# Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces

Authors: Zunwei Fu; Shanzhen Lu; Yibiao Pan; Shaoguang Shi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.


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## Body

## 1. Introduction and Main Results

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein's book [1]. Many operators in harmonic analysis or partial differential equations are related to some versions of oscillatory integrals, such as the Fourier transform, the Bochner-Riesz means, and the Radon transform which has important applications in the CT technology. Among numerous papers dealing with oscillatory singular integral operators in some function spaces, we refer to [2–7] and the references therein. More generally, let us now consider a class of oscillatory integrals defined by Ricci and Stein [8]:
(1)Tf(x)=p.v.∫ℝ‍eiP(x,y)K(x-y)f(y)dy,
where P(x,y) is a real-valued polynomial defined on ℝ×ℝ and the function K∈C1(ℝ∖{0}) is a Calderón-Zygmund kernel. That means K satisfies
(2)|K(x)|≤C|x|,|∇K(x)|≤C|x|2,x≠0,(3)∫a<|x|<b‍K(x)dx=0∀a,b(0<a<b).
Throughout this paper, the letter C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.We state a celebrated result of Ricci and Stein on oscillatory integrals as follows.Theorem 1 (see [8]).
Let1<p<∞, K satisfy (2) and (3). Then for any real-valued polynomial P(x,y), the oscillatory integral operator T is of type (Lp,Lp) and its norm depends on the total degree of P, but not on the coefficients of P in other respects.Weighted inequalities arise naturally in harmonic analysis, but their use is best justified by the variety of applications in which they appear. It is worth pointing out that many authors are interested in the inequalities when the weight functions belong to the Muckenhoupt classes ([9]), which are denoted by Ap(1<p<∞) classes for simplicity. This class consists of positive locally integrable functions (weight functions) w for which
(4)supI(1|I|∫I‍w(x)dx)(1|I|∫I‍w(x)1-p′dx)p-1<∞,
where the supremum is taken over all intervals I⊂ℝ and 1/p+1/p′=1.In 1992, Lu and Zhang [10] established the weighted version of Theorem 1.Theorem 2.
Letp,P(x,y) and K be as in Theorem 1. Then the oscillatory singular integral operator T is of type (Lp(w), Lp(w)) with w∈Ap. Here its operator norm is bounded by a constant depending on the total degree of P, but not on the coefficients of P in other respects.We point out that Theorems1 and 2 also hold for dimension n≥2. We choose the results for n=1 here in order to introduce the one-sided operators which were defined on ℝ. Theorems 1 and 2 are also true for more general kernels, that is, nonconvolution kernels, under the L2-boundedness assumption on the corresponding Calderón-Zygmund singular integral operators:
(5)T~(x)=p.v.∫ℝn‍K(x-y)f(y)dy.
However, this topic exceeds the scope of this paper. For more information about this work, see [8, 10], for example.The study of weights for one-sided operators was motivated not only by the generalization of the theory of both-sided ones, but also by their natural appearance in harmonic analysis; for example, they are required when we treat the one-sided Hardy-Littlewood maximal operator [11]:
(6)M+f(x)=suph>01h∫xx+h‍|f(y)|dy,M-f(x)=suph>01h∫x-hx‍|f(y)|dy,
arising in the ergodic maximal function. Sawyer first introduced the classical one-sided weight Ap+ classes in [11]. The general definitions of Ap+ and Ap- were introduced in [12] as
(7)Ap+:supa<b<c1(c-a)p∫ab‍w(x)dx(∫bc‍w(x)1-p′dx)p-1≤C,Ap-:supa<b<c1(c-a)p∫bc‍w(x)dx(∫ab‍w(x)1-p′dx)p-1≤C,
where 1<p<∞, 1/p+1/p′=1; also, for p=1,
(8)A1+:M-w≤Cw,A1-:M+w≤Cw.The smallest constantC for which the above inequalities are satisfied will be denoted by Ap+(w) and Ap-(w), p≥1. Ap+(w) (Ap-(w)) will be called the Ap+ (resp., Ap-) constant of w. By Lebesgue's differentiation theorem, we can easily prove A1+(w) (resp., A1-(w))≥1. In [13], the class A∞+ was introduced as A∞+=⋃p<∞‍Ap+ (see also [14]). It is easy to see that for 1≤p≤∞, Ap⊂Ap+, Ap⊂Ap-, and Ap=Ap+⋂‍Ap-.Theorem 3 (see [11]).
