𝑛𝑒^{𝑖𝜙(𝑦)}Ω𝑦𝑦^𝑛𝑓𝑥−𝑦𝑑𝑦. (4) If 𝜙(𝑥)≡0, the operator 𝑇_𝜙 becomes the singular integral operator 𝑇_Ω.

When 𝜙(𝑥)=𝑃(𝑥) is a real polynomial, the 𝐿^𝑝 boundedness of 𝑇_𝜙 was first studied by Ricci and Stein [1] with Ω∈𝐶¹(𝑆^𝑛−1), and Hu and Pan [2] obtained the weighted 𝐻¹ boundedness of 𝑇_𝜙. When Ω∈𝐿^𝑟(𝑆^𝑛−1), 𝑟>1, Lu and Zhang proved the 𝐿^𝑝 boundedness [3] and this was extended to the case of Ω∈𝐿ln⁺𝐿(𝑆^𝑛−1) by Ojanen [4] and the case of Ω∈𝐻¹(𝑆^𝑛−1) by Fan and Pan [5].

Grafakos and Stefanov [6] introduced a class of kernel functions 𝐹_𝛼(𝑆^𝑛−1) which contains all Ω(𝑦)∈𝐿¹(𝑆^𝑛−1) satisfying (3) and sup_{𝜉∈𝑆𝑛−1𝑆𝑛−1}Ω𝑦ln𝑦⋅𝜉^−11+𝛼𝑑𝜎𝑦<∞, (5) where 𝛼>0 is a fixed constant. This kernel condition has been considered by many authors [7–13].

The singular integral along surfaces which is defined by 𝑇_𝜙,Ω𝑓𝑥,𝑥_𝑛+1=p.v._ℝ𝑛Ω𝑦𝑦^𝑛𝑓𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦𝑑𝑦 (6) was also studied by many authors [14–18]. Under the condition Ω∈𝐹_𝛼(𝑆^𝑛−1), Pan et al. [16] established the following Theorem.

Theorem A (see [16]). Let 𝜙(𝑡)∈𝐶¹([0,∞)), 𝜙(0)=𝜙^′(0)=0, and 𝜙^′ is a convex increasing function for 𝑡>0, Ω∈𝐹_𝛼(𝑆^𝑛−1) for some 𝛼>0; then, 𝑇_𝜙,Ω is bounded on 𝐿^𝑝(ℝ^𝑛+1) for (2+2𝛼)/(1+2𝛼)<𝑝<2+2𝛼.

Later, Cheng and Pan [14] improved the result for 𝑛=2 by removing the condition 𝜙^′(0)=0.

Theorem B (see [14]). Let 𝜙(𝑡)∈𝐶¹([0,∞)), 𝜙(0)=0, and 𝜙^′ is a convex increasing function for 𝑡>0, Ω∈𝐹_𝛼(𝑆^𝑛−1) for some 𝛼>0; then, 𝑇_𝜙,Ω is bounded on 𝐿^𝑝(ℝ³) for (2+2𝛼)/(1+2𝛼)<𝑝<2+2𝛼.

It has been proved that the boundedness of 𝑇_𝜙 on 𝐿^𝑝(ℝ^𝑛) can be obtained from the 𝐿^𝑝(ℝ^𝑛+1) boundedness of 𝑇_𝜙,Ω (see [5]).

For a function 𝑏∈𝐿_loc(ℝ^𝑛), let 𝐴 be a linear operator on some measurable function space; the commutator between 𝐴 and 𝑏 is defined by [𝑏,𝐴]𝑓(𝑥)∶=𝑏(𝑥)𝐴𝑓(𝑥)−𝐴(𝑏𝑓)(𝑥).

It has been proved by Hu [19] that Ω∈𝐿(log𝐿)²(𝑆^𝑛−1) is a sufficient condition for the commutator to be bounded on 𝐿^𝑝(ℝ^𝑛), which is defined by 𝑏,𝑇_Ω𝑓(𝑥)=p.v._ℝ𝑛Ω𝑥−𝑦𝑥−𝑦^𝑛𝑏(𝑥)−𝑏𝑦𝑓𝑦𝑑𝑦. (7) Recently, Chen and Ding [20] established the 𝐿^𝑝 boundedness of the commutator of singular integrals with the kernel condition Ω∈𝐹_𝛼(𝑆^𝑛−1).

It is natural to ask whether the similar result holds for the commutators of oscillatory singular integrals, which is defined by 𝑏,𝑇_𝜙𝑓(𝑥)=p.v._ℝ𝑛𝑒^{𝑖𝜙(𝑦)}Ω𝑦𝑦^𝑛𝑏(𝑥)−𝑏𝑥−𝑦×𝑓𝑥−𝑦𝑑𝑦. (8) In this paper, we will give a positive answer to the above question by imposing some conditions on 𝜙.

We first prove the boundedness of the commutator of singular integral along surfaces, which is defined by 𝑏,𝑇_𝜙,Ω𝑓𝑥,𝑥_𝑛+1=p.v._ℝ𝑛Ω𝑦𝑦^𝑛𝑏𝑥,𝑥_𝑛+1−𝑏𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦×𝑓𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦𝑑𝑦. (9)

Theorem 1. Let Ω be a function in 𝐿¹(𝑆^𝑛−1) satisfying (2) and (3), 𝑏∈𝐵𝑀𝑂(ℝ^𝑛+1), radial function 𝜙∈𝐶¹([0,∞)) with 𝜙(0)=𝜙^′(0)=0, and 𝜙^′ is a convex increasing function. If Ω∈𝐹_𝛼(𝑆^𝑛−1) for some 𝛼>1, then [𝑏,𝑇_𝜙,Ω] is bounded on 𝐿²(ℝ^𝑛+1).

