# Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Authors: Zhiheng Wang; Zengyan Si
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

LetL be the infinitesimal generator of an analytic semigroup on L2(Rn) with Gaussian kernel bounds, and let L-α/2 be the fractional integrals of L for 0<α<n. For any locally integrable function b, the commutators associated with L-α/2 are defined by [b,L-α/2](f)(x)=b(x)L-α/2(f)(x)-L-α/2(bf)(x). When b∈BMO(ω) (weighted BMO space) or b∈BMO, the authors obtain the necessary and sufficient conditions for the boundedness of [b,L-α/2] on weighted Morrey spaces, respectively.


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## Body

## 1. Introduction and Main Results

Morrey [1] introduced the classical Morrey spaces to investigate the local behavior of solutions to second order elliptic partial differential equations. Chiarenza and Frasca [2] established the boundedness of the Hardy-Littlewood maximal operator, the fractional operator, and a singular integral operator on the Morrey spaces. On the other hand, Coifman and Fefferman [3] and Muckenhoupt [4] studied the boundedness of these operators on weighted Lp spaces. Motivated by these works, Komori and Shirai [5] introduced the following weighted Morrey space and investigated the boundedness of classical operators in harmonic analysis, that is, the Hardy-Littlewood maximal operator, a Calderón-Zygmund operator, the fractional integral operator, and so forth.Let1≤p<∞ and 0≤k<1. Then for two weights μ and ν, the weighted Morrey space is defined by
(1)Lp,k(μ,ν)={f∈Lloc⁡p(μ):∥f∥Lp,k(μ,ν)<∞},
where
(2)∥f∥Lp,k(μ,ν)=sup⁡Q(1ν(Q)k∫Q‍|f(x)|pμ(x)dx)1/p,
and the supremum is taken over all balls Q in Rn.Ifμ=ν, then we have the classical Morrey space Lp,k(μ) with measure μ. When k=0, then Lp,k(μ,ν)=Lp(μ) is the Lebesgue space with measure μ.Suppose thatL is a linear operator on L2(Rn) which generates an analytic semigroup e-tL with a kernel pt(x,y) satisfying a Gaussian upper bound, that is,
(3)|pt(x,y)|≤Ctn/2e-c(|x-y|2/t)
for x,y∈Rn and all t>0.For0<α<n, the fractional integral L-α/2 of the operator L is defined by
(4)L-α/2f(x)=1Γ(α/2)∫0∞‍e-tL(f)dtt-α/2+1(x).
Note that if L=-Δ is the Laplacian on Rn, then L-α/2 is the classical fractional integral Iα which plays important roles in many fields. It is well known that Iα is bounded from Lp(Rn) to Lq(Rn) for all p>1, 1/q=1/p-α/n>0 and is also of weak type (1,n/(n-α)).Let1≤p<∞ and ω be a weight function. A locally integrable function b is said to be in BMOp(ω) if
(5)∥b∥BMOp(ω)=sup⁡Q(1ω(Q)∫Q‍|b(x)-bQ|pω(x)1-pdx)1/p≤C<∞,
where bQ=(1/|Q|)∫Q‍b(y)dy and the supremum is taken over all balls Q∈Rn.Letω∈A1; García-Cuerva [6] proved that the spaces BMOp(ω) coincide, and the norms of ∥·∥BMOp(ω) are equivalent with respect to different values provided that 1≤p<∞.Letb be a locally integrable function on Rn; we consider the commutator [b,L-α/2] defined by
(6)[b,L-α/2](f)(x)=b(x)L-α/2(f)(x)-L-α/2(bf)(x).Chanillo [7] proved that the commutator [b,Iα] of the multiplication operator by b∈BMO is bounded on Lp for 1<p<∞.Duong and Yan [8] proved that [b,L-α/2] is bounded from Lp to Lq, where b∈BMO,   1<p<n/α, 1/q=1/p-α/n, 0<α<n.Mo and Lu [9] proved that the multilinear commutator generated by b→ and L-α/2 is bounded from Lp to Lq, where 1<p<n/α,1/q=1/p-α/n,0<α<1, b→=(b1,…,bm),bi∈BMO, for i=1,…,m.Lu et al. [10] proved that [b,Iα] is bounded from Lp to Lq if and only if b∈BMO.Wang [11] proved that [b,Iα] is bounded from Lp,k(ω) to Lq,kq/p(ω1-(1-α/n)q,ω), where b∈BMO(ω), 0<α<n,1<p<n/α,1/q=1/p-α/n,0<k<p/q, and ωq/p∈A1.Inspired by the above results, we study the boundedness properties of the commutator[b,L-α/2] on weighted Morrey spaces in this work. The main theorems are stated as follows.Theorem 1.