Let1<p<∞. Then(1)
M+ is bounded in Lp(w) if and only if w∈Ap+;(2)
M- is bounded in Lp(w) if and only if w∈Ap-.The one-sided weight classes are of interest, not only because they control the boundedness of the one-sided Hardy-Littlewood maximal operator, but also because they are the right classes for the weighted estimates of one-sided Calderón-Zygmund singular integral operators [15], which are defined by
(9)T~+f(x)=limε→0+∫x+ε∞‍K(x-y)f(y)dy,T~-f(x)=limε→0+∫-∞x-ε‍K(x-y)f(y)dy,
where K is the one-sided Calderón-Zygmund kernel with support in ℝ-=(-∞,0) and ℝ+=(0,+∞), respectively. We say a function K is a one-sided Calderón-Zygmund kernel if K satisfies (2) and
(10)|∫a<|x|<b‍K(x)dx|≤C,0<a<b
with support in ℝ-=(-∞,0) or ℝ+=(0,+∞). An example of such a kernel is
(11)K(x)=sin(log|x|)(xlog|x|)χ(-∞,0)(x),
where χE denotes the characteristic function of a set E.Theorem 4 (see [15]).
Let1<p<∞ and K be a one-sided Calderón-Zygmund kernel. Then(1)
T~+ is bounded in Lp(w) if and only if w∈Ap+;(2)
T~- is bounded in Lp(w) if and only if w∈Ap-.Theorem4 is the one-sided version of weighted norm inequality of singular integral due to Coifman and Fefferman [9].Highly inspired by the above statements for oscillatory singular integral operators and one-sided operator theory, in [16], the authors had introduced the one-sided oscillatory singular integral operators and studied the weighted weak type (1,1) norm inequalities for these operators. In this paper, we will further study the one-sided Muckenhoupt weight classes and give the one-sided version of Theorem 2. It is well known that the property of the one-sided Muckenhoupt weight classes is worse than the Muckenhoupt weight classes (see also [17]). For example, both the reverse Hölder inequality and the doubling condition are not true for the one-sided case. Therefore, some new methods are needed to deal with some new difficulties.We first recall the definition of one-sided oscillatory integral operator as(12)T+f(x)=limε→0+∫x+ε∞‍eiP(x,y)K(x-y)f(y)dy,T-f(x)=limε→0+∫-∞x-ε‍eiP(x,y)K(x-y)f(y)dy,
where P(x,y) is a real-valued polynomial defined on ℝ×ℝ and the kernel K is a one-sided Calderón-Zygmund kernel with support in ℝ- and ℝ+, respectively. Now, we formulate our results as follows.Theorem 5.
Let1<p<∞ and K be a one-sided Calderón-Zygmund kernel. Then for any real-valued polynomial P(x,y),(1)
there exists constantC>0 such that
(13)∥T+f∥Lp(w)≤C∥f∥Lp(w),   where w∈Ap+ and the operator norm depend on the total degree of P and Ap+(w), but not on the coefficients of P in other respects;(2)
there exists constantC>0 such that
(14)∥T-f∥Lp(w)≤C∥f∥Lp(w),   where w∈Ap- and the operator norm depend on the total degree of P and Ap-(w), but not on the coefficients of P in other respects.The rest of this paper is devoted to the argument for Theorem5. Section 2 contains some preliminaries which are essential to our proof. In Section 3, we will give the proof of Theorem 5.


## 2. Preliminaries

Lemma 6 (see [11, 18]).
Let1<p<∞ and w≥0 be locally integrable. Then the following statements are equivalent:(1)
w∈Ap+;(2)
w1-p′∈Ap′-;(3)
there existw1∈A1+ and w2∈A1- such that w=w1(w2)1-p.According to the definition ofAp+, we can easily obtain the following lemma.Lemma 7.
