Double Points Local Hardy-Littlewood Maximal Operator

Abstract

A double points local Hardy-Littlewood maximal operator 𝑀_{{𝑎,𝑏},𝑘,loc} is defined and investigated in Euclidean spaces. It is proved that 𝑀_{{𝑎,𝑏},𝑘,loc} is bounded on 𝐿^𝑝(𝑤) when 𝑝>1 and from 𝐿¹(𝑤) to 𝐿^1,∞(𝑤) with weight function 𝑤∈𝐴_{{𝑎,𝑏},𝑘,loc}, the class of double points local 𝐴_𝑝 weights which is larger than the Muckenhoupt 𝐴_𝑝 class and the local 𝐴_𝑝 weights defined by Lin and Stempak.

1. Introduction

The Hardy-Littlewood maximal operator𝑀is defined on the class of the locally integrable functions onℝ^𝑑by 𝑀𝑓(𝑥)=sup_𝑄∋𝑥1|𝑄|_𝑄𝑓(𝑥)𝑑𝑥,𝑥∈ℝ^𝑑, (1) where the supremum is taken over all cubes𝑄⊂ℝ^𝑑containing𝑥, 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 variable𝐿^𝑝spaces. 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 considerℝ^𝑑,𝑑≥1, equipped with the metric induced by the norm ‖𝑥‖=max𝑥₁,𝑥₂,…,𝑥_𝑑,𝑥=𝑥₁,𝑥₂,…,𝑥_𝑑. (2)

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∈ℝ^𝑑∶‖𝑦−𝑥‖≤𝑟}denotes the cube centered at𝑥∈ℝ^𝑑with “radius”𝑟>0.|𝐸|denotes the Lebesgue measure of a measurable set𝐸⊂ℝ^𝑑.

For given𝑎,𝑏∈ℝ^𝑑and𝑘>0, denote by𝒪_{{𝑎,𝑏},𝑘}(𝑦)the family of all cubes centered at𝑦with radius𝑟≤𝑘min{‖𝑦−𝑎‖,‖𝑦−𝑏‖}, and define 𝒪_{{𝑎,𝑏},𝑘}=_{𝑦∈ℝ𝑑}𝒪_{{𝑎,𝑏},𝑘}𝑦. (3) Note that for every𝑄∈𝒪_{{𝑎,𝑏},𝑘},0<𝑘<1, we have𝑎,𝑏∉𝑄. In particular,𝒪_{{𝑎,𝑏},𝑘}=𝒪_𝑘when𝑎is the origin and𝑏tends to∞, where𝒪_𝑘=⋃_{𝑦∈ℝ𝑑}𝒪_𝑘(𝑦)and𝒪_𝑘(𝑦)is the collection of all the cubes centered at𝑦with radius𝑟≤𝑘‖𝑦‖. The cubes in𝒪_𝑘are away from the origin, while the ones in𝒪_{{𝑎,𝑏},𝑘}are away from two points𝑎and𝑏.

Given0<𝑘<1, we define the (noncentered) double points local Hardy-Littlewood maximal operator 𝑀_{{𝑎,𝑏},𝑘,loc}𝑓(𝑥)=sup_{𝑥∈𝑄∈𝒪{𝑎,𝑏},𝑘}1|𝑄|_𝑄𝑓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑑⧵{𝑎,𝑏}, (4) where the supremum is taken over all cubes𝑄∈𝒪_{{𝑎,𝑏},𝑘}containing𝑥. It generalizes the local maximal operator𝑀_𝑘,locdefined in [4] for locally integrable functions onℝ^𝑑⧵{0}by 𝑀_𝑘,loc𝑓(𝑥)=sup_{𝑥∈𝑄∈𝒪𝑘}1|𝑄|_𝑄𝑓(𝑡)𝑑𝑡,𝑥∈ℝ^𝑑⧵{0}. (5)

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^𝑝_{{𝑎,𝑏},𝑘,loc}of double points local𝐴_𝑝weights is given in Section 2, as well as the coincidence of𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}for0<𝑘<1. That is,𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}=𝐴^𝑝_{{𝑎,𝑏},𝑚,loc}for0<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 local𝐴_𝑝condition, the operator𝑀_{{𝑎,𝑏},𝑘,loc}is bounded on𝐿^𝑝(𝑤),1<𝑝<∞and bounded from𝐿¹(𝑤)to𝐿^1,∞(𝑤).

