# Boundedness of Marcinkiewicz Integrals and Their Commutators on Generalized Weighted Morrey Spaces

Authors: Runqing Cui; Zhang Li
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We will study the boundedness of Marcinkiewicz integrals with rough kernel on
the generalized weighted Morrey spaces. We will also prove that the commutator operators formed by aBMORn function and Marcinkiewicz integrals are also bounded on the generalized weighted Morrey
spaces.


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## Body

## 1. Introduction and Results

Suppose thatSn-1 is the unit sphere in Rn(n≥2) equipped with the normalized Lebesgue measure dσ. Let Ω∈Lq(Sn-1) with 1<q≤∞ be homogeneous of degree zero and satisfy the cancellation condition
(1)∫Sn-1‍Ωx′dσx′=0,
where x′=x/|x| for any x≠0. The Marcinkiewicz integral of higher dimension μΩ is defined by
(2)μΩfx=∫0∞‍FΩ,tx2dtt31/2,
where
(3)FΩ,tx=∫x-y≤t‍Ωx-yx-yn-1fydy.
We will also consider the commutator generated by Marcinkiewicz integral μΩ and b is defined as follows:
(4)b,μΩfx=∫0∞‍∫x-y≤t‍Ωx-yx-yn-1·bx-byfydy∫x-y≤t‍Ωx-yx-yn-12dtt31/2.The following results concerning the boundedness of Marcinkiewicz integrals and their commutators on weightedLp space are known.Theorem 1 (see [1]).
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞. Then, for every q′<p<∞ and ω∈Ap/q′, there is a constant C independent of f such that
(5)μΩfLp(ω,Rn)≤CfLp(ω,Rn).Theorem 2 (see [2]).
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞ and b∈BMO(Rn). Then, for every q′<p<∞ and ω∈Ap/q′, there is a constant C independent of f such that
(6)b,μΩfLp(ω,Rn)≤Cb*fLp(ω,Rn).The classical Morrey spaces, introduced by Morrey [3] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations; they appeared to be quite useful in the study of local behavior of the solutions of elliptic differential equations and describe local regularity more precisely than Lebesgue spaces. See [4–7] for details.Givenf∈Lloc⁡p(Rn), 1≤p<∞ and 0≤λ≤n. Morrey spaces are defined by
(7)fLp,λ(Rn)=sup⁡x∈Rn,r>0r-λ/pfLp(B(x,r))<∞.Note thatLp,0(Rn)=Lp(Rn) and Lp,n(Rn)=L∞(Rn). If λ<0 or λ>n, then Lp,λ(Rn)=Θ, where Θ is the set of all functions equivalent to 0 on Rn.In [8] Chiarenza and Frasca show the boundedness of the Hardy-Littlewood maximal operator in Lp,λ(Rn) that allows them to prove continuity of fractional and classical Calderón-Zygmund operators in these spaces. Recall that integral operators of that kind appear in the representation formulae of the solutions of elliptic/parabolic equations and systems. Thus the continuity of the Calderón-Zygmund integrals implies regularity of the solutions in the corresponding spaces.LetΦ(r), r>0, be a growth function, that is, a positive increasing function in (0,∞), and satisfy doubling condition
(8)Φ2r≤DΦr,∀r>0,
where D=D(Φ)≥1 is a doubling constant independent of r. In [9] Mizuhara gave generalization Morrey spaces Lp,Φ(Rn) considering Φ(r) instead of rλ in (7). He studied also a continuity in Lp,Φ(Rn) of some classical integral operators.Komori and Shirai [10] introduced a version of the weighted Morrey space Lp,κ(ω,Rn), which is a natural generalization of the weighted Lebesgue space Lp(ω,Rn). Let 1≤p<∞, 0<κ<1 and let ω be a weight function. Then the spaces Lp,κ(ω,Rn) are defined by
(9)fLp,κ(ω,Rn)=sup⁡x∈Rn,r>01ω(B(x,r))κ∫Bx,r‍fypωydy1/p<∞.Recently, Aliev and Guliev in [11] introduced another generalization of the Morrey spaces. Let 1≤p<∞ and let φ be a positive measurable function on Rn×(0,∞). We denote by Mφp(Rn) the generalized Morrey space, the space of all functions f∈Lloc⁡p(Rn) with finite norm
(10)fMφp(Rn)=sup⁡x∈Rn,r>01φ(x,r)1Bx,rfLp(B(x,r))p1/p.
