# Boundedness of Fractional Oscillatory Integral Operators and Their Commutators in Vanishing Generalized Weighted Morrey Spaces

Authors: Bilal Çekiç; Ayşegül Çelik Alabalık
Journal: Journal of Function Spaces
 (2017)

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## Abstract

In this article, we give the boundedness conditions in terms of Zygmund-type integral inequalities for oscillatory integral operators and fractional oscillatory integral operators on the vanishing generalized weighted Morrey spaces. Moreover, we investigate corresponding commutators.


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## Body

## 1. Introduction

The classical Morrey spacesMp,λ that play important role in the theory of partial differential equations were introduced by Morrey [1] in 1938. Since then Morrey spaces have been studied by various authors. We refer readers to the survey [2] and to the elegant book [3] for further information about these spaces and references on recent developments in this field.So far various generalizations of Morrey spaces have been defined. Mizuhara [4] introduced the generalized Morrey space Mp,φ and Komori and Shirai [5] defined the weighted Morrey spaces Lp,κ(w). Guliyev [6] gave the notion of generalized weighted Morrey space Mp,φw which can be accepted as an extension of Mp,φ and Lp,κ(w). Eroglu [7] proved the boundedness of oscillatory integral operators, fractional oscillatory integral operators, and the corresponding commutators on Mp,φ and Shi et al. [8] proved the boundedness of these operators and commutators on Lp,κ(w). In [9], Lu et al. obtained the boundedness of sublinear operators with rough kernels on Mp,φ and in [10], Shi and Fu showed the boundedness of these operators on Lp,κ(w). The boundedness of some sublinear operators and their commutators on Lp,κ(w) was obtained by Shi et al. [11] and the boundedness of sublinear operators on Mp,φw was proved by Mustafayev [12].Vanishing Morrey spacesVMp,λ(Rn) are subspaces of functions in Morrey spaces which were introduced by Vitanza [13] satisfying the condition(1)limr→0⁡supx∈Rn0<t<r⁡t-λ/pfLpBx,t=0.The properties and applications of vanishing Morrey Spaces were given in [14]. On vanishing Morrey spaces, the boundedness of commutators of the multidimensional Hardy type operators was proved in [15]. The vanishing generalized Morrey spaces VMp,φRn were introduced and studied by Samko in [16].In this study, as distinct from [8], we focus on vanishing generalized weighted Morrey spaces and give Zygmund-type conditions to prove the boundedness of oscillatory and the fractional oscillatory integral operators and their commutators in these spaces. In Section 2 we recall some definitions and necessary preliminaries and in Section 3 we give our main results; namely, we prove our theorems on vanishing generalized weighted Morrey spaces.Throughout this paper,C,c,ci and so on are used as positive constant that can change from one line to another. A≲B means that A≤cB with some positive constant c. If A≲B and B≲A, then we say A≈B which means A and B are equivalent.


## 2. Preliminaries

Letw be a weight function on Rn, such that w(x)>0 for almost every x∈Rn. E is a measurable set with Lebesgue measure notated by E and we define wE=∫Ewxdx.For1≤p<∞ and Ω⊂Rn we denote the weighted Lebesgue space by Lp,wΩ with the norm(2)fLp,wΩ=∫Ωfxpwxdx1/p<∞.Let1≤p<∞, φx,r be a positive continuous function on Rn×0,∞ and let w be a weight function on Rn. We show the generalized weighted Morrey space by Mp,φRn;w=Mp,φw, which is space of all functions f∈Lp,wlocRn with finite quasinorm(3)fMp,φw=supx∈Rn,r>01φ1/px,rfLp,wBx,r,where Bx,r is the open ball centered at x of radius r.The fractional integral operator (Riesz potential)Iα and fractional Maximal operator Mα, which play important roles in real and harmonic analysis, are defined by(4)Iαfx=∫Rnfyx-yn-αdy,0<α<n,Mαfx=supr>0⁡1rn-α∫Bx,rfydy,0≤α<n,where f∈L1locRn. If α=0, then M≡M0 is the Hardy-Littlewood maximal operator.The class ofAp weights was introduced by Muckenhoupt in [17] to show that Hardy-Littlewood maximal function M is bounded on weighted Lebesgue spaces Lp,w(Rn) if and only if w∈Ap,1<p<∞.Now we define Muckenhoupt classAp. Let 1≤p<∞. A weight w is said to be an Ap weight, if there exists a positive constant cp such that, for every ball B⊂Rn,(5)∫Bwxdx∫Bwx1-p′dxp-1≤cpBp,when 1<p<∞, and for p=1(6)∫Bwydy≤c1wxBfor almost everywhere x∈B. The smallest cp is shown by wAp. We define A∞=⋃p≥1Ap.A weightw belongs to Ap,q for 1<p<q<∞ if there exists C>1 such that(7)∫Bwxqdx1/q∫Bwx-p′dx1/p′≤CB1+1/q-1/p,where 1/p+1/p′=1. Ap,q class was introduced by Muckenhoupt and Wheeden [18] to study weighted norm inequalities for fractional integral operators.


### 2.1. Vanishing Generalized Weighted Morrey Spaces

LetΩ be an open set in Rn and let Π be an arbitrary subset of Ω. Let also φx,r be a measurable nonnegative function on Π×0,ll=diam⁡Ω and positive for all (x,t)∈Π×(0,l). Let w be a weight function on Ω; then we notate by VMΠp,φΩ;w≔VMΠp,φw the vanishing weighted Morrey spaces which are defined as the spaces of functions f∈Lp,wlocΩ with finite quasi norm:(8)fVMΠp,φw=supx∈Π,r>0⁡1φ1/px,rfLp,wB~x,rsuch that(9)limr→0⁡supx∈Π⁡1φ1/px,rfLp,wB~x,r=0,where B~x,r=Bx,r∩Ω and 1≤p≤∞.Naturally, it is suitable to impose the functionφx,r on the following conditions:(10)limr→0⁡supx∈Π⁡wL1B~x,rφx,r=0,infr>1⁡supx∈Π⁡φx,r>0.which makes VMΠp,φw nontrivial, since bounded functions which have compact support belong to this space.Henceforth we denote byφ∈Bw if φx,r is a nonnegative measurable function on Π×0,l and positive for all (x,t)∈Π×(0,l) and satisfies conditions (10).


### 2.2. Oscillatory Integral Operators and Fractional Oscillatory Integral Operators

Oscillatory integral operators appear in many fields of mathematics and physics. Furthermore oscillatory integrals have been an essential part of harmonic analysis. Many important operators in harmonic analysis are some versions of oscillatory integrals, such as the Fourier transform, Bochner-Riesz means, and Radon transform. Properties of oscillatory integral operators have been studied by Stein in [19].A distribution kernelK is called a standard Calderón-Zygmund (in short C-Z) kernel when it satisfies the following conditions:(11)Kx,y≤Cx-yn,x≠y∇xKx,y+∇yKx,y≤Cx-yn+1,x≠y.C-Z integral operator T and the oscillatory integral operator S are defined by(12)Tfx=p.v.∫RnKx,yfydy,Sfx=p.v.∫RneiPx,yKx,yfydy,where Px,y is a real valued polynomial defined on Rn×Rn. Lu and Zhang [20] used L2-boundedness of T to get Lp-boundedness of S with 1<p<∞.Ricci and Stein [21] introduced the standard fractional C-Z kernel, Kα with 0<α<n, where(13)Kαx,y≤Cx-yn-α,∇xKαx,y+∇yKαx,y≤Cx-yn+1-α,x≠y.The fractional oscillatory integral operator is defined in [22] as(14)Sαfx=p.v.∫RneiPx,yKαx,yfydy,where Px,y is a real valued polynomial and defined on Rn×Rn and Kα is a standard fractional C-Z kernel; note that when α=0, Sα=S and Kα=K.Lemma 1 (see [23]).
IfK is a standard C-Z kernel and the C-Z singular integral operator T is of type L2Rn,L2Rn, then for any real polynomial P(x,y) and w∈Ap(1<p<∞), there exists constants C>0 independent of the coefficients of P such that SfLp,wRn≤CfLp,wRn.Lemma 2 (see [24]).
Let0<α<n,1<p<n/α,1/q=1/p-α/n, w∈Ap,q, φ∈Bwp, and ψ∈Bwq. Then the operators Mα and Iα are bounded from VMΠp,φRn;wp to VMΠq,ψRn;wq, if (15)∫δ∞supx∈Rn⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and(16)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x∈Rn and r.


### 2.3. The Commutator OperatorsSb and Sα,b

Letf∈Lloc1Rn; we define(17)f∗=supx∈Rn,r>0⁡1Bx,r∫Bx,rfy-fBx,rdy<∞,where(18)fBx,r=1Bx,r∫Bx,rfydy.In harmonic analysis a function of bounded mean oscillation, also known as a BMO function, is a real valued function whose mean oscillation is bounded. BMORn is the set of all locally integrable functions f on Rn with f∗<∞.Next we shall introduce the commutators of oscillatory integral operators and fractional oscillatory integral operators.Letb be a locally integrable function; the commutator operators Sb and Sα,b which are formed by S and b are defined by(19)Sbfx=Sfxbx-Sbfx,Sα,bfx=Sαfxbx-Sαbfx.Lemma 3 (see [25]).
Suppose thatK is a standard C-Z kernel, w∈Ap(1<p<∞), and the operator T is of type L2Rn,L2Rn. Then for any b∈BMORn, there exists constants C>0 independent on the coefficients of P such that(20)SbfLp,wRn≤Cb∗fLp,wRn.Lemma 4 (see [18]).
Letw∈A∞. Then the norm of BMO(Rn) is equivalent to the norm of BMO(w), where(21)BMOw=b:b∗,w=supx∈Rn,r>0⁡1wBx,r∫Bx,rby-bBx,r,wwydy<∞,bBx,r,w=1wBx,r∫Bx,rbywydy.Remark 5 (the John-Nirenberg inequality).
There are possible constantsC1,C2>0 such that for all b∈BMORn and β>0(22)x∈B:bx-bB>β≤C1Be-C2β/b∗,for all B⊂Rn. The John-Nirenberg inequality implies that(23)b∗≈supx∈Rn,r>0⁡1wBx,r∫Bx,rby-bBx,rpwydy1/pfor 1≤p<∞ and w∈A∞.Lemma 6 (see [26]).
(i) Letw∈A∞ and let b be a function in BMO(Rn). Let also 1≤p<∞, x∈Rn, and r1,r2>0. Then (24)1wBx,r1∫Bx,r1by-bBx,r2,wpwydy1/p≤C1+ln⁡r1r2b∗,where C>0 is independent of f, x, r1, and r2.
(ii) Letw∈Ap and let b be a function in BMO(Rn). Let also 1<p<∞, x∈Rn, and r1,r2>0. Then(25)1w1-p′Bx,r1∫Bx,r1by-bBx,r2,wp′wy1-p′dy1/p′≤C1+ln⁡r1r2b∗,where C>0 is independent of f, x, r1, and r2.Lemma 7 (see [24]).
Let1<p<∞,b∈BMORn,0<α<n/p,1/q=1/p-α/n,w∈Ap,q, φ∈B(wp), and ψ∈B(wq). Then Mα,b and Iα,b are bounded from VMΠp,φRn;wp to VMΠq,ψRn;wq if(26)sup0<r<δ∫δ∞ln⁡e+trsupx∈Rn⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and (27)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x and r.


## 2.1. Vanishing Generalized Weighted Morrey Spaces

LetΩ be an open set in Rn and let Π be an arbitrary subset of Ω. Let also φx,r be a measurable nonnegative function on Π×0,ll=diam⁡Ω and positive for all (x,t)∈Π×(0,l). Let w be a weight function on Ω; then we notate by VMΠp,φΩ;w≔VMΠp,φw the vanishing weighted Morrey spaces which are defined as the spaces of functions f∈Lp,wlocΩ with finite quasi norm:(8)fVMΠp,φw=supx∈Π,r>0⁡1φ1/px,rfLp,wB~x,rsuch that(9)limr→0⁡supx∈Π⁡1φ1/px,rfLp,wB~x,r=0,where B~x,r=Bx,r∩Ω and 1≤p≤∞.Naturally, it is suitable to impose the functionφx,r on the following conditions:(10)limr→0⁡supx∈Π⁡wL1B~x,rφx,r=0,infr>1⁡supx∈Π⁡φx,r>0.which makes VMΠp,φw nontrivial, since bounded functions which have compact support belong to this space.Henceforth we denote byφ∈Bw if φx,r is a nonnegative measurable function on Π×0,l and positive for all (x,t)∈Π×(0,l) and satisfies conditions (10).


## 2.2. Oscillatory Integral Operators and Fractional Oscillatory Integral Operators

Oscillatory integral operators appear in many fields of mathematics and physics. Furthermore oscillatory integrals have been an essential part of harmonic analysis. Many important operators in harmonic analysis are some versions of oscillatory integrals, such as the Fourier transform, Bochner-Riesz means, and Radon transform. Properties of oscillatory integral operators have been studied by Stein in [19].A distribution kernelK is called a standard Calderón-Zygmund (in short C-Z) kernel when it satisfies the following conditions:(11)Kx,y≤Cx-yn,x≠y∇xKx,y+∇yKx,y≤Cx-yn+1,x≠y.C-Z integral operator T and the oscillatory integral operator S are defined by(12)Tfx=p.v.∫RnKx,yfydy,Sfx=p.v.∫RneiPx,yKx,yfydy,where Px,y is a real valued polynomial defined on Rn×Rn. Lu and Zhang [20] used L2-boundedness of T to get Lp-boundedness of S with 1<p<∞.Ricci and Stein [21] introduced the standard fractional C-Z kernel, Kα with 0<α<n, where(13)Kαx,y≤Cx-yn-α,∇xKαx,y+∇yKαx,y≤Cx-yn+1-α,x≠y.The fractional oscillatory integral operator is defined in [22] as(14)Sαfx=p.v.∫RneiPx,yKαx,yfydy,where Px,y is a real valued polynomial and defined on Rn×Rn and Kα is a standard fractional C-Z kernel; note that when α=0, Sα=S and Kα=K.Lemma 1 (see [23]).
IfK is a standard C-Z kernel and the C-Z singular integral operator T is of type L2Rn,L2Rn, then for any real polynomial P(x,y) and w∈Ap(1<p<∞), there exists constants C>0 independent of the coefficients of P such that SfLp,wRn≤CfLp,wRn.Lemma 2 (see [24]).
Let0<α<n,1<p<n/α,1/q=1/p-α/n, w∈Ap,q, φ∈Bwp, and ψ∈Bwq. Then the operators Mα and Iα are bounded from VMΠp,φRn;wp to VMΠq,ψRn;wq, if (15)∫δ∞supx∈Rn⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and(16)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x∈Rn and r.