That𝑤is a weight onℝ^𝑑means that𝑤is a nonnegative locally integrable function and finite almost everywhere.𝐿^𝑝(𝑤)and𝐿^1,∞(𝑤)denote the class of weighed𝐿^𝑝space and weighed weak𝐿¹space, with the norm‖𝑓‖^𝑝_{𝐿𝑝(𝑤)}=∫_ℝ𝑑|𝑓(𝑥)|^𝑝𝑤(𝑥)𝑑𝑥and‖𝑓‖_{𝐿1,∞(𝑤)}=sup_𝑡>0(𝑡∫_|𝑓|>𝑡𝑤(𝑥)𝑑𝑥), respectively.

2. Double Points Local𝐴_𝑝Weights

For a weight𝑤onℝ^𝑑, we say that𝑤∈𝐴_𝑝for1<𝑝<∞if there is a constant𝐶>0such that sup_{𝑄⊂ℝ𝑑}1|𝑄|_𝑄𝑤(𝑥)1𝑑𝑥|𝑄|_𝑄𝑤(𝑥)^1−𝑝′𝑑𝑥^𝑝−1≤𝐶, (6) here and below1/𝑝+1/𝑝^′=1. We say that𝑤∈𝐴₁if there is a constant𝐶>0such that𝑀𝑤(𝑥)≤𝐶𝑤(𝑥). The smallest constant𝐶is called the𝐴_𝑝constant of𝑤and denoted by‖𝑤‖_𝐴𝑝. The definition of𝐴_𝑝weights 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∈𝐴_𝑝,𝑝≥1, the operator𝑀is bounded on𝐿^𝑝(𝑤)when𝑝>1and bounded from𝐿¹(𝑤)into𝐿^1,∞(𝑤)when𝑝=1; moreover, we have 𝑀𝑓_{𝐿𝑝(𝑤)}≤𝐶‖𝑤‖^𝑝′_𝐴𝑝𝑓_{𝐿𝑝(𝑤)},𝑓∈𝐿^𝑝(𝑤),𝑝>1,𝑀𝑓_𝐿1,∞(𝑤)≤𝐶‖𝑤‖_𝐴1𝑓_𝐿1(𝑤),𝑓∈𝐿¹(𝑤), (7) where the constant𝐶is independent of𝑤(cf. [6]).

The class of the local𝐴^𝑝_𝑘,locweights is defined in [4] by considering the cubes in𝒪_𝑘and the corresponding constant is denoted as‖𝑤‖_𝐴^𝑝_𝑘,loc. Lin and Stempak proved that for𝑤∈𝐴^𝑝_𝑘,loc, the local Hardy-Littlewood maximal operator is bounded from𝐿^𝑝(𝑤)to𝐿^𝑝(𝑤)when𝑝>1and from𝐿¹(𝑤)to𝐿^1,∞(𝑤).

In order to prove the coincidence of the classes𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}for0<𝑘<1, we need the following lemma.

