# Variableλ-Central Morrey Space Estimates for the Fractional Hardy Operators and Commutators

Authors: Amjad Hussain; Muhammad Asim; Fahd Jarad
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper aims to show that the fractional Hardy operator and its adjoint operator are bounded on central Morrey space with variable exponent. Similar results for their commutators are obtained when the symbol functions belong toλ-central bounded mean oscillation (λ-central BMO) space with variable exponent.


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## Body

## 1. Introduction

The boundedness of operators on function spaces is one of the core issues in harmonic analysis [1–3]. It is mainly because many problems in the theory of partial differential equations, in their simplified form, are reduced to the boundedness of operators on function spaces. It stimulates the research community to embark on such problems in this field. In this paper, we mainly obtain the boundedness of fractional Hardy operators [4]:(1)Hβgz=1zn−β∫t≤zgtdt,Hβ∗gz=∫t>zgttn−βdt,0≤β<n,z∈ℝn∖0,on variable exponent central Morrey spaces. In addition, commutators of these operators(2)b,Hβg=bHβg−Hβbg,b,Hβ∗g=bHβ∗g−Hβ∗bg,with symbol functions b in variable λ-central BMO spaces are shown bounded on central Morrey spaces with variable exponent. However, before stating our main results, we need to introduce the reader to some basic definitions and preliminary results regarding variable exponent function spaces.Notably, the function spaces with variable exponents have considerable importance in Harmonic analysis as well. Back in 1931, Orlicz [5] started the theory of variable exponent Lebesgue space. Musielak Orlicz spaces were defined and studied in [6]. The study of Sobolev and Lebesgue spaces with variable exponents in [7–11] further stimulated the subject. In the meantime, λ−central Morrey space, central BMO space, and associated function spaces have attractive applications by exploring estimates for operators along with their commutators [12–20]. Mizuta et al. defined the variable exponent nonhomogeneous λ-central Morrey space in [21]. The central BMO space first appeared in [22]. Meanwhile, the authors in [23] gave the definition of variable exponent central Morrey and λ-central BMO space along with some important results regarding the estimation of some operators. Recently, some publications [24–26] discussing the continuity of multilinear integral operators on these function spaces have added substantially to the existing literature on this topic.The one-dimensional Hardy operator was firstly defined by Hardy in [27] and is considered a classical operator in operator theory. Its mathematical form can be obtained from (1) by taking n=1 and β=0. Later on, different authors extended the definition of the one-dimensional Hardy operator to multidimensions in [28, 29]. As stated earlier, the fractional Hardy operator and its adjoint operator were introduced first in [4]. Following these publications, a flux of new results emerged discussing the boundedness of Hardy-type operators and their commutators on different function spaces [30–35]. The commutator operator also enjoyed a lot of attention from different zones of the globe [4,20,36–40]. However, the continuity of Hardy-type operators and their commutators on variable exponent function spaces took less attention by the research community worldwide [41–44]. The same is the case with central Morrey space with variable exponent. The present article aims to fill this gap by proving the boundedness of the fractional Hardy operator and its adjoint operator in this space. In addition, this article also includes new results discussing the boundedness of commutators generated by Hβ (or Hβ∗) and the λ-central BMO function b on the variable central Morrey space.Let us describe the framework of this paper. In Section2, we will remind some lemmas and propositions related to variable exponent function spaces. In Section 3 of this article, we will demonstrate the boundedness for Hardy operators and their commutators on central Morrey space with variable exponent. In Section 4, we shall investigate the similar estimates for the adjoint fractional Hardy operator and its commutators.


