Double Points Local Hardy-Littlewood Maximal Operator

and Applied Analysis 3 where for a given s ∈ Z, each Ij is one of the intervals ±[k −s , k −s−1 ], ±[0, ((1 − k)/(1 + k))k], 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 (if 0 ≤ 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 that Ga is of the first category, otherwise of the second category. For Ga ∈ Ga, define G 󸀠 a to be the union of the cubes in O{a},k satisfying Q ∩ Ga ̸ = 0. If Ga is of the first category, we define G a to be the least cube containing G 󸀠 a with the largest distance from the point a along each axis. If Ga is of the second category, we defineG a,j to be the least cube containing G 󸀠 a,j = G 󸀠 a ∩ j,Ga with the largest distance from the point a along each axis, where for every j = 1, 2, . . . , d, j,Ga = {{{{{{{{{ {{{{{{{{{ { {y ∈ R : yj ≥ 1 − k 1 + k k −s + aj} , when Ij ⊂ (0, +∞) , {y ∈ R : yj ≤ − 1 − k 1 + k k −s + aj} , when Ij ⊂ (−∞, 0) . (12) According to the definitions above, if Ga is of the first category, we have Ga ⊂ G 󸀠 a ⊂ G 󸀠󸀠 a , G 󸀠 a ⊂ {y ∈ R d : |yj − aj| ≥ ((1 − k)/(1 + k))k −s } and a ∉ G a . If Ga is of the second category, then Ga ⊂ G 󸀠 a,j, a ∉ G 󸀠󸀠 a,j, j = 1, 2, . . . , d and G 󸀠 a ∩∏ d j=1E c j,Ga = 0 which equals G a = ⋃ d j=1 G 󸀠 a,j. For a point b ∈ R, repeat the process above, we can obtain the corresponding sets Gb, G 󸀠 b, G 󸀠󸀠 b , G 󸀠 b,j, G 󸀠󸀠 b,j, andGb. For a, b ∈ R(a ̸ = b), R d \ {a, b} = ( ⋃ Ga∈Ga Ga) ∩ ( ⋃ Gb∈Gb Gb) = ⋃ Ga∈Ga,Gb∈Gb (Ga ∩ Gb) . (13) Define Ga,b to be the collection of Ga,b = Ga ∩ Gb, where Ga ∈ Ga and Gb ∈ Gb and G 󸀠 a,b to be the union of the cubes Q ∈ O{a,b},k such thatQ∩Ga,b ̸ = 0. In the proof ofTheorem 6, we will use a result; that is, G a,b ⊂ (G 󸀠 a ∩ G 󸀠 b). In fact, for every z ∈ G a,b, there exists a cubeQ0 ∈ O{a,b},k, Q0 ∩ Ga,b ̸ = 0 such that z ∈ Q0. The definition of O{a,b},k gives that there exists a y ∈ R \ {a, b} such that Q0 is the cube centered at y with radius r ≤ k‖y − a‖. Then Q0 ∈ O{a},k and Q0 ⊂ G 󸀠 a; consequently z ∈ G 󸀠 a. Similarly, z ∈ G 󸀠 b. Thus z ∈ G 󸀠 a ∩ G 󸀠 b. Lemma 4. Given 0 < k < 1, there exists k0 = k0(d, k) ∈ (0, 1) such that for each Ga,b = Ga ∩ Gb ∈ Ga,b, G a ∈ O{a},k0 (or G 󸀠󸀠 b ∈ O{b},k0) if Ga (or Gb) is of the first category, and G 󸀠󸀠 a,j ∈ O{a},k0 (or G 󸀠󸀠 b,j ∈ O{b},k0), j = 1, 2, . . . , d, if Ga (or Gb) is of the second category. Proof. The constants ka and kb associated with the points a and b, respectively, can be obtained by the same procedure as the one in the proof of Lemma 3.1 in [4]. Then we have G 󸀠󸀠 a ∈ O{a},ka and G 󸀠󸀠 b ∈ O{b},kb when Ga and Gb are of the first category. Let k0 = max{ka, kb}, thus G 󸀠󸀠 a ∈ O{a},k0 and G 󸀠󸀠 b ∈ O{b},k0 by the definition of O{a},k0 and O{b},k0 . The case when Ga and Gb are of the second category is similar to the case when Ga and Gb are of the first category. Lemma 5. