Weighted Estimates for Oscillatory Singular Integrals

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 space 𝐻¹(𝐒^𝑛−1), while such results were known previously only for kernels in 𝐿 log 𝐿(𝐒^𝑛−1), a proper subspace of 𝐻¹(𝐒^𝑛−1). One of our results established a 𝐿^𝑝(𝑤)→𝐿^𝑝(𝑤) bound for certain weights. At the same time, it provides a solution to an open problem in Lu (2005).

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.

Let 𝑛≥2 and 𝐒^𝑛−1 denote the unit sphere in 𝐑^𝑛 equipped with the induced Lebesgue measure 𝜎. For an integrable function Ω∶𝐒^𝑛−1→𝐂 satisfying _𝐒𝑛−1Ω𝑑𝜎=0, (1)

we define 𝐾(𝑥)=Ω𝑥^′|𝑥|^𝑛, (2) where 𝑥^′=𝑥/|𝑥| for 𝑥∈𝐑^𝑛⧵{0}. For 𝑑,𝑚∈𝐍, let 𝒫()=𝑑,𝑚𝑃∶𝐑^𝑚⟶𝐑∶𝑃beapolynomialwithdeg(𝑃)≤𝑑. (3)

Type I. An oscillatory integral of type I is given by 𝐼_𝑛(Ω,𝑃)=p.v._𝐑𝑛𝑒^{𝑖𝑃(𝑥)}𝐾(𝑥)𝑑𝑥, (4) where 𝐾 is given by (2) and 𝑃 is a polynomial on 𝐑^𝑛. For a given Ω∶𝐒^𝑛−1→𝐂 and 𝑑∈𝐍 the main concern is to establish a bound for sup_{()𝑃∈𝒫𝑑,𝑛}𝐼_𝑛(Ω,𝑃). (5)

Previous results in this regard include Stein [1] for Ω∈𝐿^∞(𝐒^𝑛−1) and Papadimitrakis and Parissis [2] for Ω∈𝐿log𝐿(𝐒^𝑛−1) which improved Stein's result.

Type II. A type II oscillatory singular integral is actually an integral operator of the form 𝑇_Ω,𝑄∶𝑓⟶p.v._𝐑𝑛𝑒^{𝑖𝑄(𝑥,𝑦)}𝐾𝑥−𝑦𝑓𝑦𝑑𝑦, (6) where 𝐾 is given by (2) and 𝑄 is a real-valued polynomial on 𝐑^𝑛×𝐑^𝑛. Ricci and Stein [3] showed that, if Ω∈𝐶¹(𝐒^𝑛−1), 𝑇_Ω,𝑄 is bounded on 𝐿^𝑝(𝐑^𝑛). Subsequently Lu and Zhang [4] and Jiang and Lu [5] established the same bounds for ‖𝑇_Ω,𝑄‖_𝑝,𝑝 under the weaker conditions Ω∈𝐿^1+𝜖(𝐒^𝑛−1) and Ω∈𝐿log𝐿(𝐒^𝑛−1), respectively.

We will now state our main results, beginning with oscillatory singular integrals of Type II.

A set 𝑅 in 𝐑^𝑛 is called a rectangle if there is an orthonormal basis {𝑒₁,…,𝑒_𝑛} of 𝐑^𝑛 (which may depend on 𝑅) such that 𝑅=^𝑛_𝑗=1𝑥_𝑗𝑒_𝑗∶𝑎_𝑗≤𝑥_𝑗≤𝑏_𝑗,for1≤𝑗≤𝑛. (7)

In other words, what we call a rectangle in 𝐑^𝑛 is simply any rotation of an arbitrary 𝑛-cell [𝑎₁,𝑏₁]×⋯×[𝑎_𝑛,𝑏_𝑛]. Let ℛ_𝑛 denote the collection of all rectangles in 𝐑^𝑛.