Let1<p<∞ and w∈Ap+. Then Ap+(δλ(w))=Ap+(w), where δλ(w)(x)=w(λx) for all λ>0.Proof.
For1<p<∞, if w∈Ap+, then
(15)supa<b<c1(c-a)p∫ab‍w(x)dx(∫bc‍w(x)1-p′dx)p-1≤C.
For λ>0, a′=λa, b′=λb, c′=λc, and d′=λd, we have
(16)1(c-a)p∫ab‍w(λx)dx(∫bc‍w(λx)1-p′dx)p-1=1(c-a)p∫aλbλ‍w(x)λ-1dx(∫bλcλ‍w(x)1-p′λ-1dx)p-1=1(λ(c-a))p∫aλbλ‍w(x)dx(∫bλcλ‍w(x)1-p′dx)p-1=1(c′-a′)p∫a′b′‍w(x)dx(∫b′c′‍w(x)1-p′dx)p-1≤C.
The proof is complete.We say a weightw satisfies the one-sided reverse Hölder RHr+ condition [18] if there exists C>0 such that for any a<b and 1<r<∞,
(17)∫ab‍w(x)rdx≤C(M(wχ(a,b))(b))r-1∫ab‍w(x)dx,
where M is the classical Hardy-Littlewood maximal operator. The smallest such constant will be called the RHr+ constant of w and will be denoted by RHr+(w). Corresponding to the classical reverse Hölder inequality, (17) is named the weak reverse Hölder inequality. For r=∞, we say a weight w satisfies the one-sided reverse Hölder RH∞+ condition if there exists C>0 such that w(x)≤Cm+w(x) for almost all x∈ℝ where m+ is the one-sided minimal operator defined as
(18)m+f(x)=infh>01h∫xx+h‍|f|dy.
The smallest such constant will be called the RH∞+ constant of w and will be denoted by RH∞+(w). It is clear that RH∞+(w)≥1. In [18], the authors give several characterizations of RHr+ where the constants C are not necessary the same.Lemma 8.
Leta<b<c<d, 1<r<∞, and w≥0 be locally integrable. Then the following statements are equivalent:(1)
∫ab‍w(x)rdx≤C(M(wχ(a,b))(b))r-1∫ab‍w(x)dx;(2)
(1/(b-a))∫ab‍w(x)rdx≤C((1/(c-b))∫bc‍w(x)dx)r with b-a=2(c-b);(3)
(1/(b-a))∫ab‍w(x)rdx≤C((1/(d-c))∫cd‍w(x)dx)r with b-a=d-b=2(d-c);(4)
(1/(b-a))∫ab‍w(x)rdx≤C((1/(c-b))∫bc‍w(x)dx)r with b-a=c-b;(5)
(1/(b-a))∫ab‍w(x)rdx≤C((1/(d-c))∫cd‍w(x)dx)r with b-a=d-c=γ(d-a), 0<γ≤1/2.Lemma 9 (see [18]).
A weightw∈Ap+ for p>1 if and only if there exist 0<γ<1/2 and a constant Cγ such that for b-a=d-c=γ(d-a) with a<b<c<d, the following inequality holds:
(19)∫ab‍w(x)dx(∫cd‍w(x)1-p′dx)p-1≤Cγ(b-a)p.Combining the results in [12, 15, 18, 19], we can deduce Lemma 10. In what follows, we will include its proof with slight modifications for the sake of completeness.Lemma 10.
Letw∈Ap+. Then there exists ε>0 such that w1+ε∈Ap+.Proof.
By Lemma6, we have w=w1w21-p with w1∈A1+ and w2∈A1-. For fixed interval I=(a,b), we next claim that w1∈RHr+ for all 1<r<C/(C-1) with C=max{A1+(w1),A1-(w1)}>1. In fact, we consider the truncation of w at height H defined by wH=min{w1,H} which also satisfies A1+ condition (with a constant CH≤C). Therefore, if λI=M(wHχI)(b) and Sλ={x∈I:wH(x)>λ}, then we have
(20)∫Sλ‍wH(x)dx≤CHλ|Sλ|,λ≥λI.