They also studied in [11] the boundedness of μΩ and their commutator on generalized Morrey space Mφp(Rn) when Ω∈L∞(Sn-1).Let1≤p<∞ and let φ be a positive measurable function on Rn×(0,∞) and let ω be a nonnegative measurable function on Rn. Following [12], we denote by Mφp(ω,Rn) the generalized weighted Morrey space, the space of all functions f∈Lloc⁡p(ω,Rn) with finite norm
(11)fMφp(w,Rn)=sup⁡x∈Rn,r>01φ(x,r)1w(B(x,r))fLp(ω,B(x,r))p1/p,
where
(12)fLp(ω,B(x,r))=∫B(x,r)‍fypwydy.Ifω=1 and φ(x,r)=r(λ-n)/p with 0≤λ≤n, then Mφp(ω,Rn)=Lp,λ(Rn) is the classical Morrey space; if φ(x,r)=ω(B(x,r))(κ-1)/p, 0<κ<1, then Mφp(ω,Rn)=Lp,κ(ω,Rn) is the weighted Morrey space; if ω=1, φ(x,r)p=Φ(r)r-n, then Mφp(ω,Rn)=Lp,Φ(Rn). If ω=1, then Mφp(ω,Rn)=Mφp(Rn).The purpose of this paper is to discuss the boundedness properties of Marcinkiewicz integrals with rough kernel and their commutators on the generalized weighted Morrey spacesMφp(ω,Rn). Our main results can be formulated as follows.Theorem 3.
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞. Let q′<p<∞, ω∈Ap/q′, and (φ1,φ2) satisfy the condition
(13)∫s∞‍essinf⁡r<t<∞φ1x,tωBx,t1/pωBx,r1/pdrr≤Cφ2x,s,
where C does not depend on x or r. Then there is a constant C>0 independent of f such that
(14)μΩfMφ2p(ω,Rn)≤CfMφ1p(ω,Rn).Theorem 4.
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞. Let q′<p<∞, ω∈Ap/q′, and (φ1,φ2) satisfy the condition
(15)∫s∞‍1+ln⁡rsessinf⁡r<t<∞φ1x,tωBx,t1/pωBx,r1/pdrr≤Cφ2x,s.
If b∈BMO(Rn), then there is a constant C>0 independent of f such that
(16)b,μΩfMφ2p(ω,Rn)≤Cb*fMφ1p(ω,Rn).Corollary 5.
LetΩ∈Lq(Sn-1) with 1<q≤∞ and q′<p<∞. (i)
Suppose(φ1,φ2) satisfy the condition
(17)∫s∞‍essinf⁡r<t<∞φ1x,ttn/pr1+(n/p)dr≤Cφ2x,s.
Then there is a constant C>0 independent of f such that
(18)μΩfMφ2p(Rn)≤CfMφ2p(Rn).(ii)
Suppose(φ1,φ2) satisfy the condition
(19)∫s∞‍1+ln⁡rsessinf⁡r<t<∞φ1x,ttn/pr1+(n/p)dr≤Cφ2x,s,
and b∈BMO(Rn). Then there is a constant C>0 independent of f such that
(20)b,μΩfMφ2p(Rn)≤Cb*fMφ2p(Rn).Remark 6.
(i) Letφ1(x,t)=φ2(x,t)=ω(B(x,t))(κ-1)/p, 0<κ<1 and w∈Ap(Rn)(1≤p<∞); then (φ1,φ2) satisfy conditions (13) and (15).