Let0<α<n,1<p<n/α,1/q=1/p-α/n,0≤k<p/q,ωq/p∈A1, and rω>((1-k)/(p/q-k)), where rω denotes the critical index of ω for the reverse Hölder condition. Then the following conditions are equivalent. (a)
b∈BMO(ω).(b)
[b,L-α/2] is bounded from Lp,k(ω) to Lq,kq/p(ω1-(1-α/n)q,ω).In particular, whenk=0 in Theorem 1, we get the following.Corollary 2.
Let0<α<n,1<p<n/α,1/q=1/p-α/n,ωq/p∈A1, and rω>q/p, where rω denotes the critical index of ω for the reverse Hölder condition. Then the following conditions are equivalent. (a)
b∈BMO(ω).(b)
[b,L-α/2] is bounded from Lp(ω) to Lq(ω1-(1-α/n)q).
Furthermore, ifL=-Δ is the Laplacian, then the following conditions are equivalent. (a′)
b∈BMO(ω).(b′)
[b,Iα] is bounded from Lp(ω) to Lq(ω1-(1-α/n)q).Theorem 3.
Let0<α<n,0≤k<p/q,1/q=1/p-α/n, and 1<r,s<∞ such that 1<rs<p<n/α, ωrs∈Ap/rs,q/rs. Then the following conditions are equivalent. (a)
b∈BMO.(b)
[b,L-α/2] is bounded from Lp,k(ωp,ωq) to Lq,kq/p(ωq).In particular, whenk=0 in Theorem 3, we obtain the following.Corollary 4.
Let0<α<n, 1/q=1/p-α/n, and 1<r, s<∞ such that 1<rs<p<n/α, ωrs∈Ap/rs,q/rs. Then the following conditions are equivalent. (a)
b∈BMO.(b)
[b,L-α/2] is bounded from Lp(ωp) to Lq(ωq).
Furthermore, ifL=-Δ is the Laplacian, then the following conditions are equivalent. (a′)
b∈BMO.(b′)
[b,Iα] is bounded from Lp(ωp) to Lq(ωq).Remark 5.
It is easy to see that our results extend the results in [7, 8, 10, 11] significantly.


## 2. Prerequisite Material

Let us first recall some definitions.Definition 6.
The Hardy-Littlewood maximal operatorM is defined by
(7)M(f)(x)=sup⁡x∈Q1|Q|∫Q‍|f(y)|dy.
Let ω be a weight. The weighted maximal operator Mω is defined by
(8)Mω(f)(x)=sup⁡x∈Q1ω(Q)∫Q‍|f(y)|ω(y)dy.
A variant of this maximal operator will become the main tool in our scheme; for 1<r<∞,
(9)Mr,ω(f)(x)=Mω(|f|r)1/r(x).
For 0<α<n,r≥1, the fractional maximal operator Mα,r is defined by
(10)Mα,r(f)(x)=sup⁡x∈Q(1|Q|1-αr/n∫Q‍|f(y)|rdy)1/r,
and the fractional weighted maximal operator Mα,r,ω is defined by
(11)Mα,r,ω(f)(x)=sup⁡x∈Q(1ω(Q)1-αr/n∫Q|f(y)|r‍ω(y)dy)1/r.
