Boundedness of Oscillatory Integrals with Variable Calderón-Zygmund Kernel on Weighted Morrey Spaces

Abstract

Oscillatory integral operators play a key role in harmonic analysis. In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on the weighted Morrey spaces 𝐿^𝑝,𝑘(𝜔). Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

1. Introduction and Main Results

Suppose that 𝑘 is the standard Calderón-Zygmund kernel. That is, 𝑘∈𝐶^∞(ℝ^𝑛⧵{0}) is homogeneous of degree 𝑛, and ∫_Σ𝑘(𝑥)𝑑𝜎_𝑥=0, where Σ={𝑥∈ℝ^𝑛∶|𝑥|=1}. The oscillatory integral operator 𝑇_𝜆 is defined by 𝑇_𝜆𝑓(𝑥)=𝑝⋅𝑣⋅_ℝ𝑛𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘𝑥−𝑦𝜑𝑥,𝑦𝑓𝑦𝑑𝑦, (1) where 𝜆∈ℝ,𝜑∈𝐶^∞₀(ℝ^𝑛×ℝ^𝑛), where 𝐶^∞₀(ℝ^𝑛×ℝ^𝑛) is the space of infinitely differentiable functions on ℝ^𝑛×ℝ^𝑛 with compact supports, and Φ is a real-analytic function or a real-𝐶^∞(ℝ^𝑛×ℝ^𝑛) function satisfying that, for any (𝑥₀,𝑦₀)∈supp𝜑, there exists (𝑗₀,𝑘₀), 1≤𝑗₀, 𝑘₀≤𝑛, such that 𝜕²Φ(𝑥₀,𝑦₀)/𝜕𝑥_𝑗0𝜕𝑦_𝑘0 does not vanish up to infinite order. These operators have arisen in the study of singular integrals supported on lower dimensional varieties and the singular Radon transform. In [1], Pan proved that 𝑇_𝜆 are uniform in 𝜆 bounded on 𝐿^𝑝(ℝ^𝑛)(1<𝑝<∞). Lu et al. [2] proved the weighted 𝐿^𝑝 boundedness of 𝑇_𝜆 defined by (1).

Let 𝑘(𝑥,𝑦) be a variable Calderón-Zygmund kernel. That means, for a.e. 𝑥∈ℝ^𝑛,𝑘(𝑥,⋅) is a standard Calderón-Zygmund kernel and max_{𝑗≤2𝑛,𝑗∈ℤ}𝜕^|𝑗|𝑘𝜕𝑦^𝑗_{𝐿∞(ℝ𝑛×Σ)}=𝐴<∞. (2) Define the oscillatory integral operator with variable Calderón-Zygmund kernel 𝑇^∗_𝜆 by 𝑇^∗_𝜆𝑓(𝑥)=𝑝⋅𝑣⋅_ℝ𝑛𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘𝑥,𝑥−𝑦𝜑𝑥,𝑦𝑓𝑦𝑑𝑦, (3) where 𝜆, 𝜑, and Φ satisfy the same assumptions as those in the operator defined by (1).

Lu et al. [2] investigated the 𝐿^𝑝 and weighted 𝐿^𝑝 boundedness about this class of oscillatory integral operators.

The classical Morrey space 𝐿^𝑝,𝜆 was first introduced by Morrey in [3] to study the local behavior of solutions to second order elliptic partial differential equations. In 2009, Komori and Shirai [4] first defined the weighted Morrey spaces 𝐿^𝑝,𝜅(𝜔) which could be viewed as an extension of weighted Lebesgue spaces. They studied the boundedness of the fractional integral operator, the Hardy-Littlewood maximal operator, and the Calderón-Zygmund singular integral operator on the space. The boundedness results about some operators on these spaces can be see in ([5–17]). Recently, Shi et al. [18] obtained the boundedness of a class of oscillatory integrals with Calderón-Zygmund kernel and polynomial phase on weighted Morrey spaces. Their results are stated as follows.