Definition 1. Let 𝑝∈(1,∞), and let 𝑤 be a nonnegative, locally integrable function on 𝐑^𝑛. We say that 𝑤 is in the weight class 𝑉_𝑝 if sup_{𝑅∈ℛ𝑛}1|𝑅|_𝑅𝑤(𝑥)1𝑑𝑥|𝑅|_𝑅𝑤(𝑥)^{−1/(𝑝−1)}𝑑𝑥^𝑝−1<∞. (8)

It is easy to see that 𝑉_𝑝 is a subcollection of the well-known weight class 𝐴_𝑝 of Muckenhoupt [6, 7]. Examples of weights in 𝑉_𝑝 include all weights of the form |𝐺(𝑥)|^𝛼, where 𝐺(𝑥) is a polynomial in 𝐑^𝑛 and −1<𝛼deg(𝐺)<𝑝−1.

Theorem 2. Let 𝑄(𝑥,𝑦) be a real-valued polynomial on 𝐑^𝑛×𝐑^𝑛. Suppose that 𝑤∈𝑉_𝑝, Ω∈𝐻¹(𝐒^𝑛−1) and Ω satisfies (1). Then the operator 𝑇_Ω,𝑄 is bounded on 𝐿^𝑝(𝐑^𝑛,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 space 𝐻¹(𝐒^𝑛−1) is the Hardy space on the unit sphere. Since 𝐿log𝐿(𝐒^𝑛−1) is a proper subspace of 𝐻¹(𝐒^𝑛−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 that Ω∈𝐻¹(𝐒^𝑛−1) and Ω satisfies (1). Then sup_{()𝑃∈𝒫𝑑,𝑛}𝐼_𝑛(Ω,𝑃)≤𝑐_𝑛1+log𝑑‖Ω‖_𝐻¹_(𝐒^𝑛−1₎, (9) where 𝑐_𝑛 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]: sup_{()𝑃∈𝒫𝑑,𝑛}𝐼_𝑛(Ω,𝑃)≤𝑐1+log𝑑‖Ω‖_{𝐿log𝐿(𝐒}^𝑛−1₎+1. (10)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 Theorems 2 and 3

We will begin by recalling the atomic decomposition for 𝐻¹(𝐒^𝑛−1).

Definition 4. A measurable function 𝑎(⋅) on 𝐒^𝑛−1 is called a regular 𝐻¹ atom if it satisfies the following:(i)∫_𝐒𝑛−1𝑎(𝑦)𝑑𝜎(𝑦)=0,(ii)supp(𝑎)⊆𝐒^𝑛−1∩𝐵(𝜃₀,𝜌) for some 𝜃₀∈𝐒^𝑛−1 and 𝜌>0, where 𝐵(𝜃₀,𝜌)={𝑦∈𝐑^𝑛∶|𝑦−𝜃₀|<𝜌},(iii)‖𝑎‖_∞≤𝜌^−𝑛+1.
An exceptional atom is just an 𝐿^∞ function 𝑎(⋅) on 𝐒^𝑛−1 satisfying ‖𝑎‖_∞≤1.

The following result is from [9, 10].

Lemma 5. For every ℎ∈𝐻¹(𝐒^𝑛−1) there exist {𝜆_𝑘}⊂𝐂 and 𝐻¹ atoms (both regular and exceptional) {𝑎_𝑘(⋅)} such that ℎ=_𝑘𝜆_𝑘𝑎_𝑘, (11)and ‖ℎ‖_𝐻¹_(𝐒^𝑛−1₎≈∑_𝑘|𝜆_𝑘|.

