# Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

Authors: Xukui Shao; Shuangping Tao
Journal: Journal of Function Spaces
 (2019)

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## Abstract

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent functionp(·) equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.


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## Body

## 1. Introduction and Main Results

LetΩ(x,z′)∈L∞(Rn)×Ls(Sn-1)(1<s≤∞). It satisfies(1)ΩL∞Rn×LsSn-1=supx∈Rn⁡∫Sn-1Ωx,z′sdz′1/s<∞;(2)Ωx,μz=Ωx,z,∫Sn-1Ωx,z′dz′=0,for  anyx,z∈Rn,μ>0,where Sn-1=x∈Rn:x=1 is equipped with the Lebesgue measure dz′.In 1955, Calderón and Zygmund [1] investigated the Lp boundedness of the singular integral operator with variable kernels. They found that these operators connect closely with the problem about the second-order linear elliptic equations with variable coefficients. Muckenhoupt and Wheeden [2] subsequently introduced the fractional integral operator with variable kernels, which is defined by(3)TΩ,αfx=∫RnΩx,x-yx-yn-αfydy.Muckenhoupt and Wheeden [2] also gave the (Lp,Lq) boundedness with the power weight of TΩ,α.Theorem A (see [2]).
Let0<α<n, 1<p<n/α, and 1/q=1/p-α/n. Suppose that Ω(x,z)∈L∞(Rn)×Ls(Sn-1) with s>p′. Then there exists a constant C>0 independent of f such that(4)TΩ,αfLq≤CΩL∞Rn×LsSn-1fLp.It is well known that the fractional integral operators play an important role in harmonic analysis, which greatly promotes the process of the intersection and integration of harmonic analysis and other disciplines.Given a local integrable functionb, the corresponding m-order commutator is defined by(5)TΩ,α,bmfx=∫RnΩx,x-yx-yn-αbx-bymfydy.In recent years, the boundedness of singular integral operators with variable kernels has been widely concerned. For example, Ding Lin and Shao [3] obtained the Lp boundedness of Marcinkiewicz integral operator μΩ with variable kernels; Wang [4] proved the boundedness properties of singular integral operators TΩ, fractional integral TΩ,α, and parametric Marcinkiewicz integral μΩρ with variable kernels on the Hardy spaces Hp(Rn) and weak Hardy spaces WHp(Rn). For the related results of the singular integral operator with variable kernels, the reader is refereed to [5–8].After the paper [9], the variable exponent space theory has been rapidly developed in the past 20 years due to its extensive application in the fields of fluid dynamics and differential equations with nongrowth conditions. For example, in [10], the authors considered the boundedness of higher order commutators of Marcinkiewicz integral on the Lebesgue space with variable exponent. Ho [11] has given some sufficient conditions for the boundedness of fractional integral operators and singular integral operators in Morrey space with variable exponent Mp(·),u; he also obtained the weak type estimates of fractional integral operators on Morrey space with variable exponent and singular integral operators on Morrey-Banach space (see [12, 13]). In 2016, Tao and Li [14] proved the boundedness of