Theorem 2. Let Ω be a function in 𝐿¹(𝑆¹) satisfying (2) and (3), 𝑏∈𝐵𝑀𝑂(ℝ³), radial function 𝜙(|𝑡|)=|𝑡|. If Ω∈𝐹_𝛼(𝑆¹) for some 𝛼>1, then [𝑏,𝑇_𝜙,Ω] is bounded on 𝐿^𝑝(ℝ³) for (𝛼+1)/𝛼<𝑝<𝛼+1.

Remark 3. However, for 𝑛≥3, we can not prove the 𝐿^𝑝(ℝ^𝑛+1) boundedness of [𝑏,𝑇_𝜙,Ω] by our method using Lemma 11, since the conditions imposed on 𝜙 in Theorem 1 conflict with Lemma 11. Only when 𝑛=2 by removing the condition 𝜙^′(0)=0 in Theorem 1 can we eliminate the conflict, and 𝜙(|𝑡|)=|𝑡| is a feasible function. Also, by another method, it is hard to give the boundedness of the maximal operator defined by 𝑏,𝑀_𝜙,Ω𝑓𝑥,𝑥_𝑛+1=sup_{𝑗∈ℤ2𝑗<𝑦<2𝑗+1}Ω𝑦𝑦^𝑛×𝑏𝑥,𝑥_𝑛+1×−𝑏𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦×𝑓𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦𝑑𝑦. (10)
Then we give the boundedness of the commutators of oscillatory singular integral [𝑏,𝑇_𝜙].

Let 𝑏(𝑥)∈BMO(ℝ^𝑛), 𝑥=(𝑥,𝑥_𝑛+1)∈ℝ^𝑛+1, 𝐵(𝑥)=𝑏(𝑥), and we have the following result.

Theorem 4. If [𝐵,𝑇_𝜙,Ω] is bounded on 𝐿^𝑝(ℝ^𝑛+1) with bound 𝐶‖𝐵‖_{𝐵𝑀𝑂(ℝ}^𝑛+1₎, then [𝑏,𝑇_𝜙] is bounded on 𝐿^𝑝(ℝ^𝑛) with bound 𝐶‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛₎.

Combining Theorem 4 with Theorems 1 and 2, respectively, we can get the following two theorems immediately.

Theorem 5. Let Ω be a function in 𝐿¹(𝑆^𝑛−1) satisfying (2) and (3), 𝑏∈𝐵𝑀𝑂(ℝ^𝑛), radial function 𝜙∈𝐶¹([0,∞)) with 𝜙(0)=𝜙^′(0)=0, and 𝜙^′ is a convex increasing function. If Ω∈𝐹_𝛼(𝑆^𝑛−1) for some 𝛼>1, then [𝑏,𝑇_𝜙] is bounded on 𝐿²(ℝ^𝑛).

Theorem 6. Let Ω be a function in 𝐿¹(𝑆¹) satisfying (2) and (3), 𝑏∈𝐵𝑀𝑂(ℝ²), radial function 𝜙(|𝑡|)=|𝑡|. If Ω∈𝐹_𝛼(𝑆¹) for some 𝛼>1, then [𝑏,𝑇_𝜙] is bounded on 𝐿^𝑝(ℝ²) for (𝛼+1)/𝛼<𝑝<𝛼+1.

In above theorems, the phase functions are radial. But when Ricci and Stein first studied the oscillatory singular integral 𝑇_𝜙, they take 𝜙(𝑥)=𝑃(𝑥), apparently nonradial. In Theorem 7, we will take 𝜙(𝑥)=𝑃(𝑥)=∑^𝑚_|𝛼|/2=1𝑎_𝛼𝑥^𝛼, and this condition was mentioned in [21].

Theorem 7. Let Ω be a function in 𝐿¹(𝑆^𝑛−1) satisfying (2) and (3), 𝑏∈𝐵𝑀𝑂(ℝ^𝑛). If Ω∈𝐹_𝛼(𝑆^𝑛−1) is an odd kernel for some 𝛼>1, 𝜙(𝑥)=∑^𝑚_|𝛼|/2=1𝑎_𝛼𝑥^𝛼 is an even phase; then, [𝑏,𝑇_𝜙] extends to a bounded operator from 𝐿^𝑝(ℝ^𝑛) into itself for (𝛼+1)/𝛼<𝑝<𝛼+1.

2. Lemmas

We give some lemmas which will be used in the proof of Theorems 1 and 2.

Lemma 8. Let 𝑚_𝛿(𝜉̃)∈𝐶¹(ℝ^𝑛+1)(0<𝛿<∞) be a family of multipliers such that supp𝑚_𝛿⊂{𝜉̃∶|𝜉̃|≤𝛿}, ∇_𝜉𝑚_𝛿=(𝜕𝑚_𝛿/𝜕𝜉₁,...,𝜕𝑚_𝛿/𝜕𝜉_𝑛), and for some constants 𝐶,0<𝐴≤1/2, and 𝛼>0𝑚_𝛿∞≤𝐶min𝐴𝛿,log^−(𝛼+1)(,∇2+𝛿)_𝜉𝑚_𝛿∞≤𝐶. (11) Let 𝑇_𝛿 be the multiplier operator defined by 𝑇_𝛿𝑓(𝜉̃)=𝑚_𝛿(𝜉̃)𝑓(𝜉̃), 𝜉̃=(𝜉,𝜉_𝑛+1). For 𝑏∈𝐵𝑀𝑂(ℝ^𝑛+1), denote by [𝑏,𝑇_𝛿] the commutator of 𝑇_𝛿. Then for any 0<𝜈<1, there exists a positive constant 𝐶=𝐶(𝑛,𝜈) such that 𝑏,𝑇_𝛿𝑓₂≤𝐶‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛+1₎(𝐴𝛿)^𝜈1log𝐴𝑓₂,𝑖𝑓𝛿<10√𝐴;𝑏,𝑇_𝛿𝑓₂≤𝐶‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛+1₎log^{−(𝛼+1)𝜈+1}(2+𝛿)𝑓₂,1𝑖𝑓𝛿>√𝐴. (12)

