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

Authors: Yali Pan; Changwen Li; Xinsong Wang
Journal: Journal of Function Spaces and Applications
 (2013)

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## 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 spacesLp,k(ω). Meanwhile, the corresponding results for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.


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## Body

## 1. Introduction and Main Results

Suppose thatk is the standard Calderón-Zygmund kernel. That is, k∈C∞(ℝn∖{0}) is homogeneous of degree n, and ∫Σ‍k(x)dσx=0, where Σ={x∈ℝn:|x|=1}. The oscillatory integral operator Tλ is defined by
(1)Tλf(x)=p·v·∫ℝn‍eiλΦ(x,y)k(x-y)φ(x,y)f(y)dy,
where λ∈ℝ,φ∈C0∞(ℝn×ℝn), where C0∞(ℝn×ℝn) is the space of infinitely differentiable functions on ℝn×ℝn with compact supports, and Φ is a real-analytic function or a real-C∞(ℝn×ℝn) function satisfying that, for any (x0,y0)∈suppφ, there exists (j0,k0), 1≤j0, k0≤n, such that ∂2Φ(x0,y0)/∂xj0∂yk0 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 Tλ are uniform in λ bounded on Lp(ℝn)(1<p<∞). Lu et al.  [2] proved the weighted Lp boundedness of Tλ defined by (1).Letk(x,y) be a variable Calderón-Zygmund kernel. That means, for a.e. x∈ℝn,k(x,·) is a standard Calderón-Zygmund kernel and
(2)max|j|≤2n,j∈ℤ∥∂|j|k∂yj∥L∞(ℝn×Σ)=A<∞.
Define the oscillatory integral operator with variable Calderón-Zygmund kernel Tλ* by
(3)Tλ*f(x)=p·v·∫ℝn‍eiλΦ(x,y)k(x,x-y)φ(x,y)f(y)dy,
where λ, φ, and Φ satisfy the same assumptions as those in the operator defined by (1).Lu et al. [2] investigated the Lp and weighted Lp boundedness about this class of oscillatory integral operators.The classical Morrey spaceLp,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 Lp,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.LetP(x,y) be a real valued polynomial defined on ℝn×ℝn and let k satisfy the following hypotheses:
(4)|k(x,y)|≤C|x-y|n,∀x≠y,|∇xk(x,y)|+|∇yk(x,y)|≤C|x-y|n+1,∀x≠y.
We define
(5)Sf(x)=p·v·∫ℝn‍k(x,y)f(y)dy,Rf(x)=p·v·∫ℝn‍eiP(x,y)k(x,y)f(y)dy.Theorem A (see [18]).
Let1<p<∞, 0<κ<1, and ω∈Ap. If S is of type (L2,L2), then, for any real polynomial P(x,y), there exists a constant C>0 such that
(6)∥Rf∥oLp,κ(ω)≤C∥f∥Lp,κ(ω).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.
Letλ∈ℝ, φ∈C0∞(ℝn×ℝn), and Φ a real-C∞(ℝn×ℝn) function satisfying that, for any (x0,y0)∈suppφ, there exists (j0,k0), 1≤j0, k0≤n, such that ∂2Φ(x0,y0)/∂xj0∂yk0 does not vanish up to infinite order. Assume that k is a standard Calderón-Zygmund kernel and Tλ is defined as in (1). Then for any 1<p<∞,  0<κ<1, and ω∈Ap, Tλ is bounded on Lp,κ(ω).Theorem 2.
Letλ∈ℝ, φ∈C0∞(ℝn×ℝn), and Φ a real-C∞(ℝn×ℝn) function satisfying that, for any (x0,y0)∈suppφ, there exists (j0,k0), 1≤j0, k0≤n, such that ∂2Φ(x0,y0)/∂xj0∂yk0 does not vanish up to infinite order. Assume that k is a variable Calderón-Zygmund kernel and Tλ* is defined as in (3). Then for any 1<p<∞, 0<κ<1, and ω∈Ap, Tλ* is bounded on Lp,κ(ω).