(ii) Letφ1(x,t)=φ2(x,t)=(Φ(t)t-n)1/p. If 1≤D(Φ)≤2n, then (φ1,φ2) also satisfy conditions (13) and (15).We verify only (i). (ii) can be verified similarly. In fact, from (36) in Section 2 we have constant δ>0 such that
(21)ωBx,2ks≥C2δωBx,s.
Then
(22)∫s∞‍essinf⁡r<t<∞φ1x,tωBx,t1/pωBx,r1/pdrr≤∫s∞‍1+ln⁡rsessinf⁡r<t<∞φ1x,tωBx,t1/pωBx,r1/pdrr=∫s∞‍1+ln⁡rsωBx,r(k-1)/pdrr≤∑k=0∞‍1+k∫2ks2k+1s‍ωBx,r(k-1)/pdrr≤C∑k=0∞‍1+kωBx,2k+1s(k-1)/p≤C∑k=0∞‍1+k2-(k+1)δ((1-κ)/p)ωBx,s(k-1)/p≤CωBx,s(k-1)/p=Cφ2x,s.Corollary 7.
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞ and 0<κ<1. If q′<p<∞, ω∈Ap/q′, then there is a constant C>0 independent of f such that
(23)μΩfLp,κ(ω,Rn)≤CfLp,κ(ω,Rn).
If q′<p<∞, ω∈Ap/q′, and b∈BMO(Rn), then there is a constant C>0 independent of f such that
(24)b,μΩfLp,κ(ω,Rn)≤Cb*fLp,κ(ω,Rn).Corollary 8.
Suppose thatΩ∈Lq(Sn-1) with 1<q≤∞. If q′<p<∞, 1≤D(Φ)≤2n, then there is a constant C>0 independent of f such that
(25)μΩfLp,Φ(Rn)≤CfLp,Φ(Rn).
If q′<p<∞, 1≤D(Φ)≤2n, and b∈BMO(Rn), then there is a constant C>0 independent of f such that
(26)b,μΩfLp,Φ(Rn)≤Cb*fLp,Φ(Rn).


## 2. Some Preliminaries

We begin with some properties ofAp weights which play a great role in the proofs of our main results.A weightω is a nonnegative, locally integrable function on Rn. Let B=B(x0,rB) denote the ball with the center x0 and radius rB. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted measure ω(E)=∫E‍ω(x)dx. For any given weight function ω on Rn,X⫅Rn, and 0<p<∞, denote by Lp(ω,X) the space of all function f satisfying
(27)fLp(ω,X)=∫X‍fxpωxdx1/p<∞.A weightω is said to belong to Ap(Rn) for 1<p<∞, if there exists a constant
(28)1B∫B‍ωxdx1B∫B‍ωx1-p′dxp-1≤C,
where p′ is the dual of p such that 1/p+1/p′=1. The class A1(Rn) is defined by
(29)1B∫B‍wydy≤C·essinf⁡x∈BwxforeveryballB⊂Rn.
A weight ω is said to belong to A∞(Rn) if there are positive numbers C and δ so that
(30)ωEωB≤CEBδ
for all balls B and all measurable E⊂B. It is well known that
(31)A∞Rn=⋃1≤p<∞‍ApRn.
By (28), we have
(32)ω-(p′/p)B1/p′=ω-(1/p)Lp′(B)≤CBωB-(1/p)
for 1<p<∞. Note that
(33)essinf⁡x∈Efx-1=esssup⁡x∈E1fx
is true for any real-valued nonnegative function f and measurable on E (see [13] page 143) and (29); we get
(34)ω-1L∞(B)=esssup⁡x∈B1wx=1essinf⁡x∈Bwx≤CBωB-1.The classicalAp(Rn) weight theory was first introduced by Muckenhoupt in the study of weighted Lp-boundedness of Hardy-Littlewood maximal function in [14].Lemma 9 (see [14, 15]).