Proof. We assume that ‖𝑏‖_BMO(ℝ^𝑛+1₎=1. Let 𝑥=(𝑥,𝑥_𝑛+1) and let Ψ(𝑥) be a radial function such that suppΨ⊂{𝑥∶1/4≤|𝑥|≤4}, and _𝑙∈ℤΨ2^−𝑙𝑥=1 (13) for |𝑥|>0. Set Ψ₀(𝑥)=∑⁰_𝑙=−∞Ψ(2^−𝑙𝑥) and Ψ_𝑙(𝑥)=Ψ(2^−𝑙𝑥) for positive integer 𝑙. Let 𝐾_𝛿(𝑥)=𝑚^∨_𝛿(𝑥) the inverse Fourier transform of 𝑚_𝛿. Split 𝐾_𝛿 as 𝐾_𝛿(𝑥)=𝐾_𝛿(𝑥)Ψ₀(𝑥)+^∞_𝑙=1𝐾_𝛿(𝑥)Ψ_𝑙(𝑥)=^∞_𝑙=0𝐾_𝛿,𝑙(𝑥). (14)
Let 𝑇_𝛿,𝑙 be the convolution operator whose kernel is 𝐾_𝛿,𝑙; that is, 𝑇_𝛿,𝑙𝑓=𝐾_𝛿,𝑙∗𝑓. Recall that supp𝑚_𝛿⊂{𝜉̃∶|𝜉̃|≤𝛿}. Trivial computation shows that ‖𝐾_𝛿,𝑙‖_∞≤‖𝐾_𝛿‖_∞≤‖𝑚_𝛿‖₁≤𝐶𝛿^𝑛+1. This via the Young inequality says that 𝑇_𝛿,𝑙𝑓_∞≤𝐶𝛿^𝑛+1𝑓₁. (15) Note that ∫_ℝ𝑛+1Ψ(𝜂)𝑑𝜂=0. Thus 𝐾_𝛿,𝑙∞=_ℝ𝑛+1𝑚_𝛿𝜉−2^−𝑙𝜂,𝜉_𝑛+1−2^−𝑙𝜂_𝑛+1−𝑚_𝛿×𝜉,𝜉_𝑛+1−2^−𝑙𝜂_𝑛+1Ψ𝜂𝑑𝜂_∞≤𝐶2^−𝑙∇_𝜉𝑚_{𝛿∞ℝ𝑛+1}𝜂Ψ𝜂𝑑𝜂≤𝐶2^−𝑙∇_𝜉𝑚_{𝛿∞ℝ𝑛+1}𝜂Ψ𝜂𝑑𝜂≤𝐶2^−𝑙. (16) On the other hand, by the Yong inequality, we have 𝐾_𝛿,𝑙∞≤𝐾_𝛿∞Ψ_𝑙1≤𝐶min𝐴𝛿,log^−(𝛼+1)(2+𝛿). (17) Then, using the same argument of the proof of Lemma 2 in [22] we can prove Lemma 8.

Let the measure 𝜎_𝑗 on ℝ^𝑛+1 be defined by _ℝ𝑛+1𝑓𝑦,𝑦_𝑛+1𝑑𝜎_𝑗=_ℝ𝑛𝑓𝑦Ω𝑦𝑦,𝜙^′𝑦^𝑛𝜒_{{2𝑗<|𝑦|≤2𝑗+1}}𝑑𝑦 (18) for all 𝑗∈ℤ. Define the maximal operator in ℝ^𝑛+1 by 𝜎^∗𝑓=sup_𝑗∈ℤ|𝜎_𝑗|∗|𝑓|.

Lemma 9 (see [18]). Suppose 𝜎^∗ is bounded on 𝐿^𝑞(ℝ^𝑛+1) for all 1<𝑞<∞. Then, for arbitrary functions 𝑔_𝑗, the following vector valued inequality: _𝑗𝜎_𝑗∗𝑔_𝑗^21/2_{𝐿𝑞(ℝ𝑛+1)}≤𝐶_𝑗𝑔_𝑗^21/2_{𝐿𝑞(ℝ𝑛+1)} (19) holds with any 1<𝑞<∞.

The maximal function in ℝ² is defined by 𝑀_𝜙𝑓𝑥₁,𝑥₂=sup_𝑘∈ℤ12^𝑘2𝑘+1₂^𝑘𝑓𝑥₁−𝑡,𝑥₂−𝜙(𝑡)𝑑𝑡. (20) We know that the 𝐿^𝑞(ℝ^𝑛+1) boundedness of 𝜎^∗ is deduced from the 𝐿^𝑞(ℝ²) boundedness of 𝑀_𝜙 by method of rotations, and if 𝜙 is as in Theorem 1 or Theorem 2, 𝑀_𝜙 is a bounded operator on 𝐿^𝑞(ℝ²) for all 1<𝑞<∞ (see [23, 24]).

Let 𝜑∈𝒮(ℝ^𝑛) be a radial function satisfying 0≤𝜑≤1 with its support in the unit ball and 𝜑(𝜉)=1 for |𝜉|≤1/2. The function 𝜑₀(𝜉)=𝜑(𝜉/2)−𝜑(𝜉)∈𝒮(ℝ^𝑛) satisfies ∑_𝑗∈ℤ𝜑₀(2^−𝑗𝜉)=1 for 𝜉≠0. For 𝑗∈ℤ, denote by Δ_𝑗 and 𝐺_𝑗 the convolution operators whose symbols are 𝜑₀(2^−𝑗𝜉) and 𝜑(2^−𝑗𝜉), respectively.

Lemma 10 (see [20]). For the multiplier 𝐺_𝑘(𝑘∈ℤ), 𝑏∈𝐵𝑀𝑂(ℝ^𝑛), and any fixed 0<𝜏<1/2, we have 𝐺_𝑘𝑏(𝑥)−𝐺_𝑘𝑏𝑦2≤𝐶^𝑘𝜏𝜏𝑥−𝑦^𝜏‖𝑏‖_𝐵𝑀𝑂, (21) where 𝐶 is independent of 𝑘 and 𝜏.