Let 𝑃(𝑥,𝑦) be a real valued polynomial defined on ℝ^𝑛×ℝ^𝑛 and let 𝑘 satisfy the following hypotheses: 𝑘≤𝐶𝑥,𝑦𝑥−𝑦^𝑛∇,∀𝑥≠𝑦,_𝑥𝑘𝑥,𝑦+∇_𝑦𝑘≤𝐶𝑥,𝑦𝑥−𝑦^𝑛+1,∀𝑥≠𝑦. (4) We define 𝑆𝑓(𝑥)=𝑝⋅𝑣⋅_ℝ𝑛𝑘𝑓𝑦(𝑥)𝑥,𝑦𝑑𝑦,𝑅𝑓=𝑝⋅𝑣⋅_ℝ𝑛𝑒^{𝑖𝑃(𝑥,𝑦)}𝑘𝑥,𝑦𝑓𝑦𝑑𝑦. (5)

Theorem A (see [18]). Let 1<𝑝<∞, 0<𝜅<1, and 𝜔∈𝐴_𝑝. If 𝑆 is of type (𝐿²,𝐿²), then, for any real polynomial 𝑃(𝑥,𝑦), there exists a constant 𝐶>0 such that 𝑅𝑓_𝐿^𝑝,𝜅_(𝜔)≤𝐶𝑓_{𝐿𝑝,𝜅(𝜔)}. (6)

The purpose of this paper is to generalize the above results to the case with real-𝐶^∞ or analytic phase functions. Our main results in this paper are formulated as follows.

Theorem 1. Let 𝜆∈ℝ, 𝜑∈𝐶^∞₀(ℝ^𝑛×ℝ^𝑛), and Φ a real-𝐶^∞(ℝ^𝑛×ℝ^𝑛) function satisfying that, for any (𝑥₀,𝑦₀)∈supp𝜑, there exists (𝑗₀,𝑘₀), 1≤𝑗₀, 𝑘₀≤𝑛, such that 𝜕²Φ(𝑥₀,𝑦₀)/𝜕𝑥_𝑗0𝜕𝑦_𝑘0 does not vanish up to infinite order. Assume that 𝑘 is a standard Calderón-Zygmund kernel and 𝑇_𝜆 is defined as in (1). Then for any 1<𝑝<∞, 0<𝜅<1, and 𝜔∈𝐴_𝑝, 𝑇_𝜆 is bounded on 𝐿^𝑝,𝜅(𝜔).

Theorem 2. Let 𝜆∈ℝ, 𝜑∈𝐶^∞₀(ℝ^𝑛×ℝ^𝑛), and Φ a real-𝐶^∞(ℝ^𝑛×ℝ^𝑛) function satisfying that, for any (𝑥₀,𝑦₀)∈supp𝜑, there exists (𝑗₀,𝑘₀), 1≤𝑗₀, 𝑘₀≤𝑛, such that 𝜕²Φ(𝑥₀,𝑦₀)/𝜕𝑥_𝑗0𝜕𝑦_𝑘0 does not vanish up to infinite order. Assume that 𝑘 is a variable Calderón-Zygmund kernel and 𝑇^∗_𝜆 is defined as in (3). Then for any 1<𝑝<∞, 0<𝜅<1, and 𝜔∈𝐴_𝑝, 𝑇^∗_𝜆 is bounded on 𝐿^𝑝,𝜅(𝜔).

2. Notations and Preliminary Lemmas

Let 𝐵=𝐵(𝑥₀,𝑟) be the ball with the center 𝑥₀ and radius 𝑟. Given a ball 𝐵 and 𝜆>0, 𝜆𝐵 denotes the ball with the same center as 𝐵 whose radius is 𝜆 times that of 𝐵.