Supposeω∈Ap(Rn) and the following statements hold. (i)
For any1≤p<∞, there is a positive number C such that
(35)ωBkωBj≤C2npk-jfork>j.(ii)
For any1≤p<∞, there are positive numbers C and δ such that
(36)ωBkωBj≥C2δk-jfork>j.(iii)
For any1<p<∞, one has ω1-p′∈Ap′(Rn).Following [16], a locally integrable function b is said to be in BMO(Rn) if
(37)sup⁡B⊂Rn1B∫B‍bx-bBdx=b*<∞,
where
(38)bB=1B∫B‍bydy.Lemma 10 (see [12]).
Supposeω∈A∞(Rn) and b∈BMO(Rn). Then, for any 1≤p<∞ and r1,r2>0, one has
(39)1ωBx0,r1∫Bx0,r1‍bx-bB(x0,r2)pωxdx1/p≤C1+ln⁡r1r2b*.


## 3. Proof of Theorem3

We first prove the following conclusions.Theorem 11.
Suppose thatΩ∈Lq(Sn-1), 1<q≤∞. Then, for every q′<p<∞, ω∈Ap/q′, there is a constant C independent of f such that
(40)μΩfLp(ω,B(x0,s))≤CωBx0,s1/p·∫2s∞fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.Proof.
We representf as f=f1+f2, where f1(y)=f(y)χB(x0,2s)(y),χB(x0,2s) denotes the characteristic function of B(x0,2s). Then
(41)μΩfLp(ω,B(x0,s))≤μΩf1Lp(ω,B(x0,s))+μΩf2Lp(ω,B(x0,s)).
Since f1∈Lp(ω,Rn) and from the boundedness of μΩ on Lp(ω,Rn)(q′<p<∞) (Theorem 1) it follows that
(42)μΩf1Lp(ω,B(x0,s))≤μΩf1Lp(ω,Rn)≤Cf1Lp(ω,Rn)=CfLp(ω,B(x0,2s)).
By Hölder inequality,
(43)Bx0,s≤CωBx0,s1/pω-(1/p)Lp′(B(x0,s)).
Then, for q′<p<∞,
(44)μΩf1Lp(ω,B(x0,s))≤CBx0,sfLp(ω,B(x0,2s))∫2s∞‍drrn+1≤CBx0,s∫2s∞‍fLp(ω,B(x0,r))drrn+1≤CωBx0,s1/pω-(1/p)Lp′(B(x0,s))·∫2s∞‍fLp(ω,B(x0,r))drrn+1≤CωBx0,s1/p·∫2s∞‍fLp(ω,B(x0,r))ω-(1/p)Lp′(B(x0,r))drrn+1.
By (32), we get
(45)μΩf1Lp(ω,B(x0,s))≤CωBx0,s1/p·∫2s∞‍fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
To estimateμΩ(f2)Lp(ω,B(x0,s)), observe that when x∈B(x0,s) and y∈Bx0,2j+1s∖B(x0,2js)(j≥1),
(46)t≥x-y≥y-x0-x-x0≥2j-1s.
Therefore, by Minkowski’s inequality, we have
(47)μΩf2x=∫0∞‍∫B(x0,2s)c∩{y:|x-y|≤t}‍Ωx-yx-yn-1fydy2dtt31/2=∫0∞∑j=1∞‍∫(B(x0,2j+1l)∖B(x0,2jl))c∩{y:|x-y|≤t}‍Ωx-yx-yn-1‍·fydy∑j=1∞‍2dtt31/2≤C∑j=1∞‍∫Bx0,2j+1s∖Bx0,2js‍Ωx-yx-yn-1fydy·∫2j-1s∞‍dtt31/2≤C∑j=1∞‍2j+1s-1∫B(x0,2j+1s)∖B(x0,2js)‍Ωx-yx-yn-1fydy.
