# Boundedness of Sublinear Operators with Rough Kernels on Weighted Morrey Spaces

Authors: Shaoguang Shi; Zunwei Fu
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The aim of this paper is to get the boundedness of a class of sublinear operators with rough kernels on weighted Morrey spaces under generic size conditions, which are satisfied by most of the operators in classical harmonic analysis. Applications to the corresponding commutators formed by certain operators and
BMO functions are also obtained.


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## Body

## 1. Introduction and Main Results

Given a functionΩ over the unit sphere Sn-1 of ℝn(n≥2) equipped with the normalized Lebesgue measure dσ and x′=x/|x|, a Calderón-Zygmund singular integral operator with rough kernel was given by
(1)TΩf(x)=p.v.∫ℝn‍Ω(x-y)|x-y|nf(y)dy
and a related maximal operator
(2)MΩf(x)=supr>01rn∫B(x,r)‍Ω(x-y)f(y)dy,
where Ω is homogeneous of degree zero and satisfies
(3)Ω∈Lr(Sn-1),1<r≤∞(4)∫Sn-1‍Ω(x′)dx′=0.
When Ω is a smooth kernel and TΩ a standard Calderón-Zygmund singular integral operator which has been fully studied by many papers, a classical survey work; see, for example, [1].For simplicity of notation,Ω is always homogeneous of degree zero and satisfies (3) and (4) throughout this paper if there are no special instructions. Here and in what follows, for x0∈ℝn, r>0, and λ>0, B=B(x0,r) denotes the ball centered at x0 with radius r and λB=B(x0,λr). When Ω satisfies some size conditions, the kernel of the operator TΩ has no regularity, and so the operator TΩ is called rough singular integral operator. In recent years, a variety of operators related to the singular integrals for Calderón-Zygmund, but lacking the smoothness required in the classical theory, have been studied. Duoandikoetxea [2] studied the norm inequalities for TΩ in homogeneous case on weighted Lp(1<p<∞) spaces. For more corresponding works, we refer the reader to [3–8] and the references therein.In [9], Hu et al. considered some more general sublinear operators with rough kernels which satisfy
(5)|𝒯Ωf(x)|≤C∫ℝn‍|Ω(x-y)f(y)||x-y|ndy,x∉suppf
for f∈L1(ℝn) with compact support. Condition (5) was first introduced by Soria and Weiss [10]. Inequality (5) is satisfied by many operators with rough kernels in classical harmonic analysis, such as TΩ (see [11]) and the oscillatory singular integral operator
(6)T¯Ωf(x)=p.v.∫ℝn‍eiP(x,y)Ω(x-y)|x-y|nf(y)dy,x∉suppf,
where the phase is a polynomial. The boundedness of T¯Ω on weighted Lp(ℝn)(1≤p<∞) spaces was fully studied by Ojanen in his doctoral dissertation [12].LetDk={x∈ℝn:|x|≤2k} and let Ak=Dk∖Dk-1 for k∈Z. Throughout this paper, we will denote by χE the characteristic function of the set E. Inspired by the works of [6, 13], in this paper, we consider some sublinear operators under some size conditions (the following (7) and (8)) which are more general than (5):
(7)|𝒯Ωf(x)|≤C|x|-n∫ℝn‍|Ω(x-y)f(y)|dy,
when suppf⊆Ak and |x|≥2k+1 with k∈ℤ and
(8)|𝒯Ωf(x)|≤C2-kn∫ℝn‍|Ω(x-y)f(y)|dy,