For any f∈Lp(Rn),p≥1, the sharp maximal function ML♯f associated with the generalized approximations to the identity {e-tL}t>0 is given by
(12)ML♯f(x)=sup⁡x∈Q1|Q|∫Q‍|f(y)-e-tQLf(y)|dy,
where tQ=rQ2 and rQ is the radius of the ball Q.
In the above definitions, the supremum is taken over all ballsQ containing x.Definition 7.
A weight functionω is said to be in the Muckenhoupt class Ap with 1<p<∞ if, for every ball Q in Rn, there exists a positive constant C which is independent of Q such that
(13)(1|Q|∫Q‍ω(x)dx)(1|Q|∫Q‍ω(x)-1/(p-1)dx)p-1≤C.
When p=1,ω∈A1, if
(14)(1|Q|∫Q‍ω(x)dx)≤Cessinf⁡x∈Q⁡ω(x).
Whenp=∞,ω∈A∞, if there exist positive constants δ and C such that, given a ball Q and a measurable subset E of Q,
(15)ω(E)ω(Q)≤C(|E||Q|)δ.Definition 8.
A weight functionω belongs to Ap,q for 1<p<q<∞ if, for every ball Q in Rn, there exists a positive constant C which is independent of Q such that
(16)(1|Q|∫Q‍ω(x)qdx)1/q(1|Q|∫Q‍ω(x)-p′dx)1/p′≤C,
where p′ denotes the conjugate exponent of p>1, that is, 1/p+1/p′=1.Definition 9.
A weight functionω belongs to the reverse Hölder class RHr if there exist two constants r>1 and C>0 such that the reverse Hölder inequality
(17)(1|Q|∫Q‍ω(x)rdx)1/r≤C(1|Q|∫Q‍ω(x)dx)
holds for every ball Q in Rn.It is well known that ifω∈Ap with 1≤p<∞, then there exists r>1 such that ω∈RHr. It follows from Hölder's inequality that ω∈RHr implies ω∈RHs for all 1<s<r. Moreover, if ω∈RHr,r>1, then we have ω∈RHr+ϵ for some ϵ>0. We thus write rw=sup⁡{r>1:ω∈RHr} to denote the critical index of ω for the reverse Hölder condition.We will make use of the following lemmas. We first provide a weighted version of the local goodλ inequality for ML♯ which allows us to obtain an analog of the classical Fefferman-Stein (see [3, 12]) estimate on weighted Morrey spaces.Lemma 10 (see [13]).
Assume that the semigroupe-tL has a kernel pt(x,y) which satisfies the upper bound (3). Take λ>0,f∈L01(Rn), and a ball Q0 such that there exists x0∈Q0 with Mf(x0)≤λ. Then, for every ω∈A∞,0<η<1, one can find γ>0 (independent of λ,Q0,f,x0) and constant Cω,r>0 (which only depend on ω), such that
(18)ω{x∈Q0:Mf(x)>Aλ,ML♯f(x)≤γλ}≤Cωηrω(Q0),
where A>1 is a fixed constant which depends only on n.As a consequence, by using the standard arguments, we have the following estimates.For everyf∈Lp,k(μ,ν), with 1<p<∞. if μ,ν∈A∞,1<p<∞,0≤k<1, then
(19)∥f∥Lp,k(μ,ν)≤∥Mf∥Lp,k(μ,ν)≤C∥ML♯f∥Lp,k(μ,ν).
In particular, when μ=ν=ω and ω∈A∞, we have
(20)∥f∥Lp,k(ω)≤∥Mf∥Lp,k(ω)≤C∥ML♯f∥Lp,k(ω).Lemma 11 (see [11]).