WhenΩ∈L∞(Sn-1), then by assumption we have ω∈Ap, 1<p<∞. It follows from the Hölder inequality and (32) that
(48)sup⁡x∈B(x0,s)|μΩf2x|≤CΩL∞Sn-1∑j=1∞‍2j+1s-n∫B(x0,2j+1s)‍fydy≤C∑j=1∞‍2j+1s-nfLp(ω,B(x0,2j+1s))·ω-(1/p)Lp′(B(x0,2j+1s))≤C∑j=1∞‍∫2j+1s2j+2s‍2j+1s-(1+n)fLp(ω,B(x0,r))·ω-(1/p)Lp′(B(x0,r))dr≤C∫2s∞‍fLp(ω,B(x0,r))ω-(1/p)Lp′(B(x0,r))drr1+n≤C∫2s∞‍fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
Then
(49)μΩf2Lp(ω,B(x0,s))≤CωBx0,s1/p∫2s∞‍fLp(ω,B(x0,r))·ωBx0,r-(1/p)drr.
WhenΩ∈Lq(Sn-1), 1<q<∞. It follows from the Hölder inequality that
(50)∫B(x0,2j+1s)∖B(x0,2js)‍Ωx-yx-yn-1fydy≤C∫B(x0,2j+1s)∖B(x0,2js)‍Ωx-y′qdy1/q·∫B(x0,2j+1s)∖B(x0,2js)‍fyq′|x-y|(n-1)q′dy1/q′.
When x∈B(x0,s) and y∈B(x0,2j+1s)∖B(x0,2js), then by a direct calculation we can see that 2j-1s≤|y-x|<2j+1s. Hence
(51)∫B(x0,2j+1s)∖B(x0,2js)‍Ωx-y′qdy1/q≤CΩLqSn-1Bx0,2j+1s1/q.
We also note that if x∈B(x0,s), y∈B(x0,2s)c, then |y-x|~|y-x0|. Consequently
(52)∫B(x0,2j+1s)∖B(x0,2js)‍fyq′x-yn-1q′dy1/q′≤2j+1s1-n∫B(x0,2j+1s)‍fyq′dy1/q′.
It follows from (47), (50), (51), and (52) that
(53)sup⁡x∈B(x0,s)|μΩf2x|≤C∑j=1∞‍2j+1s-(n/q′)∫B(x0,2j+1s)‍fyq′dy1/q′.
Since q′<p<∞, ν=p/q′>1. Thus from the Hölder inequality we get that
(54)sup⁡x∈B(x0,s)μΩf2x≤C∑j=1∞‍2j+1s-n/q′fLp(ω,B(x0,2j+1s))·ω-(1/p)Lq′ν′(B(x0,2j+1s))≤C∑j=1∞‍∫2j+1s2j+2s‍2j+1s-(1+n/q′)·fLp(ω,B(x0,r))ω-(1/p)Lq′ν′(B(x0,r))dr≤∫2s∞‍fLp(ω,B(x0,r))·ω-(1/p)Lq′ν′(B(x0,r))drr1+n/q′.
Note that ω∈Aν. It follows from the Aν condition that
(55)ωBx0,rω1-ν′Bx0,rν-1≤C.
That is,
(56)ω-(1/p)Lq′ν′(B(x0,r))≤rn/q′ωBx0,r-(1/p).
Then
(57)sup⁡x∈B(x0,s)μΩf2x≤C∫2s∞fLpω,Bx0,rωBx0,r-(1/p)drr.
Therefore
(58)μΩf2Lp(ω,B(x0,s))≤CωBx0,s1/p·∫2s∞fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
Combining (45), (49), and (58), the proof of Theorem 11 is completed.Proof of Theorem3.
Sincef∈Mφ1p(w,Rn), from (33) and the fact fLp(ω,B(x0,r)) is a nondecreasing function of r, we get
(59)fLp(ω,B(x0,r))essinf⁡0<r<t<∞φ1x0,tωBx0,t1/p≤esssup⁡0<r<t<∞⁡fLp(ω,B(x0,r))φ1x0,tωBx0,t1/p≤esssup⁡0<t<∞⁡fLp(ω,B(x0,t))φ1x0,tωBx0,t1/p≤fMφ1p(w,Rn).