The classical 𝐴_𝑝 weighted theory was first introduced by Muckenhoupt in [19]. A weight 𝜔 is a locally integrable function on ℝ^𝑛, which takes values in (0,∞) a.e. For a given weight function 𝜔, we denote the Lebesgue measure of 𝐵 by |𝐵| and the weighted measure of 𝐸 by 𝜔(𝐸); that is, 𝜔(𝐸)=∫_𝐸𝜔(𝑥)𝑑𝑥. Given a weight 𝜔, we say that 𝜔 satisfies the doubling condition if there exists a constant 𝐷>0 such that, for any ball 𝐵, we have 𝜔(2𝐵)≤𝐷𝜔(𝐵).

We say 𝜔∈𝐴_𝑝 with 1<𝑝<∞, if there exists a constant 𝐶>0, such that 1|𝐵|_𝐵𝜔(𝑥)1𝑑𝑥|𝐵|_𝐵𝜔(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1≤𝐶, (7) for every ball 𝐵⊆ℝ^𝑛. When 𝑝=1, 𝜔∈𝐴₁ if there exists 𝐶>0, such that 1|𝐵|_𝐵𝜔(𝑥)𝑑𝑥≤𝐶essinf_𝑥∈𝐵𝜔(𝑥), (8) for almost every 𝑥∈ℝ^𝑛. We define 𝐴_∞=⋃_𝑝≥1𝐴_𝑝. A weight function 𝜔 is said to belong to the reverse Hölder class 𝑅𝐻_𝑟 if there exist two constants 𝑟>0 and 𝐶>0 such that the following reverse Hölder inequality holds: 1|𝐵|_𝐵𝜔(𝑥)^𝑟𝑑𝑥^1/𝑟1≤𝐶|𝐵|_𝐵𝜔(𝑥)𝑑𝑥, (9) for every ball 𝐵⊆ℝ^𝑛.

It is well known that, if 𝜔∈𝐴_𝑝 with 1≤𝑝<∞, then there exists 𝑟>1 such that 𝜔∈𝑅𝐻_𝑟.

Lemma 3 (see [20]). Let 𝜔∈𝐴_𝑝, 𝑝≥1, and 𝑟>0. Then for any ball 𝐵 and 𝜆>1, 𝜔()(𝐵),𝜔()2𝐵≤𝐶𝜔𝜆𝐵≤𝐶𝜆^𝑛𝑝𝜔(𝐵), (10) where 𝐶 does not depend on 𝐵 nor on 𝜆.

Lemma 4 (see [21]). Let 𝜔∈𝑅𝐻_𝑟 with 𝑟>1. Then there exists a constant 𝐶 such that 𝜔(𝐸)𝜔(𝐵)|𝐸|≤𝐶|𝐵|^{(𝑟−1)/𝑟}, (11) for any measurable subset 𝐸 of a ball 𝐵.

The weighted Morrey spaces were defined as follows.

Definition 5 (see [4]). Let 1≤𝑝<∞, 0<𝜅<1, and 𝜔 a weight function. Then the weighted Morrey space is defined by 𝐿^𝑝,𝜅(𝜔)=𝑓∈𝐿^𝑝_loc(𝜔)∶𝑓_𝐿^𝑝,𝜅_(𝜔)<∞, (12) where 𝑓_{𝐿𝑝,𝜅(𝜔)}=sup_𝐵1𝜔(𝐵)^𝜅_𝐵𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥^1/𝑝, (13) and the supremum is taken over all balls 𝐵 in ℝ^𝑛. The space 𝐿^𝑝_loc(𝜔) is defined by 𝐿^𝑝_loc(𝜔)=𝑓∶𝑓𝜒_𝐾∈𝐿^𝑝(𝜔),foreverycompactset𝐾⊆ℝ^𝑛. (14) Our argument is based heavily on the following results.

Lemma 6 (see [2]). Assume that 𝑇_𝜆 is defined as in (1). Then for any 1<𝑝<∞ and 𝜔∈𝐴_𝑝, one has 𝑇_𝜆𝑓_𝐿^𝑝_(𝜔)≤𝐶𝑛,𝑝,Φ,𝜑,𝐶_𝑝,𝜔𝐵𝑓_𝐿^𝑝_(𝜔), (15) where 𝐶(𝑛,𝑝,Φ,𝜑,𝐶_𝑝,𝜔) is independent of 𝜆, 𝑘, and 𝑓 and 𝐵=‖𝑘‖_𝐶¹_(Σ).