Indeed, it is straightforward if Sλ=I since
(21)wH(Sλ)=∫ab‍wH(x)dx≤λI(b-a)≤CHλ|Sλ|.
We now assume Sλ≠I and fix ε>0 and an open set O such that Sλ⊂O⊂I with |O|≤ε+|Sλ|. Let Oi=(c,d), which is connected. There are two cases; that is, a≤c<d<b and a≤c<d=b. In the first case, it is easy to check that d is not contained in Sλ. By the definition of Sλ, w1+, we have ∫cd‍wH(x)dx≤CHλ(d-c), while the second case is handled as the case Sλ=I since ∫cd‍wH(x)dx≤C(b-c). Thus wH(Oi)≤CHλ|Oi|. Adding up with i, we get
(22)wH(Sλ)≤wH(O)≤CHλ|Oi|≤CHλ(ε+|Sλ|).
Therefore, we obtain (20). For fixed θ>-1, multiply both sides of (20) by λθ and integrate from λI to infinity; we can obtain
(23)1θ+1∫I‍(wHθ+2-λIθ+1)(x)dx≤CHθ+2∫I‍wHθ+2(x)dx.
Now if r=θ+2<CH/(CH-1), then 1/(θ+1)-CH/(θ+2)>0, which implies
(24)∫I‍wHr(x)dx≤CHλIr-1∫I‍wH(x)dx=CH(M(wHχI)(b))r-1∫I‍wH(x)dx.
The inequalityCH≤C implies CH/(CH-1)≥C/(C-1). Therefore, if r≤C/(C-1), then we have
(25)∫I‍wHr(x)dx=CH(M(wHχI)(b))r-1∫ab‍wH(x)dx=C(M(w1χ(a,b))(b))r-1∫ab‍wH(x)dx.
Hence w1∈RHr+ by the monotone convergence theorem. Since w2∈A1-, we next claim that w21-p∈RH∞+. In fact, for any interval I=(a,b), we have
(26)(1|I|∫I‍w2(x)dx)1-p≤1|I|∫I‍w2(x)1-pdx
by Hölder's inequality and the A1- condition. For almost every x∈I-=(2a-b,a), we have
(27)Cw2≥1|I|∫I‍w2(x)dx.
Thus,
(28)w2(x)1-p≤C(1|I|∫I‍w2(x)dx)1-p≤C1|I|∫I‍w2(x)1-pdx≤C1b-x∫xb‍w2(x)1-pdx,
which implies our claim. Hence,
(29)1|I|∫I‍wr≤1|I|∫I‍w1rsupI(w2-r(p-1))≤C(1I1∫I1‍w1)r(1I1∫I1‍w21-p)r≤C(infI1w1)r(supI1w21-p)r≤C(infw1)r(1I2∫I2‍w21-p)r≤C(1I2∫I2‍w)r,
where I1=(b,2b-a) and I2=(2b-a,3b-2a). By Lemma 8, we obtain w∈RHr+. Hence, w1-p′∈RHr- for all 1<r<C/(C-1) by Lemma 6.
Let us fixa<d and choose b,c such that b-a=d-c=(d-a)/4 (e.g., we choose b=(d+3a)/4, c=(3d+a)/4). Following from the five points a,b,(b+c)/2,c,d, we have four intervals, namely,
(30)I1=(a,b),I2=(b,(b+c)2),I3=((b+c)2,c),I4=(c,d).
By Lemma 8, we have
(31)1|I1|∫I‍wr(1|I4|∫I4‍wr(1-p′))p-1≤(1|I2|∫I2‍w)r(1|I3|∫I3‍w(1-p′))r(p-1)≤Cr.
Thus, wr∈Ap+ by Lemma 9. Choosing 0<ε=r-1<1/(C-1), then we complete the proof of the lemma.To prove Theorem5, we still need a celebrated interpolation theorem of operators with change of measures.Lemma 11 (see [20]).
Suppose thatu0,v0,u1,v1 are positive weight functions and 1<p0, p1<∞. Assume sublinear operator S satisfies
(32)∥Sf∥Lp0(u0)≤C0∥f∥Lp0(v0),∥Sf∥Lp1(u1)≤C1∥f∥Lp1(v1).