Let0<α<n,1<p<n/α, 1/q=1/p-α/n, and ωq/p∈A1. Then if 0<k<p/q and rω>(1-k)/(p/q-k), one has
(21)∥Mα,1f∥Lq,kq/p(ωq/p,ω)≤C∥f∥Lp,k(ω).
The same conclusion still holds for Iα.Lemma 12 (see [11]).
Let0<α<n,1<p<n/α,1/q=1/p-α/n, and ωq/p∈A1. Then if  0<k<p/q,1<r<p, and rω>(1-k)/(p/q-k), one has
(22)∥Mr,ωf∥Lq,kq/p(ωq/p,ω)≤C∥f∥Lq,kq/p(ωq/p,ω).Lemma 13 (see [11]).
Consider0<α<n,1<p<n/α,1/q=1/p-α/n, 0<k<p/q, and ω∈A∞. For any 1<r<p, one has
(23)∥Mα,r,ωf∥Lq,kq/p(ω)≤C∥f∥Lp,k(ω).Remark 14.
By checking the proof of Lemmas11–13, we know that the three lemmas above still hold when k=0.Lemma 15.
Let0<α<n,1<p<n/α,1/q=1/p-α/n, and ωq/p∈A1. Then if 0≤k<p/q and rω>(1-k)/(p/q-k), one has
(24)∥L-α/2f∥Lq,kq/p(ωq/p,ω)≤C∥f∥Lp,k(ω).Proof.
Since the semigroupe-tL has a kernel pt(x,y) which satisfies the upper bound (3), it is easy to check that L-α/2(f)(x)≤CIα(|f|)(x) for all x∈Rn. Using the boundedness property of Iα on weighted Morrey space (see Lemma 11), we have
(25)∥L-α/2f∥Lq,kq/p(ωq/p,ω)≤∥Iαf∥Lq,kq/p(ωq/p,ω)≤C∥f∥Lp,k(ω),
where 1<p<n/α and 1/q=1/p-α/n.Remark 16.
SinceIα is of weak type (1,n/(n-α)). from the proof of Lemma 15, we can get that L-α/2 is also of weak type (1,n/(n-α)).Lemma 17 (see [8, 14]).
Assume that the semigroupe-tL has a kernel pt(x,y) which satisfies the upper bound (3). Then for 0<α<n, the difference operator L-α/2-e-tLL-α/2 has an associated kernel Kα,t(x,y) which satisfies
(26)|Kα,t(x,y)|≤C|x-y|n-αt|x-y|2,
for some positive constant C.Lemma 18.
Assume that the semigroupe-tL has a kernel pt(x,y) which satisfies the upper bound (3), and let b∈BMO(ω),ω∈A1. Then, for every function f∈Lp(Rn),p>1, and for all x∈Rn, one has
(27)sup⁡x∈Q1|Q|∫Q‍|e-tQL(b(y)-bQ)f(y)|dy≤C∥b∥BMO(ω)ω(x)Mr,ω(f)(x),
where tQ=rQ2,rQ being the radius of Q.Proof.
For anyf∈Lp(Rn),1<p<∞ and x∈Q. We have
(28)1|Q|∫Q‍|e-tQL((b(·)-bQ)f)(y)|dy≤1|Q|∫Q‍∫Rn‍|ptQ(y,z)||(b(z)-bQ)f(z)|dzdy≤1|Q|∫Q‍∫2Q‍|ptQ(y,z)||(b(z)-bQ)f(z)|dzdy+1|Q|∫Q‍∑k=1∞‍∫2k+1Q∖2kQ‍|ptQ(y,z)|×|(b(z)-bQ)f(z)|dzdy≐M+N.
For any y∈Q and z∈2Q. We have
(29)|ptQ(y,z)|≤CtQ-n/2≤C1|2Q|.