Lemma 7 (see [2]). Assume that 𝑇^∗_𝜆 is defined as in (3). Then for any 1<𝑝<∞ and 𝜔∈𝐴_𝑝, one has 𝑇^∗_𝜆𝑓_𝐿^𝑝_(𝜔)≤𝐶𝑛,𝑝,Φ,𝜑,𝐶_𝑝,𝜔𝐴𝑓_𝐿^𝑝_(𝜔), (16) where 𝐶(𝑛,𝑝,Φ,𝜑,𝐶_𝑝,𝜔) is independent of 𝜆, 𝑘, and 𝑓. 𝐴 is defined in (2).

Definition 8 (see [4]). The Hardy-Littlewood maximal operator 𝑀 is defined by 𝑀𝑓(𝑥)=sup_𝐵∋𝑥1|𝐵|_𝐵𝑓𝑦𝑑𝑦,𝑓∈𝐿_locℝ^𝑛. (17)

Lemma 9 (see [4]). If 1<𝑝<∞, 0<𝜅<1, and 𝜔∈𝐴_𝑝 then the Hardy-Littlewood maximal operator 𝑀 is bounded on 𝐿^𝑝,𝜅(𝜔).

Lemma 10 (see [22]). Denote by ℋ_𝑚 the spaces of spherical harmonic functions of degree 𝑚. Then(a)𝐿²(Σ)=⊕^∞_𝑚=0ℋ_𝑚, and 𝑔_𝑚=dimℋ_𝑚≤𝐶(𝑛)𝑚^𝑛−2 for any 𝑚∈ℕ;(b)for any 𝑚=0,1,2,…, there exists an orthogonal system {𝑌_𝑗𝑚}^𝑔𝑚_𝑗=1 of ℋ_𝑚 such that ‖𝑌_𝑗𝑚‖_𝐿∞(Σ)≤𝐶(𝑛)𝑚^𝑛/2−1, 𝑌_𝑗𝑚=(−𝑚)^−𝑛(𝑚+𝑛−2)^−𝑛Λ^𝑛𝑌_𝑗𝑚, 𝑗=1,…,𝑔_𝑚, and Λ is the Beltrami-Laplace operator on Σ.

In the following the letter 𝐶 will denote a constant which may vary at each occurrence.