For q′<p<∞, since (φ1,φ2) satisfy (13), we have
(60)∫s∞‍fLp(ω,B(x0,r))ωBx0,r-1/pdrr≤∫s∞‍fLp(ω,B(x0,r))essinf⁡r<t<∞φ1x0,tωBx0,t1/p·essinf⁡r<t<∞φ1x0,tωBx0,t1/pωBx0,r1/pdrr≤CfMφ1p(w,Rn)·∫s∞‍essinf⁡r<t<∞φ1x0,tωBx0,t1/pωBx0,r1/pdrr≤CfMφ1p(w,Rn)φ2(x0,s).
Then by (40), we get
(61)μΩfMφ2p(w,Rn)≤Csup⁡x0∈Rn,s>01φ2x0,s·1wBx0,s∫B(x0,s)‍μΩfypwydy1/p≤Csup⁡x0∈Rn,s>01φ2x0,s·∫s∞‍fLpω,Bx0,rωBx0,r-1/pdrr≤CfMφ1p(w,Rn).
This completes the proof of Theorem 3.


## 4. Proof of Theorem4

As in the proof of Theorem3, it suffices to prove the following result.Theorem 12.
Assume thatΩ∈Lq(Sn-1) with 1<q≤∞ and b∈BMO(Rn). Then, for every q′<p<∞, ω∈Ap/q′, there is a constant C independent of f such that
(62)b,μΩfLp(ω,B(x0,s))≤Cb*ωBx0,s1/p·∫2s∞‍1+ln⁡rsfLp(ω,B(x0,r))ωBx0,r-1/pdrr.Proof.
We representf as
(63)fy=f1+f2,f1y=fyχB(x0,2s)y,f2y=fyχ(B(x0,2s))cy.
Then
(64)b,μΩfLp(ω,B(x0,s))≤b,μΩf1Lp(ω,B(x0,s))+b,μΩf2Lp(ω,B(x0,s)).
By Theorem 2, we have
(65)b,μΩf1Lp(ω,B(x0,s))≤Cb*fLp(ω,B(x0,2s)).
As the proof of (45), we get
(66)b,TΩf1Lp(ω,B(x0,s))≤Cb*ωBx0,s1/p·∫2s∞‍fLpω,Bx0,rωBx0,r-1/pdrr.
We now turn to deal with the term[b,μΩ](f2)Lp(ω,B(x0,s)). For any given x∈B(x0,s), we have
(67)b,μΩf2x≤Cbx-bBx0,sμΩf2x+CμΩb-bB(x0,s)f2x=I1+I2.
By (48) and (57), for any Ω∈Lq(Sn-1), 1<q≤∞, we have
(68)I1≤Cby-bB(x0,s)·∫2s∞‍fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
Then from Lemma 10 we get
(69)I1Lp(ω,B(x0,s))≤Cb*ωBx0,s1/p·∫2s∞‍fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
WhenΩ∈L∞(Sn-1), by assumption, we have ω∈Ap, 1<p<∞. It follows from the Hölder inequality that
(70)sup⁡x∈B(x0,s)μΩb-bB(x0,s)f2x≤CΩL∞(Sn-1)∑j=1∞‍2j+1s-n·∫B(x0,2j+1s)‍by-bB(x0,s)fydy≤C∑j=1∞‍2j+1s-nfLpω,Bx0,2j+1s·b·-bBx0,sLp′(ω1-p′,B(x0,2j+1s))≤C∑j=1∞‍∫2j+1s2j+2s‍2j+1s-(1+n)fLp(ω,B(x0,r))·b·-bB(x0,s)Lp′(ω1-p′,B(x0,r))dr≤C∫2s∞‍fLp(ω,B(x0,r))·b·-bB(x0,s)Lp′(ω1-p′,B(x0,r))drr1+n.