Marcinkiewicz integral and it is commutators on Morrey space with variable exponent. In [15], the (Lp(·)(Rn),Lp(·)(Rn))-boundedness of the parameterized Littlewood-Paley operators and their commutators was given by Wang and Tao.Motivated by the above research, in this paper, we will consider the boundedness of the fractional integral operators and their commutators with variable kernels on variable exponent weak Morrey spaces, where the smoothness condition onΩ has been removed.Before stating the main results of this article, we first recall some necessary definitions and notations.For0<β≤1, the Lipschitz space Lipβ(Rn) is defined as(6)LipβRn=f:fLipβ=supx,y∈Rn;x≠y⁡fx-fyx-yβ<∞.B M O ( R n ) space is defined as(7)BMORn=f∈Lloc1Rn:fBMORn=supQ⁡1Q∫Qfx-fQdx<∞,where the supreme is taken over all cubes Q⊂Rn, and fQ=(1/Q)∫Qf(y)dy.For anyx∈Rn and r>0, let B(x,r)=z∈Rn:z-x≤r. E denotes the Lebesgue measure of E⊂Rn and χE its characteristic function.DefineP(Rn) to be the set of p(·):Rn→[1,∞) such that(8)1≤p-≔essinfx∈Rn⁡px,p+≔esssupx∈Rn⁡px<∞.Letp(·)∈P(Rn). The Lebesgue space with variable exponent Lp(·)(Rn) consists of all Lebesgue measurable function f satisfying(9)fLp·Rn=inf⁡η>0:∫Rnfxηpxdx≤1<∞.Lp(·)(Rn) becomes a Banach function space when equipped with the Luxemburg-Nakano norm above.The weak Lebesgue space with variable exponentWLp(·)(Rn) consists of all Lebesgue measurable function f satisfying(10)fWLp·Rn=supη>0⁡ηχx:f>ηLp·Rn.It is easy to see that·WLp(·)(Rn) is a quasi-norm; that is, for any g1,g2∈WLp(·)(Rn), we have(11)g1+g2WLp·Rn≤2g1WLp·Rn+g2WLp·Rn.Letf∈Lloc1(Rn), the Hardy-Littlewood maximum operator is defined by(12)Mfx=supr>0⁡1Bx,r∫Bx,rfydy.LetB(Rn) denote the set of p(·)∈P(Rn) which satisfies the following conditions:(13)px-py≤C-log⁡x-y,x-y≤12and(14)px-py≤Clog⁡x+e,y≥x.It is proved that the Hardy-Littlewood maximal operator M is bounded on Lp(·)(Rn) as p(·)∈B(Rn) satisfies p->1 in [16].Remark 1.
For anyp(·)∈B(Rn) and λ>1, by Jensen’s inequality, we have λp(·)∈B(Rn). See Remark 2.13 in [17].
We say an order pair of variable exponents function(p(·),q(·))∈Bα(Rn), if p(·)∈P(Rn), 0<α<n/p+, and(15)1p·-1q·=αnwith q(·)(n-α)/n∈B(Rn).Remark 2.
(1) The condition((n-α)/n)q(·)∈B(Rn) is equivalent to saying that there exists r0 with n/(n-α)<r0<∞ such that q(·)/q0∈B(Rn).
(2)((n-α)/n)q(·)∈B(Rn) implies q(·)∈B(Rn).Definition 3 (see [18]).
Letq(·)∈P(Rn) and u(x,r):Rn×(0,∞)→(0,∞) be a Lebesgue measurable function; we say u∈Wq(·) if there exists a constant C>0 such that for any x∈Rn and r>0, u fulfills(16)∑j=0∞χBx,rLq·RnχBx,2j+1rLq·Rnux,2j+1r<Cux,r.Definition 4 (see [18]).
Letp(·)∈P(Rn) and u(x,r):Rn×(0,∞)→(0,∞). The Morrey space with variable exponent Mp(·),u(Rn) is defined by(17)Mp·,uRn=f:fMp·,u=supz∈Rn,r>0⁡1uz,rχBz,rfLp·Rn<∞.The weak Morrey space with variable exponent WMp(·),u(Rn) is defined by(18)WMp·,uRn=f:fWMp·,u=supz∈Rn,r>0⁡1uz,rχBz,rfWLp·Rn<∞.
For anyf,g∈WMp(·),u(Rn), one has(19)f+gWMp·,uRn≤2fWMp·,uRn+gWMp·,uRn.That is, ·WMp(·),u(Rn) is a quasi-norm.The main results of this paper are stated as follows.Theorem 5.