## 2.3. The Commutator OperatorsSb and Sα,b

Letf∈Lloc1Rn; we define(17)f∗=supx∈Rn,r>0⁡1Bx,r∫Bx,rfy-fBx,rdy<∞,where(18)fBx,r=1Bx,r∫Bx,rfydy.In harmonic analysis a function of bounded mean oscillation, also known as a BMO function, is a real valued function whose mean oscillation is bounded. BMORn is the set of all locally integrable functions f on Rn with f∗<∞.Next we shall introduce the commutators of oscillatory integral operators and fractional oscillatory integral operators.Letb be a locally integrable function; the commutator operators Sb and Sα,b which are formed by S and b are defined by(19)Sbfx=Sfxbx-Sbfx,Sα,bfx=Sαfxbx-Sαbfx.Lemma 3 (see [25]).
Suppose thatK is a standard C-Z kernel, w∈Ap(1<p<∞), and the operator T is of type L2Rn,L2Rn. Then for any b∈BMORn, there exists constants C>0 independent on the coefficients of P such that(20)SbfLp,wRn≤Cb∗fLp,wRn.Lemma 4 (see [18]).
Letw∈A∞. Then the norm of BMO(Rn) is equivalent to the norm of BMO(w), where(21)BMOw=b:b∗,w=supx∈Rn,r>0⁡1wBx,r∫Bx,rby-bBx,r,wwydy<∞,bBx,r,w=1wBx,r∫Bx,rbywydy.Remark 5 (the John-Nirenberg inequality).
There are possible constantsC1,C2>0 such that for all b∈BMORn and β>0(22)x∈B:bx-bB>β≤C1Be-C2β/b∗,for all B⊂Rn. The John-Nirenberg inequality implies that(23)b∗≈supx∈Rn,r>0⁡1wBx,r∫Bx,rby-bBx,rpwydy1/pfor 1≤p<∞ and w∈A∞.Lemma 6 (see [26]).
(i) Letw∈A∞ and let b be a function in BMO(Rn). Let also 1≤p<∞, x∈Rn, and r1,r2>0. Then (24)1wBx,r1∫Bx,r1by-bBx,r2,wpwydy1/p≤C1+ln⁡r1r2b∗,where C>0 is independent of f, x, r1, and r2.
(ii) Letw∈Ap and let b be a function in BMO(Rn). Let also 1<p<∞, x∈Rn, and r1,r2>0. Then(25)1w1-p′Bx,r1∫Bx,r1by-bBx,r2,wp′wy1-p′dy1/p′≤C1+ln⁡r1r2b∗,where C>0 is independent of f, x, r1, and r2.Lemma 7 (see [24]).
Let1<p<∞,b∈BMORn,0<α<n/p,1/q=1/p-α/n,w∈Ap,q, φ∈B(wp), and ψ∈B(wq). Then Mα,b and Iα,b are bounded from VMΠp,φRn;wp to VMΠq,ψRn;wq if(26)sup0<r<δ∫δ∞ln⁡e+trsupx∈Rn⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and (27)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x and r.


## 3. Main Results

### 3.1. Boundedness ofS and Sα in VMΠp,φ(Rn;w)

Lemma 8.
Let1<p<∞, w∈Ap and K is a standard C-Z kernel and C-Z singular operator T is of type L2Rn,L2Rn. Then for any ball B=Bx0,r in Rn and any real polynomial P(x,y) the following inequality holds:(28)SfLp,wB≤cb∗wL1B1/p∫r∞fLp,wBx0,twL1Bx0,t-1/pdtt,where the constant c>0 does not depend on B and f.Proof.
We splitf into two parts in a neighbourhood of point x0 as follows:(29)f=f1+f2,f1y=fyχ2By,f2y=fyχRn∖2By.By linearity of the operator S we have(30)SfLp,wB≤Sf1Lp,wB+Sf2Lp,wB.From the boundedness of Sf on Lp,wRn (see Lemma 1), it follows that(31)Sf1Lp,wB≤Sf1Lp,wRn≤cf1Lp,wRn=cfLp,w2B,where constant c>0 is independent of B and f.