Let 𝜉̃=(𝜉,𝜉_𝑛+1)∈ℝ^𝑛+1 and let 𝜓(𝜉̃)∈𝐶^∞₀(ℝ^𝑛+1) be a radial function such that 0≤𝜓≤1, supp𝜓⊂{1/2≤|𝜉̃|≤2}, and ∑_𝑙∈ℤ𝜓³(2^−𝑙𝜉̃)=1, |𝜉̃|≠0. Define the multiplier operator 𝑆_𝑙 by 𝑆_𝑙𝑓(𝜉̃)=𝜓(2^−𝑙|𝜉̃|)𝑓(𝜉̃).

Lemma 11. For any 𝑗∈ℤ, define the operator 𝑇_𝑗 by 𝑇_𝑗𝑓=𝜎_𝑗∗𝑓, and 𝜙 is monotonic and satisfies condition (1) or (2):(1)|𝜙(|𝑦|)|≤𝐶|𝑦|;(2)|𝜙(|𝑦|)|≥𝐶|𝑦|, |𝜙(𝑎)𝜙(𝑏)|≤𝐶|𝜙(𝑎𝑏)| for ∀𝑎,𝑏>0, and |𝜙(|𝑦|)|≤𝐶|𝑦|^𝑘1, 𝑘₁>1 if |𝑦|>1, |𝜙(|𝑦|)|≤𝐶|𝑦|^𝑘2, 0<𝑘₂<1 if |𝑦|≤1. Let 𝑏∈𝐵𝑀𝑂(ℝ^𝑛+1), and denote by [𝑏,𝑆_𝑙−𝑗𝑇_𝑗𝑆²_𝑙−𝑗] the commutator of 𝑆_𝑙−𝑗𝑇_𝑗𝑆²_𝑙−𝑗. Suppose Ω∈𝐿¹(𝑆^𝑛−1) satisfying (2). Then for any fixed 0<𝜏<1/2, 1<𝑝<∞, _𝑗∈ℤ𝑏,𝑆_𝑙−𝑗𝑇_𝑗𝑆²_𝑙−𝑗𝑓(𝑥)_𝐿𝑝≤𝐶‖𝑏‖_𝐵𝑀𝑂2max^𝜏𝑙𝜏,2^{𝜏𝑘1𝑙}𝜏,2^{𝜏𝑘2𝑙}𝜏,2𝑓_𝐿𝑝. (22)

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], 𝑀₁=‖∑_𝑗∈ℤ𝑆_𝑙−𝑗[𝜋_{(𝑇𝑗𝑆}²_{𝑙−𝑗𝑓)}(𝑏)−𝑇_𝑗(𝜋_(𝑆²_{𝑙−𝑗𝑓)}(𝑏))]‖_𝐿𝑝, and 𝜋_𝑓(𝑔)=∑_𝑗∈ℤ(Δ_𝑗𝑓)(𝐺_𝑗−3𝑔) is the paraproduct of Bony [25] between two functions 𝑓 and 𝑔. In the estimate of 𝑀₁, we will use the following formulas: 𝐺_𝑖−3𝑏,𝑇_𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1=𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1𝑇_𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1−𝑇_𝑗𝐺_𝑖−3𝑏Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1=_{2𝑗<|𝑦|≤2𝑗+1}Ω𝑦𝑦^𝑛×𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1−𝐺_𝑖−3𝑏×𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦⋅Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦𝑑𝑦≤𝐶_{2𝑗<|𝑦|≤2𝑗+1}Ω𝑦𝑦^𝑛×𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1−𝐺_𝑖−3𝑏×𝑥−𝑦,𝑥_𝑛+1𝑦⋅Δ−𝜙_𝑖𝑆²_𝑙−𝑗𝑓𝑥−𝑦,𝑥_𝑛+1−𝜙𝑦𝑑𝑦, (23)by Lemma 10, 𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1−𝐺_𝑖−3𝑏𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙≤𝐶^𝑖𝜏𝜏𝑦,𝜙𝑦^𝜏‖𝑏‖_BMO2=𝐶^𝑖𝜏𝜏𝑦²+𝜙²𝑦^𝜏‖𝑏‖_BMO. (24) If 𝜙 satisfies condition (1), we have 𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1−𝐺_𝑖−3𝑏𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙≤𝐶^𝑖𝜏𝜏𝑦^𝜏‖𝑏‖_BMO. (25) Thus 𝐺_𝑖−3𝑏,𝑇_𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+12≤𝐶^𝑖𝜏𝜏‖𝑏‖_BMO×_{2𝑗<|𝑦|≤2𝑗+1}Ω𝑦𝑦^𝑛𝑦^𝜏×Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙𝑑𝑦≤𝐶^{(𝑖+𝑗)𝜏}𝜏‖𝑏‖_BMO×_{2𝑗<|𝑦|≤2𝑗+1}Ω𝑦𝑦^𝑛×Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙𝑑𝑦=𝐶^{(𝑖+𝑗)𝜏}𝜏‖𝑏‖_BMO𝑇_|Ω|,𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1. (26) If 𝜙 satisfies condition (2), we have 𝐺_𝑖−3𝑏𝑥,𝑥_𝑛+1−𝐺_𝑖−3𝑏𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙≤𝐶^𝑖𝜏𝜏𝜙^𝜏𝑦‖𝑏‖_BMO𝜙≤𝐶^𝜏2^𝑖𝜏𝜙^𝜏𝑦‖𝑏‖_BMO. (27) Thus if |𝑦|>1, 𝐺_𝑖−3𝑏,𝑇_𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1𝜙≤𝐶^𝜏2^(𝑖+𝑗)𝜏‖𝑏‖_BMO×_{2𝑗<|𝑦|≤2𝑗+1}Ω𝑦𝑦^𝑛×Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥−𝑦,𝑥_𝑛+1𝑦2−𝜙𝑑𝑦≤𝐶^{(𝑖+𝑗)𝑘1𝜏}𝜏‖𝑏‖_BMO𝑇_|Ω|,𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1, (28) and if |𝑦|≤1, 𝐺_𝑖−3𝑏,𝑇_𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+12≤𝐶^{(𝑖+𝑗)𝑘2𝜏}𝜏‖𝑏‖_BMO𝑇_|Ω|,𝑗Δ_𝑖𝑆²_𝑙−𝑗𝑓𝑥,𝑥_𝑛+1. (29)