Thus,
(30)M≤C1|2Q|∫2Q|(b(z)-bQ)f(z)|‍dz≤C1|2Q|(∫2Q|b(z)-bQ|r′‍ω(z)1-r′dz)1/r′×(∫2Q|f(z)|r‍ω(z)dz)1/r≤C∥b∥BMO(ω)ω(2Q)|2Q|(1ω(2Q)∫2Q|f(z)|r‍ω(z)dz)1/r≤C∥b∥BMO(ω)ω(x)Mr,ωf(x).
Moreover, for any y∈Q and z∈2k+1Q∖2kQ, we have |y-z|≥2k-1rQ and |ptQ|≤C(e-c22(k-1)2(k+1)n/|2k+1Q|):
(31)N=1|Q|∫Q∑k=1∞∫2k+1Q∖2kQ|ptQ(y,z)||(b(z)-bQ)f(z)|dzdy‍≤C∑k=1∞e-c22(k-1)2(k+1)n|2k+1Q|‍∫2k+1Q|(b(z)-bQ)f(z)|dz‍≤C∑k=1∞e-c22(k-1)2(k+1)n|2k+1Q|‍∫2k+1Q|(b(z)-b2k+1Q)f(z)|dz‍+C∑k=1∞e-c22(k-1)2(k+1)n|2k+1Q|‍∫2k+1Q|(b2k+1Q-bQ)f(z)|dz‍≐N1+N2.
We estimate each term in turn. For N1, we apply Hölder's inequalities with exponent r. Then we have
(32)N1≤C∑k=1∞e-c22(k-1)2(k+1)n|2k+1Q|×(∫2k+1Q|b(z)-b2k+1Q|r′‍ω(z)1-r′dz)1/r′×(∫2k+1Q|f(z)|r‍ω(z)dz)1/r≤C∑k=1∞2(k+1)ne-c22(k-1)‍∥b∥BMO(ω)ω(2k+1Q)|2k+1Q|×(1ω(2k+1Q)∫2k+1Q|f(z)|r‍ω(z)dz)1/r≤C∥b∥BMO(ω)ω(x)Mr,ωf(x).
Sinceω∈A1, then |b2k+1Q-bQ|≤Ckω(x)∥b∥BMO(ω). This fact together with Hölder's inequality implies(33)N2≤C∑k=1∞2(k+1)ne-c22(k-1)k|2k+1Q|ω(x)‍×∥b∥BMO(ω)∫Q|f(z)|dz‍≤C∑k=1∞k2(k+1)ne-c22(k-1)ω(x)‍×∥b∥BMO(ω)(1|2k+1Q|∫2k+1Q|f(z)|rdz‍)1/r=C∑k=1∞k2(k+1)ne-c22(k-1)ω(x)‍×∥b∥BMO(ω)(ω(2k+1Q)|2k+1Q|1ω(2k+1Q)∫2k+1Q|f(z)|rdz‍)1/r≤C∑k=1∞k2(k+1)ne-c22(k-1)ω(x)‍×∥b∥BMO(ω)(1ω(2k+1Q)∫2k+1Q|f(z)|rω(x)dz‍)1/r≤C∥b∥BMO(ω)ω(x)Mr,ωf(x).Then Lemma 18 is proved.Lemma 19.
Let0<α<n,ω∈A1, and b∈BMO(ω). Then for all r>1 and for all x∈Rn, one has
(34)ML♯([b,L-α/2]f)(x)≤C∥b∥BMO(ω)(ω(x)Mr,ω(L-α/2f)(x)+ω(x)1-α/n×Mα,r,ω(f)(x)+ω(x)Mα,1(f)(x)(L-α/2f)(x)+ω(x)1-(α/n)).Proof.
For any givenx∈Rn, fix a ball Q=Q(x0,rQ) which contains x. We decompose f=f1+f2, where f1=fχ2Q. Observe that
(35)[b,L-α/2]f(x)=(b-bQ)L-α/2f-L-α/2(b-bQ)f1-L-α/2(b-bQ)f2,e-tQL([b,L-α/2]f)=e-tQL[(b-bQ)L-α/2f-L-α/2(b-bQ)f1-L-α/2(b-bQ)f2].