By Lemma 9, we have ω1-p′∈Ap′(Rn). Then by Lemma 10 and (32),
(71)b·-bB(x0,s)Lp′(ω1-p′,B(x0,r))≤C∫B(x0,r)‍bz-bB(x0,s)p′ω1-p′zdz1/p′≤Cb*1+ln⁡rsω1-p′Bx0,r1/p′=Cb*1+ln⁡rsω-(1/p)Lp′(B(x0,r))≤Cb*rn1+ln⁡rsωBx0,r-(1/p).
Then
(72)sup⁡x∈B(x0,s)μΩb-bB(x0,s)f2x≤C∫2s∞‍1+ln⁡rsfLp(ω,B(x0,r))ωBx0,r-(1/p)dr.
Therefore,
(73)I2Lpω,Bx0,s≤Cb*ωBx0,s1/p·∫2s∞‍fLp(ω,B(x0,r))ωBx0,r-(1/p)drr.
WhenΩ∈Lq(Sn-1), it follows from (50), (51), and (52) that
(74)I2≤C∑j=1∞‍2j+1s-(n/q′)·∫B(x0,2j+1s)‍by-bB(x0,s)fyq′dy1/q′.
Set s=p/q′. Since ω∈As, from Lemma 9, we know ω1-s′∈As′. By Hölder inequality
(75)∫B(x0,2j+1s)‍by-bB(x0,s)q′fyq′dy1/q′≤CfLp(ω,B(x0,2j+1s))·b·-bB(x0,s)Lq′s′(ω1-s′,B(x0,2j+1s)).
Consequently,
(76)I2≤∑j=1∞‍∫2j+1s2j+2s‍2j+1s-(1+(n/q′))·fLp(ω,B(x0,2j+1s))·b·-bB(x0,s)Lq′s′(ω1-s′,B(x0,2j+1s))dr≤C∫2s∞‍fLp(ω,B(x0,r))·b·-bB(x0,s)Lq′s′(ω1-s′,r)drr1+n/q′.
Since ω1-s′∈As′, from (32), we know
(77)ω1-s′Bx0,r1/q′s′≤Crn/q′Bx0,r-(1/p).
Using Lemma 10 and the fact that ω∈As, we thus obtain
(78)∫B(x0,2j+1s)‍by-bB(x0,s)q′s′ω1-s′ydy1/q′s′≤Cb*1+ln⁡rsω1-s′Bx0,r1/q′s′≤Cb*rn/q′1+ln⁡rsωBx0,r-(1/p).
Hence
(79)I2≤Cb*·∫2s∞‍1+ln⁡rsfLp(ω,B(x0,r))·ωBx0,r-(1/p)drr.
Therefore
(80)I2Lp(ω,B(x0,s))≤Cb*ωBx0,s1/p·∫2s∞‍1+ln⁡rsfLp(ω,B(x0,r))·ωBx0,r-(1/p)drr.Proof of Theorem4.
Sincef∈Mφ1p(w,Rn) and (φ1,φ2) satisfy (15), for q′<p<∞, we have
(81)∫s∞‍1+ln⁡rsfLp(ω,B(x0,r))ωBx0,r-(1/p)drr≤∫s∞‍1+ln⁡rsfLp(ω,B(x0,r))essinf⁡r<t<∞φ1x0,tωBx0,t1/p·essinf⁡r<t<∞φ1x0,tωBx0,t1/pωBx0,r1/pdrr≤CfMφ1p(w,Rn)∫s∞‍1+ln⁡rs·essinf⁡r<t<∞φ1x0,tωBx0,t1/pωBx0,r1/pdrr≤CfMφ1p(w,Rn)φ2x0,s.
Then from (62) we get
(82)b,μΩfMφ2p(w,Rn)≤Csup⁡x0∈Rn,s>01φ2x0,s·1wBx0,s∫B(x0,s)‍b,μΩfypwydy1/p≤Cb*sup⁡x0∈Rn,s>01φ2x0,s·∫s∞‍1+ln⁡rsfLp(ω,B(x0,r))·ωBx0,r-(1/p)drr≤Cb*fMφ1p(w,Rn).



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*Source: 450145-2015-01-15.xml*