Let0<α<n, p(·)∈P(Rn) satisfying (13), (14), and (15) with 1=p-≤p+<n/α. If u∈Wq(·) and Ω(x,z) satisfies (1) and (2), then there exists a constant C>0 such that for any f∈Mp(·),u,(20)TΩ,αfWMq·,u≤CfMp·,u.In particular, we have the strong type result as TΩ,α: Mp(·),u→Mq(·),u(1<p-≤p+<n/α).Theorem 6.
Letb∈Lipβ(Rn) with 0<β≤1. Then under the hypotheses of Theorem 5, there exists a constant C>0 such that for any f∈Mp(·),u,(21)TΩ,α,bmfWMq·,u≤CbLipβRnmfMp·,u.In particular, TΩ,α,bm is bounded from Mp(·),u to Mq(·),u as 1<p-≤p+<n/α.Theorem 7.
Suppose that∈BMO(Rn), 0<α<n, p(·)∈P(Rn) satisfying (13), (14), and (15) with 1=p-≤p+<n/α. If Ω(x,z) satisfies (1) and (2) and u meets with the following condition:(22)∑j=1∞j+1mχBx,rLq·RnχBx,2j+1rLq·Rnux,2j+1r<Cux,r.Then there exists a constant C>0 such that for any f∈Mp(·),u,(23)TΩ,α,bmfWMq·,u≤CbBMORnmfMp·,u.In particular, TΩ,α,bm is bounded from Mp(·),u to Mq(·),u as 1<p-≤p+<n/α.Throughout this paper, the letterC stands for a positive constant that is independent of the essential variables and not necessarily the same one in each occurrence.


## 2. Preliminaries Lemmas

In this section we shall give some lemmas which will be used in the proofs of our main theorems.Lemma 8 (see [19]).
Letp(·)∈B(Rn). Define p′(·) by 1/p(·)+1/p′(·)=1; then there exists a constant C>0 such that for any ball B, we have(24)C-1≤1BχBLp·RnχBLp′·Rn≤C.Lemma 9 (see [20]).
Letp(·)∈P(Rn) and s>p+. Defined q~(·) by 1/p(·)=1/s+1/q~(·), for all measurable functions f and g; we have(25)fgLp·Rn≤CfLq~·RngLsRn.Lemma 10 (see [16]).
Letp(·)∈P(Rn). Then fLp·Rn<C1 if and only if(26)∫Rnfxpxdx<C2.In particular, if either constant equals 1 we can take the other equal to 1 as well.By applying the similar method used in the proof of [21, Lemma 4], we can obtain the following result.Lemma 11.
Suppose that0<θ<min⁡α,n-α, x∈Rn. Then (27)TΩ,αfx≤Cn,α,θMΩ,α+θfx1/2MΩ,α-θfx1/2,where(28)MΩ,αfx=supr>0⁡1rn-α∫y≤rΩx,yfx-ydy.Lemma 12.
Let0<α<n, p(·)∈B(Rn)1=p-≤p+<n/α. Suppose that Ω(x,z) satisfies (1) and (2); q(·) is defined by (15). Then there exists a constant C>0 such that for any f∈Lp(·)(Rn),(29)TΩ,αfWLq·Rn≤CfLp·Rn.Proof.
Letf∈Lp(·)(Rn). Without loss of generality we may assume that fLp(·)(Rn)=1. Noting that 1=p-≤p+<n/α<∞, we only need to prove that, for any η>0,(30)supη>0⁡ηχTΩ,αf>ηLq·Rn≤C.Since q+<∞, by Lemma 10 it will suffice to prove that(31)∫TΩ,αf>ηηqxdx≤C.
Fix aθ with 0<θ<min{α,n-α} satisfies 1+(θ/n)q+<2. Let r(x)=2/(1+θq(x)/n). Then r->1. Thus, we have(32)1p·-2r·q·=α-θn,1p·-2r′·q·=α+θn.By Lemma 11 and Young’s inequality, it has (33)TΩ,αfx≤CMΩ,α+θfx1/2MΩ,α-θfx1/2≤CMΩ,α+θfxrx/2rx+CMΩ,α-θfxr′x/2r′x≤Cr+/2MΩ,α+θfxrx/2+Cr-/2MΩ,α-θfxr′x/2.Noting that r->1, r+′<∞, then(34)∫TΩ,αf>ηηqxdx≤∫MΩ,α+θfrx/2>η/2Cηqxdx+∫MΩ,α-θfr′x/2>η/2Cηqxdx.