Then
(36)1|Q|∫Q‍|[b,L-α/2]f(y)-e-tQL[b,L-α/2]f(y)|dy≤1|Q|∫Q‍|(b(y)-bQ)L-α/2f(y)|dy+1|Q|∫Q‍|L-α/2(b-bQ)f1(y)|dy+1|Q|∫Q‍|e-tQL((b-bQ)L-α/2f)(y)|dy+1|Q|∫Q‍|e-tQLL-α/2((b-bQ)f1(y))|dy+1|Q|∫Q‍|(L-α/2-e-tQLL-α/2)((b-bQ)f2)(y)|dy≐I+II+III+IV+V.
We estimate each term separately.
Sinceω∈A1, then it follows from Hölder's inequality that
(37)I≤1|Q|∫Q|(b(y)-bQ)L-α/2f(y)|‍dy≤1|Q|(∫Q|b(y)-bQ|r′‍ω(y)1-r′dy)1/r′×(∫Q|L-α/2f(y)|r‍ω(y)dy)1/r≤C∥b∥BMO(ω)ω(Q)|Q|(1ω(Q)∫Q|L-α/2f(y)|r‍ω(y)dy)1/r≤C∥b∥BMO(ω)ω(x)Mr,ω(L-α/2f)(x).
Applying Kolmogorov's inequality (see [15, page 485]), Hölder's inequality, and the continuity of L-α/2, we thus have
(38)II=1|Q|∫Q‍|L-α/2(b-bQ)f1(y)|dy≤C1|Q|1-α/n∥L-α/2(b-bQ)f1∥Ln/(n-α),∞≤C1|Q|1-α/n∫Q‍(b-bQ)f1(y)dy≤C1|Q|1-α/n(∫Q‍|b(y)-bQ|r′ω(y)1-r′dy)1/r′×(∫Q‍|f(y)|rω(y)dy)1/r≤C∥b∥BMO(ω)w(Q)1-α/n|Q|1-α/n×(1w(Q)1-rα/n∫Q|f(y)|r‍ω(y)dy)1/r≤C∥b∥BMO(ω)ω(x)1-α/nMα,r,ω(f)(x).
By Lemma 18, we have
(39)III≤C∥b∥BMO(ω)ω(x)Mr,ω(L-α/2f)(x).
For IV, using the estimate obtained in II, we get
(40)IV≤1|Q|∫Q∫2Q|ptQ(y,z)||b(z)-bQ|‍‍|f(z)|dzdy≤1|2Q|∫2Q|L-α/2((b(z)-bQ))f(z)|‍dz≤C∥b∥BMO(ω)ω(x)1-α/nMα,r,ω(f)(x).
By virtue of Lemma17, we have(41)V≤1|Q|∫Q∫(2Q)c|Kα,tQ(y,z)||(b(z)-bQ)f(z)|‍‍dzdy≤C∑k=1∞∫2krQ≤|x0-z|<2k+1rQ1|x0-z|n-αtQ|x0-z|2×|(b(z)-bQ)f(z)|dz≤C∑k=1∞2-2k1|2k+1Q|1-α/n∫2k+1Q|(b(z)-bQ)f(z)|dz‍‍≤C∑k=1∞2-2k1|2k+1Q|1-α/n∫2k+1Q|(b(z)-b2k+1Q)f(z)|dz‍‍+C∑k=1∞2-2k(b2k+1Q-bQ)1|2k+1Q|1-α/n‍∫2k+1Q|f(z)|dz‍≐VI+VII.
ForVI, applying the same arguments as in II, we get
(42)VI≤C∥b∥BMO(ω)∑k=1∞2-2kω(x)1-α/nMα,r,ω(f)(x)‍≤C∥b∥BMO(ω)ω(x)1-α/nMα,r,ω(f)(x).