SinceK(x,y)≤C/x-yn, x≠y, we have (32)Sf2x≤∫Rn∖2BeiPx,yKx,yf2ydy≤C∫Rnf2ydyx-ynfor x∈Bx0,r. It is obvious that x∈B and y∈Rn∖2B implies that x0-y<x-y. Thus we get(33)Sf2x≤C∫Rn∖2Bfydyx0-yn.Since the right-hand side does not depend on x we get(34)Sf2Lp,wB≲wL1B1/p∫Rn∖2Bfyx0-yndy.By Fubini’s theorem we have(35)∫Rn∖2Bfydyx0-yn≈∫Rn∖2Bfy∫x0-y∞dttn+1dy≈∫2r∫2r≤x0-y≤t∞fydydttn+1≲∫2r∞∫Bx0,tfydydttn+1.
By Hölder’s inequality, we get(36)∫Rn∖2Bfydyx0-yn≲∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≤wAp1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.Thus,(37)Sf2Lp,wB≲wL1B1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.On the other hand,(38)fLp,w2B≈rnfLp,w2B∫2r∞dttn+1≲rn∫2r∞fLp,wBx0,tdttn+1≲wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1.
By (31) and (38) we write(39)Sf1Lp,wB≲wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≈wL1B1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.
Then by (37) and (39) we get the inequality (29).Remark 9.
Note that Lemma8 was proved in [7] in the case w≡1.Theorem 10.
Let1<p<∞, w∈Ap, and φ,ψ∈Bw. The oscillatory integral operator S is bounded from VMΠp,φRn;w to VMΠp,ψRn;w for any real polynomial P(x,y) if K is a standard C-Z kernel, the C-Z singular operator T is of type L2Rn,L2Rn: (40)cδ≔∫δ∞supx∈Π⁡φ1/px,twL1Bx,t1/pdtt<∞for every δ>0, and(41)∫r∞φ1/px,twL1Bx,t1/pdtt≤c0ψ1/px,rwL1Bx,r1/p,where c0 does not depend on x∈Π and r>0.Proof.
To estimate the norm of the operator we use (28) and get(42)SfVMΠp,Ψw=supx∈Π,r>01ψ1/px,rSfLp,wBx,r≤csupx∈Π,r>01ψ1/px,rwL11/p∫r∞t-1fLp,wBx,twL1Bx,t-1/pdt.By using (41) we can write(43)SfVMΠp,φw≤csupx∈Π,r>0⁡1φ1/px,rfLp,wBx,r.Thus,(44)SfVMΠp,φw≤cfVMΠp,φw.Now we prove that it belongs to VMΠp,ψw; that is, (45)limr→0⁡supx∈Π⁡1φ1/px,rfLp,wBx,r=0⟹limr→0⁡supx∈Π⁡1ψ1/px,rSfLp,wBx,r=0.To show that supx∈Π⁡(1/ψ1/px,r)SfLp,wBx,r<ε for small r, we split the right-hand side of (28):(46)1ψ1/px,rSfLp,wBx,r≤CIδ0x,r+Jδ0x,r,where δ0>0 (we may take δ0<1) and(47)Iδ0x,r≔wL1Bx,r1/pψ1/px,r∫rδ0φ1/px,twL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdtt,Jδ0x,r≔wL1Bx,r1/pψ1/px,r∫δ0∞φx,twL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdttand it is supposed that r<δ0.
Now we fixδ0>0 such that supx∈Π⁡sup0<r<t⁡1/φ1/px,rfLp,wBx,r<ε/2Cc0, where c0 and C are constants from (41) and (46), respectively. Then we can write(48)supx∈Π⁡CIδ0x,r<ε2,0<r<δ0.By choosing r sufficiently small and considering (40) we have(49)Jδ0x,r≤cδ0wL1Bx,r1/pψ1/px,rfVMΠp,φw,where cδ0 is the constant from (40). Then by (10) we choose r small enough such that(50)supx∈Π⁡wL1Bx,rψx,r<ε2Ccδ0fVMΠp,φwpwhich completes the proof.Theorem 11.
Let1<p<∞,0<α<n/p,1/q=1/p-α/n,Px,y be real polynomial; w∈Ap,q, φ∈Bwp, and ψ∈Bwq. Then Sα is bounded from VMΠp,φwp to VMΠq,ψwq if (51)∫δ∞supx∈Π⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and(52)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C is not depending on x∈Π and r.Proof.
SinceSαfx≤Iαfx and thanks to Lemma 2 proof is completed.