## 2. Notations and Preliminary Lemmas

LetB=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 classicalAp weighted theory was first introduced by Muckenhoupt in [19]. A weight ω is a locally integrable function on ℝn, 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 sayω∈Ap with 1<p<∞, if there exists a constant C>0, such that
(7)(1|B|∫B‍ω(x)dx)(1|B|∫B‍ω(x)-1/(p-1)dx)p-1≤C,
for every ball B⊆ℝn. When p=1, ω∈A1 if there exists C>0, such that
(8)1|B|∫B‍ω(x)dx≤Cessinfx∈Bω(x),
for almost every x∈ℝn. We define A∞=⋃‍p≥1Ap. A weight function ω is said to belong to the reverse Hölder class RHr if there exist two constants r>0 and C>0 such that the following reverse Hölder inequality holds:
(9)(1|B|∫B‍ω(x)rdx)1/r≤C(1|B|∫B‍ω(x)dx),
for every ball B⊆ℝn.It is well known that, ifω∈Ap with 1≤p<∞, then there exists r>1 such that ω∈RHr.Lemma 3 (see [20]).
Letω∈Ap, p≥1, and r>0. Then for any ball B and λ>1,
(10)ω(2B)≤Cω(B),ω(λB)≤Cλnpω(B),
where C does not depend on B nor on λ.Lemma 4 (see [21]).
Letω∈RHr with r>1. Then there exists a constant C such that
(11)ω(E)ω(B)≤C(|E||B|)(r-1)/r,
for any measurable subset E of a ball B.The weighted Morrey spaces were defined as follows.Definition 5 (see [4]).
Let1≤p<∞, 0<κ<1, and ω a weight function. Then the weighted Morrey space is defined by
(12)Lp,κ(ω)={f∈Llocp(ω):∥f∥oLp,κ(ω)<∞},
where
(13)o∥f∥Lp,κ(ω)=supB(1ω(B)κ∫B‍|f(x)|pω(x)dx)1/p,
and the supremum is taken over all balls B in ℝn. The space Llocp(ω) is defined by
(14)Llocp(ω)={f:fχK∈Lp(ω),foreverycompactsetK⊆ℝn}.
Our argument is based heavily on the following results.Lemma 6 (see [2]).
Assume thatTλ is defined as in (1). Then for any 1<p<∞ and ω∈Ap, one has
(15)∥Tλf∥oLp(ω)≤C(n,p,Φ,φ,Cp,ω)B∥f∥oLp(ω),
where C(n,p,Φ,φ,Cp,ω) is independent of λ, k, and f and B=∥k∥C1(Σ).Lemma 7 (see [2]).
Assume thatTλ* is defined as in (3). Then for any 1<p<∞ and ω∈Ap, one has
(16)∥Tλ*f∥oLp(ω)≤C(n,p,Φ,φ,Cp,ω)A∥f∥oLp(ω),
where C(n,p,Φ,φ,Cp,ω) is independent of λ, k, and f. A is defined in (2).Definition 8 (see [4]).
The Hardy-Littlewood maximal operatorM is defined by
(17)Mf(x)=supB∋x1|B|∫B‍|f(y)|dy,f∈Lloc(ℝn).Lemma 9 (see [4]).
If1<p<∞, 0<κ<1, and ω∈Ap then the Hardy-Littlewood maximal operator M is bounded on Lp,κ(ω).Lemma 10 (see [22]).
Denote byℋm the spaces of spherical harmonic functions of degree m. Then(a)
L2(Σ)=⊕m=0∞ℋm, and gm=dimℋm≤C(n)mn-2 for any m∈ℕ;(b)
for anym=0,1,2,…, there exists an orthogonal system {Yjm}j=1gm of ℋm such that ∥Yjm∥L∞(Σ)≤C(n)mn/2-1, Yjm=(-m)-n(m+n-2)-nΛnYjm, j=1,…,gm, and Λ is the Beltrami-Laplace operator on Σ.In the following the letterC will denote a constant which may vary at each occurrence.