3. The Proof of Theorems 1 and 2

Proof of Theorem 1. Let 𝜉̃=(𝜉,𝜉_𝑛+1)∈ℝ^𝑛+1 and let 𝜓(𝜉̃)∈𝐶^∞₀(ℝ^𝑛+1) be a radial function such that 0≤𝜓≤1, supp𝜓⊂{1/2≤|𝜉̃|≤2}, and _𝑙∈𝑍𝜓³2^−𝑙𝜉̃=1,𝜉̃≠0. (30) Define the multiplier operator 𝑆_𝑙 by 𝑆_𝑙𝑓𝜉̃=𝜓2^−𝑙𝜉̃𝑓𝜉̃. (31) Let the measure 𝜎_𝑗 on ℝ^𝑛+1 be defined by _ℝ𝑛+1𝑓𝑦,𝑦_𝑛+1𝑑𝜎_𝑗=_ℝ𝑛𝑓𝑦Ω𝑦𝑦,𝜙^′𝑦^𝑛𝜒_{{2𝑗<|𝑦|≤2𝑗+1}}𝑑𝑦 (32) for all 𝑗∈ℤ. Since 𝜎_𝑗∗𝑓=_ℝ𝑛𝑓𝑥−𝑦,𝑥_𝑛+1𝑦Ω𝑦−𝜙^′𝑦^𝑛𝜒_{{2𝑗<|𝑦|≤2𝑗+1}}𝑇𝑑𝑦,_𝜙,Ω𝑓=_ℝ𝑛𝑓𝑥−𝑦,𝑥_𝑛+1𝑦Ω𝑦−𝜙^′𝑦^𝑛𝑑𝑦, (33) we get 𝑇_𝜙,Ω𝑓=_𝑗∈ℤ𝜎_𝑗∗𝑓. (34) Define the operator 𝑇_𝑗𝑓(𝑥)=𝜎_𝑗∗𝑓(𝑥), where 𝑥=(𝑥,𝑥_𝑛+1)∈ℝ^𝑛+1 and the multiplier 𝑇^𝑙_𝑗𝑓𝜉̃=𝑇_𝑗𝑆_𝑙−𝑗𝑓𝜉̃=𝜓2^𝑗−𝑙𝜉̃𝜎_𝑗𝜉̃𝑓𝜉̃. (35) From the above notation, it is easy to see that 𝑏,𝑇_𝜙,Ω𝑓(𝑥)=_{𝑙∈ℤ𝑗∈ℤ}𝑏,𝑆_𝑙−𝑗𝑇_𝑗𝑆²_𝑙−𝑗𝑓(𝑥)=_{𝑙∈ℤ𝑗∈ℤ}𝑏,𝑆_𝑙−𝑗𝑇^𝑙_𝑗𝑆_𝑙−𝑗𝑓(𝑥)∶=_𝑙∈ℤ𝑉_𝑙𝑓(𝑥), (36) where 𝑉_𝑙𝑓(𝑥)=_𝑗∈ℤ𝑏,𝑆_𝑙−𝑗𝑇^𝑙_𝑗𝑆_𝑙−𝑗𝑓(𝑥). (37) Then by the Minkowski inequality, we get 𝑏,𝑇_𝜙,Ω𝑓_{𝐿2(ℝ𝑛+1)}≤^√[log2]_𝑙=−∞𝑉_𝑙𝑓_{𝐿2(ℝ𝑛+1)}+^∞_{√𝑙=[log2]+1}𝑉_𝑙𝑓_{𝐿2(ℝ𝑛+1)}. (38) For ‖∑^√[log2]_𝑙=−∞𝑉_𝑙𝑓‖_{𝐿2(ℝ𝑛+1)}, we recall 𝜎_𝑗𝜉,𝜉_𝑛+1=_𝑆𝑛−1Ω(𝜃)^2𝑗+1_2𝑗𝑒^{−𝑖(𝑠𝜃⋅𝜉+𝜙(|𝑠|)𝜉𝑛+1)}𝑑𝑠𝑠𝑑𝜎(𝜃). (39) By Lemma 2.3 of [16], we have 𝜎_𝑗𝜉,𝜉_𝑛+1≤𝐶‖Ω‖_𝐿¹2^𝑗𝜉. (40) Denote by ∇_𝜉𝜎_𝑗 the before 𝑛 components truncation of ∇𝜎_𝑗; that is, ∇_𝜉𝜎_𝑗=𝜕𝜎_𝑗𝜕𝜉₁𝜕𝜎,...,^𝑗𝜕𝜉_𝑛. (41) Since 𝜎_𝑗𝜉,𝜉_𝑛+1=_ℝ𝑛Ω𝑦^′𝑦^𝑛𝜒_{{2𝑗<|𝑦|≤2𝑗+1}}𝑒^{−𝑖(𝑦⋅𝜉+𝜙(|𝑦|)𝜉𝑛+1)}𝑑𝑦, (42) we get ∇_𝜉𝜎_𝑗≤𝐶2^𝑗‖Ω‖_𝐿¹. (43) Set 𝑚_𝑗(𝜉̃)=𝜎_𝑗(𝜉̃), 𝑚^𝑙_𝑗(𝜉̃)=𝑚_𝑗(𝜉̃)𝜓(2^𝑗−𝑙|𝜉̃|). Recall that 𝑇^𝑙_𝑗 by 𝑇^𝑙_𝑗𝑓(𝜉̃)=𝑚^𝑙_𝑗(𝜉̃)𝑓(𝜉̃). Straightforward computations lead to 𝑚^𝑙_𝑗(2^−𝑗𝜉̃)_𝐿∞≤𝐶‖Ω‖_𝐿¹2^𝑙. (44) Since supp𝑚^𝑙_𝑗2^−𝑗𝜉̃⊂𝜉̃≤2^𝑙+2, (45) we get ∇_𝜉𝑚^𝑙_𝑗2^−𝑗𝜉̃_𝐿∞≤𝐶‖Ω‖_𝐿¹. (46) Let 𝑇^𝑙_𝑗 be the operator defined by 𝑇^𝑙_𝑗𝑓(𝜉̃)=𝑚^𝑙_𝑗(2^−𝑗𝜉̃)𝑓(𝜉̃). Denote by 𝑇^𝑙_𝑗,𝑏,1𝑓=[𝑏,𝑇^𝑙_𝑗]𝑓 and 𝑇^𝑙_𝑗,𝑏,0𝑓=𝑇^𝑙_𝑗𝑓. Similarly, denote