Referring to the argument used in the proofs of [16, Theorems 1.8 and 1.9], we can obtain the following inequalities:(35)∫MΩ,α+θfrx/2>η/2Cηqxdx≤C,∫MΩ,α-θfr′x/2>η/2Cηqxdx≤C.
Then, we have(36)∫TΩ,αf>ηηqxdx≤C.Thus(37)supη>0⁡ηχTΩ,αf>ηLq·Rn≤CfLp·Rn.
Lemma12 is proved.Lemma 13.
Suppose thatb∈Lloc(Rn), θ>0 with 0<α-θ<α+θ<n. Then, for any x∈Rn,(38)TΩ,α,bmfx≤Cn,α,θMΩ,α+θ,bfx1/2MΩ,α-θ,bfx1/2,where(39)MΩ,α,bfx=supr>0⁡1rn-α∫y≤rΩx,yfx-ybx-bymdy.With the similar argument in the proof of [22, Lemma 2], it is easy to draw the above conclusion; the details are omitted here.Lemma 14.
Letb∈Lipβ(Rn), 0<β≤1, p(·)∈B(Rn) with 0<(α+mβ)/n≤1/p+. q(·) is defined as(40)1px-1qx=α+mβn.If Ω(x,z) satisfies (1) and (2), then there exists a constant C>0 such that for any f∈Lp(·)(Rn),(41)TΩ,α,bmfWLq·Rn≤CbLipβRnmfLp·Rn.Lemma 15.
Letb∈BMO(Rn),p(·)∈B(Rn) with 0<α/n≤1/p+. If Ω(x,z) satisfies (1) and (2), q(·) as defined in (15), then there exists a constant C>0 such that for any f∈Lp(·)(Rn),(42)TΩ,α,bmfWLq·Rn≤CbBMORnmfLp·Rn.By applying Lemma13, we can prove Lemmas 14 and 15 with the similar way used in the proof of Lemma 12. Thus, we omit the details here.Lemma 16 (see [23]).
Let0<α<n; if (p(·),q(·))∈Bα(Rn), then there exists a constant C>0 such that for all balls B,(43)χBLp′·RnχBLq·RnB1-α/n≤C.Lemma 17 (see [24]).
Supposeb∈BMO(Rn),p(·)∈B(Rn), m to be a positive integer, i,j∈Z with i<j, and then(44)C-1bBMOm≤supB⁡b-bBmχBLq·RnχBLq·Rn≤bBMOm,b-bBimχBjLq·Rn≤Cj-imbBMOmχBLq·Rn,where B=B(x,r),Bi=B(x,2ir).


## 3. Proofs of Theorems5–7

Proof of Theorem5.
Letf∈Mp(·),u(Rn). For any z∈Rn, j∈N∖0 and r>0, write (45)fx=f0x+∑j=1∞fjx,where f0=fχB(z,2r) and fj=fχB(z,2j+1r)∖B(z,2jr).
Lemma12 immediately implies that (46)χBz,rTΩ,αf0WLq·Rn≤Cf0Lp·Rn=CfχBz,2rLp·Rn.
Note that there exists a constantC>0 such that (47)χBz,2r≤CMχBz,r.In the view of 1≤p-≤p+<∞, we have 1<q-≤q+<∞, so the Hardy-Littlewood maximal operator M is bounded on Lq(·)(Rn); it follows that (48)χBz,2rLq·Rn≤CMχBz,rLq·Rn≤CχBz,rLq·Rn.Since(49)uz,2r<Cuz,r,then we have(50)1uz,rχBz,rTΩ,αf0WLq·Rn≤C1uz,2rχBz,2rfLp·Rn≤CfMp·,u.