when suppf⊆Ak and |x|≤2k-1 with k∈ℤ, respectively. It is worth pointing out that MΩ satisfies conditions (7) and (8). Also, condition (5) implies the size conditions (7) and (8) since |x-y|>|x|/2 when |x|≥2k+1 and suppf⊆Ak while suppf⊆Ak and |x|≤2k-1 imply |x-y|>|y|/2.The topic of this paper is intended as an attempt to study the boundedness of sublinear operators with rough kernels which satisfy (7) and (8) on weighted Morrey spaces. We first recall some definitions and notations for weighted spaces. The Muckenhoupt classes Ap and A(p,q) [14] contain the functions w which satisfy
(9)Ap:supB(1|B|∫B‍w(x)dx)×(1|B|∫B‍w(x)1-p′dx)p-1≤C,1<p<∞,A(p,q):supB(1|B|∫B‍w(x)qdx)1/q×(1|B|∫B‍w(x)-p′dx)1/p′≤C,1<p,q<∞,
respectively, where 1/p+1/p′=1. For p=1, the A1 and A(1,q)(1<q<∞) weights are defined by
(10)Mw(x)≤Cw(x),A(1,q):supB(1|B|∫B‍w(x)qdx)1/q×(esssupB1w(x))≤C,
respectively. Here esssup and the following essinf are the abbreviations of essential supremum and essential infimum, respectively. Clearly, w∈A1 if and only if there is a constant C>0 such that
(11)1|B|∫B‍w(x)dx≤CessinfBw(x).In [15], Komori and Shirai introduced a weighted Morrey space, which is a natural generalization of weighted Lebesgue space, and investigated the boundedness of classical operators in harmonic analysis. Let 1≤p<∞,  0<λ<1 and let w be a weight function. Then the weighted Morrey space Mp,λ(w) is defined by
(12)Mp,λ(w)=(1w(B)λ∫B‍|f(x)|pw(x)dx)1p{(1w(B)λ∫B‍|f(x)|pw(x)dx)1pf:∥f∥Mp,λ(w)=supB(1w(B)λ∫B‍|f(x)|pw(x)dx)1/p<∞(1w(B)λ∫B‍|f(x)|pw(x)dx)1p},
where w(B)=∫B‍w(x)dx. For w∈Ap(1≤p<∞), if λ=0, then Mp,0(w)=Lp(w) while λ=1 implies Mp,1(w)=L∞(w).Now, we formulate our major results of this paper as follows.Theorem 1.
Let0<λ<1, 1<r≤∞, and r′≤p<∞ and let a sublinear operator 𝒯Ω satisfy (7) and (8). If 𝒯Ω is bounded on Lp(w) with w∈Ap/r′, then 𝒯Ω is bounded on Mp,λ(w).Whenp=1, we have the following theorem.Theorem 2.
Let1<r<∞and1/r+λ<1 and let 𝒯Ω satisfy (7) and (8). Then if 𝒯Ω is bounded from L1(w) to L1,∞(w) with w∈A1, there exists a constant C>0 such that for all μ>0 and all balls B,
(13)w({x∈B:𝒯Ωf(x)>μ})≤Cμ-1∥f∥M1,λ(w)w(B)λ.In the fractional case, we need to consider a weighted Morrey space with two weights which is also introduced by Komori and Shirai in [15]. Let 1≤p<∞,  0<λ<1. For two weights w1 and w2,
(14)Mp,λ(w1,w2)=(1w2(B)λ∫B‍|f(x)|pw1(x)dx){f:∥f∥Mp,k(w1,w2)=supB(1w2(B)λ∫B‍|f(x)|pw1(x)dx)1/p<∞∥f∥Mp,k(w1,w2)}.
If w1=w2=w, we write Mp,λ(w1,w1)=Mp,λ(w2,w2)=Mp,λ(w).We can get similar results for fractional integrals following the line of Theorems1 and 2.Theorem 3.
Let0<α<n, 1≤r<∞, and 0<λ<1. Suppose that a sublinear operator 𝒯α,Ω satisfies the size conditions
(15)|𝒯α,Ωf(x)|≤C|x|-(n-α)∫ℝn‍|Ω(x-y)f(y)|dy
when suppf⊆Ak and |x|≥2k+1 with k∈Z and
(16)|𝒯α,Ωf(x)|≤C2-k(n-α)∫ℝn‍|Ω(x-y)f(y)|dy
when suppf⊆Ak and |x|≤2k-1 with k∈Z. Then one has the following.(a)
If𝒯α,Ω maps Lp(wp) into Lq(wq) with w∈A(p/r′,q), then 𝒯α,Ω is bounded from Mp,λ(wp,wq) to Mq,qλ/p(wq), where r′≤p<n/α,1/q=1/p-α/n and p≤q≤∞.(b)
If𝒯α,Ω is bounded from L1(w) to Lq,∞(wq) with w∈A(1,q) and 1/r+λ<1, then there exists a constant C>0 such that for all μ>0 and all balls B,
(17)w({x∈B:𝒯α,Ωf(x)>μ})1/q≤Cμ-1∥f∥M1,λ(w,wq)w(B)λ,where 1<q<∞.We emphasize that (15) and (16) are weaker conditions than the following condition:
(18)|𝒯α,Ωf(x)|≤C∫ℝn‍|Ω(x-y)f(y)||x-y|n-αdy,0<α<n
for any integral function f with compact support. Condition (18) is satisfied by most fractional integral operators with rough kernels, such as the fractional integral operators of Muckenhoupt and Wheeden [16]:
(19)𝒯α,Ωf(x)=∫ℝn‍Ω(x-y)f(y)|x-y|n-αdy,0<α<n.