## 2. Function Spaces with Variable Exponents

In this section, we are going to introduce some notations and definitions related to the variable exponent function spaces. Throughout this article, we denote byB and χB the Lebesgue measure and characteristic function of a measurable set B⊂ℝn, respectively. Also, Bj=B0,2j=x∈ℝn:x≤2j with Aj=x∈ℝn:2j−1<x≤2j and χj=χAj for j∈ℤ. The notation g≈f implies that there exist two positive constants C1 and C2 such that C1f≤g≤C2f. Furthermore, E⊆ℝn represents an open set and p·:E⟶1,∞ is a measurable function, and p′· denotes the conjugate exponent of p· which satisfies(3)1p′·+1p·=1.The setPE consists of all p· and p′· such that(4)1<p−=essinfpx:x∈E<p+=esssuppx:x∈E<∞.The spaceLp· is a set of all measurable function f on the open set E, in such a way that, for positive η,(5)∫Efxηpxdx<∞,which becomes a Banach function space when equipped with the Luxemburg-norm(6)fLp·E=infη>0:∫Efxηpxdx≤1.Local version of variable exponent Lebesgue space is denoted byLlocp·E and is defined by(7)Llocp·E=f:f∈Lp·F∀compact subsetF⊂E.We useBℝn to denote a set containing p·∈Pℝn satisfying the condition that the Hardy-Littlewood maximal operator M: (8)Mf=supr>01Br∫Brfdy,where Br=y∈ℝn:x−y<r is bounded on Lp·ℝn.Proposition 1 (see [8,45]).
LetE denote an open set and p·∈PE fulfill the following inequalities:(9)px−py≤−Clogx−y,12≥x−y,(10)px−py≤Clogx+e,x≤y,then p·∈BE, where C is a positive constant independent of x and y.Lemma 1 (see [7]) (generalized Hölder inequality).
Letp·,p1·,p2·∈PE.(a)
Ifg∈Lp·E and f∈Lp′·E, then we have(11)∫Egxfx≤rpgLp·EfLp′·E,whererp=1+1/p−−1/p+.(b)
Ifg∈Lp1·E,f∈Lp2·E, and 1/p·=1/p1·+1/p2·, then we have(12)gfLp·E≤rp,p1gLp1·EfLp2·E,whererp,p1=1+1/p1−−1/p1+1/p−.Lemma 2 (see [46]).
Ifp·∈Bℝn, then there exist constants 0<δ<1 and a positive constant C such that for all balls B in ℝn and all measurable subsets S⊂B,(13)χBLp·ℝnχSLp·ℝn≤CBS,χSLp·ℝnχBLp·ℝn≤CSBδ.Remark 1.
Letp·∈Pℝn and meet conditions (9) and (10) in Proposition 1, then so does p′·. This implies that p·,p′·∈Bℝn. Therefore, using Lemma 2, we have a constant δ1∈0,1/p2+ such that the inequality(14)χSLp2·ℝnχBLp2·ℝn≤CSBδ1,is satisfied for all balls B⊂ℝn and for S⊂B. Similarly, if p1·,p1′·∈Bℝn, then by Lemma 2, we have constants δ2∈0,1/p1+, δ3∈0,1/p1′+ such that(15)χSLp1·ℝnχBLp1·ℝn≤CSBδ2,χSLp1′·ℝnχBLp1′·ℝn≤CSBδ3,for all balls B⊂ℝn and for S⊂B.Lemma 3 (see [46]).
Assuming thatp·∈Bℝn, for all balls B⊂ℝn and for a positive constant C, the following inequality holds:(16)C−1≤1BχBLp·ℝnχBLp′·ℝn≤C.Definition 1 (see [47]).
Letf∈Lloc1ℝn, set(17)bBMOℝn=supB1B∫Bbx−bBdx,where supremum is taken all over the ball B⊂ℝn and bB=B−1∫Bbydy. The function b is known as bounded mean oscillation if bBMOℝn<∞ and BMOℝn consist of all f∈Lloc1ℝn with BMOℝn<∞.Lemma 4 (see [48]).
Letp·∈Pℝn, then for all b∈BMOℝn and all l,i∈ℤ with l>i, we have(18)C−1bBMOℝn≤supB:Ball1χBLp·ℝnb−bBχBLp·ℝn≤CbBMOℝn,b−bBiχBlLp·ℝn≤Cl−ibBMOℝnχBlLp·ℝn.Definition 2 (see [23]).
Letp·∈Pℝn and λ∈ℝ. Then, the variable exponent central Morrey space B˙p·,λℝn is defined as(19)B˙p·,λℝn=f∈LLocp·ℝn:fB˙p·,λℝn<∞,where(20)fB˙p·,λℝn=supR>0fχB0,RLp·ℝnB0,RλχB0,RLp·ℝn.Definition 3 (see [23]).
Letp·∈Pℝn and λ<1/n. Then, the variable exponent λ-central BMO space CBMOp·,λℝn is defined as(21)CBMOp·,λℝn=f∈LLocp·ℝn:fCBMOp·,λℝn<∞,where(22)fCBMOp·,λℝn=supR>0f−fB0,RχB0,RLp·ℝnB0,RλχB0,RLp·ℝn.
While proving our main results, we control the boundedness of the fractional Hardy operator using the boundedness of the fractional integral operatorIβ:(23)Iβgz=∫ℝngtz−tn−βdt,on variable Lebesgue space. In this regard, we need the following proposition.Proposition 2 (see [49]).
Letp1·∈Pℝn,0<β<n/p1+ and define p2· by(24)1p2·=1p1·−βn.Then,(25)IβfLp2·ℝn≤CfLp1·ℝn.Proposition2 is useful in establishing the following Lemma (see [50]).Lemma 5.
Supposeβ, p1·p2· be as defined in Proposition 2, then(26)χBjLp2·ℝn≤C2−jβχBjLp1·ℝn,for all balls Bj=x∈ℝn:x≤2j with j∈ℤ.