On the other hand, for anyj≥1 and x∈B(z,r), by Hölder’s inequality, we have (51)TΩ,αfjx=∫Bz,2j+1r∖Bz,2jrΩx,x-yx-yn-αfjydy≤C2jr-n-α∫Bz,2j+1r∖Bz,2jrΩx,x-yfydy≤C2jr-n-αχBz,2j+1rfLp·RnΩx,x-yχBz,2j+1r∖Bz,2jrLp′·Rn≕C2-jn-αr-n-αχBz,2j+1rfLp·RnIj.
According to Lemmas9, 16, and BLp(·)≈B1/p(·) in [20, Theorem 4.5.9], define 1/p′~·=1/p′·-1/s; it yields that (52)χBz,2j+1rLp′~·Rn≈Bz,2j+1r1/p′~·≈χBz,2j+1rLp′·RnBz,2j+1r-1/s≤CBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn.Thus (53)Ij=Ωx,x-yχBz,2j+1r∖Bz,2jrLp′·Rn≤Ωx,x-yχBz,2j+1r∖Bz,2jrLsRnχBz,2j+1rLp′~·Rn≤CΩx,x-yχBz,2j+1r∖Bz,2jrLsRnBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn≤C∫Bz,2j+1r∖Bz,2jrΩx,x-ysdy1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn≤C∫Bx-z,2j+1r∖Bx-z,2jrΩx,ysdy1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn.
Noting that for anyx∈B(x-z,2j+1r)∖B(x-z,2jr), x∈B(z,r), we have (54)y≤y-x-z+x+z≤y-x-z+x+z≤2j+2rand (55)y≥y-x-z+x+z≥y-x-z-x+z≥2j-1r.Therefore (56)Bx-z,2j+1r∖Bx-z,2jrx⊂B0,2j+2r∖B0,2j-1r.
For anyj≥1, we can get (57)Ij≤C∫B0,2j+2r∖B0,2j-1rΩx,x-ysdy1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn=C∫2j-1r2j+2r∫Sn-1Ωx,y′sλn-1dy′dλ1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn≤C∫2j-1r2j+2rλn-1∫Sn-1Ωx,y′sdy′dλ1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn.
SinceΩ(x,z)∈L∞(Rn)×Ls(Sn-1), from (1) and (2), it follows that (58)Ij≤C2jrn/sΩL∞Rn×LsSn-1Bz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn=C2jrn/s2j+1r-n/s-αΩL∞Rn×LsSn-1Bz,2j+1rχBz,2j+1rLq·Rn≤C2jr-αΩL∞Rn×LsSn-1Bz,2j+1rχBz,2j+1rLq·Rn.
Then(59)TΩ,αfjx≤C2-jn-αr-n-α2jr-αχBz,2j+1rfLp·RnBz,2j+1rχBz,2j+1rLq·Rn≤C2-jnr-nχBz,2j+1rfLp·RnBz,2j+1rχBz,2j+1rLq·Rn≤CχBz,2j+1rfLp·RnχBz,2j+1rLq·Rn.Therefore,(60)χBz,rx∑j=1∞TΩ,αfjx≤CχBz,rx∑j=1∞χBz,2j+1rfLp·χBz,2j+1rLq·.