Definition 1. Let0<𝑘<1,1≤𝑝<∞,𝑎,𝑏∈ℝ^𝑑,and𝑤∈𝐿_loc(ℝ^𝑑⧵{𝑎,𝑏})a nonnegative function. If there exists a constant𝐶>0such that sup_𝑄1|𝑄|_𝑄𝑤(𝑥)𝑑𝑥^1/𝑝1|𝑄|_𝑄𝑤(𝑥)^{−𝑝′/𝑝}𝑑𝑥^1/𝑝′≤𝐶 (8) for1<𝑝<∞, where the supremum takes over all the cubes𝑄∈𝒪_{{𝑎,𝑏},𝑘}, then𝑤is called the double points local𝐴_𝑝weight associated with𝑎and𝑏. And for𝑝=1, the class𝐴¹_{{𝑎,𝑏},𝑘,loc}is defined by 1|𝑄|_𝑄𝑤(𝑥)𝑑𝑥≤𝐶essin_𝑥∈𝑄f𝑤(𝑥), (9) for all𝑄∈𝒪_{{𝑎,𝑏},𝑘}. Denote the collection of all the double points local𝐴_𝑝weights as𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}=𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}(ℝ^𝑑). The smallest constant𝐶is called the double points local𝑘−𝐴_𝑝constant and denoted as‖𝑤‖_𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}.

According to the definition above, it is easy to see that𝐴_𝑝⊂𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}. Furthermore, if𝑤∈𝐴_𝑝, then‖𝑤‖_𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}≤‖𝑤‖_𝐴𝑝. But𝐴_𝑝≠𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}. In fact, let𝑤(𝑥)=min{‖𝑥−𝑎‖,‖𝑥−𝑏‖}; then for any𝑡∈ℝ, we have𝑤(𝑥)^𝑡∈𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}, while𝑤(𝑥)^𝑡∈𝐴_𝑝if and only if−𝑑<𝑡<𝑑(𝑝−1)when𝑝>1and−𝑑<𝑡≤0when𝑝=1.

On the other hand, define𝒪_{𝑎},𝑘=⋃_{𝑦∈ℝ𝑑}𝒪_{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 in𝒪_𝑘along𝑎mentioned in [4]. Replacing𝒪_{{𝑎,𝑏},𝑘}with𝒪_{𝑎},𝑘in Definition 1 gives the definition of the class𝐴^𝑝_{{𝑎},𝑘,loc}. It is also easy to see that𝐴^𝑝_{{𝑎},𝑘,loc}⊂𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}. The same process gives𝒪_{𝑏},𝑘and consequently the class𝐴^𝑝_{{𝑏},𝑘,loc}. We also have the relation𝐴^𝑝_{{𝑏},𝑘,loc}⊂𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}.

Lemma 2. Let0<𝑘<1; for every0<𝑟<1, there exists a𝑛=𝑛(𝑘,𝑟)satisfying the following property: if𝑅and𝑅are 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 most𝑛cubes𝑄_𝑗⊂𝑅and𝑄_𝑗∈𝒪_{{𝑎,𝑏},𝑘},1≤𝑗≤𝑛, covering𝑅⧵𝑅and|𝑄_𝑗∩𝑅|≥𝐶|𝑅|, where𝑄_𝑗,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 𝐴^𝑝_{{}𝑎,𝑏,𝑘,loc}=𝐴^𝑝_{{}𝑎,𝑏,𝑚,loc}. (10)

The proofs of Lemma 2 and Proposition 3 are analogous to the proofs of Lemma2.1and Proposition2.2in [4], respectively, so we omit them here.

By Proposition 3, the class𝐴^𝑝_{{𝑎,𝑏},𝑘,loc}is independent of the choice of𝑘∈(0,1)and we will denote it by𝐴^𝑝_{{𝑎,𝑏},loc}.

3. Preparatory Lemmas

For a given0<𝑘<1,𝑎∈ℝ^𝑑,𝑎=(𝑎₁,𝑎₂,…,𝑎_𝑑), considering a grid ofℝ^𝑑⧵{𝑎}based on the sequence{±𝑘^−𝑠}_𝑠∈ℤ, which results in the collection𝒢_𝑎of all the rectangles 𝐺_𝑎=𝐼₁+𝑎₁×𝐼₂+𝑎₂×⋯×𝐼_𝑑+𝑎_𝑑, (11) where for a given𝑠∈ℤ, each𝐼_𝑗is 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 that𝐺_𝑎is of the first category, otherwise of the second category.