## 3. Fractional Hardy Operator and Commutator

In this section, we present theorems on the boundedness of the fractional Hardy operator and commutators on central Morrey space with their proofs.Theorem 1.
Letp1·∈Pℝn and satisfy conditions (9) and (10) in Proposition 1. Define the variable exponent p2· by(27)1p2x=1p1x−βn.Ifλ2=λ1+β/n and λ2>−δ1+δ3, where δ1 and δ3 are the same constants as appeared in inequalities (14) and (15), then(28)HβfB˙p2·,λ2≤CfB˙p1·,λ1.Proof. By definition of the fractional Hardy operator and Lemma 1, it is easy to see that(29)Hβfxχkx≤1xn−β∫Bkftdtχkx≤C2−kn−β∑j=−∞kfjLp1·ℝnχjLp1′·ℝnχkx.Taking theLp2·ℝn norm on both sides, we have(30)HβfχkLp2·ℝn≤C2−kn−β∑j=−∞kfjLp1·ℝnχjLp1′·ℝnχkLp2·ℝn≤C2−kn−β∑j=−∞kfjLp1·ℝnχBjLp1′·ℝnχBkLp2·ℝn.Through the use of Lemma3 and the inequality (15), it is easy to see that(31)HβfχkLp2·ℝn≤C2kβ∑j=−∞kfjLp1·ℝnχBjLp1′·ℝnχBkLp2′·ℝn−1≤C2kβ∑j=−∞kfjLp1·ℝnχBjLp1′·ℝnχBkLp1′·ℝnχBkLp1′·ℝnχBkLp2′·ℝn−1≤C2kβ∑j=−∞k2nδ3j−kfjLp1·ℝnχBkLp1′·ℝnχBkLp2′·ℝn−1.In view of the condition1/p1′x=1/p2′x−β/n and Lemma 5, the last inequality reduces to the following inequality:(32)HβfχkLp2·ℝn≤CfB˙p1·,λ1ℝn∑j=−∞k2nδ3j−kBjλ1χjLp1·ℝn.Since(33)χjLp1·ℝn≈B1/p1·≈B1/p2·+β/n≈Bβ/nχjLp2·ℝn,therefore, from the inequality (32), we infer that(34)HβfχkLp2·ℝn≤CfB˙p1·,λ1ℝn∑j=−∞k2nδ3j−kBjλ1+β/nχjLp2·ℝn=CfB˙p1·,λ1ℝn∑j=−∞k2nδ3j−kBkλ2Bjλ2Bkλ2χkLp2·ℝnχjLp2·ℝnχkLp2·ℝn≤CfB˙p1·,λ1ℝn|Bk|λ2χkLp2·ℝn∑j=−∞k2nδ3j−kBjλ2Bkλ2χjLp2·ℝnχkLp2·ℝn.Finally, inequality (14) helps us to have(35)HβfB˙p2·,λ2ℝn≤CfB˙p1·,λ1ℝn∑j=−∞k2nj−kδ3+δ1+λ2.Sinceδ3+δ1+λ2>0, so we get(36)HβfB˙p2·,λ2ℝn≤CfB˙p1·,λ1ℝn.Theorem 2.
Let0<β<n and let p·,q·,r·∈Pℝn and satisfying conditions (9) and (10) in Proposition 1 with p·<n/β,p′·<r· and(37)1q·=1p·+1r·−βn.Let0<ν<1/n and −1/q+<μ. If μ=ν+λ+β/n, with max−ν+1,−δ1+δ3+β/n<λ, where δ1,δ3 are the same constants as appeared in inequalities (14) and (15), and b∈bCBMOr·,ν, then(38)b,HβfB˙q·,μ≤CbCBMOr·,νfB˙p·,λ.(39)b,Hβfx·χBx≤1xn−β∫y≤xbx−byfydy·χBx≤1xn−β∫y≤xbx−bBfydy·χBx+1xn−β∫y≤xby−bBfydy·χBx=A1+A2.Proof. We decompose the integral appearing in the commutator operator asLet us first estimateA1. By taking the variable Lebesgue space norm on both sides, we get(40)A1Lq·ℝn=b·−bBHβf·χB·Lq·ℝn.Taking into consideration the condition1/q·=1/s·+1/r·,1/s·=1/p·−β/n, the generalized Hölder inequality gives us the following estimation of