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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 637314, 5 pages
Double Points Local Hardy-Littlewood Maximal Operator
Mathematics Science College, Harbin Normal University, Harbin 150025, China
Received 3 July 2013; Accepted 7 November 2013
Academic Editor: Ademir Fernando Pazoto
Copyright © 2013 Futao Song and Na Ju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A double points local Hardy-Littlewood maximal operator is defined and investigated in Euclidean spaces. It is proved that is bounded on when and from to with weight function , the class of double points local weights which is larger than the Muckenhoupt class and the local weights defined by Lin and Stempak.
The Hardy-Littlewood maximal operatoris defined on the class of the locally integrable functions onby where the supremum is taken over all cubescontaining, 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.  gave the estimates of Hardy-Littlewood operator on the multilinear spaces. Lerner  showed that Hardy-Littlewood maximal operator was bounded on variablespaces. Gallardo  characterized the pairs of weightsfor which the Hardy-Littlewood maximal operatorsatisfies a weak type integral inequality.
In this paper we consider,, equipped with the metric induced by the norm
By a cube we always mean the usual closed cube with sides parallel to the coordinate axes; in terms of the metric,denotes the cube centered atwith “radius”.denotes the Lebesgue measure of a measurable set.
For givenand, denote bythe family of all cubes centered atwith radius, and define Note that for every,, we have. In particular,whenis the origin andtends to, whereandis the collection of all the cubes centered atwith radius. The cubes inare away from the origin, while the ones inare away from two pointsand.
Given, we define the (noncentered) double points local Hardy-Littlewood maximal operator where the supremum is taken over all cubescontaining. It generalizes the local maximal operatordefined in  for locally integrable functions onby
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 classof double points localweights is given in Section 2, as well as the coincidence offor. That is,for. Section 3 includes some lemmas that need to prove the main result Theorem 6 given in Section 4. We prove that for a weightsatisfying the double points localcondition, the operatoris bounded on,and bounded fromto.
Thatis a weight onmeans thatis a nonnegative locally integrable function and finite almost everywhere.anddenote the class of weighedspace and weighed weakspace, with the normand, respectively.
2. Double Points LocalWeights
For a weighton, we say thatforif there is a constantsuch that here and below. We say thatif there is a constantsuch that. The smallest constantis called theconstant ofand denoted by. The definition ofweights was first introduced by Muckenhoupt  for. Muckenhoupt proved the characterization of the weighed typefor Hardy-Littlewood maximal operator. That is, for,, the operatoris bounded onwhenand bounded fromintowhen; moreover, we have where the constantis independent of(cf. ).
The class of the localweights is defined in  by considering the cubes inand the corresponding constant is denoted as. Lin and Stempak proved that for, the local Hardy-Littlewood maximal operator is bounded fromtowhenand fromto.
In order to prove the coincidence of the classesfor, we need the following lemma.
Definition 1. Let,,anda nonnegative function. If there exists a constantsuch that for, where the supremum takes over all the cubes, thenis called the double points localweight associated withand. And for, the classis defined by for all. Denote the collection of all the double points localweights as. The smallest constantis called the double points localconstant and denoted as.
According to the definition above, it is easy to see that. Furthermore, if, then. But. In fact, let; then for any, we have, whileif and only ifwhenandwhen.
On the other hand, define, whereis the collection of the cubes centered atwith radius no more than. Then the cubes inare just the translations of the cubes inalongmentioned in . Replacingwithin Definition 1 gives the definition of the class. It is also easy to see that. The same process givesand consequently the class. We also have the relation.
Lemma 2. Let; for every, there exists asatisfying the following property: ifandare the cubes centered atwith radiusand, respectively, there exist at mostcubesand,, coveringand, where,are the cubes with centers located onand radiusand.
Proposition 3. Letand. Then
By Proposition 3, the classis independent of the choice ofand we will denote it by.
3. Preparatory Lemmas
For a given,,, considering a grid ofbased on the sequence, which results in the collectionof all the rectangles where for a given, eachis one of the intervals,or(called intervals of the first, second, or third type, resp.), but at least one of the intervals must be of the first type (if,denotes the intervalwhileis just) If one of the intervals is of the second type, we say thatis of the first category, otherwise of the second category.
For, defineto be the union of the cubes insatisfying. Ifis of the first category, we defineto be the least cube containingwith the largest distance from the pointalong each axis. Ifis of the second category, we defineto be the least cube containingwith the largest distance from the pointalong each axis, where for every,
According to the definitions above, ifis of the first category, we have,and. Ifis of the second category, then,,andwhich equals.
For a point, repeat the process above, we can obtain the corresponding sets,,,,and.
Defineto be the collection of, whereandandto be the union of the cubessuch that. In the proof of Theorem 6, we will use a result; that is,.
In fact, for every, there exists a cube,such that. The definition ofgives that there exists asuch thatis the cube centered atwith radius. Thenand; consequently. Similarly,. Thus.
Lemma 4. Given, there existssuch that for each,oriforis of the first category, andor,, iforis of the second category.
Proof. The constantsandassociated with the pointsand, respectively, can be obtained by the same procedure as the one in the proof of Lemmain . Then we haveandwhenandare of the first category. Let, thusandby the definition ofand. The case whenandare of the second category is similar to the case whenandare of the first category.
Lemma 5. Given. Letandbe the cubes insatisfying. If; then.
4. The Main Results
Now we come to the main results of the paper.
Theorem 6. Letand. If, then the double points local maximal operatoris bounded onwhenand bounded frominto. Moreover, the corresponding constant depends only on the double points localconstant offor some,.
Proof. Denotebyfor short. The definition ofgives thatfor. First, we consider the case
Sinceis sublinear and, we have
Translating the cubes alongand, respectively, in the corresponding part of proof for Theoremin  gives
Note that the familiesandhave the finite overlapping property; that is, there exists a constantsuch that everybelongs to at mostsets fromor. Combining this with (16) gives where.
As for the case, we define the level sets
Since, the weak weighed boundedness of the maximal operatorand the similar arguments as the casegive that where. Thus the proof is finished.
Theorem 7. Letand,be a weight function on. The conditionis necessary for the double points local maximal operatorto be bounded frominto; that is,
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).
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