Then,
(33)∥Sf∥Lp(u)≤C∥f∥Lp(v)
holds for any 0<θ<1 and 1/p=θ/p0+(1-θ)/p1, where u=u0pθ/p0u1p(1-θ)/p1, v=v0pθ/p0v1p(1-θ)/p1, and C≤C0θC11-θ.Lemmas10 and 11 are the main tools in proving Theorem 5.


## 3. Proof of Theorem5

In this section, we will prove Theorem5 by induction, which is partly motivated by [8, 10]. We begin with the proof of (1). For any nonzero real polynomial P(x,y) in x and y, there are k,l,m≥0 such that
(34)P(x,y)=aklxkyl+R(x,y)
with akl≠0 and
(35)R(x,y)=∑0≤α<k,0≤β≤m‍aαβxαyβ+∑0≤β<l‍akβxkyβ.
We will write dx(P)=k and dy(P)=l. Below we will carry out the argument by using a double induction on k and l.Ifdx(P)=0 and dy(P) is arbitrary, then P(x,y)=P(y) and T+f can be written as
(36)T+f(x)=limε→0+∫x+ε∞‍K(x-y)g(y)dy,
where g(y)=eiP(y)f(y). Therefore, the conclusion of Theorem 5 follows from Theorem 4.Letk≥1 and assume that the conclusion of Theorem 5 holds for all P(x,y) with dx(P)≤k-1 and dy(P) arbitrary.We will now prove that the conclusion of Theorem5 holds for all P(x,y) with dx(P)=k and dy(P) arbitrary.Ifdx(P)=k and dy(P)=0, then
(37)P(x,y)=ak0xk+Q(x,y)
with dx(Q)≤k-1. By taking the factor eiak0xk out of the integral sign, we see that this case follows from the above inductive hypothesis.Supposel≥1 and the desired bound holds when dx(P)=k and dy(P)≤l-1.Now, letP(x,y) be a polynomial with dx(P)=k and dy(P)=l, as given in (34).Case 1 (|akl|=1).
Write(38)T+f(x)=∫x1+x‍eiP(x,y)K(x-y)f(y)dy+∑j=1∞‍∫2j-1+x2j+x‍eiP(x,y)K(x-y)f(y)dy=:T0+f(x)+∑j=1∞‍Tj+f(x).
Take any h∈ℝ, and write
(39)P(x,y)=akl(x-h)k(y-h)l+R(x,y,h),
where the polynomial R(x,y,h) satisfies the induction assumption and the coefficients of R(x,y,h) depend on h.
We consider first the estimates forT0+. It is easy to check that
(40)T0+f(x)=∫x1+x‍ei(R(x,y,h)+akl(y-h)k+l)K(x-y)f(y)dy+∫x1+x‍{eiP(x,y)-ei(R(x,y,h)+akl(y-h)k+l)}×K(x-y)f(y)dy=:T01+f(x)+T02+f(x).
Now we split f into three parts as
(41)f(y)=f(y)χ{|y-h|<1/2}(y)+f(y)χ{1/2≤|y-h|<5/4}(y)+f(y)χ{|y-h|≥5/4}(y)=:f1(y)+f2(y)+f3(y).
Observe that if|x-h|<1/4, then
(42)T01+f1(x)=∫x1+x‍ei(R(x,y,h)+akl(y-h)k+l)K(x-y)f1(y)dy.
Thus, it follows from the induction assumption that
(43)∫|x-h|<1/4‍|T01+f1(x)|pw(x)dx≤C∫|y-h|<1/2‍|f(y)|pw(y)dy,
where C is independent of h and the coefficients of P(x,y).
Notice that if|x-h|<1/4,1/2≤|y-h|<5/4, then y-x>1/4. Thus,
(44)|T01+f2(x)|≤C∫x+1/4x+1‍|K(x-y)f2(y)|dy≤CM+(f2)(x).
So we have
(45)∫|x-h|<1/4‍|T01+f2(x)|pw(x)dx≤C∫|y-h|<5/4‍|f(y)|pw(y)dy,
where C is independent of h and the coefficients of P(x,y).
