# Double Points Local Hardy-Littlewood Maximal Operator

Authors: Futao Song; Na Ju
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A double points local Hardy-Littlewood maximal operatorMa,b,k,loc is defined and investigated in Euclidean spaces. It is proved that Ma,b,k,loc is bounded on Lpw when p>1 and from L1w to L1,∞w with weight function w∈Aa,b,k,loc, the class of double points local Ap weights which is larger than the Muckenhoupt Ap class and the local Ap weights defined by Lin and Stempak.


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## Body

## 1. Introduction

The Hardy-Littlewood maximal operatorMis defined on the class of the locally integrable functions onℝdby
(1)Mf(x)=supQ∋x1|Q|∫Q‍|f(x)|dx,x∈ℝd,
where the supremum is taken over all cubesQ⊂ℝdcontainingx, with sides parallel to the coordinate axes. It is one of the most fundamental and important operators in Fourier analysis and often used to majorize other important operators. Many papers are devoted to study the Hardy-Littlewood operator and its generalizations. Li et al. [1] gave the estimates of Hardy-Littlewood operator on the multilinear spaces. Lerner [2] showed that Hardy-Littlewood maximal operator was bounded on variableLpspaces. Gallardo [3] characterized the pairs of weights(u,w)for which the Hardy-Littlewood maximal operatorMsatisfies a weak type integral inequality.In this paper we considerℝd,d≥1, equipped with the metric induced by the norm
(2)∥x∥=max{|x1|,|x2|,…,|xd|},x=(x1,x2,…,xd).By a cube we always mean the usual closed cube with sides parallel to the coordinate axes; in terms of the metric,Q(x,r)={y∈ℝd:∥y-x∥≤r}denotes the cube centered atx∈ℝdwith “radius”r>0.|E|denotes the Lebesgue measure of a measurable setE⊂ℝd.For givena,b∈ℝdandk>0, denote by𝒪{a,b},k(y)the family of all cubes centered atywith radiusr≤kmin{∥y-a∥,∥y-b∥}, and define
(3)𝒪{a,b},k=⋃y∈ℝd𝒪{a,b},k(y).
Note that for everyQ∈𝒪{a,b},k,0<k<1, we havea,b∉Q. In particular,𝒪{a,b},k=𝒪kwhenais the origin andbtends to∞, where𝒪k=⋃y∈ℝd‍𝒪k(y)and𝒪k(y)is the collection of all the cubes centered atywith radiusr≤k∥y∥. The cubes in𝒪kare away from the origin, while the ones in𝒪{a,b},kare away from two pointsaandb.Given0<k<1, we define the (noncentered) double points local Hardy-Littlewood maximal operator
(4)M{a,b},k,locf(x)=supx∈Q∈𝒪{a,b},k1|Q|∫Q|f(t)|dt,hhhhhhhhhhhhhhhhhhhhh..x∈ℝd∖{a,b},
where the supremum is taken over all cubesQ∈𝒪{a,b},kcontainingx. It generalizes the local maximal operatorMk,locdefined in [4] for locally integrable functions onℝd∖{0}by
(5)Mk,locf(x)=supx∈Q∈𝒪k1|Q|∫Q‍|f(t)|dt,x∈ℝd∖{0}.The purpose of this paper is to investigate the double points local Hardy-Littlewood maximal operator. The paper was organized as follows.The definition of the classA{a,b},k,locpof double points localApweights is given in Section 2, as well as the coincidence ofA{a,b},k,locpfor0<k<1. That is,A{a,b},k,locp=A{a,b},m,locpfor0<k,m<1. Section 3 includes some lemmas that need to prove the main result Theorem 6 given in Section 4. We prove that for a weightwsatisfying the double points localApcondition, the operatorM{a,b},k,locis bounded onLp(w),1<p<∞and bounded fromL1(w)toL1,∞(w).Thatwis a weight onℝdmeans thatwis a nonnegative locally integrable function and finite almost everywhere.Lp(w)andL1,∞(w)denote the class of weighedLpspace and weighed weakL1space, with the norm∥f∥Lp(w)p=∫ℝd‍|f(x)|pw(x)dxand∥f∥L1,∞(w)=supt>0(t∫|f|>t‍w(x)dx), respectively.