by 𝑇^𝑙_𝑗,𝑏,1𝑓=[𝑏,𝑇^𝑙_𝑗]𝑓 and 𝑇^𝑙_𝑗,𝑏,0𝑓=𝑇^𝑙_𝑗𝑓. Thus via the Plancherel theorem and Lemma 8 it is stated that for any fixed 0<𝑣<1, 𝑘∈{0,1}, 𝑇^𝑙_{𝑗,𝑏,𝑘}𝑓_𝐿2≤𝐶‖𝑏‖^𝑘_BMOℝ𝑛+1‖Ω‖_𝐿¹2^𝑣𝑙𝑓_𝐿2,𝑙≤log√2. (47) Dilation-invariance says that 𝑇^𝑙_{𝑗,𝑏,𝑘}𝑓_𝐿2≤𝐶‖𝑏‖^𝑘_BMOℝ𝑛+1‖Ω‖_𝐿¹2^𝑣𝑙𝑓_𝐿2,𝑙≤log√2. (48) By the proof of Theorem 1 in [20], we can get 𝑉_𝑙𝑓_𝐿2≤𝐶‖𝑏‖_BMOℝ𝑛+12^𝑣𝑙‖Ω‖_𝐿¹𝑓_𝐿2,𝑙≤log√2. (49) So, we have ^√[log2]_𝑙=−∞𝑉_𝑙𝑓_{𝐿2(ℝ𝑛+1)}≤𝐶^√[log2]_𝑙=−∞2^𝑣𝑙‖𝑏‖_{BMO(ℝ𝑛+1)}𝑓_{𝐿2(ℝ𝑛+1)}≤𝐶‖𝑏‖_BMOℝ𝑛+1𝑓_{𝐿2(ℝ𝑛+1)}. (50) For ‖∑^∞_{√𝑙=1+[log2]}𝑉_𝑙𝑓‖_{𝐿2(ℝ𝑛+1)}, by Lemma 2.3 of [16], if 𝜙 satisfies the hypotheses in Theorem 1, we have 𝜎_𝑗𝜉,𝜉_𝑛+1≤𝐶log^−𝛼−12^𝑗𝜉+2,∇_𝜉𝜎_𝑗≤𝐶2^𝑗. (51) When 𝜙(|𝑡|)=|𝑡|, if 𝑛=2, we also have the above estimates (see [14]). Set 𝑚_𝑗(𝜉̃)=𝜎_𝑗(𝜉̃), 𝑚^𝑙_𝑗(𝜉̃)=𝑚_𝑗(𝜉̃)𝜓(2^𝑗−𝑙|𝜉̃|). Recall 𝑇^𝑙_𝑗 by 𝑇^𝑙_𝑗𝑓(𝜉̃)=𝑚^𝑙_𝑗(𝜉̃)𝑓(𝜉̃). Straightforward computations lead to 𝑚^𝑙_𝑗2^−𝑗𝜉̃_𝐿∞≤𝐶log^−𝛼−12+2^𝑙,∇_𝜉𝑚^𝑙_𝑗2^−𝑗𝜉̃_𝐿∞𝑚≤𝐶,supp^𝑙_𝑗2^−𝑗𝜉̃⊂𝜉̃≤2^𝑙+2. (52) Let 𝑇^𝑙_𝑗 be the operator defined by 𝑇^𝑙_𝑗𝑓(𝜉̃)=𝑚^𝑙_𝑗(2^−𝑗𝜉̃)𝑓(𝜉̃). Denote by 𝑇^𝑙_𝑗,𝑏,1𝑓=[𝑏,𝑇^𝑙_𝑗]𝑓 and 𝑇^𝑙_𝑗,𝑏,0𝑓=𝑇^𝑙_𝑗𝑓. Similarly, denote by 𝑇^𝑙_𝑗,𝑏,1𝑓=[𝑏,𝑇^𝑙_𝑗]𝑓 and 𝑇^𝑙_𝑗,𝑏,0𝑓=𝑇^𝑙_𝑗𝑓. Thus via the Plancherel theorem and Lemma 8 it is stated that for any fixed 0<𝑣<1, 𝑘∈{0,1}, 𝑇^𝑙_{𝑗,𝑏,𝑘}𝑓_𝐿2≤𝐶‖𝑏‖^𝑘_{BMO(ℝ𝑛+1)}log^{(−𝛼−1)𝑣+1}2+2^𝑙𝑓_𝐿2,𝑙≥1+log√2. (53) Dilation-invariance says that 𝑇^𝑙_{𝑗,𝑏,𝑘}𝑓_𝐿2≤𝐶‖𝑏‖^𝑘_{BMO(ℝ𝑛+1)}log^{(−𝛼−1)𝑣+1}2+2^𝑙𝑓_𝐿2,𝑙≥1+log√2. (54) By the proof of Theorem 1 in [20], we can get 𝑉_𝑙𝑓_𝐿2≤𝐶‖𝑏‖_BMOℝ𝑛+1log^{(−𝛼−1)𝑣+1}2+2^𝑙𝑓_𝐿2,𝑙≥1+log√2. (55) So take 𝑣→1, and we have ^∞_{√𝑙=1+[log2]}𝑉_𝑙𝑓_{𝐿2(ℝ𝑛+1)}≤𝐶‖𝑏‖_BMOℝ𝑛+1^∞_{√𝑙=1+log2}𝑙^{(−𝛼−1)𝑣+1}𝑓_{𝐿2(ℝ𝑛+1)}≤𝐶‖𝑏‖_{BMO(ℝ𝑛+1)}𝑓_{𝐿2(ℝ𝑛+1)}. (56) Then, by (50) and (56) we obtain Theorem 1.