Applying the quasi-norm·WLq(·) on both sides of the above inequality, then (61)χBz,r∑j=1∞TΩ,αfjWLq·≤CχBz,rWLq·∑j=1∞fχBz,2j+1rLp·χBz,2j+1rLq·≤C∑j=1∞uz,2j+1rχBz,rWLq·χBz,2j+1rLq·fMp·,u.Note that(62)χx∈Rn:χEx>ηLp·=χELp·,0<η<1,0,η>1.Then(63)χEWLp·=supη>0⁡ηχx∈Rn:χEx>ηLp·=sup0<η<1⁡ηχELp·=χELp·.According to (63), for some C>0 independent of B(z,r), we have that(64)1uz,rχBz,r∑j=1∞TΩ,αfjWLq·≤C∑j=1∞uz,2j+1ruz,rχBz,rWLq·χBz,2j+1rLq·fMp·,u≤C∑j=1∞uz,2j+1ruz,rχBz,rLq·χBz,2j+1rLq·fMp·,u.Remark 1 and (50) yield that (65)1uz,rχBz,rTΩ,αfWLq·≤C1uz,rχBz,rTΩ,αf0WLq·+1uz,rχBz,r∑j=1∞TΩ,αfjWLq·≤C∑j=0∞uz,2j+1ruz,rχBz,rLq·χBz,2j+1rLq·fMp·,u≤CfMp·,u.By taking the supremum over B(z,r), we obtain (66)TΩ,αfWMq·,u≤CfMp·,u.Proof of Theorem6.
Letb∈Lipβ and f∈Mp(·),u(Rn). Using the same decomposition of f(x) as in the proof of Theorem 5, by Lemma 14, we have (67)TΩ,α,bmf0WLq·Rn≤CbLipβmf0Lp·Rn.Thus, we get by (49) (68)1uz,rχBz,rTΩ,α,bmf0WLq·Rn≤C1uz,2rχBz,rTΩ,α,bmf0WLq·Rn≤CbLipβm1uz,2rfχBz,2rLp·Rn≤CbLipβmfMp·,uRn.
For anyj∈N∖0, b∈Lipβ, we can obtain by using Hölder’s inequality, (69)TΩ,α,bmfjx≤∫Bz,2j+1r∖Bz,2jrΩx,x-yx-yn-αbx-bymfydy≤bLipβm∫Bz,2j+1r∖Bz,2jrΩx,x-yx-yn-α-mβfydy≤CbLipβm2jr-n-α-mβ∫Bz,2j+1r∖Bz,2jrΩx,x-yfydy≤CbLipβm2jr-n-α-mβχBz,2j+1rfLp·Rn×Ωx,x-yχBz,2j+1r∖Bz,2jrLp′·Rn≕CbLipβm2jr-n-α-mβχBz,2j+1rfLp·RnIIj.Note that x∈B(x-z,2j+1r)∖B(x-z,2jr), x∈B(z,r); we have (70)Bx-z,2j+1r∖Bx-z,2jrx⊂B0,2j+2r∖B0,2j-1r.
Similar to the estimate ofIj, by the fact of 1/p(x)-1/q(x)=(α+mβ)/n, we have that (71)IIj≤∫B0,2j+2r∖B0,2j-1rΩx,x-ysdy1/sχBz,2j+1rLp′~·Rn=∫2j-1r2j+2r∫Sn-1Ωx,y′sλn-1dy′dλ1/sχBz,2j+1rLp′·RnBz,2j+1r-1/s≤∫2j-1r2j+2rλn-1∫Sn-1Ωx,y′sdy′dλ1/sχBz,2j+1rLq′·RnBz,2j+1r-1/s-α+mβ/n≤C2jr-α-mβΩL∞Rn×LsSn-1χBz,2j+1rLq′·Rn.It follows from Lemma 8 that (72)IIj≤C2jr-α-mβΩL∞Rn×LsSn-1Bz,2j+1rχBz,2j+1rLq·Rn.The above estimates imply that (73)TΩ,α,bmfjx≤CbLipβm2jr-nχBz,2j+1rfLp·RnBz,2j+1rχBz,2j+1rLq·Rn≤CbLipβmχBz,2j+1rfLp·RnχBz,2j+1rLq·Rn.Therefore (74)χBz,r∑j=1∞TΩ,α,bmfjx≤CbLipβmχBz,r∑j=1∞fχBz,2j+1rLp·χBz,2j+1rLq·.