For some mapping properties of 𝒯α,Ω on various kinds of function spaces, see [17–19] and the references therein.We end this section with the outline of this paper. Section2 contains the proofs of Theorems 1 and 3; this part is partly motivated by the methods in [20] dealing with the case of the Lebesgue measure. In Section 3, we extend the corresponding results to commutators of certain sublinear operators.


## 2. Boundedness of Sublinear Operators

Proofs of Theorems1 and 3 depend heavily on some properties of Ap weights, which can be found in any papers or any books dealing with weighted boundedness for operators in harmonic analysis, such as [1]. For the convenience of the reader we collect some relevant properties of Ap weights without proofs, thus making our exposition self-contained.Lemma 4.
Let1≤p<∞ and w∈Ap. Then the following statements are true.(a)
There exists a constantC such that(20)w(2B)≤Cw(B),wherew satisfies this condition; one says w satisfies the doubling condition.(b)
There exists a constantC>1 such that(21)w(2B)≥Cw(B),wherew satisfies this condition; one says w satisfies the reverse doubling condition.(c)
There exist two constantsC and r>1 such that the following reverse Hölder inequality holds for every ball B⊂ℝn:(22)(1|B|∫B‍w(x)rdx)1/r≤C(1|B|∫B‍w(x)dx).(d)
For allλ>1, one has(23)w(λB)≤Cλnpw(B).(e)
There exist two constantsC and δ>0 such that for any measurable set Q⊂B(24)w(Q)w(B)≤C(|Q||B|)δ,ifw satisfies (24); one says w∈A∞.(f)
For allp<q<∞, one has(25)A∞=∪pAp,Ap⊂Aq.The following lemma about the rough kernelΩ is essential to our proofs. One can find its proof in [21].Lemma 5.
LetΩ∈Lr(Sn-1) with 1≤r<∞. Then the following statements are true.(a)
Ifx∈Ak and j≥k+1, then ∫Aj‍|Ω(x-y)|rdy≤C2jn.(b)
Ify∈Ak and k≥j+1, then ∫Aj‍|Ω(x-y)|rdx≤C2k(n-1)+j.Proof of Theorem1.
Let1<r′≤p<∞, w∈Ap/r′, and 0<λ<1. Our task is to show
(26)1w(B)λ∫B‍|𝒯Ωf(x)|pw(x)dx≤C∥f∥Mp,λ(w)p.
For a fixed ball B=B(x0,r), there is no loss of generality in assuming r=1. We decompose f=fχ2B+fχ(2B)c:=f1+f2. Since 𝒯Ω is a sublinear operator, so we get
(27)1w(B)λ∫B‍|𝒯Ωf(x)|pw(x)dx≤Cw(B)λ∫B‍|𝒯Ωf1(x)|pw(x)dx+Cw(B)λ∫B‍|𝒯Ωf2(x)|pw(x)dx:=I+II.
By the assumption on𝒯Ω and (25), we can obtain
(28)I≤Cw(B)λ∫ℝn‍|𝒯Ωf1(x)|pw(x)dx≤Cw(B)λ∫2B‍|f(x)|pw(x)dx≤C∥f∥Mp,λ(w)p.
For the termII, by (8) we have
(29)II≤Cw(B)λ∫B‍|∑k=1∞‍2-kn𝒯Ω,kf(x)|pw(x)dx,
where
(30)𝒯Ω,kf(x)=∫Ak+1‍|Ω(x-y)f(y)|dy.