3. Proof of Theorems

Proof of Theorem 1. It is sufficient to prove that there exists a constant 𝐶>0 such that 1𝜔(𝐵)^𝜅_𝐵𝑇_𝜆𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (18)
Fix a ball 𝐵=𝐵(𝑥₀,𝑟_𝐵) and decompose 𝑓=𝑓₁+𝑓₂, with 𝑓₁=𝑓𝜒_2𝐵. Then we have 1𝜔(𝐵)^𝜅_𝐵𝑇_𝜆𝑓(𝑥)^𝑝𝜔(𝑥)1𝑑𝑥≤𝐶𝜔(𝐵)^𝜅_𝐵𝑇_𝜆𝑓₁(𝑥)^𝑝𝜔(𝑥)+1𝑑𝑥𝜔(𝐵)^𝜅_𝐵𝑇_𝜆𝑓₂(𝑥)^𝑝𝜔(𝑥)𝐼𝑑𝑥=𝐶₁+𝐼₂. (19)
Using Lemmas 3 and 6, we get 𝐼₁1≤𝐶𝜔(𝐵)^𝜅_2𝐵𝑓(𝑥)^𝑝𝜔(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝐿𝑝,𝜅(𝜔)}⋅𝜔(2𝐵)^𝜅𝜔(𝐵)^𝜅≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (20) We now estimate 𝐼₂. We can write 𝑇_𝜆𝑓₂(𝑥)=_(2𝐵)𝑐𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘𝑥−𝑦𝜑𝑥,𝑦𝑓𝑦𝑑𝑦. (21)
Now by an argument similar to the proof of Lemma  6 in [2], we choose 𝜙₁∈𝐶^∞₀(ℝ^𝑛) such that 𝜙₁(𝑥)≡1, when |𝑥|≤1, and 𝜙₁(𝑥)≡0 when |𝑥|>2. Let 𝜙₂=1−𝜙₁ and 𝑁∈ℕ which is large enough and will be determined later. Write 𝑘(𝑥)=𝑘¹_𝜆(𝑥)+𝑘²_𝜆(𝑥), (22) where 𝑘^𝑗_𝜆(𝑥)=𝑘(𝑥)𝜙_𝑗𝜆^1/𝑁𝑥,𝑗=1,2. (23) Then 𝑇_𝜆𝑓₂(𝑥)=𝑝⋅𝑣⋅_(2𝐵)𝑐𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘¹_𝜆𝑓𝑦𝑥−𝑦×𝜑𝑥,𝑦𝑑𝑦+𝑝⋅𝑣⋅_(2𝐵)𝑐𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘²_𝜆𝑓𝑦𝑥−𝑦×𝜑𝑥,𝑦𝑑𝑦∶=𝑇¹_𝜆𝑓₂(𝑥)+𝑇²_𝜆𝑓₂(𝑥). (24)
Let us first estimate 𝑇¹_𝜆𝑓₂(𝑥). To do so, using Taylor’s expansion and the compactness of supp𝜑, we write Φ𝑥,𝑦=Φ(𝑥,𝑥)+𝑃𝑥,𝑦+𝑟_𝑁𝑥,𝑦 (25) for(𝑥,𝑦)∈supp𝜑, where 𝑃(𝑥,𝑦) is a polynomial with deg 𝑃<𝑁 and |𝑟_𝑁(𝑥,𝑦)|≤𝐶|𝑥−𝑦|^𝑁 with 𝐶 independent