Applying the quasi-norm·WLq(·) on both sides of the above inequality, we get (75)χBz,r∑j=1∞TΩ,α,bmfjWLq·≤CbLipβmχBz,rWLq·∑j=1∞fχBz,2j+1rLp·χBz,2j+1rLq·≤CbLipβm∑j=1∞uz,2j+1rχBz,rWLq·χBz,2j+1rLq·fMp·,u.
It implies by (63) that (76)1uz,rχBz,r∑j=1∞TΩ,α,bmfjWLq·≤CbLipβm∑j=1∞uz,2j+1ruz,rχBz,rWLq·χBz,2j+1rLq·fMp·,u≤CbLipβm∑j=1∞uz,2j+1ruz,rχBz,rLq·χBz,2j+1rLq·fMp·,u.From (68), we can arrive at (77)1uz,rχBz,rTΩ,α,bmfWLq·≤CbLipβm1uz,rχBz,rTΩ,α,bmf0WLq·+1uz,rχBz,r∑j=1∞TΩ,α,bmfjWLq·≤CbLipβm∑j=0∞uz,2j+1ruz,rχBz,rLq·χBz,2j+1rLq·fMp·,u≤CbLipβmfMp·,u.By taking the supremum over z∈Rn and r>0, we conclude that (78)TΩ,α,bmfWMq·,u≤CbLipβmfMp·,u.This completes the proof of Theorem 6.Proof of Theorem7.
Letb∈BMO(Rn), f∈Mp(·),u(Rn). As in the proof in Theorem 5, write (79)fx=f0x+∑j=1∞fjx.
By Lemma15, we can obtain(80)TΩ,α,bmf0WLq·Rn≤CbBMOmf0Lp·Rn.From (49), we get (81)1uz,rχBz,rTΩ,α,bmf0WLq·Rn≤CbBMOm1uz,2rfχBz,2rLp·Rn.That is,(82)1uz,rχBz,rTΩ,α,bmf0WLq·Rn≤CbBMOmfMp·,uRn.
Furthermore, for anyj∈N∖0, we have (83)TΩ,α,bmfjx≤C∫Bz,2j+1r∖Bz,2jrΩx,x-yx-yn-αbx-bBz,rmfydy+C∫Bz,2j+1r∖Bz,2jrΩx,x-yx-yn-αbBz,r-bymfydy=J1+J2.For J1, Lemmas 9, 16 and Hölder’s inequality assure that (84)J1≤CΩx,x-yx-yn-αχBz,2j+1r∖Bz,2jrLp′·Rn×χBz,2j+1rfLp·Rnbx-bBz,rm≤CΩx,x-yx-yn-αχBz,2j+1r∖Bz,2jrLsRnχBz,2j+1rLp′~·Rn×χBz,2j+1rfLp·Rnbx-bBz,rm≤C∫B0,2j+2r∖B0,2j-1rΩx,x-ysx-ysn-αdy1/sχBz,2j+1rLp′~·Rn×χBz,2j+1rfLp·Rnbx-bBz,rm≤C∫2j-1r2j+2r∫Sn-1Ωx,y′sλsn-α-n-1dy′dλ1/sχBz,2j+1rLp′~·Rn×χBz,2j+1rfLp·Rnbx-bBz,rm≤C2j-1(n/s-n-αr-n-α+n/sΩL∞Rn×LsSn-1χBz,2j+1rLp′~·Rn×χBz,2j+1rfLp·Rnbx-bBz,rm≤C2j-1(n/s-n-αrα-n+n/sΩL∞Rn×LsSn-1χBz,2j+1rLp′·Rn×Bz,2j+1r-1/sχBz,2j+1rfLp·Rnbx-bBz,rm≤C2j-1(n/s-n-αrα-n+n/sΩL∞Rn×LsSn-1Bz,2j+1r1-α/nχBz,2j+1rLq·Rn×Bz,2j+1r-1/sχBz,2j+1rfLp·Rnbx-bBz,rm≤CχBz,2j+1rfLp·RnχBz,2j+1rLq·Rnbx-bBz,rm.Note that (85)bx-bBz,rm=1Bz,r∫Bz,rbx-bBz,rmdy≤1Bz,rbx-bBz,rmχBz,rLq·RnχBz,rLq′·Rn.Then according to Lemma 17, we have (86)bx-bBz,rm≤1Bz,rbx-bBz,rmχBz,rLq·RnχBz,rLq′·Rn≤CbBMOm1Bz,rχBz,rLq·RnχBz,rLq′·Rn≤CbBMOm.