A1:(41)A1Lq·ℝn≤b−bBχBLr·ℝnHβfχBLs·ℝn=CbCBMOr·,νBνχBLr·ℝnHβfB˙s·,σBσχBLs·ℝn,where σ=λ+β/n. Using the result of Theorem 1, we obtain(42)A1Lq·ℝn≤CbCBMOr·,νBν+σχBLr·ℝnχBLs·ℝnfB˙p·,λ.Here, it is easy to see that(43)χBLr·ℝnχBLs·ℝn≈B1/r·B1/s·=B1/q·≈χBLq·ℝn.Therefore, on account of the conditionμ=ν+σ, for A1, we have(44)A1Lq·ℝn≤CBμχBLq·ℝnbCBMOr·,νfB˙p·,λ.Next, we considerA2 for approximation:(45)A2=1xn−β∫y≤xby−bBfydy·χBx,which can be decomposed further as(46)A2=∑k=−∞01xn−β∑j=−∞k∫2jB∖2j−1Bby−bBfydy·χ2kB∖2k−1Bx≤∑k=−∞01xn−β∑j=−∞k∫2jB∖2j−1Bby−b2jBfydy·χ2kB∖2k−1Bx+∑k=−∞01xn−β∑j=−∞k∫2jB∖2j−1BbB−b2jBfydy·χ2kB∖2k−1Bx=A21+A22,where(47)A21=∑k=−∞02kBβ/n−1∑j=−∞k∫2jB∖2j−1Bby−b2jBfydy·χ2kB∖2k−1Bx.We define a new variablet· such that 1/t·=1/p′·−1/r·, then by the generalized Hölder inequality, we have(48)A21≤C∑k=−∞02kBβ/n−1χ2kB∖2k−1Bx∑j=−∞kby−b2jBχ2jBLr·fχ2jBLp·χ2jBLt·=C∑k=−∞02kBβ/n−1χ2kB∖2k−1Bx∑j=−∞kbCBMOr·,ν2jBνχ2jBLr·fB˙p·,λ2jBλχ2jBLp·χ2jBLt·=C∑k=−∞02kBβ/n−1χ2kB∖2k−1BxbCBMOr·,νfB˙p·,λ∑j=−∞k2jBν+λ2jB1/t·+1/p·+1/r·=CbCBMOr·,νfB˙p·,λ∑k=−∞02kBμχ2kB∖2k−1Bx.With the Lebesgue space with variable exponent norm on both sides, the above inequality takes the following form:(49)A21Lq·≤CbCBMOr·,νfB˙p·,λ∑k=−∞02kBμχ2kBLq·=CbCBMOr·,νfB˙p·,λ|B|1/q·|B|μ∑k=−∞02kμ+1/q·=CbCBMOr·,νfB˙p·,λχBLq·Bμ∑k=−∞02kμ+1/q+.Hence,(50)A21Lq·≤CbCBMOr·,νfB˙p·,λχBLq·Bμ.Finally, consider(51)A22=∑k=−∞02kBβ/n−1∑j=−∞k∫2jB∖2j−1BbB−b2jBfydy·χ2kB∖2k−1Bx.The factor bB−b2jB in the above inequality needs to be dealt with first. So,(52)bB−b2jB=∑i=j−1b2i+1B−b2iB=∑i=j−112iB∫2iBby−b2i+1Bdy≤C∑i=j−112iBb−b2i+1Bχ2i+1BLr·χ2i+1BLr′·.Next, Lemma 3 helps us to write(53)bB−b2jB≤C∑i=j−112iBb−b2i+1bχ2i+1BLr·2i+1Bχ2i+1BLr·≤C∑i=j−1bCBMOr·,ν2j+1Bν≤CbCBMOr·,ν∑i=j−12j+1Bν≤CbCBMOr·,ν2j+1Bνj.In turn,A22 satisfies the below inequality:(54)A22≤C∑k=−∞0χ2kB∖2k−1Bx2kBβ/n−1∑j=−∞kbCBMOr·,ν2j+1Bνjfχ2jBLp·χ2jBLp′·≤CbCBMOr·,ν∑k=−∞0χ2kB∖2k−1Bx2kBβ/n−1∑j=−∞k2j+1Bνjfχ2jBLp·2jBχ2jBLp·≤CbCBMOr·,ν∑k=−∞0χ2kB∖2k−1Bx2kBβ/n−1∑j=−∞k2jBν+1jfB˙p·,λ2jBλ≤CbCBMOr·,νfB˙p·,λ∑k=−∞0χ2kB∖2k−1Bx2kBβ/n−1∑j=−∞k2jBν+λ+1j≤CbCBMOr·,νfB˙p·,λ∑k=−∞0k2kBμχ2kB∖2k−1Bx.Ultimately, our last step would be applying the norm on both sides to get(55)A22Lq·≤CbCBMOr·,μfB˙p·,λ∑k=−∞0k2kBμχ2kBLq·≤CbCBMOr·,μfB˙p·,λ∑k=−∞0k2kBμ2kB1/q·≤CbCBMOr·,μfB˙p·,λ∑k=−∞0k2kμ+1/q·Bμ+1/q·≤CbCBMOr·,μfB˙p·,λBμχBLq·.Combining all the approximations ofA1, A2, A21, A22, we obtained the required result(56)b,HβfχBLq·ℝn≤CbCBMOr·,νfB˙p·,λBμχBLq·ℝn.