Sinceω∈A1, then |b2k+1Q-bQ|≤Ckω(x)∥b∥BMO(ω). Thus,
(43)VII≤C∥b∥BMO(ω)∑k=1∞2-2kkω(x)Mα,1(f)(x)‍≤C∥b∥BMO(ω)ω(x)Mα,1(f)(x).
Then
(44)V≤C∥b∥BMO(ω)×(ω(x)1-α/nMα,r,ω(f)(x)+ω(x)Mα,1(f)(x)).
Combining the above estimates I–V, we get (34). The proof of Lemma 19 is complete.


## 3. Proofs of the Main Results

In this section we prove our main results. We start with the proof of Theorem1.Proof.
(a)⇒(b): Applying Lemmas 10 and 19, we get
(45)∥[b,L-α/2]f∥Lq,kq/p(ω1-(1-α/n)q,ω)≤∥ML♯([b,L-α/2]f)∥Lq,kq/p(ω1-(1-α/n)q,ω)≤C∥b∥BMO(ω)×(∥ωMr,ω(L-α/2f)∥Lq,kq/p(ω1-(1-α/n)q,ω)+∥ω1-α/nMα,r,ωf∥Lq,kq/p(ω1-(1-α/n)q,ω)+∥ωMα,1f∥Lq,kq/p(ω1-(1-α/n)q,ω)∥ωMr,ω(L-α/2f)∥Lq,kq/p(ω1-(1-α/n)q,ω))≤C∥b∥BMO(ω)×(∥Mr,ω(L-α/2f)∥Lq,kq/p(ωq/p,ω)+∥Mα,r,ωf∥Lq,kq/p(ω)+∥Mα,1f∥Lq,kq/p(ωq/p,ω)∥ωMr,ω(L-α/2f)∥Lq,kq/p(ω1-(1-α/n)q,ω)).
Since0≤k<p/q,ωq/p∈A1, and rω>(1-k)/(p/q-k), by making use of Lemmas 11–13, then we obtain
(46)∥[b,L-α/2]f∥Lq,kq/p(ω1-(1-α/n)q,ω)≤C∥b∥BMO(ω)(∥L-α/2f∥Lq,kq/p(ωq/p,ω)+∥f∥Lp,k(ω))≤C∥b∥BMO(ω)∥f∥Lp,k(ω).
The last inequality follows from Lemma 15. This completes the proof of (a)⇒(b).
(b)⇒(a): Let L=-Δ be the Laplacian on Rn; then L-α/2 is the classical fractional integral Iα. Choose Z0∈Rn so that |Z0|=3. For x∈Q(Z0,2),   |x|-α+n can be written as the absolutely convergent Fourier series |x|-α+n=∑m∈Zn‍amei〈νm,x〉 with ∑m‍|am|<∞ since |x|-α+n∈C∞(Q(Z0,2)). For any x0∈Rn and ρ>0, let Q=Q(x0,ρ) and QZ0=Q(x0+Z0ρ,ρ),(47)∫Q|b(x)-bQZ0|‍dx=1|QZ0|∫Q|∫QZ0(b(x)-b(y))dy‍|‍dx=1ρn∫Qs(x)(∫QZ0(b(x)-b(y))‍|x-y|α-n|x-y|n-αdy)dx,where s(x)=sgn⁡(∫QZ0‍(b(x)-b(y))dy). Fix x∈Q and y∈QZ0 and we have (y-x)/ρ∈Q(Z0,2); hence, we have(48)ρ-α+nρn∫Qs(x)(∫QZ0(b(x)-b(y))|x-y|α-n(|x-y|ρ)n-αdy‍)‍dx=ρ-α∑m∈Znam‍×∫Qs(x)(∫QZ0(b(x)-b(y))×|x-y|α-nei〈νm,y/ρ〉dy∫QZ0(b(x)-b(y)))e-i〈νm,x/ρ〉dx≤ρ-α|∑m∈Zn|am|‍∫Qs(x)‍[b,L-α/2]×(χQZ0ei〈νm,·/ρ〉)χQ(x)e-i〈νm,x/ρ〉dx∑m∈Zn|am||≤ρ-α∑m∈Zn|am|‍∥[b,L-α/2](χQZ0ei〈νm,·/ρ〉)∥Lq,0(ω1-(1-(α/n))q,ω)×(∫Qω(x)q′[(1-(α/n))-1/q]‍dx)1/q′≤Cρ-α∑m∈Zn|am|‍∥χQZ0∥Lp,0(ω)(∫Qω(x)q′(1/q′-α/n)‍dx)1/q′≤Cω(Q)1/p+1/q′-α/n=Cω(Q).