We distinguish two cases according to the size ofp and r to get the estimates for 𝒯Ω,k.
Case1   (p>r′). In this case, w∈Ap/r′ implies that
(31)∫B‍w-r′/(p-r′)dy≤|B|p/(p-r′)w(B)r′/(p-r′).
By (31), Hölder’s inequality, and Lemma 5, we have
(32)𝒯Ω,kf(x)≤C(∫Ak+1‍|Ω(x-y)|rdy)1/r×(∫2k+1B‍|f(y)|r′dy)1/r′≤C2(k+1)n/r×(∫2k+1B‍|f(y)|r′w(y)r′/pw(y)-r′/pdy)1/r′≤C2(k+1)n/r(∫2k+1B‍|f(y)|pw(y)dy)1/p×(∫2k+1B‍w(y)-r′/(p-r′)dy)(p-r′)/(pr′)≤C∥f∥Mp,λ(w)2(k+1)n/r|2k+1B|1/r′w(2k+1B)(1-λ)/p≤C∥f∥Mp,λ(w)2(k+1)nw(2k+1B)(1-λ)/p.
Case2  (p=r′). In this case, w∈A1 implies that
(33)(essinfx∈2k+1Bw(x))-1≤|2k+1B|w(2k+1B),
which in combination with the Hölder inequality and Lemma 5 yields that
(34)𝒯Ω,kf(x)≤C2(k+1)n/r×(∫2k+1B‍|f(y)|pw(y)w(y)-1dy)1/p≤C∥f∥Mp,λ(w)2(k+1)nw(2k+1B)(1-λ)/p.
Substituting (32) and (34) into (29), we can assert that
(35)II≤C∥f∥Mp,λ(w)p(∑k=1∞‍w(B)(1-λ)/pw(2k+1B)(1-λ)/p)p≤C∥f∥Mp,λ(w)p,
where we have used (21) in the last inequality. Combining (28) and (29), we obtain the proof of Theorem 1.Proof of Theorem2.
The task is now to show the following inequality:(36)supμ>0μw(B)λw({x∈B:|𝒯Ωf(x)|>μ})≤C∥f∥Mp,λ(w)p.
In order to get this inequality, it will be necessary to decompose f=fχ2B+fχ(2B)c:=f1+f2 with B as in Theorem 1. Since 𝒯Ω is a sublinear operator, we can rewrite
(37)w({x∈B:|𝒯Ωf(x)|>μ})≤w({x∈B:|𝒯Ωf1(x)|>μ2})+w({x∈B:|𝒯Ωf2(x)|>μ2}):=J+JJ.
An application of (20) and the weighted weak type estimates for 𝒯Ω yield that
(38)J≤Cμ-1∥f∥M1,λ(w)w(B)μ.
To estimate the termJJ, we note that
(39)JJ≤Cμ∫{x∈B:|𝒯Ωf(x)|>μ/2}‍×|∑k=1∞‍2-kn𝒯Ω,kf(x)|w(x)dx.
By (22), (33), and the Hölder inequality, we can estimate JJ as
(40)JJ≤Cμ∑k=1∞‍2-kn×∫2k+1B‍∫B‍|Ω(x-y)|w(x)dx|f(y)|dy≤Cw(B)μ∑k=1∞‍2-kn+k(n-1)/r×∫2k+1B‍|f(y)|w(y)w(y)-1dy≤Cw(B)μ∥f∥M1,λ(w)×∑k=1∞‍w(2k+1B)λ(essinfx∈2k+1Bw)-1≤Cμ∥f∥M1,λ(w)×∑k=1∞‍2kn/r-kn(1-λ)w(B)λ≤Cμ∥f∥M1,λ(w)w(B)λ.
Combining these inequalities for J and JJ, we have completed the proof of Theorem 2.Proof of Theorem3.
We can use the similar arguments as in the proof of Theorem1 and Theorem 2. For the proof of (a), it suffices to show that
(41)1wq(B)qλ/p∫B‍|𝒯α,Ωf(x)|qw(x)qdx≤C∥f∥Mp,λ(wp,wq)q.
