# Weighted Estimates for Oscillatory Singular Integrals

Authors: H. Al-Qassem; L. Cheng; A. Fukui; Y. Pan
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We establish uniform bounds for oscillatory singular integrals as well as oscillatory singular integral operators. We allow the singular kernel to be given by a function in the Hardy spaceH1(Sn−1), while such results were known previously only for kernels in L log L(Sn−1), a proper subspace of H1(Sn−1). One of our results established a Lp(w)→Lp(w) bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005).


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## Body

## 1. Introduction

In this paper we establish uniform bounds for oscillatory singular integrals. We consider two types of oscillatory singular integrals, which will be described later.Letn≥2 and Sn-1 denote the unit sphere in Rn equipped with the induced Lebesgue measure σ. For an integrable function Ω:Sn-1→C satisfying
(1)∫Sn-1Ωdσ=0,we define(2)K(x)=Ω(x′)|x|n,
where x′=x/|x| for x∈Rn∖{0}. For d,m∈N, let
(3)𝒫(d,m)={P:Rm⟶R:Pbeapolynomialwithdeg(P)≤d}.Type I.An oscillatory integral of type I is given by
(4)In(Ω,P)=p.v.∫RneiP(x)K(x)dx,
where K is given by (2) and P is a polynomial on Rn. For a given Ω:Sn-1→C and d∈N the main concern is to establish a bound for
(5)supP∈𝒫(d,n)|In(Ω,P)|.Previous results in this regard include Stein [1] for Ω∈L∞(Sn-1) and Papadimitrakis and Parissis [2] for Ω∈LlogL(Sn-1) which improved Stein's result.Type II. A type II oscillatory singular integral is actually an integral operator of the form
(6)TΩ,Q:f⟶p.v.∫RneiQ(x,y)K(x-y)f(y)dy,
where K is given by (2) and Q is a real-valued polynomial on Rn×Rn. Ricci and Stein [3] showed that, if Ω∈C1(Sn-1), TΩ,Q is bounded on Lp(Rn). Subsequently Lu and Zhang [4] and Jiang and Lu [5] established the same bounds for ∥TΩ,Q∥p,p under the weaker conditions Ω∈L1+ϵ(Sn-1) and Ω∈LlogL(Sn-1), respectively.We will now state our main results, beginning with oscillatory singular integrals of Type II.A setR in Rn is called a rectangle if there is an orthonormal basis {e1,…,en} of Rn (which may depend on R) such that
(7)R={∑j=1nxjej:aj≤xj≤bj,for1≤j≤n}.In other words, what we call a rectangle inRn is simply any rotation of an arbitrary n-cell [a1,b1]×⋯×[an,bn]. Let ℛn denote the collection of all rectangles in Rn.Definition 1.
Letp∈(1,∞), and let w be a nonnegative, locally integrable function on Rn. We say that w is in the weight class Vp if
(8)supR∈ℛn(1|R|∫R‍w(x)dx)(1|R|∫R‍w(x)-1/(p-1)dx)p-1<∞.It is easy to see thatVp is a subcollection of the well-known weight class Ap of Muckenhoupt [6, 7]. Examples of weights in Vp include all weights of the form |G(x)|α, where G(x) is a polynomial in Rn and -1<αdeg(G)<p-1.Theorem 2.
LetQ(x,y) be a real-valued polynomial on Rn×Rn. Suppose that w∈Vp, Ω∈H1(Sn-1) and Ω satisfies (1). Then the operator TΩ,Q is bounded on Lp(Rn,w) for 1<p<∞, with a bound on its norm which may depend on the degree of Q but is otherwise independent of the coefficients of Q.The spaceH1(Sn-1) is the Hardy space on the unit sphere. Since LlogL(Sn-1) is a proper subspace of H1(Sn-1), Theorem 2 represents an improvement over results mentioned earlier. By taking w=1, it answers an open question in [8, page 52] in the affirmative.Our second result has the same flavor as the first, but it concerns Type I oscillatory singular integrals instead.Theorem 3.
Suppose thatΩ∈H1(Sn-1) and Ω satisfies (1). Then
(9)supP∈𝒫(d,n)|In(Ω,P)|≤cn(1+logd)∥Ω∥H1(Sn-1),
where cn is a constant independent of d and Ω.Our result in this regard is built on the work of Papadimitrakis and Parissis who gave the following bound in [2]:
(10)supP∈𝒫(d,n)|In(Ω,P)|≤c(1+logd)(∥Ω∥LlogL(Sn-1)+1).They also showed the logarithmic growth of the bound in d to be best possible. Our bound, while dependent on the dimension n, provides an improvement over the factor (∥Ω∥LlogL+1).