For𝐺_𝑎∈𝒢_𝑎, define𝐺^′_𝑎to be the union of the cubes in𝒪_{𝑎},𝑘satisfying𝑄∩𝐺_𝑎≠∅. If𝐺_𝑎is of the first category, we define𝐺^′′_𝑎to be the least cube containing𝐺^′_𝑎with the largest distance from the point𝑎along each axis. If𝐺_𝑎is of the second category, we define𝐺^′′_𝑎,𝑗to be the least cube containing𝐺^′_𝑎,𝑗=𝐺^′_𝑎∩𝐸_{𝑗,𝐺𝑎}with the largest distance from the point𝑎along each axis, where for every𝑗=1,2,…,𝑑, 𝐸_{𝑗,𝐺𝑎}=𝑦∈ℝ^𝑑∶𝑦_𝑗≥1−𝑘𝑘1+𝑘^−𝑠+𝑎_𝑗,when𝐼_𝑗⊂(0,+∞),𝑦∈ℝ^𝑑∶𝑦_𝑗≤−1−𝑘𝑘1+𝑘^−𝑠+𝑎_𝑗,when𝐼_𝑗⊂(−∞,0). (12)

According to the definitions above, if𝐺_𝑎is of the first category, we have𝐺_𝑎⊂𝐺^′_𝑎⊂𝐺^′′_𝑎,𝐺^′_𝑎⊂{𝑦∈ℝ^𝑑∶|𝑦_𝑗−𝑎_𝑗|≥((1−𝑘)/(1+𝑘))𝑘^−𝑠}and𝑎∉𝐺^′′_𝑎. If𝐺_𝑎is of the second category, then𝐺_𝑎⊂𝐺^′_𝑎,𝑗,𝑎∉𝐺^′′_𝑎,𝑗,𝑗=1,2,…,𝑑and𝐺^′_𝑎∩∏^𝑑_𝑗=1𝐸^𝑐_{𝑗,𝐺𝑎}=∅which equals𝐺^′_𝑎=⋃^𝑑_𝑗=1𝐺^′_𝑎,𝑗.

For a point𝑏∈ℝ^𝑑, repeat the process above, we can obtain the corresponding sets𝐺_𝑏,𝐺^′_𝑏,𝐺^′′_𝑏,𝐺^′_𝑏,𝑗,𝐺^′′_𝑏,𝑗,and𝒢_𝑏.

For𝑎,𝑏∈ℝ^𝑑(𝑎≠𝑏), ℝ^𝑑⧵{𝑎,𝑏}=_{𝐺𝑎∈𝒢𝑎}𝐺_𝑎∩_{𝐺𝑏∈𝒢𝑏}𝐺_𝑏=_{𝐺𝑎∈𝒢𝑎,𝐺𝑏∈𝒢𝑏}𝐺_𝑎∩𝐺_𝑏. (13)

Define𝒢_𝑎,𝑏to be the collection of𝐺_𝑎,𝑏=𝐺_𝑎∩𝐺_𝑏, where𝐺_𝑎∈𝒢_𝑎and𝐺_𝑏∈𝒢_𝑏and𝐺^′_𝑎,𝑏to be the union of the cubes𝑄∈𝒪_{{𝑎,𝑏},𝑘}such that𝑄∩𝐺_𝑎,𝑏≠∅. In the proof of Theorem 6, we will use a result; that is,𝐺^′_𝑎,𝑏⊂(𝐺^′_𝑎∩𝐺^′_𝑏).

In fact, for every𝑧∈𝐺^′_𝑎,𝑏, there exists a cube𝑄₀∈𝒪_{{𝑎,𝑏},𝑘},𝑄₀∩𝐺_𝑎,𝑏≠∅such that𝑧∈𝑄₀. The definition of𝒪_{{𝑎,𝑏},𝑘}gives that there exists a𝑦∈ℝ^𝑑⧵{𝑎,𝑏}such that𝑄₀is the cube centered at𝑦with radius𝑟≤𝑘‖𝑦−𝑎‖. Then𝑄₀∈𝒪_{𝑎},𝑘and𝑄₀⊂𝐺^′_𝑎; consequently𝑧∈𝐺^′_𝑎. Similarly,𝑧∈𝐺^′_𝑏. Thus𝑧∈𝐺^′_𝑎∩𝐺^′_𝑏.