of 𝑥 and 𝑦. Define 𝑅𝑓(𝑥)=𝑝⋅𝑣⋅_(2𝐵)𝑐𝑒^{𝑖𝜆𝑃(𝑥,𝑦)}𝑘¹_𝜆𝑥−𝑦𝜑𝑥,𝑦𝑓𝑦𝑑𝑦. (26) Therefore 𝑒^{−𝑖𝜆Φ(𝑥,𝑥)}𝑇¹_𝜆𝑓₂(𝑥)(𝑥)=−𝑅𝑓_{|𝑥−𝑦|≤2𝜆−1/𝑁}𝑒^{𝑖𝜆𝑃(𝑥,𝑦)}𝑒^{𝑖𝜆𝑟𝑁(𝑥,𝑦)}×𝑘−1_1𝜆𝜑𝑓𝑦=𝑥−𝑦𝑥,𝑦𝑑𝑦_{∞𝑗=02−𝑗𝜆−1/𝑁<|𝑥−𝑦|≤2−𝑗+1𝜆−1/𝑁}𝑒^{𝑖𝜆𝑃(𝑥,𝑦)}𝑒^{𝑖𝜆𝑟𝑁(𝑥,𝑦)}−1×𝑘¹_𝜆𝜑𝑓𝑦≡𝑥−𝑦𝑥,𝑦𝑑𝑦_∞𝑗=0𝑇¹_𝜆,𝑗𝑓₂(𝑥). (27) On 𝑇¹_𝜆,𝑗𝑓₂(𝑥), by the properties of 𝑟_𝑁 and 𝑘, we have 𝑇¹_𝜆,𝑗𝑓₂(𝑥)≤𝐶2^−𝑗𝑁𝑀𝑓(𝑥). (28)
So we have 𝑇¹_𝜆𝑓₂(𝑥)≤𝐶^∞_𝑗=02^−𝑗𝑁𝑀𝑓(𝑥)+𝐶𝑅𝑓(𝑥). (29) By Theorem A and Lemma 9, we have 𝑇¹_𝜆𝑓_{2𝐿𝑝,𝜅(𝜔)}≤𝐶𝑓_{𝐿𝑝,𝜅(𝜔)}. (30)
Now, let us turn to estimate 𝑇²_𝜆𝑓₂(𝑥). We consider the following two cases.
Case 1 (𝜆≤1). Similar to that estimate of 𝑇²_𝜆 in Lemma 6 in [2], we have 𝑇²_𝜆𝑓₂(𝑥)≤𝐶𝑀𝑓(𝑥). (31)
By Lemma 9 we have 𝑇²_𝜆𝑓_{2𝐿𝑝,𝜅(𝜔)}≤𝐶𝑓_{𝐿𝑝,𝜅(𝜔)}. (32)
Case 2 (𝜆>1). We choose 𝜑₀∈𝐶^∞₀(ℝ^𝑛) such that supp𝜑₀⊆𝑥∈ℝ^𝑛|𝑥|,𝜙∶1<≤2₂(𝑥)=^∞_𝑗=0𝜑₀2^−𝑗𝑥. (33) Let 𝑘²_𝜆,𝑗(𝑥)=𝑘(𝑥)𝜑₀2^−𝑗𝜆^1/𝑁𝑥. (34) Then 𝑇²_𝜆𝑓₂(𝑥)=_(2𝐵)𝑐𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘²_𝜆𝑥−𝑦𝜑𝑥,𝑦𝑓𝑦𝑑𝑦,_{∞𝑗=0(2𝐵)𝑐}𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘²_𝜆,𝑗𝜑𝑓𝑦≡𝑥−𝑦𝑥,𝑦𝑑𝑦_∞𝑗=0𝑇²_𝜆,𝑗𝑓₂(𝑥). (35) For 𝑇²_𝜆,𝑗, by its definition, we can get 𝑇²_𝜆,𝑗𝑓₂(𝑥)≤𝐶_{2𝑗𝜆−1/𝑁<|𝑥−𝑦|≤2𝑗+1𝜆−1/𝑁}×1𝑥−𝑦^𝑛𝑓𝑦𝑑𝑦≤𝐶𝑀𝑓(𝑥). (36) The inequality (36) also can be seen in [2]; we omit the details here.
By Lemma 9, we have 𝑇²_𝜆𝑓_{2𝐿𝑝,𝜅(𝜔)}≤𝐶𝑓_{𝐿𝑝,𝜅(𝜔)}. (37) Therefore 𝐼₂≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (38) This finishes the proof of Theorem 1.