## 4. Adjoint Fractional Hardy Operator and Commutator

In this last section, we first establish the boundedness of adjoint fractional Hardy operator and then use it to prove the boundedness of commutator generated by this operator andλ-central BMO function b. The first result is as follows.Theorem 3.
Letp1·∈Pℝn and satisfying conditions (9) and (10) in Proposition 1, define the variable exponent p2· by(57)1p2x=1p1x−βn.Ifλ2=λ1+β/n and λ1<δ2−1−β/n, where δ2 is the same constant as appeared in inequality (15), then(58)Hβ∗fB˙p2·,λ2≤CfB˙p1·,λ1.Proof. Since(59)Hβ∗fxχkx≤∫ℝn∖Bkfttβ−ndtχkx≤C∑j=k+1∞2jβ−nfjLp1·ℝnχjLp1′·ℝnχkx,from which we infer that(60)Hβ∗fxχkLp2·ℝn≤C∑j=k+1∞2jβ−nfjLp1·ℝnχjLp1′·ℝnχkLp2·ℝn.Lemma3 guides us to have the following inequality:(61)Hβ∗fxχkLp2·ℝn≤C∑j=k+1∞2jβfjLp1·ℝnχjLp1·ℝn−1χkLp2·ℝn,which by Lemma 5 reduces to the following one:(62)Hβ∗fxχkLp2·ℝn≤C∑j=k+1∞2jβfjLp1·ℝnχjLp1·ℝn−12−kβχkLp1·ℝn≤C∑j=k+1∞2βj−kfjLp1·ℝnχkLp1·ℝnχjLp1·ℝn≤C∑j=k+1∞2j−kβ−nδ2fjLp1·ℝn,where we made use of inequality (15) in the last step of the above result. Hence, we obtain(63)Hβ∗fB˙p2·,λ2ℝn≤CfB˙p1·,λ1ℝn∑j=k+1∞2j−kβ−nδ2Bjλ1Bkλ2χBjLp1·ℝnχBkLp2·ℝn.Utilizing the condition λ2=λ1+β/n, a result similar to (33) and Lemma 2, we obtain(64)Hβ∗fB˙p2·,λ2ℝn≤CfB˙p1·,λ1ℝn∑j=k+1∞2nj−kβ/n+λ1−δ2+1.Finally, the series is convergent due to the fact thatλ1<δ2−1−β/n, and hence the result.Keeping in view the analysis made in the previous section, we only outline the proof of the following theorem without going into many details.Theorem 4.
Letp·,q·,r·∈Pℝn and satisfying conditions (9) and (10) in Proposition 1 with p·<n/β,p′·<r· and(65)1q·=1p·+1r·−βn.
Let0<ν<1/n and −1/q+<μ<0. If μ=ν+λ+β/n, with λ<minδ2−1−β/n,−ν+β/n, where δ2 is the same constant as appeared in inequality (14) and b∈bCBMOr·,ν, then(66)b,Hβ∗fB˙q·,μ≤CbCBMOr·,νfB˙p·,λ.
Proof. As in the previous section, we start from decomposing the integral:(67)b,Hβ∗fx·χBx≤∫y>xbx−byfyyn−βdy·χBx≤∫y>xbx−bBfyyn−βdy·χBx+∫y>xby−bBfyynβdy·χBx=D1+D2.