## 2. Double Points LocalApWeights

For a weightwonℝd, we say thatw∈Apfor1<p<∞if there is a constantC>0such that
(6)supQ⊂ℝd(1|Q|∫Qw(x)dx)(1|Q|∫Q‍w(x)1-p′dx)p-1≤C,
here and below1/p+1/p′=1. We say thatw∈A1if there is a constantC>0such thatMw(x)≤Cw(x). The smallest constantCis called theApconstant ofwand denoted by∥w∥Ap. The definition ofApweights was first introduced by Muckenhoupt [5] ford=1. Muckenhoupt proved the characterization of the weighed type(p,p)for Hardy-Littlewood maximal operator. That is, forw∈Ap,p≥1, the operatorMis bounded onLp(w)whenp>1and bounded fromL1(w)intoL1,∞(w)whenp=1; moreover, we have
(7)∥Mf∥Lp(w)≤C∥w∥App′∥f∥Lp(w),f∈Lp(w),p>1,∥Mf∥L1,∞(w)≤C∥w∥A1∥f∥L1(w),f∈L1(w),
where the constantCis independent ofw(cf. [6]).The class of the localAk,locpweights is defined in [4] by considering the cubes in𝒪kand the corresponding constant is denoted as∥w∥Ak,locp. Lin and Stempak proved that forw∈Ak,locp, the local Hardy-Littlewood maximal operator is bounded fromLp(w)toLp(w)whenp>1and fromL1(w)toL1,∞(w).In order to prove the coincidence of the classesA{a,b},k,locpfor0<k<1, we need the following lemma.Definition 1.
Let0<k<1,1≤p<∞,a,b∈ℝd,andw∈Lloc(ℝd∖{a,b})a nonnegative function. If there exists a constantC>0such that
(8)supQ(1|Q|∫Q‍w(x)dx)1/p(1|Q|∫Q‍w(x)-p′/pdx)1/p′≤C
for1<p<∞, where the supremum takes over all the cubesQ∈𝒪{a,b},k, thenwis called the double points localApweight associated withaandb. And forp=1, the classA{a,b},k,loc1is defined by
(9)1|Q|∫Q‍w(x)dx≤Cessinx∈Qfw(x),
for allQ∈𝒪{a,b},k. Denote the collection of all the double points localApweights asA{a,b},k,locp=A{a,b},k,locp(ℝd). The smallest constantCis called the double points localk-Apconstant and denoted as∥w∥A{a,b},k,locp.According to the definition above, it is easy to see thatAp⊂A{a,b},k,locp. Furthermore, ifw∈Ap, then∥w∥A{a,b},k,locp≤∥w∥Ap. ButAp≠A{a,b},k,locp. In fact, letw(x)=min{∥x-a∥,∥x-b∥}; then for anyt∈ℝ, we havew(x)t∈A{a,b},k,locp, whilew(x)t∈Apif and only if-d<t<d(p-1)whenp>1and-d<t≤0whenp=1.On the other hand, define𝒪{a},k=⋃y∈ℝd‍𝒪{a},k(y), where𝒪{a},k(y)is the collection of the cubes centered atywith radius no more thank∥x-a∥. Then the cubes in𝒪{a},kare just the translations of the cubes in𝒪kalongamentioned in [4]. Replacing𝒪{a,b},kwith𝒪{a},kin Definition 1 gives the definition of the classA{a},k,locp. It is also easy to see thatA{a},k,locp⊂A{a,b},k,locp. The same process gives𝒪{b},kand consequently the classA{b},k,locp. We also have the relationA{b},k,locp⊂A{a,b},k,locp.Lemma 2.
Let0<k<1; for every0<r<1, there exists an=n(k,r)satisfying the following property: ifRandR^are the cubes centered atx(≠a,b)with radiusrmin{∥x-a∥,∥x-b∥}and(r+k(1-r))min{∥x-a∥,∥x-b∥}, respectively, there exist at mostncubesQj⊂R^andQj∈𝒪{a,b},k,1≤j≤n, coveringR^∖Rand|Qj∩R|≥C|R|, whereQj,1≤j≤nare the cubes with centers located on∂Rand radiusk(1-r)min{∥x-a∥,∥x-b∥}andC=min{1,(k(1-r)/2r)d}.Proposition 3.
Let0<k,m<1,and1≤p<∞. Then
(10)A{a,b},k,locp=A{a,b},m,locp.The proofs of Lemma2 and Proposition 3 are analogous to the proofs of Lemma2.1and Proposition2.2in [4], respectively, so we omit them here.By Proposition3, the classA{a,b},k,locpis independent of the choice ofk∈(0,1)and we will denote it byA{a,b},locp.