Again observe that if|x-h|<1/4 and |y-h|≥5/4, then y-x>1. Thus,
(46)T01+f3(x)=0.
Combining (43), (45), and (46), we get
(47)∫|x-h|<1/4‍|T01+f(x)|pw(x)dx≤C∫|y-h|<5/4‍|f(y)|pw(y)dy,
where C is independent of h and the coefficients of P(x,y).
Evidently, if|x-h|<1/4 and 0<y-x<1, then
(48)|eiP(x,y)-ei(R(x,y,h)+akl(y-h)k+l)|≤C|akl||x-y|=C(y-x).
Therefore, when |x-h|<1/4, we have
(49)|T02+f(x)|≤C∫xx+1‍|f(y)|dx≤CM+(f(·)χB(h,5/4)(·))(x).
It follows from Theorem 3 that
(50)∫|x-h|<1/4‍|T02+f(x)|pw(x)dx≤C∫|y-h|<5/4‍|f(y)|pw(y)dy,
where C is independent of h and the coefficients of P(x,y).
From (47) and (50), it follows that the inequality
(51)∫|x-h|<1/4‍|T0+f(x)|pw(x)dx≤C∫|y-h|<5/4‍|f(y)|pw(y)dy
holds uniformly in h∈ℝ+, which implies
(52)∥T0+f∥Lp(w)≤C∥f∥Lp(w),
where C is independent of the coefficients of P(x,y) and w∈Ap+.
We proceed with the proof of Theorem5 with the estimates for Tj+f. Because of the size condition (2), we observe that for j≥1(53)|Tj+f(x)|≤∫2j-1+x2j+x‍|f(y)||x-y|dy≤CM+(f)(x),
where C is independent of j. By Lemma 10, we know that there exists ε>0 such that w1+ε∈Ap+. Thus we have
(54)∥Tj+f∥Lp(w1+ε)≤C∥f∥Lp(w1+ε),
where C is independent of j. We now only need to recall Lemma 3.7 in [16] to see that
(55)∥Tj+f∥Lp≤C2-jδ∥f∥Lp,
where C depends only on the total degree of P(x,y) and δ>0. It follows from (54), (55), and Lemma 11 that
(56)∥Tj+f∥Lp(w)≤C2-jθδ∥f∥Lp(w),
where 0<θ<1, θ is independent of j, and C depends only on the total degree of P(x,y).
From (52) and (56), it is clear that when w∈Ap+,
(57)∥T+f∥Lp(w)≤C∥f∥Lp(w),
where C depends only on the total degree of P(x,y).Case 2 (|akl|≠1).
In this case, we writeλ=|akl|1/(k+l) and
(58)P(x,y)=λ-(k+l)akl(λx)k(λy)l+R(λxλ,λyλ)=Q(λx,λy).
Therefore,
(59)T+f(x)=p.v.∫‍eiQ(λx,λy)K(x-y)f(y)dy=p.v.∫‍eiQ(λx,y)K(λxλ-yλ)f(yλ)λ-1dy=Tλ+(f(·λ))(λx),
where Kλ(x-y)=λ-1K(x/λ-y/λ) and
(60)Tλ+f(x)=p.v.∫‍eiQ(x,y)Kλ(x-y)f(y)dy.
It is easy to check that Kλ satisfies (2) and (10). We have thus established that
(61)∥Tλ+f∥Lp(w)≤C∥f∥Lp(w)
with similar statements as in Case 1. By Lemma 7, we have
(62)∫|T+f(x)|pw(x)dx=∫‍|Tλ+f(·λ)(λx)|pw(x)dx=λ-1∫‍|Tλ+f(·λ)(x)|pw(xλ)dx≤C∫‍|f(xλ)|pw(xλ)dx=C∫‍|f(x)|pw(x)dx;
that is,
(63)∥T+f∥Lp(w)≤C∥f∥Lp(w),
where C depends on the total degree of P(x,y) but not on the coefficients of P(x,y).(2) We omit the details, since they are very similar to that of the proof of (1) withw∈Ap- instead of w∈Ap+.



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