## 2. Proofs of Theorems2 and 3

We will begin by recalling the atomic decomposition forH1(Sn-1).Definition 4.
A measurable functiona(·) on Sn-1 is called a regular H1 atom if it satisfies the following:(i)
∫Sn-1‍a(y)dσ(y)=0,(ii)
supp(a)⊆Sn-1∩B(θ0,ρ) for some θ0∈Sn-1 and ρ>0, where B(θ0,ρ)={y∈Rn:|y-θ0|<ρ},(iii)
∥a∥∞≤ρ-n+1.
An exceptional atom is just anL∞ function a(·) on Sn-1 satisfying ∥a∥∞≤1.The following result is from [9, 10].Lemma 5.
For everyh∈H1(Sn-1) there exist {λk}⊂C and H1 atoms (both regular and exceptional) {ak(·)} such that
(11)h=∑k‍λkak,and ∥h∥H1(Sn-1)≈∑k‍|λk|.Proof of Theorem2.
Letd=deg(Q). It suffices to show that, for 1<p<∞, there exists a C(n,d,p,w)>0 such that
(12)∥TΩ,Qf∥Lp(Rn,w)≤C(n,d,p,w)∥Ω∥H1(Sn-1)∥f∥Lp(Rn,w),for all f∈C0∞(Rn). Since the sum in (11) converges in the sense of distribution, by Lemma 5, we only need to prove
(13)∥TΩ,Q∥Lp(Rn,w)→Lp(Rn,w)≤C(n,d,p,w),
whenΩ is a regular atom.
Below we will assume thatΩ(·)=a(·) satisfies Conditions (i)–(iii) in Definition 4. Obviously we may also assume that ρ∈(0,1/4). We also extend Ω to be a homogeneous function of degree 0 by setting Ω(x)=Ω(x/|x|) for x∈Rn∖{0}. Let M be an n×n orthogonal matrix such that θ0Mt=(0,…,0,1)=e. We define the linear transformation Γ on Rn by
(14)Γy=y(ρIn-1001)M,
where In-1 denotes the (n-1)×(n-1) identity matrix and y=(y1,…,yn)=(y~,yn). By letting Ψ(x,y)=Q(Γx,Γy), fΓ(x)=f(Γx), wΓ(x)=ρn-1w(Γx), and
(15)h(x)=ρn-1(|x|nΩ(Γx)|Γx|n),
we get
(16)(TΩ,Qf)(Γx)=(Th,ΨfΓ)(x).
Ifh(x)≠0 for some x=(x~,xn)∈Rn∖{0}, then by (i) we have
(17)|(ρx~,xn)|(ρx~,xn)|-e|<ρ,
which implies that
(18)xn>(1-ρ22)|(ρx~,xn)|.
Thus,
(19)|x|=ρ-1(|(ρx~,xn)|2+(ρ2-1)xn2)1/2<ρ-1[1+(ρ2-1)(1-ρ22)2]1/2×|(ρx~,xn)|≤2|Γx|.
Therefore we have
(20)∥h∥∞≤2n/2.
By its definition and a well-known argument,h is homogeneous of degree 0 and satisfies (1). Also observe that deg(Ψ)=deg(Q) and wΓ is an Ap weight with an Ap bound independent of Γ. Thus, by Theorem  5 in [11] and Theorem  5 of [5], there is a C(n,d,p,w)>0 such that
(21)∥TΩ,Q∥Lp(Rn,w)→Lp(Rn,w)=∥Th,Ψ∥Lp(Rn,wΓ)→Lp(Rn,wΓ)≤C(n,d,p,w).
This proves Theorem 2.Proof of Theorem3.
LetP∈𝒫(d,n), and let Q(x,y)=P(x-y). For a Ω∈H1(Sn-1) which satisfies (1), we write
(22)Ω=∑j‍λjaj,
where {aj} are regular H1 atoms and ∥Ω∥H1≈∑j‍|λj|. By the proof of Theorem 2, for each j, there exist a Pj∈𝒫(d,n) and a function ωj on Sn-1 which satisfies (1) and ∥ωj∥∞≤2n/2 such that
(23)∥TΩ,Q∥2,2≤∑j‍|λj|∥Tωj,Qj∥2,2,
where Qj(x,y)=Pj(x-y). By (23) and Theorem 1 in [2], we have
(24)|In(Ω,P)|≤∥TΩ,Q∥2,2≤∑j‍|λj|∥Tωj,Qj∥2,2≤c(1+logd)(∑j‍|λj|∥ωj∥LlogL(Sn-1))≤cn(1+logd)∥Ω∥H1(Sn-1),
which proves Theorem 3.



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