### 3.2. Boundedness ofSb and Sα,b in the Spaces VMΠp,φw

Lemma 12.
Let1≤p<∞,b∈BMORn,w∈Ap, K is a standard C-Z kernel, and the C-Z integral operator T is type of L2Rn,L2Rn. Then for 1<p<∞ and polynomial Px,y the inequality(53)SbfLp,wB≤Cb∗wL1B1/p∫r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdttholds for any ball B=Bx0,r and for all f∈Lplocw.Proof.
Let1<p<∞. For any x0∈Rn, we split f into two parts in a neighbourhood of point x0 such that(54)f=f1+f2,f1y=fyχ2By,f2y=fyχRn∖2By,where r>0, and by linearity of the operator Sb we have(55)SbfLp,wB≤Sbf1Lp,wB+Sbf2Lp,wB.Moreover, from the boundedness of Sb in Lp,wRn (see Lemma 3), it follows that(56)Sbf1Lp,wB≤Sbf1Lp,wRn≤Cb∗f1Lp,wRn=Cb∗fLp,w2B,where constant C>0 is independent of f.
Forx∈Bx0,r we have(57)Sbf2x≲∫Rnby-bxx-ynf2ydy≈∫Rn∖2Bby-bxx0-ynfydy.Then(58)Sbf2Lp,wB≤C∫B∫Rn∖2Bby-bxx0-ynfydypwxdx1/p≲∫B∫Rn∖2Bby-bB,wx0-ynfydypwxdx1/p+∫B∫Rn∖2Bbx-bB,wx0-ynfydypwxdx1/p=I+II.Let us estimate I(59)I=CwL1B1/p∫Rn∖2Bby-bB,wx0-ynfydy=CwL1B1/p∫Rn∖2Bby-bB,wfy∫x0-y∞dttn+1dy=CwL1B1/p∫2r∞∫2r≤x0-y≤tby-bB,wfydydttn+1≤CwL1B1/p∫2r∞∫Bx0,tby-bB,wfydydttn+1.Using Hölder’s inequality and (25), we have(60)I≲wL1B1/p∫2r∞∫Bx0,tby-bBx0,r,wp′wy1-p′dy1/p′fLp,wBx0,tdttn+1≲wAp1/pb∗wL1B1/p∫2r∞1+ln⁡trw-1/pLp′Bx0,tfLp,wBx0,tdttn+1≲wAp1/pb∗wL1B1/p∫2r∞e+lntrwBx0,t-1/pfLp,wBx0,tdtt.In order to estimate II note that(61)II=∫Bbx-bB,wpwxdx1/p∫Rn∖2Bfyx0-yndy.By (23) we get(62)II≲b∗wL1B1/p∫Rn∖2Bfyx0-yndy.Applying Hölder’s inequality, we get(63)∫Rn∖2Bfyx0-yndy≲∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≤wAp1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Thus by (63) we write(64)II≲b∗wL1B1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Summing up I and II, for all 1<p<∞ we get(65)Sbf2Lp,wBx,r≲b∗wL1B1/p∫2r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdtt.On the other hand,(66)fLp,w2B≈BfLp,w2B∫2r∞dttn+1≤B∫2r∞fLp,wBx0,tdttn+1≤wL1B1/pw-1/pLp′B∫2r∞fLp,wBx0,tdttn+1≤wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bdttn+1≤wAp1/pwL1B1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Thus,(67)SbfLp,wB≲b∗fLp,w2B+b∗wL1B1/p∫2r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdtt,which along with (66) and (67) lead us to (53).Theorem 13.
Let1<p<∞ and b∈BMORn,w∈Ap and φ,ψ∈B(w). Then Sb is bounded from VMΠp,φRn;w to VMΠp,ψRn;w if(68)cδ≔sup0<r<δ⁡∫δ∞ln⁡e+trsupx∈Rn⁡φ1/px,twL1Bx,t1/pdtt<∞for every δ>0, and(69)∫r∞ln⁡e+trφ1/px,twL1Bx,t1/pdtt≤c0ψ1/px,rwL1Bx,r1/p,where c0 does not depend on x∈Rn and r>0.Proof.
Boundedness follows from Lemma12 and the same procedure is argued in the proof of Theorem 10. We have to prove that(70)limr→0⁡supx∈Rn⁡1φ1/px,rfLp,wBx,r=0⟹limr→0⁡supx∈Rn⁡1ψ1/px,rSbfLp,wBx,r=0.We suppose that 0<r<δ0. In view of (53) we can write(71)1ψ1/px,rSbfLp,wBx,r≤Cb∗wL1Bx,r1/pψ1/px,r∫r∞ln⁡e+trfLp,wBx,twL1Bx,t-1/pdtt.To show that supx∈Rn⁡1/ψ1/px,rSbfLp,wBx,r<ε for small r, we split the right side of (71) (72)1ψ1/px,rSbfLp,wBx,r≤CIδ0x,r+Jδ0x,r,where (73)Iδ0x,r≔b∗wL1B1/pψ1/px,r∫rδ0φ1/px,tln⁡e+t/rtwL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdt,Jδ0x,r≔b∗wL1Bx,t1/pψ1/px,r∫δ0∞φ1/px,tln⁡e+t/rtwL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdt.Now we choose any fixed δ0>0 such that(74)supx∈Rn⁡sup0<r<t1φ1/px,rfLp,wBx,r<ε2Cc0b∗,where C and c0 are constants appearing in (72) and (69), respectively. This allows to estimate the first term uniformly in r∈0,δ0:(75)supx∈Rn⁡CIδ0x,r<ε2,0<r<δ0.
The estimation of the second term can be obtained by choosingr sufficiently small. Indeed, by (10), we have(76)Jδ0x,r≤b∗cδ0fVMΠp,φwwL1B1/pψ1/px,r,where cδ0 is the constant from (68). Then by (10) it suffices to choose r small enough such that(77)supx∈Rn⁡wL1Bx,tψx,r≤ε2Ccδ0b∗fVMp,φwpwhich completes the proof.Theorem 14.
Let1<p<∞,b∈BMORn,0<α<n/p,1/q=1/p-α/n,Px,y be a real valued polynomial, φ∈B(wp) and ψ∈B(wq). Then Sα,b is bounded from VMΠp,φwp to VMΠq,ψwq if(78)sup0<r<δ⁡∫δ∞ln⁡e+trsupx∈Π⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and (79)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x and r.Proof.
SinceSα,bfx≤Iα,bfx, from the boundedness of Iα,b (see Lemma 7) proof is completed.