## 3. Preparatory Lemmas

For a given0<k<1,a∈ℝd,a=(a1,a2,…,ad), considering a grid ofℝd∖{a}based on the sequence{±k-s}s∈ℤ, which results in the collection𝒢aof all the rectangles
(11)Ga=(I1+{a1})×(I2+{a2})×⋯×(Id+{ad}),
where for a givens∈ℤ, eachIjis one of the intervals±[k-s,k-s-1],±[0,((1-k)/(1+k))k-s],or±[((1-k)/(1+k))k-s,k-s](called intervals of the first, second, or third type, resp.), but at least one of the intervals must be of the first type (if0≤u<v<+∞,-[u,v]denotes the interval[-v,-u],while+[u,v]is just[u,v]) If one of the intervals is of the second type, we say thatGais of the first category, otherwise of the second category.ForGa∈𝒢a, defineGa′to be the union of the cubes in𝒪{a},ksatisfyingQ∩Ga≠∅. IfGais of the first category, we defineGa′′to be the least cube containingGa′with the largest distance from the pointaalong each axis. IfGais of the second category, we defineGa,j′′to be the least cube containingGa,j′=Ga′∩Ej,Gawith the largest distance from the pointaalong each axis, where for everyj=1,2,…,d,
(12)Ej,Ga={{y∈ℝd:yj≥1-k1+kk-s+aj},whenIj⊂(0,+∞),{y∈ℝd:yj≤-1-k1+kk-s+aj},whenIj⊂(-∞,0).According to the definitions above, ifGais of the first category, we haveGa⊂Ga′⊂Ga′′,Ga′⊂{y∈ℝd:|yj-aj|≥((1-k)/(1+k))k-s}anda∉Ga′′. IfGais of the second category, thenGa⊂Ga,j′,a∉Ga,j′′,j=1,2,…,dandGa′∩∏j=1d‍Ej,Gac=∅which equalsGa′=⋃j=1d‍Ga,j′.For a pointb∈ℝd, repeat the process above, we can obtain the corresponding setsGb,Gb′,Gb′′,Gb,j′,Gb,j′′,and𝒢b.Fora,b∈ℝd(a≠b),
(13)ℝd∖{a,b}=(⋃Ga∈𝒢a‍Ga)∩(⋃Gb∈𝒢b‍Gb)=⋃Ga∈𝒢a,Gb∈𝒢b(Ga∩Gb).Define𝒢a,bto be the collection ofGa,b=Ga∩Gb, whereGa∈𝒢aandGb∈𝒢bandGa,b′to be the union of the cubesQ∈𝒪{a,b},ksuch thatQ∩Ga,b≠∅. In the proof of Theorem 6, we will use a result; that is,Ga,b′⊂(Ga′∩Gb′).In fact, for everyz∈Ga,b′, there exists a cubeQ0∈𝒪{a,b},k,Q0∩Ga,b≠∅such thatz∈Q0. The definition of𝒪{a,b},kgives that there exists ay∈ℝd∖{a,b}such thatQ0is the cube centered atywith radiusr≤k∥y-a∥. ThenQ0∈𝒪{a},kandQ0⊂Ga′; consequentlyz∈Ga′. Similarly,z∈Gb′. Thusz∈Ga′∩Gb′.Lemma 4.
Given0<k<1, there existsk0=k0(d,k)∈(0,1)such that for eachGa,b=Ga∩Gb∈𝒢a,b,Ga′′∈𝒪{a},k0(orGb′′∈𝒪{b},k0)ifGa(orGb)is of the first category, andGa,j′′∈𝒪{a},k0(orGb,j′′∈𝒪{b},k0),j=1,2,…,d, ifGa(orGb)is of the second category.Proof.
The constantskaandkbassociated with the pointsaandb, respectively, can be obtained by the same procedure as the one in the proof of Lemma3.1in [4]. Then we haveGa′′∈𝒪{a},kaandGb′′∈𝒪{b},kbwhenGaandGbare of the first category. Letk0=max{ka,kb}, thusGa′′∈𝒪{a},k0andGb′′∈𝒪{b},k0by the definition of𝒪{a},k0and𝒪{b},k0. The case whenGaandGbare of the second category is similar to the case whenGaandGbare of the first category.Lemma 5.
Given0<k<1. LetQandQ~be the cubes inℝdsatisfyingQ⊂Q~. IfQ~∈𝒪{a},k(or𝒪{b},k); thenQ∈𝒪{a},k(or𝒪{b},k).Lemma5 can be obtained from Lemma3.3in [4] by translating the cubes in it.