Proof of Theorem 2. Let 𝑑=deg(𝑄). It suffices to show that, for 1<𝑝<∞, there exists a 𝐶(𝑛,𝑑,𝑝,𝑤)>0 such that 𝑇_Ω,𝑄𝑓_{𝐿𝑝(𝐑𝑛,𝑤)}≤𝐶𝑛,𝑑,𝑝,𝑤‖Ω‖_𝐻¹_(𝐒^𝑛−1₎𝑓_{𝐿𝑝(𝐑𝑛,𝑤)}, (12)for all 𝑓∈𝐶^∞₀(𝐑^𝑛). Since the sum in (11) converges in the sense of distribution, by Lemma 5, we only need to prove 𝑇_{Ω,𝑄𝐿𝑝(𝐑𝑛,𝑤)→𝐿𝑝(𝐑𝑛,𝑤)}≤𝐶𝑛,𝑑,𝑝,𝑤, (13)
when Ω is a regular atom.
Below we will assume that Ω(⋅)=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∈𝐑^𝑛⧵{0}. Let 𝑀 be an 𝑛×𝑛 orthogonal matrix such that 𝜃₀𝑀^𝑡=(0,…,0,1)=𝐞. We define the linear transformation Γ on 𝐑^𝑛 by Γ𝑦=𝑦𝜌𝐼_𝑛−1001𝑀, (14) where 𝐼_𝑛−1 denotes the (𝑛−1)×(𝑛−1) identity matrix and 𝑦=(𝑦₁,…,𝑦_𝑛)=(𝑦,𝑦_𝑛). By letting Ψ(𝑥,𝑦)=𝑄(Γ𝑥,Γ𝑦), 𝑓_Γ(𝑥)=𝑓(Γ𝑥), 𝑤_Γ(𝑥)=𝜌^𝑛−1𝑤(Γ𝑥), and ℎ(𝑥)=𝜌^𝑛−1|𝑥|^𝑛Ω(Γ𝑥)|Γ𝑥|^𝑛, (15) we get 𝑇_Ω,𝑄𝑓(Γ𝑥)=𝑇_ℎ,Ψ𝑓_Γ(𝑥). (16)
If ℎ(𝑥)≠0 for some 𝑥=(𝑥,𝑥_𝑛)∈𝐑^𝑛⧵{0}, then by (i) we have 𝜌𝑥,𝑥_𝑛𝜌𝑥,𝑥_𝑛−𝐞<𝜌, (17) which implies that 𝑥_𝑛>𝜌1−²2𝜌𝑥,𝑥_𝑛. (18) Thus, |𝑥|=𝜌⁻¹(𝜌𝑥,𝑥_𝑛)²+𝜌²−1𝑥²_𝑛^1/2<𝜌⁻¹1+𝜌²−11−𝜌²2^21/2×𝜌𝑥,𝑥_𝑛≤√2|Γ𝑥|. (19) Therefore we have ‖ℎ‖_∞≤2^𝑛/2. (20)
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 𝑤_Γ is an 𝐴_𝑝 weight with an 𝐴_𝑝 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 𝑇_{Ω,𝑄𝐿𝑝(𝐑𝑛,𝑤)→𝐿𝑝(𝐑𝑛,𝑤)}=𝑇_{ℎ,Ψ𝐿𝑝(𝐑𝑛,𝑤Γ)→𝐿𝑝(𝐑𝑛,𝑤Γ)}≤𝐶𝑛,𝑑,𝑝,𝑤. (21) This proves Theorem 2.

Proof of Theorem 3. Let 𝑃∈𝒫(𝑑,𝑛), and let 𝑄(𝑥,𝑦)=𝑃(𝑥−𝑦). For a Ω∈𝐻¹(𝐒^𝑛−1) which satisfies (1), we write Ω=_𝑗𝜆_𝑗𝑎_𝑗, (22) where {𝑎_𝑗} are regular 𝐻¹ atoms and ‖Ω‖_𝐻¹≈∑_𝑗|𝜆_𝑗|. By the proof of Theorem 2, for each 𝑗, there exist a 𝑃_𝑗∈𝒫(𝑑,𝑛) and a function 𝜔_𝑗 on 𝐒^𝑛−1 which satisfies (1) and ‖𝜔_𝑗‖_∞≤2^𝑛/2 such that 𝑇_Ω,𝑄2,2≤_𝑗𝜆_𝑗𝑇_{𝜔𝑗,𝑄𝑗2,2}, (23) where 𝑄_𝑗(𝑥,𝑦)=𝑃_𝑗(𝑥−𝑦). By (23) and Theorem 1 in [2], we have 𝐼_𝑛(Ω,𝑃)≤𝑇_Ω,𝑄2,2≤_𝑗𝜆_𝑗𝑇_{𝜔𝑗,𝑄𝑗2,2}≤𝑐1+log𝑑_𝑗𝜆_𝑗𝜔_{𝑗𝐿log𝐿(𝐒𝑛−1)}≤𝑐_𝑛1+log𝑑‖Ω‖_𝐻¹_(𝐒^𝑛−1₎, (24) which proves Theorem 3.