This impliesb∈BMO(ω). Thus Theorem 1 is proved.Similarly, to prove Theorem3, we need the following lemmas.Lemma 20.
Let0<α<n,1<r,s<∞ such that rs<p<n/α and b∈BMO. Then for all r>1 and for all x∈Rn, one has
(49)ML♯([b,L-α/2]f)(x)≤C∥b∥BMO(Mr(L-α/2f)(x)+Mα,rs(f)(x)),
where Mr(f)(x)=M(|f|r)1/r(x).Proof.
The case0<α<1 was proved by Duong and Yan (see [8] for details). The general case 0<α<n follows by repeating the same steps as in Lemma 19. Since the main steps and the ideas are almost the same, here we omit the proof.Lemma 21 (see [5]).
If0<α<n,1<p<n/α,1/q=1/p-α/n,0<k<p/q, and ω∈Ap,q, then the fractional maximal operator Mα,1 is bounded from Lp,k(ωp,ωq) to Lq,kq/p(ωq).Lemma 22 (see [5]).
If0<α<n,1<p<n/α,1/q=1/p-α/n,0<k<p/q, and ω∈Ap,q, then the fractional maximal operator Iα is bounded from Lp,k(ωp,ωq) to Lq,kq/p(ωq).Lemma 23 (see [5]).
If1<p<∞,0<k<1, and ω∈Ap, then M is bounded on Lp,k(ω).Remark 24.
By applying the same argument as in Lemma15, we know that the conclusion in Lemma 22 still holds for L-α/2. We omit the proof here.Remark 25.
By checking the proof of Lemmas21–23, we know that the three lemmas above still hold when k=0.Now we prove Theorem3.Proof.
(a)⇒(b): Since ωrs∈Ap/rs,q/rs, then we get ωq∈Aq/rs and ωp∈Ap/rs. Applying Lemmas 10 and 20–23, we get(50)∥[b,L-α/2]f∥Lq,kq/p(ωq)≤∥ML♯([b,L-α/2]f)∥Lq,kq/p(ωq)≤C∥b∥BMO(∥Mr(L-α/2f)∥Lq,kq/p(ωq)+∥Mα,rs(f)∥Lq,kq/p(ωq))≤C∥b∥BMO(∥L-α/2f∥Lq,kq/p(ωq)+∥f∥Lp,k(ωp,ωq))≤C∥b∥BMO∥f∥Lp,k(ωp,ωq).In the last inequality, we used the fact that L-α/2 is bounded from Lp,k(ωp,ωq) to Lq,kq/p(ωq) (see Remark 24).
(b)⇒(a): Let L=-Δ be the Laplacian on Rn; then L-α/2 is the classical fractional integral Iα. Let k=0 and weight ω≡1, and then Lp,k(ωp,ωq)=Lp and Lq,kq/p(ωq)=Lq. From [10] we deduce that the (Lp,Lq) boundedness of [b,Iα] implies b∈BMO. Thus Theorem 3 is proved.



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*Source: 413716-2014-06-01.xml*