## 3.  Proof of Theorems

Proof of Theorem1.
It is sufficient to prove that there exists a constantC>0 such that
(18)1ω(B)κ∫B‍|Tλf(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p.
Fix a ballB=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have
(19)1ω(B)κ∫B‍|Tλf(x)|pω(x)dx≤C{1ω(B)κ∫B‍|Tλf1(x)|pω(x)dx+1ω(B)κ∫B‍|Tλf2(x)|pω(x)dx}=C{I1+I2}.
Using Lemmas3 and 6, we get
(20)I1≤C1ω(B)κ∫2B‍|f(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p·ω(2B)κω(B)κ≤C∥f∥Lp,κ(ω)p.
We now estimate I2. We can write
(21)|Tλf2(x)|=|∫(2B)c‍eiλΦ(x,y)k(x-y)φ(x,y)f(y)dy|.
Now by an argument similar to the proof of Lemma  6 in [2], we choose ϕ1∈C0∞(ℝn) 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
(22)k(x)=kλ1(x)+kλ2(x),
where
(23)kλj(x)=k(x)ϕj(λ1/Nx),j=1,2.
Then
(24)Tλf2(x)=p·v·∫(2B)c‍eiλΦ(x,y)kλ1(x-y)×φ(x,y)f(y)dy+p·v·∫(2B)c‍eiλΦ(x,y)kλ2(x-y)×φ(x,y)f(y)dy:=Tλ1f2(x)+Tλ2f2(x).
Let us first estimateTλ1f2(x). To do so, using Taylor’s expansion and the compactness of suppφ, we write
(25)Φ(x,y)=Φ(x,x)+P(x,y)+rN(x,y)
for(x,y)∈suppφ, where P(x,y) is a polynomial with deg P<N and |rN(x,y)|≤C|x-y|N with C independent of x and y. Define
(26)Rf(x)=p·v·∫(2B)c‍eiλP(x,y)kλ1(x-y)φ(x,y)f(y)dy.
Therefore
(27)e-iλΦ(x,x)Tλ1f2(x)-Rf(x)=∫|x-y|≤2λ-1/N‍eiλP(x,y)[eiλrN(x,y)-1]×kλ1(x-y)φ(x,y)f(y)dy=∑j=0∞‍∫2-jλ-1/N<|x-y|≤2-j+1λ-1/N‍eiλP(x,y)[eiλrN(x,y)-1]×kλ1(x-y)φ(x,y)f(y)dy≡∑j=0∞‍Tλ,j1f2(x).
On Tλ,j1f2(x), by the properties of rN and k, we have
(28)|Tλ,j1f2(x)|≤C2-jNMf(x).
So we have(29)|Tλ1f2(x)|≤C∑j=0∞‍2-jN|Mf(x)|+C|Rf(x)|.
By Theorem A and Lemma 9, we have
(30)∥Tλ1f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).
Now, let us turn to estimateTλ2f2(x). We consider the following two cases.
Case 1 (λ≤1). Similar to that estimate of Tλ2 in Lemma 6 in [2], we have
(31)|Tλ2f2(x)|≤CM(f)(x).
By Lemma9 we have
(32)∥Tλ2f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).
Case 2 (λ>1). We choose φ0∈C0∞(ℝn) such that
(33)suppφ0⊆{x∈ℝn:1<|x|≤2},ϕ2(x)=∑j=0∞‍φ0(2-jx).
Let
(34)kλ,j2(x)=k(x)φ0(2-jλ1/Nx).
Then