Proof of Theorem 2. By (36), we have 𝑏,𝑇_𝜙,Ω𝑓_{𝐿𝑝(ℝ3)}≤^√[log2]_𝑙=−∞𝑉_𝑙𝑓_{𝐿𝑝(ℝ3)}+^∞_{√𝑙=[log2]+1}𝑉_𝑙𝑓_{𝐿𝑝(ℝ3)}. (57) For ‖∑^√[log2]_𝑙=−∞𝑉_𝑙𝑓‖_{𝐿𝑝(ℝ3)}, recall 𝑇^𝑙_𝑗𝑓(𝑥)=𝑇_𝑗𝑆_𝑙−𝑗𝑓(𝑥); then, 𝑉_𝑙𝑓(𝑥)=∑_𝑗∈ℤ[𝑏,𝑆_𝑙−𝑗𝑇_𝑗𝑆²_𝑙−𝑗]𝑓(𝑥). 𝜙(|𝑡|)=|𝑡|, and applying Lemma 11, we get for 1<𝑝<∞𝑉_𝑙𝑓_𝐿𝑝≤𝐶‖𝑏‖_BMO(ℝ^𝑛+1₎𝑓_𝐿𝑝,𝑙≤log√2. (58) Interpolating between (49) and (58) with 𝑛=2, as the proof of Theorem 1 in [20], we can get ^√[log2]_𝑙=−∞𝑉_𝑙𝑓_{𝐿𝑝(ℝ3)}≤𝐶‖𝑏‖_BMO(ℝ³₎𝑓_{𝐿𝑝(ℝ3)}. (59) For ‖∑^∞_{√𝑙=1+[log2]}𝑉_𝑙𝑓‖_{𝐿𝑝(ℝ3)}, 𝜙(|𝑡|)=|𝑡|, and applying Lemma 11, we get for any fixed 0<𝜏<1/2, 1<𝑝<∞, 𝑉_𝑙𝑓_𝐿𝑝≤𝐶‖𝑏‖_BMO(ℝ^𝑛+1₎2^𝜏𝑙𝜏𝑓_𝐿𝑝,𝑙≥1+log√2. (60) Take 𝜏=1/𝑙; then, we get 𝑉_𝑙𝑓_𝐿𝑝≤𝐶𝑙‖𝑏‖_BMO(ℝ^𝑛+1₎𝑓_𝐿𝑝,𝑙≥1+log√2. (61) For 𝜙(|𝑡|)=|𝑡|, (55) can be established only when 𝑛=2, so interpolating between (55) and (61) with 𝑛=2, as the proof of Theorem 1 in [20], we get ^∞_{√𝑙=1+[log2]}𝑉_𝑙𝑓_{𝐿𝑝(ℝ3)}≤𝐶‖𝑏‖_BMO(ℝ³₎𝑓_{𝐿𝑝(ℝ3)}. (62) Then, by (59) and (62) we obtain Theorem 2.

4. The proof of Theorems 4 and 7

We begin with a lemma, which plays an important role in proving Theorem 4.

Lemma 12. Let 𝑏(𝑥)∈𝐵𝑀𝑂(ℝ^𝑛), 𝑥=(𝑥,𝑥_𝑛+1)∈ℝ^𝑛+1, and 𝐵(𝑥)=𝑏(𝑥); then, 𝐵(𝑥)∈𝐵𝑀𝑂(ℝ^𝑛+1) and ‖𝐵‖_{𝐵𝑀𝑂(ℝ}^𝑛+1₎=‖𝑏‖_{𝐵𝑀𝑂(ℝ}^𝑛₎.