Hence(87)J1≤CbBMOmχBz,2j+1rfLp·RnχBz,2j+1rLq·Rn.
Now turn to estimateJ2. By Lemmas 9, 17 and Hölder’s inequality, we have (88)J2≤CΩx,x-yx-yn-αby-bBz,rmχBz,2j+1r∖Bz,2jrLp′·Rn×χBz,2j+1rfLp·Rn≤CΩx,x-yx-yn-αχBz,2j+1r∖Bz,2jrLsRnby-bBz,rmχBz,2j+1rp′~·Rn×χBz,2j+1rfLp·Rn≤C∫Bz,2j+1r∖Bz,2jrΩx,x-ysx-ysn-αdy1/sj+1mbBMOmχBz,2j+1rp′~·Rn×χBz,2j+1rfLp·Rn≤Cj+1mbBMOm∫B0,2j+2r∖B0,2j-1rΩx,x-ysx-ysn-αdy1/sχBz,2j+1rLp′·Rn×Bz,2j+1r-1/sχBz,2j+1rfLp·Rn.Thus applying Lemma 16 leads to (89)J2≤Cj+1mbBMOm∫2j-1r2j+2r∫Sn-1Ωx,y′sλsn-α-n-1dy′dλ1/sBz,2j+1r1-1/s-α/nχBz,2j+1rLq·Rn×χBz,2j+1rfLp·Rn≤Cj+1mbBMOmΩL∞Rn×LsSn-12j-1(n/s-n-αrα-n+n/s×Bz,2j+1r1-1/s-α/nχBz,2j+1rLq·RnχBz,2j+1rfLp·Rn≤Cj+1mbBMOmχBz,2j+1rfLp·RnχBz,2j+1rLq·Rn.
The estimates ofJ1 and J2 imply that (90)χBz,r∑j=1∞TΩ,α,bmfjx≤CbBMOmχBz,r∑j=1∞j+1mfχBz,2j+1rLp·χBz,2j+1rLq·.We apply the quasi-norm ·WLq(·) on both sides of the above inequality (91)χBz,r∑j=1∞TΩ,α,bmfjWLq·≤CbBMOmχBz,rWLq·∑j=1∞j+1mfχBz,2j+1rLp·χBz,2j+1rLq·≤CbBMOm∑j=1∞j+1muz,2j+1rχBz,rWLq·χBz,2j+1rLq·fMp·,u.Denote I=(1/u(z,r))χB(z,r)∑j=1∞TΩ,α,bmfjWLq(·). Then we have (92)I≤CbBMOm∑j=1∞j+1muz,2j+1ruz,rχBz,rWLq·χBz,2j+1rLq·fMp·,u≤CbBMOmfMp·,u.It follows from (82) that (93)1uz,rχBz,rTΩ,α,bmfWLq·≤CbBMOm1uz,rχBz,rTΩ,α,bmf0WLq·+1uz,rχBz,r∑j=1∞TΩ,α,bmfjWLq·≤CbBMOmfMp·,u.Thus, by taking the supremum over z∈Rn and r>0, we obtain (94)TΩ,α,bmfWMq·,u≤CbBMOmfMp·,u.
The proof of Theorem7 is completed.



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*Source: 7152346-2019-06-27.xml*