## 4. The Main Results

Now we come to the main results of the paper.Theorem 6.
Let0<k<1and1≤p<∞. Ifw∈A{a,b},locp, then the double points local maximal operatorM{a,b},k,locis bounded onLp(w)whenp>1and bounded fromL1(w)intoL1,∞(w). Moreover, the corresponding constant depends only on the double points localk0-Apconstant ofwfor somek0,0<k0<1.Proof.
DenoteM{a,b},k,locbyMk,locfor short. The definition ofGa,b′gives thatMk,locf(x)=Mk,loc(fχGa,b′)(x)forx∈Ga,b. First, we consider the casep>1(14)∫ℝd‍|Mk,locf(x)|pw(x)dx=∑Ga,b∈𝒢a,b‍∫Ga,b‍|Mk,locf(x)|pw(x)dx=∑Ga,b∈𝒢a,b‍∫Ga,b‍|Mk,loc(fχGa,b′)(x)|pw(x)dx≤∑Ga,b∈𝒢a,b‍∫Ga,b‍|M(fχGa,b′)(x)|pw(x)dx.
SinceMis sublinear andGa,b′⊂(Ga′∩Gb′), we have
(15)∫ℝd‍|Mk,locf(x)|pw(x)dx≤2p∑Ga,b∈𝒢a,b‍(∫Ga,b‍|M(fχGa′)(x)|pw(x)dx+∫Ga,b‍|M(fχGb′)(x)|pw(x)dx)=2p(∫ℝd|M(fχGa′)(x)||p‍w(x)dx+∫ℝd‍|M(fχGb′)(x)|pw(x)dx)=2p(∑Ga∈𝒢a‍∫Ga|M(fχGa′)(x)|p‍w(x)dx+∑Gb∈𝒢b‍∫Gb‍|M(fχGb′)(x)|pw(x)dx).
Translating the cubes alongaandb, respectively, in the corresponding part of proof for Theorem4.1in [4] gives
(16)∫Ga‍|M(fχGa′)(x)|pw(x)dx≤C∥w∥A{a},k0,locppp′∫Ga′‍|f(x)|pw(x)dx,∫Gb‍|M(fχGb′)(x)|pw(x)dx≤C∥w∥A{b},k0,locppp′∫Gb′‍|f(x)|pw(x)dx.
Note that the families{Ga′}Ga∈𝒢aand{Gb′}Gb∈𝒢bhave the finite overlapping property; that is, there exists a constantm=m(d,k)such that everyx∈ℝd∖{a,b}belongs to at mostmsets from{Ga′}Ga∈𝒢aor{Gb′}Ga∈𝒢b. Combining this with (16) gives
(17)∫ℝd|Mk,locf(x)|p‍w(x)dx≤2p(∑Ga∈𝒢a‍C∥w∥A{a},k0,locppp′∫Ga′|f(x)|p‍w(x)dx+∑Gb∈𝒢b‍C∥w∥A{b},k0,locppp′∫Gb′‍|f(x)|pw(x)dx)≤CBk0,p(∫ℝd‍|f(x)|p(∑Ga∈𝒢a‍χGa′)w(x)dx+∫ℝd‍|f(x)|p(∑Gb∈𝒢b‍χGb′)w(x)dx)≤CmBk0,p∫ℝd‍|f(x)|pw(x)dx,
whereBk0,p=max{∥w∥A{a},k0,locppp′,∥w∥A{b},k0,locppp′}.
As for the casep=1, we define the level sets
(18)Eλ={x∈ℝd:Mk,locf(x)>λ},Eλ,Ga,b′={x∈ℝd:Mk,loc(fχGa,b′)(x)>λ},Fλ,Ga,b′={x∈ℝd:M(fχGa,b′)(x)>λ},Fλ,Ga′={x∈ℝd:M(fχGa′)(x)>λ},Fλ,Gb′={x∈ℝd:M(fχGb′)(x)>λ}.
SinceFλ,Ga,b′⊂(Fλ,Ga′∪Fλ,Gb′), the weak weighed boundedness of the maximal operatorMand the similar arguments as the casep>1give that
(19)∫Eλ‍w(x)dx=∑Ga,b∈𝒢a,b‍∫Ga,b‍χEλ,Ga,b′(x)w(x)dx≤∑Ga,b∈𝒢a,b‍∫Ga,b‍χFλ,Ga,b′(x)w(x)dx≤∑Ga∈𝒢a‍∫Ga‍χFλ,Ga′(x)w(x)dx+∑Gb∈𝒢b‍∫Gb‍χFλ,Gb′(x)w(x)dx≤CBk0,1λ-1(∑Ga∈𝒢a‍∫Ga′‍|f(x)|w(x)dx+∑Gb∈𝒢b‍∫Gb′‍|f(x)|w(x)dx)≤CmBk0,1λ-1∫ℝd‍|f(x)|w(x)dx,
whereBk0,1=max{∥w∥A{a},k0,loc1,∥w∥A{b},k0,loc1}. Thus the proof is finished.Theorem 7.
Let0<k<1and1≤p<∞,wbe a weight function onℝd. The conditionw∈A{a,b},locpis necessary for the double points local maximal operatorM{a,b},k,locto be bounded fromLp(w)intoLp,∞(w); that is,
(20)w({x∈ℝd:M{a,b},k,locf(x)>λ})≤Cλp∫ℝd‍|f(x)|pw(x)dx,λ>0.The proof of Theorem7 is analogous to that of Theorem4.2in [4], and we omit it here.



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