Proof. We know ‖𝑏‖_BMO(ℝ^𝑛₎=sup_{𝑄⊂ℝ𝑛}1|𝑄|_𝑄𝑏(𝑥)−𝑏_𝑄𝑑𝑥, (63) where 𝑏_𝑄=(1/|𝑄|)∫_𝑄𝑏(𝑥)𝑑𝑥 and 𝑄 is the square in ℝ^𝑛 whose edges are parallel to the axis. So ‖𝐵‖_BMO(ℝ^𝑛+1₎=sup_{𝑄⊂ℝ𝑛+1}1𝑄^𝑄𝐵(𝑥)−𝐵_𝑄𝑑𝑥, (64) where 𝑄 is the square in ℝ^𝑛+1 whose edges are parallel to the axis. Consider 𝐵_𝑄=1𝑄^𝑄𝐵(𝑥)𝑑𝑥=1𝑎|𝑄|_𝑚^𝑚+𝑎_𝑄𝑏(𝑥)𝑑𝑥𝑑𝑥_𝑛+1=1𝑎|𝑄|_𝑄𝑏(𝑥)𝑑𝑥_𝑚^𝑚+𝑎𝑑𝑥_𝑛+1=1|𝑄|_𝑄𝑏(𝑥)𝑑𝑥=𝑏_𝑄, (65) where 𝑄 is the projection on ℝ^𝑛 of 𝑄 and 𝑎 is the side length of 𝑄. Then ‖𝐵‖_BMO(ℝ^𝑛+1₎=sup_{𝑄⊂ℝ𝑛+1}1𝑄^𝑄𝐵(𝑥)−𝐵_𝑄𝑑𝑥=sup_{𝑄⊂ℝ𝑛+1}1𝑎|𝑄|_𝑚^𝑚+𝑎_𝑄𝑏(𝑥)−𝑏_𝑄𝑑𝑥𝑑𝑥_𝑛+1=sup_{𝑄⊂ℝ𝑛+1}1𝑎|𝑄|_𝑄𝑏(𝑥)−𝑏_𝑄𝑑𝑥_𝑚^𝑚+𝑎𝑑𝑥_𝑛+1=sup_{𝑄⊂ℝ𝑛+1}1|𝑄|_𝑄𝑏(𝑥)−𝑏_𝑄𝑑𝑥=sup_{𝑄⊂ℝ𝑛}1|𝑄|_𝑄𝑏(𝑥)−𝑏_𝑄𝑑𝑥=‖𝑏‖_BMO(ℝ^𝑛₎. (66)

Proof of Theorem 4. By Lemma 12, 𝐵∈BMO(ℝ^𝑛+1). Using the method in [5], for 𝑓∈𝐿^𝑝(ℝ^𝑛) and 𝑁∈ℕ, let 𝐹_𝑁(𝑥,𝑥_𝑛+1)=𝑓(𝑥)𝑒^{−𝑖𝑥𝑛+1}𝜒_{[−𝑁,𝑁]}(𝑥_𝑛+1). Then by mean value theorem of integrals and Lemma 12, we have 2𝑁_ℝ𝑛𝑏(𝑥)_ℝ𝑛Ω𝑦𝑦^𝑛𝑓𝑥−𝑦𝑒^{𝑖𝜙(|𝑦|)}𝜒_{[−𝑁,𝑁]}×𝑥_𝑛+1𝑦−−𝜙𝑑𝑦_ℝ𝑛Ω𝑦𝑦^𝑛𝑏𝑥−𝑦𝑓𝑥−𝑦𝑒^{𝑖𝜙(|𝑦|)}×𝜒_{[−𝑁,𝑁]}𝑥_𝑛+1−𝜙𝑦𝑑𝑦^𝑝≤𝑑𝑥_ℝℝ𝑛𝑏(𝑥)_ℝ𝑛Ω𝑦𝑦^𝑛𝑓𝑥−𝑦𝑒^{−𝑖(𝑥𝑛+1−𝜙(|𝑦|))}×𝜒_{[−𝑁,𝑁]}𝑥_𝑛+1𝑦−−𝜙𝑑𝑦_ℝ𝑛Ω𝑦𝑦^𝑛𝑏𝑥−𝑦𝑓𝑥−𝑦×𝑒^{−𝑖(𝑥𝑛+1−𝜙(|𝑦|))}𝜒_{[−𝑁,𝑁]}×𝑥_𝑛+1−𝜙𝑦𝑑𝑦^𝑝𝑑𝑥𝑑𝑥_𝑛+1=𝐵,𝑇_𝜙,Ω𝐹_𝑁^𝑝_{𝐿𝑝ℝ𝑛+1}≤𝐶‖𝐵‖^𝑝_BMOℝ𝑛+1𝐹_𝑁^𝑝_{𝐿𝑝(ℝ𝑛+1)}=𝐶2𝑁‖𝑏‖^𝑝_BMO(ℝ𝑛)𝑓^𝑝_{𝐿𝑝(ℝ𝑛)}. (67) Dividing both sides by 2𝑁 and letting 𝑁→∞, we obtain 𝑏,𝑇_𝜙𝑓_{𝐿𝑝(ℝ𝑛)}≤𝐶‖𝑏‖_BMO(ℝ^𝑛₎𝑓_{𝐿𝑝(ℝ𝑛)}. (68) Thus, we obtain Theorem 4.

Proof of Theorem 7. Theorem 7 can be proved by using arguments which are essentially the same as the proof of Theorem 1 in [20]. Only the following two things must be modified.(i)Instead of 𝐾_𝑗(𝑥) and 𝑇_𝑗𝑓(𝑥), we use 𝐾_𝑗,𝜙(𝑥)=𝑒^{𝑖𝜙(𝑥)}Ω(𝑥)|𝑥|^𝑛𝜒_{{2𝑗<|𝑥|≤2𝑗+1}},𝑇_𝑗,𝜙𝑓(𝑥)=𝐾_𝑗,𝜙(𝑥)=∗𝑓_{2𝑗<|𝑦|≤2𝑗+1}Ω(𝑥)|𝑥|^𝑛𝑒^{𝑖𝜙(𝑥)}𝑓𝑥−𝑦𝑑𝑦. (69)(ii)Since Ω(𝜃) is odd and 𝜙(𝜃𝑡) is even with respect to 𝜃, we get Ω(𝜃)𝑒^{𝑖𝜙(𝜃𝑡)} is odd and ∫_𝑆𝑛−1Ω(𝜃)𝑒^{𝑖𝜙(𝜃𝑡)}𝑑𝜎(𝜃)=0. So we use the estimates in [21]: Consider 𝐾_𝑗,𝜙(𝜉)≤𝐶‖Ω‖_𝐿¹2^𝑗𝜉,𝐾_𝑗,𝜙(𝜉)≤𝐶log^−𝛼−12^𝑗𝜉+2 (70)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by NCET of China (Grant: NCET-11-0574), the Fundamental Re-search Funds for the Central Universities (FRF-TP-12-006B), NSF of China (Grant: 10901017), and CSC Funding of China.