Lemma 4. Given0<𝑘<1, there exists𝑘₀=𝑘₀(𝑑,𝑘)∈(0,1)such that for each𝐺_𝑎,𝑏=𝐺_𝑎∩𝐺_𝑏∈𝒢_𝑎,𝑏,𝐺^′′_𝑎∈𝒪_{𝑎},𝑘0(or𝐺^′′_𝑏∈𝒪_{𝑏},𝑘0)if𝐺_𝑎(or𝐺_𝑏)is of the first category, and𝐺^′′_𝑎,𝑗∈𝒪_{𝑎},𝑘0(or𝐺^′′_𝑏,𝑗∈𝒪_{𝑏},𝑘0),𝑗=1,2,…,𝑑, if𝐺_𝑎(or𝐺_𝑏)is of the second category.

Proof. The constants𝑘_𝑎and𝑘_𝑏associated with the points𝑎and𝑏, respectively, can be obtained by the same procedure as the one in the proof of Lemma3.1in [4]. Then we have𝐺^′′_𝑎∈𝒪_{{𝑎},𝑘𝑎}and𝐺^′′_𝑏∈𝒪_{{𝑏},𝑘𝑏}when𝐺_𝑎and𝐺_𝑏are of the first category. Let𝑘₀=max{𝑘_𝑎,𝑘_𝑏}, thus𝐺^′′_𝑎∈𝒪_{𝑎},𝑘0and𝐺^′′_𝑏∈𝒪_{𝑏},𝑘0by the definition of𝒪_{𝑎},𝑘0and𝒪_{𝑏},𝑘0. The case when𝐺_𝑎and𝐺_𝑏are of the second category is similar to the case when𝐺_𝑎and𝐺_𝑏are of the first category.

Lemma 5. Given0<𝑘<1. Let𝑄and𝑄be the cubes inℝ^𝑑satisfying𝑄⊂𝑄. If𝑄∈𝒪_{𝑎},𝑘(𝑜𝑟𝒪_{𝑏},𝑘); then𝑄∈𝒪_{𝑎},𝑘(𝑜𝑟𝒪_{𝑏},𝑘).

Lemma 5 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^𝑝_{{𝑎,𝑏},loc}, then the double points local maximal operator𝑀_{{𝑎,𝑏},𝑘,loc}is bounded on𝐿^𝑝(𝑤)when𝑝>1and bounded from𝐿¹(𝑤)into𝐿^1,∞(𝑤). Moreover, the corresponding constant depends only on the double points local𝑘₀−𝐴_𝑝constant of𝑤for some𝑘₀,0<𝑘₀<1.