For a fixed ballB=B(x0,1), we decompose f=fχ2B+fχ(2B)c:=f1+f2. Since 𝒯α,Ω is a sublinear operator, we get
(42)1wq(B)qλ/p∫B‍|𝒯α,Ωf(x)|qw(x)qdx≤1wq(B)qλ/p∫B‍(|𝒯α,Ωf1(x)|q+|𝒯α,Ωf2(x)|q)wq(x)dx:=K+KK.
To estimate the termK, using the fact that 𝒯α,Ω is bounded from Lp(wp) to Lq(wq) with w∈A(p/r′,q), we can get
(43)∫B‍|𝒯α,Ωf1(x)|qwq(x)dx≤C∥f∥Mp,λ(wp,wq)qwq(B)qλ/p,
which implies that
(44)K≤C∥f∥Mp,λ(wp,wq).
For the termKK, by the similar arguments as that of Theorem 1, we obtain
(45)KK≤C∑k‍(2-k(n-α)∫Ak‍|Ω(x-y)||f(y)|dy)q×wq(B)1-qλ/p≤C∑k[2-k(n-α)+nk/r(|f(y)|r′dy)1/r′]q‍×wq(B)1-qλ/p≤C∥f∥Mp,λ(wp,wq)×(∑k=1∞‍wq(B)(1/q-λ/p)wq(2k+1B)(1/q-λ/p))q≤C∥f∥Mp,λ(wp,wq)q.
We have completed the proof of (a).
We will omit the proof of(b) since we can prove it by using A(1,q) condition and the weak type estimates of 𝒯α,Ω similar to the proof of Theorem 2.


## 3. Boundedness of Commutators

We say thatb is a BMO(ℝn) function if the following sharp maximal function is finite:
(46)b#(x)=supB1|B|∫B‍|b(y)-bB|dy,
where the supreme is taken over all balls B⊂ℝn and fB=(1/|B|)∫B‍f(y)dy. This means ∥b∥BMO(ℝn)=∥b#∥L∞<+∞. An early work about BMO(ℝn) space can be attributed to John and Nirenberg [22]. For 1<p<∞, there is a close relation between BMO(ℝn) and Ap weights:
(47)BMO(ℝn)={αlogw:w∈Ap,α≥0}.Given an operatorT acting on a generic function f and a function b, the commutator Tb is formally defined as
(48)Tbf=[b,T]f=bT(f)-T(bf).SinceL∞(ℝn)⊊BMO(ℝn), the boundedness of Tb is worse than T (e.g., the singularity; see also [23]). Therefore, many authors want to know whether Tb shares the similar boundedness with T. There are a lot of articles that deal with the topic of commutators of different operators with BMO functions on Lebesgue spaces. The first results for this commutator were obtained by Coifman et al. [24] in their study of certain factorization theorems for generalized Hardy spaces. In the present section, we will extend the boundedness of 𝒯Ω and 𝒯α,Ω to 𝒯Ω,b and 𝒯α,Ω,b, respectively.Theorem 6.
Letr, p, λ, and w be as in Theorem 1. Suppose that the sublinear operator 𝒯Ω satisfies condition (5) for any integral function f with compact support. If 𝒯Ω,b is bounded on Lp(w) with b∈BMO(ℝn), then 𝒯Ω,b is bounded on Mp,λ(w).Theorem 7.
Letp, r, q, α, w, andλ be as in Theorem 3(a) and let the sublinear operator 𝒯α,Ω satisfy condition (18) for any integral function f with compact support. If 𝒯α,Ω,b maps Lp(wp) into Lq(wq)with b∈BMO(ℝn), then 𝒯α,Ω,b is bounded from Mp,λ(wp,wq) to Mq,qλ/p(wq).The following lemmas aboutBMO(ℝn) functions will help us to prove Theorems 6 and 7.Lemma 8 (see [25, Theorem 3.8]).
Let1≤p<∞ and b∈BMO(ℝn). Then for any ball B⊂ℝn, the following statements are true.(a)
There exist constantsC1 and C2 such that for all α>0(49)|{x∈B:|b(x)-bB|>α}|≤C1|B|e-C2α/∥b∥BMO(ℝn).(b)
Inequality (49) is called John-Nirenberg inequality:
(50)|b2λB-bB|≤2nλ∥b∥BMO(ℝn).Lemma 9 ([1, Proposition 7.1.2] (see also [14, Theorem 5])).