## 3.1. Boundedness ofS and Sα in VMΠp,φ(Rn;w)

Lemma 8.
Let1<p<∞, w∈Ap and K is a standard C-Z kernel and C-Z singular operator T is of type L2Rn,L2Rn. Then for any ball B=Bx0,r in Rn and any real polynomial P(x,y) the following inequality holds:(28)SfLp,wB≤cb∗wL1B1/p∫r∞fLp,wBx0,twL1Bx0,t-1/pdtt,where the constant c>0 does not depend on B and f.Proof.
We splitf into two parts in a neighbourhood of point x0 as follows:(29)f=f1+f2,f1y=fyχ2By,f2y=fyχRn∖2By.By linearity of the operator S we have(30)SfLp,wB≤Sf1Lp,wB+Sf2Lp,wB.From the boundedness of Sf on Lp,wRn (see Lemma 1), it follows that(31)Sf1Lp,wB≤Sf1Lp,wRn≤cf1Lp,wRn=cfLp,w2B,where constant c>0 is independent of B and f.
SinceK(x,y)≤C/x-yn, x≠y, we have (32)Sf2x≤∫Rn∖2BeiPx,yKx,yf2ydy≤C∫Rnf2ydyx-ynfor x∈Bx0,r. It is obvious that x∈B and y∈Rn∖2B implies that x0-y<x-y. Thus we get(33)Sf2x≤C∫Rn∖2Bfydyx0-yn.Since the right-hand side does not depend on x we get(34)Sf2Lp,wB≲wL1B1/p∫Rn∖2Bfyx0-yndy.By Fubini’s theorem we have(35)∫Rn∖2Bfydyx0-yn≈∫Rn∖2Bfy∫x0-y∞dttn+1dy≈∫2r∫2r≤x0-y≤t∞fydydttn+1≲∫2r∞∫Bx0,tfydydttn+1.
By Hölder’s inequality, we get(36)∫Rn∖2Bfydyx0-yn≲∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≤wAp1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.Thus,(37)Sf2Lp,wB≲wL1B1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.On the other hand,(38)fLp,w2B≈rnfLp,w2B∫2r∞dttn+1≲rn∫2r∞fLp,wBx0,tdttn+1≲wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1.
By (31) and (38) we write(39)Sf1Lp,wB≲wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≈wL1B1/p∫2r∞fLp,wBx0,twL1Bx0,t-1/pdtt.
Then by (37) and (39) we get the inequality (29).Remark 9.
Note that Lemma8 was proved in [7] in the case w≡1.Theorem 10.
Let1<p<∞, w∈Ap, and φ,ψ∈Bw. The oscillatory integral operator S is bounded from VMΠp,φRn;w to VMΠp,ψRn;w for any real polynomial P(x,y) if K is a standard C-Z kernel, the C-Z singular operator T is of type L2Rn,L2Rn: (40)cδ≔∫δ∞supx∈Π⁡φ1/px,twL1Bx,t1/pdtt<∞for every δ>0, and(41)∫r∞φ1/px,twL1Bx,t1/pdtt≤c0ψ1/px,rwL1Bx,r1/p,where c0 does not depend on x∈Π and r>0.Proof.
To estimate the norm of the operator we use (28) and get(42)SfVMΠp,Ψw=supx∈Π,r>01ψ1/px,rSfLp,wBx,r≤csupx∈Π,r>01ψ1/px,rwL11/p∫r∞t-1fLp,wBx,twL1Bx,t-1/pdt.By using (41) we can write(43)SfVMΠp,φw≤csupx∈Π,r>0⁡1φ1/px,rfLp,wBx,r.Thus,(44)SfVMΠp,φw≤cfVMΠp,φw.Now we prove that it belongs to VMΠp,ψw; that is, (45)limr→0⁡supx∈Π⁡1φ1/px,rfLp,wBx,r=0⟹limr→0⁡supx∈Π⁡1ψ1/px,rSfLp,wBx,r=0.To show that supx∈Π⁡(1/ψ1/px,r)SfLp,wBx,r<ε for small r, we split the right-hand side of (28):(46)1ψ1/px,rSfLp,wBx,r≤CIδ0x,r+Jδ0x,r,where δ0>0 (we may take δ0<1) and(47)Iδ0x,r≔wL1Bx,r1/pψ1/px,r∫rδ0φ1/px,twL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdtt,Jδ0x,r≔wL1Bx,r1/pψ1/px,r∫δ0∞φx,twL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdttand it is supposed that r<δ0.
Now we fixδ0>0 such that supx∈Π⁡sup0<r<t⁡1/φ1/px,rfLp,wBx,r<ε/2Cc0, where c0 and C are constants from (41) and (46), respectively. Then we can write(48)supx∈Π⁡CIδ0x,r<ε2,0<r<δ0.By choosing r sufficiently small and considering (40) we have(49)Jδ0x,r≤cδ0wL1Bx,r1/pψ1/px,rfVMΠp,φw,where cδ0 is the constant from (40). Then by (10) we choose r small enough such that(50)supx∈Π⁡wL1Bx,rψx,r<ε2Ccδ0fVMΠp,φwpwhich completes the proof.Theorem 11.
Let1<p<∞,0<α<n/p,1/q=1/p-α/n,Px,y be real polynomial; w∈Ap,q, φ∈Bwp, and ψ∈Bwq. Then Sα is bounded from VMΠp,φwp to VMΠq,ψwq if (51)∫δ∞supx∈Π⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and(52)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C is not depending on x∈Π and r.Proof.
SinceSαfx≤Iαfx and thanks to Lemma 2 proof is completed.