Proof. Denote𝑀_{{𝑎,𝑏},𝑘,loc}by𝑀_𝑘,locfor short. The definition of𝐺^′_𝑎,𝑏gives that𝑀_𝑘,loc𝑓(𝑥)=𝑀_𝑘,loc(𝑓𝜒_{𝐺′𝑎,𝑏})(𝑥)for𝑥∈𝐺_𝑎,𝑏. First, we consider the case𝑝>1_ℝ𝑑𝑀_𝑘,loc𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥=_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝑀_𝑘,loc𝑓(𝑥)^𝑝𝑤(𝑥)=𝑑𝑥_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝑀_𝑘,loc𝑓𝜒_{𝐺′𝑎,𝑏}(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝑀𝑓𝜒_{𝐺′𝑎,𝑏}(𝑥)^𝑝𝑤(𝑥)𝑑𝑥. (14)
Since𝑀is sublinear and𝐺^′_𝑎,𝑏⊂(𝐺^′_𝑎∩𝐺^′_𝑏), we have _ℝ𝑑𝑀_𝑘,loc𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤2^𝑝_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝑀𝑓𝜒_𝐺′𝑎(𝑥)^𝑝𝑤(𝑥)+𝑑𝑥_{𝐺𝑎,𝑏}𝑀𝑓𝜒_𝐺′𝑏(𝑥)^𝑝𝑤(𝑥)𝑑𝑥=2^𝑝_ℝ𝑑𝑀𝑓𝜒_𝐺′𝑎(𝑥)^|𝑝𝑤(𝑥)+𝑑𝑥_ℝ𝑑𝑀𝑓𝜒_𝐺′𝑏(𝑥)^𝑝𝑤(𝑥)𝑑𝑥=2^𝑝_{𝐺𝑎∈𝒢𝑎𝐺𝑎}𝑀𝑓𝜒_𝐺′𝑎(𝑥)^𝑝𝑤(𝑥)𝑑𝑥+_{𝐺𝑏∈𝒢𝑏𝐺𝑏}𝑀𝑓𝜒_𝐺′𝑏(𝑥)^𝑝𝑤(𝑥)𝑑𝑥. (15)
Translating the cubes along𝑎and𝑏, respectively, in the corresponding part of proof for Theorem4.1in [4] gives _𝐺𝑎𝑀𝑓𝜒_𝐺′𝑎(𝑥)^𝑝𝑤(𝑥)‖𝑤‖𝑑𝑥≤𝐶^𝑝𝑝′_{𝐴𝑝0{𝑎},𝑘,loc𝐺′𝑎}𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥,_𝐺𝑏𝑀𝑓𝜒_𝐺′𝑏(𝑥)^𝑝𝑤(𝑥)‖𝑤‖𝑑𝑥≤𝐶^𝑝𝑝′_{𝐴𝑝0{𝑏},𝑘,loc𝐺′𝑏}𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥. (16)
Note that the families{𝐺^′_𝑎}_{𝐺𝑎∈𝒢𝑎}and{𝐺^′_𝑏}_{𝐺𝑏∈𝒢𝑏}have the finite overlapping property; that is, there exists a constant𝑚=𝑚(𝑑,𝑘)such that every𝑥∈ℝ^𝑑⧵{𝑎,𝑏}belongs to at most𝑚sets from{𝐺^′_𝑎}_{𝐺𝑎∈𝒢𝑎}or{𝐺^′_𝑏}_{𝐺𝑎∈𝒢𝑏}. Combining this with (16) gives _ℝ𝑑𝑀_𝑘,loc𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤2^𝑝_{𝐺𝑎∈𝒢𝑎}𝐶‖𝑤‖^𝑝𝑝′_{𝐴𝑝{𝑎}0,𝑘,loc𝐺′𝑎}𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥+_{𝐺𝑏∈𝒢𝑏}𝐶‖𝑤‖^𝑝𝑝′_{𝐴𝑝{𝑏}0,𝑘,loc𝐺′𝑏}𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥≤𝐶𝐵_{𝑘0,𝑝ℝ𝑑}𝑓(𝑥)^𝑝_{𝐺𝑎∈𝒢𝑎}𝜒_𝐺′𝑎𝑤(𝑥)+𝑑𝑥_ℝ𝑑𝑓(𝑥)^𝑝_{𝐺𝑏∈𝒢𝑏}𝜒_𝐺′𝑏𝑤(𝑥)𝑑𝑥≤𝐶𝑚𝐵_{𝑘0,𝑝ℝ𝑑}𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥, (17) where𝐵_𝑘0,𝑝=max{‖𝑤‖^𝑝𝑝′_{𝐴𝑝0{𝑎},𝑘,loc},‖𝑤‖^𝑝𝑝′_{𝐴𝑝0{𝑏},𝑘,loc}}.