Letw∈A∞ and 1<p<∞. Then the following statements are equivalent:(a)
∥b∥BMO(ℝn)~supB((1/|B|)∫B‍|b(x)-bB|pdx)1/p,(b)
∥b∥BMO(ℝn)~supBinfa∈ℝ(1/|B|)∫B‍|b(x)-a|dx,(c)
∥b∥BMO(w)=supB(1/w(B))∫B‍|b(x)-bB,w|w(x)dx, where BMO(w)={b:∥b∥BMO(w)<∞} and bB,w=(1/w(B))∫B‍b(y)w(y)dy.Lemma 10.
Letp, r, b, andw be as in Theorem 6 and let B=B(x0,1) be a generic fixed ball. Then the inequality
(51)(∫|x0-y|>2‍|Ω(x0-y)f(y)||x0-y|n|bB,w-b(y)|dy)p≤C∥f∥Mp,λ(w)pw(B)λ-1
holds for every y∈(2B)c, where (2B)c=ℝn∖(2B).Proof.
We will consider two cases.
Case
1  (P>r′). In this case, w∈Ap/r′. Using Hölder’s inequality and Lemma 5 to the left-hand side of (51), we have
(52)∫|x0-y|>2‍|Ω(x0-y)f(y)||x0-y|n|bB,w-b(y)|dy≤∑j=1∞‍∫2j<|x0-y|<2j+1‍|Ω(x0-y)f(y)||x0-y|n×|bB,w-b(y)|dy≤∑j=1∞‍1|2jB|∫Aj‍|Ω(x0-y)f(y)||bB,w-b(y)|dy≤C∥f∥Mp,λ(w)∑j=1∞‍2nj/rw(2j+1B)λ/p|2jB|×(∫2j+1B‍|bB,w-b(y)|pr′/(p-r′)×w(y)-r′/(p-r′)dy(∫2j+1B‍|bB,w-b(y)|pr′/(p-r′)|bB,w-b(y)|pr′/(p-r′))(p-r′)/pr′.
Set(53)A=(∫2j+1B‍|bB,w-b(y)|p-′r′w1-p-′dy)1/p-′r′,
where p-=p/r′>1. Thus
(54)A≤(∫2j+1B‍(|b2j+1B,w1-p-′-b(y)|+|b2j+1B,w1-p-′-bB,w|∫2j+1B‍(|b2j+1B,w1-p-′-b(y)|)p-′r′×w(y)1-p-′dy)1/p-′r′≤(∫2j+1B‍|b2j+1B,w1-p-′-b(y)|p-′r′×w(y)1-p-′dy∫2j+1B‍)1/p-′r′+|b2j+1B,w1-p-′-bB,w|×w1-p-′(2j+1B)1/p-′r′=:A1+A2.
Lemma9 implies that
(55)A1≤Cw1-p-′(2j+1B)1/p-′r′.
We are now in a position to deal withA2; by (50), we have
(56)|b2j+1B,w1-p-′-bB,w|≤|b2j+1B,w1-p-′-b2j+1B|+|b2j+1B-bB|+|bB-bB,w|≤1w1-p-′(2j+1B)×∫2j+1B‍|b(y)-b2j+1B|w(y)1-p-′dy+2n(j+1)∥b∥BMO(ℝn)+1w(B)∫B‍|b(y)-bB|w(y)dy:=A21+A22+A23.
Combining (23) with (49), we have
(57)A23=1w(B)×∫0∞‍w({x∈B:|b(y)-bB|>α})dα≤C∫0∞‍e-C2αδ/∥b∥BMO(ℝn)dα≤C.
In the same manner we can see that(58)A21≤C.
It follows immediately that
(59)A2≤C(2n(j+1)+2)w1-p-′(2j+1B)1/p-′r′.
Therefore
(60)A≤C(j+1)w1-p-′(2j+1B)1/p-′r′.