(35)Tλ2f2(x)=∫(2B)c‍eiλΦ(x,y)kλ2(x-y)φ(x,y)f(y)dy,∑j=0∞‍∫(2B)c‍eiλΦ(x,y)kλ,j2(x-y)φ(x,y)f(y)dy≡∑j=0∞‍Tλ,j2f2(x).
For Tλ,j2, by its definition, we can get
(36)|Tλ,j2f2(x)|≤C∫2jλ-1/N<|x-y|≤2j+1λ-1/N‍×1|x-y|n|f(y)|dy≤CM(f)(x).
The inequality (36) also can be seen in [2]; we omit the details here.
By Lemma9, we have
(37)∥Tλ2f2∥Lp,κ(ω)≤C∥f∥Lp,κ(ω).
Therefore
(38)I2≤C∥f∥Lp,κ(ω)p.
This finishes the proof of Theorem 1.Proof of Theorem2.
It is sufficient to prove that there exists a constantC>0 such that
(39)1ω(B)κ∫B‍|Tλ*f(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p.
Fix a ball B=B(x0,rB) and decompose f=f1+f2, with f1=fχ2B. Then we have
(40)1ω(B)κ∫B‍|Tλ*f(x)|pω(x)dx≤C{1ω(B)κ∫B‍|Tλ*f1(x)|pω(x)dx+1ω(B)κ∫B‍|Tλ*f2(x)|pω(x)dx}=C{J1+J2}.
Using Lemmas 3 and 7, we get
(41)J1≤C1ω(B)κ∫2B‍|f(x)|pω(x)dx≤C∥f∥Lp,κ(ω)p·ω(2B)κω(B)κ≤C∥f∥Lp,κ(ω)p.
We now estimate J2.
For eachm∈ℕ and j=1,…,gm, we get
(42)ajm(x)=∫Σ‍Ω(x,z)Yjm(z)dσz,
where Ω(x,z)=|z|nk(x,z). Then for a.e. x∈ℝn,
(43)Ω(x,z)=∑m=1∞‍∑j=1gm‍ajm(x)Yjm(z′),
where z′=z/|z| for any z∈ℝn∖{0}. By Lemma 10, we have that, for any x∈ℝn,
(44)|ajm(x)|=m-n(m+n-2)-n|∫Σ‍Ω(x,z)ΛnYjm(z)dσz|=m-n(m+n-2)-n|∫Σ‍ΛnΩ(x,z)Yjm(z)dσz|≤C(n)Am-2n.
By Lemma10 again, we can verify that, for any ϵ>0, N∈ℕ, and a.e. x∈ℝn, if |y-x|≥ϵ, then
(45)|∑m=1N‍∑j=1gm‍eiλΦ(x,y)ajm(x)Yjm((x-y)′)|x-y|nφ(x,y)f2(y)|≤C(ϵ)A|f2(y)|.
Therefore, from (43), (45), and the Lebesgue dominated convergence theorem, it follows that
(46)Tλ*f2(x)=limϵ→0∫|x-y|≥ϵ‍eiλΦ(x,y)k(x,x-y)φ(x,y)f2(y)dy=limϵ→0∑m=1∞‍∑j=1gm‍∫|x-y|≥ϵ‍eiλΦ(x,y)ajm(x)Yjm((x-y)′)|x-y|n×φ(x,y)f2(y)dy=limϵ→0∑m=1∞‍∑j=1gm‍ajm(x)∫|x-y|≥ϵ‍eiλΦ(x,y)Yjm((x-y)′)|x-y|n×φ(x,y)f2(y)dy.
We write
(47)Rjmf2(x)=∫|x-y|≥ϵ‍eiλΦ(x,y)Yjm((x-y)′)|x-y|nφ(x,y)f2(y)dy.
It is easy to see that Rjmf2(x) is the oscillatory integral operator defined by (1). By Theorem 1 we have that Rjm is bounded on weighted Morrey spaces. Therefore, by (44) and the above discussion we have
(48)J2≤C∥f∥Lp,κ(ω)p.
This finishes the proof of Theorem2.



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