## 3.2. Boundedness ofSb and Sα,b in the Spaces VMΠp,φw

Lemma 12.
Let1≤p<∞,b∈BMORn,w∈Ap, K is a standard C-Z kernel, and the C-Z integral operator T is type of L2Rn,L2Rn. Then for 1<p<∞ and polynomial Px,y the inequality(53)SbfLp,wB≤Cb∗wL1B1/p∫r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdttholds for any ball B=Bx0,r and for all f∈Lplocw.Proof.
Let1<p<∞. For any x0∈Rn, we split f into two parts in a neighbourhood of point x0 such that(54)f=f1+f2,f1y=fyχ2By,f2y=fyχRn∖2By,where r>0, and by linearity of the operator Sb we have(55)SbfLp,wB≤Sbf1Lp,wB+Sbf2Lp,wB.Moreover, from the boundedness of Sb in Lp,wRn (see Lemma 3), it follows that(56)Sbf1Lp,wB≤Sbf1Lp,wRn≤Cb∗f1Lp,wRn=Cb∗fLp,w2B,where constant C>0 is independent of f.
Forx∈Bx0,r we have(57)Sbf2x≲∫Rnby-bxx-ynf2ydy≈∫Rn∖2Bby-bxx0-ynfydy.Then(58)Sbf2Lp,wB≤C∫B∫Rn∖2Bby-bxx0-ynfydypwxdx1/p≲∫B∫Rn∖2Bby-bB,wx0-ynfydypwxdx1/p+∫B∫Rn∖2Bbx-bB,wx0-ynfydypwxdx1/p=I+II.Let us estimate I(59)I=CwL1B1/p∫Rn∖2Bby-bB,wx0-ynfydy=CwL1B1/p∫Rn∖2Bby-bB,wfy∫x0-y∞dttn+1dy=CwL1B1/p∫2r∞∫2r≤x0-y≤tby-bB,wfydydttn+1≤CwL1B1/p∫2r∞∫Bx0,tby-bB,wfydydttn+1.Using Hölder’s inequality and (25), we have(60)I≲wL1B1/p∫2r∞∫Bx0,tby-bBx0,r,wp′wy1-p′dy1/p′fLp,wBx0,tdttn+1≲wAp1/pb∗wL1B1/p∫2r∞1+ln⁡trw-1/pLp′Bx0,tfLp,wBx0,tdttn+1≲wAp1/pb∗wL1B1/p∫2r∞e+lntrwBx0,t-1/pfLp,wBx0,tdtt.In order to estimate II note that(61)II=∫Bbx-bB,wpwxdx1/p∫Rn∖2Bfyx0-yndy.By (23) we get(62)II≲b∗wL1B1/p∫Rn∖2Bfyx0-yndy.Applying Hölder’s inequality, we get(63)∫Rn∖2Bfyx0-yndy≲∫2r∞fLp,wBx0,tw-1/pLp′Bx0,tdttn+1≤wAp1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Thus by (63) we write(64)II≲b∗wL1B1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Summing up I and II, for all 1<p<∞ we get(65)Sbf2Lp,wBx,r≲b∗wL1B1/p∫2r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdtt.On the other hand,(66)fLp,w2B≈BfLp,w2B∫2r∞dttn+1≤B∫2r∞fLp,wBx0,tdttn+1≤wL1B1/pw-1/pLp′B∫2r∞fLp,wBx0,tdttn+1≤wL1B1/p∫2r∞fLp,wBx0,tw-1/pLp′Bdttn+1≤wAp1/pwL1B1/p∫2r∞fLp,wBx0,twBx0,t-1/pdtt.Thus,(67)SbfLp,wB≲b∗fLp,w2B+b∗wL1B1/p∫2r∞ln⁡e+trfLp,wBx0,twBx0,t-1/pdtt,which along with (66) and (67) lead us to (53).Theorem 13.
Let1<p<∞ and b∈BMORn,w∈Ap and φ,ψ∈B(w). Then Sb is bounded from VMΠp,φRn;w to VMΠp,ψRn;w if(68)cδ≔sup0<r<δ⁡∫δ∞ln⁡e+trsupx∈Rn⁡φ1/px,twL1Bx,t1/pdtt<∞for every δ>0, and(69)∫r∞ln⁡e+trφ1/px,twL1Bx,t1/pdtt≤c0ψ1/px,rwL1Bx,r1/p,where c0 does not depend on x∈Rn and r>0.Proof.
Boundedness follows from Lemma12 and the same procedure is argued in the proof of Theorem 10. We have to prove that(70)limr→0⁡supx∈Rn⁡1φ1/px,rfLp,wBx,r=0⟹limr→0⁡supx∈Rn⁡1ψ1/px,rSbfLp,wBx,r=0.We suppose that 0<r<δ0. In view of (53) we can write(71)1ψ1/px,rSbfLp,wBx,r≤Cb∗wL1Bx,r1/pψ1/px,r∫r∞ln⁡e+trfLp,wBx,twL1Bx,t-1/pdtt.To show that supx∈Rn⁡1/ψ1/px,rSbfLp,wBx,r<ε for small r, we split the right side of (71) (72)1ψ1/px,rSbfLp,wBx,r≤CIδ0x,r+Jδ0x,r,where (73)Iδ0x,r≔b∗wL1B1/pψ1/px,r∫rδ0φ1/px,tln⁡e+t/rtwL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdt,Jδ0x,r≔b∗wL1Bx,t1/pψ1/px,r∫δ0∞φ1/px,tln⁡e+t/rtwL1Bx,t1/psup0<r<t⁡1φ1/px,rfLp,wBx,rdt.Now we choose any fixed δ0>0 such that(74)supx∈Rn⁡sup0<r<t1φ1/px,rfLp,wBx,r<ε2Cc0b∗,where C and c0 are constants appearing in (72) and (69), respectively. This allows to estimate the first term uniformly in r∈0,δ0:(75)supx∈Rn⁡CIδ0x,r<ε2,0<r<δ0.
The estimation of the second term can be obtained by choosingr sufficiently small. Indeed, by (10), we have(76)Jδ0x,r≤b∗cδ0fVMΠp,φwwL1B1/pψ1/px,r,where cδ0 is the constant from (68). Then by (10) it suffices to choose r small enough such that(77)supx∈Rn⁡wL1Bx,tψx,r≤ε2Ccδ0b∗fVMp,φwpwhich completes the proof.Theorem 14.
Let1<p<∞,b∈BMORn,0<α<n/p,1/q=1/p-α/n,Px,y be a real valued polynomial, φ∈B(wp) and ψ∈B(wq). Then Sα,b is bounded from VMΠp,φwp to VMΠq,ψwq if(78)sup0<r<δ⁡∫δ∞ln⁡e+trsupx∈Π⁡φ1/px,twLqBx,tdtt<∞for every δ>0, and (79)∫r∞φ1/px,twLqBx,tdtt≤Cψ1/qx,rwLqBx,r,where C does not depend on x and r.Proof.
SinceSα,bfx≤Iα,bfx, from the boundedness of Iα,b (see Lemma 7) proof is completed.



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