Following the steps taken to approximateA1 in Theorem 2, we directly estimate D1 as follows:(68)D1Lq·ℝn≤Cb−bBχBLr·ℝnHβ∗fχBLs·ℝn=CbCBMOr·,νBνχBLr·ℝnBσχBLs·ℝnHβ∗fB˙s·,σ,where σ=λ+β/n. Now, the result (43) and Theorem 3 help us to write(69)D1Lq·ℝn≤CbCBMOr·,νBμχBLq·ℝnfB˙p·,λ,
Next, comparingD2 with A2 of Theorem 2, we arrive at(70)D2≤∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−1∫2jB∖2j−1Bby−b2jBfydy+∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−1∫2jB∖2j−1BbB−b2jBfydy=D21+D22.and for the approximation ofD21, we follow a procedure similar to the one followed in the estimation of A21. Hence, we get(71)D21≤C∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−1by−b2jBχ2jBLr·fχ2jBLp·χ2jBLt·=C∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−1bCBMOr·,ν2jBνχ2jBLr·fB˙p·,λ2jBλχ2jBLp·χ2jBLt·=CbCBMOr·,νfB˙p·,λ∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−12jBν+λ2jB1/t·+1/p·+1/r·=CbCBMOr·,νfB˙p·,λ∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBμ=CbCBMOr·,νfB˙p·,λ∑k=−∞02k+1Bμχ2kB∖2k−1Bx.Eventually, it is easy to see that(72)D21Lq·≤CbCBMOr·,νfB˙p·,λ∑k=−∞02k+1Bμχ2kBLq·≤CbCBMOr·,νfB˙p·,λ∑k=−∞02k+1Bμ2kB1/q·≤CbCBMOr·,νfB˙p·,λBμ+1/q·∑k=−∞02k+1μ+1/q·≤CbCBMOr·,νfB˙p·,λBμχBLq·.Next, by virtue of inequality (53), D22 satisfies(73)D22≤C∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−1bCBMOr·,ν2j+1Bνjfχ2jBLp·χ2jBLp′·≤CbCBMOr·,ν∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−12j+1Bνjfχ2jBLp·2jBχ2jBLp·≤CbCBMOr·,ν∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBβ/n−12j+1Bν+1jfB˙p·,λ2jBλ≤CbCBMOr·,νfB˙p·,λ∑k=−∞0χ2kB∖2k−1Bx∑j=k+1∞2jBν+λ+β/nj≤CbCBMOr·,νfB˙p·,λ∑k=−∞0χ2kB∖2k−1Bx2k+1Bμk+1≤CbCBMOr·,νfB˙p·,λ∑k=−∞0k+12k+1Bμχ2kB∖2k−1Bx.To finish the estimation, we take norm on both sides of the above inequality to obtain(74)D22Lq·≤CbCBMOr·,νfB˙p·,λ∑k=−∞0k+12k+1Bμχ2kBLq·≤CbCBMOr·,νfB˙p·,λ∑k=−∞0k+12k+1Bμ2kB1/q·≤CbCBMOr·,νfB˙p·,λ∑k=−∞0k+12k+1μ+1/q·Bμ+1/q·≤CbCBMOr·,νfB˙p·,λ|B|μχBLq·.In the end, combining all the estimates ofD1, D2, D21, D22, we arrive at the following conclusive inequality:(75)b,Hβ∗fχBLq·ℝn≤CbCBMOr·,νfB˙p·,λBμχBLq·ℝn,which is as desired.



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*Source: 5855068-2022-05-09.xml*