Proof of Theorem 2. It is sufficient to prove that there exists a constant 𝐶>0 such that 1𝜔(𝐵)^𝜅_𝐵𝑇^∗_𝜆𝑓(𝑥)^𝑝𝜔(𝑥)𝑑𝑥≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (39) Fix a ball 𝐵=𝐵(𝑥₀,𝑟_𝐵) and decompose 𝑓=𝑓₁+𝑓₂, with 𝑓₁=𝑓𝜒_2𝐵. Then we have 1𝜔(𝐵)^𝜅_𝐵𝑇^∗_𝜆𝑓(𝑥)^𝑝𝜔(𝑥)1𝑑𝑥≤𝐶𝜔(𝐵)^𝜅_𝐵𝑇^∗_𝜆𝑓₁(𝑥)^𝑝𝜔(𝑥)+1𝑑𝑥𝜔(𝐵)^𝜅_𝐵𝑇^∗_𝜆𝑓₂(𝑥)^𝑝𝜔(𝑥)𝐽𝑑𝑥=𝐶₁+𝐽₂. (40) Using Lemmas 3 and 7, we get 𝐽₁1≤𝐶𝜔(𝐵)^𝜅_2𝐵𝑓(𝑥)^𝑝𝜔(𝑥)𝑓𝑑𝑥≤𝐶^𝑝_{𝐿𝑝,𝜅(𝜔)}⋅𝜔(2𝐵)^𝜅𝜔(𝐵)^𝜅≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (41) We now estimate 𝐽₂.
For each 𝑚∈ℕ and 𝑗=1,…,𝑔_𝑚, we get 𝑎_𝑗𝑚(𝑥)=_ΣΩ(𝑥,𝑧)𝑌_𝑗𝑚(𝑧)𝑑𝜎_𝑧, (42) where Ω(𝑥,𝑧)=|𝑧|^𝑛𝑘(𝑥,𝑧). Then for a.e. 𝑥∈ℝ^𝑛, Ω(𝑥,𝑧)=^∞𝑔_𝑚=1^𝑚_𝑗=1𝑎_𝑗𝑚(𝑥)𝑌_𝑗𝑚𝑧^′, (43) where 𝑧^′=𝑧/|𝑧| for any 𝑧∈ℝ^𝑛⧵{0}. By Lemma 10, we have that, for any 𝑥∈ℝ^𝑛, 𝑎_𝑗𝑚(𝑥)=𝑚^−𝑛(𝑚+𝑛−2)^−𝑛_ΣΩ(𝑥,𝑧)Λ^𝑛𝑌_𝑗𝑚(𝑧)𝑑𝜎_𝑧=𝑚^−𝑛(𝑚+𝑛−2)^−𝑛_ΣΛ^𝑛Ω(𝑥,𝑧)𝑌_𝑗𝑚(𝑧)𝑑𝜎_𝑧≤𝐶(𝑛)𝐴𝑚^−2𝑛. (44)
By Lemma 10 again, we can verify that, for any 𝜖>0, 𝑁∈ℕ, and a.e. 𝑥∈ℝ^𝑛, if |𝑦−𝑥|≥𝜖, then ^𝑁𝑔_𝑚=1^𝑚_𝑗=1𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑎_𝑗𝑚(𝑥)𝑌_𝑗𝑚𝑥−𝑦^′𝑥−𝑦^𝑛𝜑𝑥,𝑦𝑓₂𝑦≤𝐶(𝜖)𝐴𝑓₂𝑦. (45)
Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that 𝑇^∗_𝜆𝑓₂(𝑥)=lim_{𝜖→0|𝑥−𝑦|≥𝜖}𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑘𝑥,𝑥−𝑦𝜑𝑥,𝑦𝑓₂𝑦𝑑𝑦=lim^∞_𝜖→0^𝑔_𝑚=1^𝑚_{𝑗=1|𝑥−𝑦|≥𝜖}𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑎_𝑗𝑚(𝑥)𝑌_𝑗𝑚𝑥−𝑦^′𝑥−𝑦^𝑛×𝜑𝑥,𝑦𝑓₂𝑦𝑑𝑦=lim^∞_𝜖→0^𝑔_𝑚=1^𝑚_𝑗=1𝑎_𝑗𝑚(𝑥)_{|𝑥−𝑦|≥𝜖}𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑌_𝑗𝑚𝑥−𝑦^′𝑥−𝑦^𝑛×𝜑𝑥,𝑦𝑓₂𝑦𝑑𝑦. (46) We write 𝑅_𝑗𝑚𝑓₂(𝑥)=_{|𝑥−𝑦|≥𝜖}𝑒^{𝑖𝜆Φ(𝑥,𝑦)}𝑌_𝑗𝑚𝑥−𝑦^′𝑥−𝑦^𝑛𝜑𝑥,𝑦𝑓₂𝑦𝑑𝑦. (47) It is easy to see that 𝑅_𝑗𝑚𝑓₂(𝑥) is the oscillatory integral operator defined by (1). By Theorem 1 we have that 𝑅_𝑗𝑚 is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have 𝐽₂≤𝐶𝑓^𝑝_{𝐿𝑝,𝜅(𝜔)}. (48)
This finishes the proof of Theorem 2.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11001001), Natural Science Foundation from the Education Department of Anhui Province (nos. KJ2012B166, KJ2013A235).