As for the case𝑝=1, we define the level sets 𝐸_𝜆=𝑥∈ℝ^𝑑∶𝑀_𝑘,loc𝑓(𝑥),𝐸>𝜆_{𝜆,𝐺′𝑎,𝑏}=𝑥∈ℝ^𝑑∶𝑀_𝑘,loc𝑓𝜒_{𝐺′𝑎,𝑏}𝑥),𝐹(>𝜆_{𝜆,𝐺′𝑎,𝑏}=𝑥∈ℝ^𝑑∶𝑀𝑓𝜒_{𝐺′𝑎,𝑏}𝑥),𝐹(>𝜆_{𝜆,𝐺′𝑎}=𝑥∈ℝ^𝑑∶𝑀f𝜒_𝐺′𝑎(𝑥),𝐹>𝜆_{𝜆,𝐺′𝑏}=𝑥∈ℝ^𝑑∶𝑀𝑓𝜒_𝐺′𝑏(𝑥)>𝜆. (18)
Since𝐹_{𝜆,𝐺′𝑎,𝑏}⊂(𝐹_{𝜆,𝐺′𝑎}∪𝐹_{𝜆,𝐺′𝑏}), the weak weighed boundedness of the maximal operator𝑀and the similar arguments as the case𝑝>1give that _𝐸𝜆𝑤(𝑥)𝑑𝑥=_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝜒_{𝐸′𝜆,𝐺𝑎,𝑏}(𝑥)𝑤(𝑥)𝑑𝑥≤_{𝐺𝑎,𝑏∈𝒢𝑎,𝑏𝐺𝑎,𝑏}𝜒_{𝐹′𝜆,𝐺𝑎,𝑏}(𝑥)𝑤(𝑥)𝑑𝑥≤_{𝐺𝑎∈𝒢𝑎𝐺𝑎}𝜒_{𝐹′𝑎𝜆,𝐺}(𝑥)𝑤(𝑥)𝑑𝑥+_{𝐺𝑏∈𝒢𝑏𝐺𝑏}𝜒_{𝐹′𝑏𝜆,𝐺}(𝑥)𝑤(𝑥)d𝑥≤𝐶𝐵_𝑘0,1𝜆⁻¹_{𝐺𝑎∈𝒢𝑎𝐺′𝑎}𝑓(𝑥)𝑤(𝑥)𝑑𝑥+_{𝐺𝑏∈𝒢𝑏𝐺′𝑏}𝑓(𝑥)𝑤(𝑥)𝑑𝑥≤𝐶𝑚𝐵_𝑘0,1𝜆⁻¹_ℝ𝑑𝑓(𝑥)𝑤(𝑥)𝑑𝑥, (19) where𝐵_𝑘0,1=max{‖𝑤‖_𝐴¹_{0{𝑎},𝑘,loc},‖𝑤‖_𝐴¹_{0{𝑏},𝑘,loc}}. Thus the proof is finished.

Theorem 7. Let0<𝑘<1and1≤𝑝<∞,𝑤be a weight function onℝ^𝑑. The condition𝑤∈𝐴^𝑝_{{𝑎,𝑏},loc}is necessary for the double points local maximal operator𝑀_{{𝑎,𝑏},𝑘,loc}to be bounded from𝐿^𝑝(𝑤)into𝐿^𝑝,∞(𝑤); that is, 𝑤𝑥∈ℝ^𝑑∶𝑀_{{𝑎,𝑏},𝑘,loc}𝑓(𝑥)≤𝐶>𝜆𝜆^𝑝∫_ℝ𝑑𝑓(𝑥)^𝑝𝑤(𝑥)𝑑𝑥,𝜆>0. (20)

The proof of Theorem 7 is analogous to that of Theorem4.2in [4], and we omit it here.

Acknowledgments

This paper is supported by Project of Science and Technology Research Foundation of Educational Commission of Heilongjiang Province (12511158), Foundation for Young Key Teacher of Harbin Normal University (KGB201011), and Natural Science Foundation of Heilongjiang Province (A201206).