A further use of (21) and w∈p/r′ allow us to obtain
(61)∑j=1∞‍2nj/rw(2j+1B)λ/p|2jB|×(∫2j+1B‍|b(y)-bB,w|p-′r′w(y)1-p-′dy)1/p-′r′≤∑j=1∞‍2nj/rw(2j+1B)λ/p|2jB|×(j+1)w(y)1-p-′(2j+1B)1/p-′r′≤C∑j=1∞‍2nj/r|2j+1B|1/r′(j+1)|2jB|×w(B)(1-λ)/pw(2j+1B)(1-λ)/pw(B)(λ-1)/p≤C∑j=1∞‍j+1D(j+1)(1-λ)/pw(B)(λ-1)/p≤Cw(B)(λ-1)/p,
where D>1 is a constant that appeared in (21).
Case2  (P=r′). In this case, w∈A1. We can prove (51) by a similar analysis as in the proof of Theorem 1 (in the case P=r′) and Case 1.Having disposed of the previous preliminary step, we can now return to the proofs of Theorems6 and 7.Proof of Theorem6.
The task is now to find a constantC such that for fixed ball B=B(x0,1), we can obtain
(62)1w(B)λ∫B‍|𝒯Ω,bf(x)|pw(x)dx≤C∥f∥Mp,λ(w)p.
We decomposef=fχ2B+fχ(2B)c:=f1+f2 and consider the corresponding splitting
(63)∫B‍|𝒯Ω,bf(x)|pw(x)dx≤C(∫B‍|𝒯Ω,bf1(x)|pw(x)dx+∫B‍|𝒯Ω,bf2(x)|pw(x)dx)=:L+LL.
It follows from theLp(w) boundedness of 𝒯Ω,b and w∈Ap/r′⊂Ap that
(64)L≤C∫2B‍|f(x)|pw(x)dx≤C∥f∥Mp,λ(w)pw(B)λ.
Then a further use of (5) derives that
(65)|𝒯Ω,bf2(x)|p≤C(∫ℝn‍|Ω(x-y)f2(y)||b(x)-b(y)||x-y|ndy)p≤C(∫|x0-y|>2‍|Ω(x0-y)f(y)||x0-y|n×{|b(x)-bB,w|+|bB,w-b(y)|}dy|Ω(x0-y)f(y)||x0-y|n)p.
Therefore
(66)LL≤C(∫|x0-y|>2‍|Ω(x0-y)f(y)||x0-y|ndy)p×∫B‍|b(x)-bB,w|pw(x)dx+C(∫|x0-y|>2‍|Ω(x0-y)f(y)||x0-y|n×|b(y)-bB,w|dy|Ω(x0-y)f(y)||x0-y|n)pw(B):=LL1+LL2.
By Lemma10, we have
(67)LL2≤C∥f∥Mp,λ(w)pw(B)λ.
We proceed to estimateLL1. Without loss of generality, we assume that p>r′. Taking into account (20), (22), and Lemma 9, we have
(68)LL1=(∑j=1∞‍∫2j<|x0-y|<2j+1‍|Ω(x0-y)f(y)||x0-y|ndy)p×∫B‍|b(x)-bB,w|pw(x)dx≤(∑j=1∞‍1|2jB|∫Aj‍|Ω(x0-y)|rdy)1/r×(∫2j+1B‍|f(y)|r′dy)1/r′×∫B‍|b(x)-bB,w|pw(x)dx≤C[∑j=1∞‍2nj/r|2jB|×(∫2j+1B‍|f(y)|pw(y)dy)1/p×(∫2j+1B‍w(y)-r′/(p-r′)dy)(p-r′)/pr′∑j=1∞‍]p×∫B‍|b(x)-bB,w|pw(x)dx≤C∥f∥Mp,λ(w)p×(∑j=1∞‍w(B)(1-λ)/pw(2j+1B)(1-λ)/p)pw(B)λ≤C∥f∥Mp,λ(w)pw(B)λ.
Hence
(69)LL≤C∥f∥Mp,λ(w)pw(B)λ.
According to (64) and (69), we have completed the proof of Theorem 6.Proof of Theorem7.
The proof of Theorem7 is similar to that of Theorem 6, except using w∈A(p,q). We omit its proof here.



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*Source: 784983-2013-04-09.xml*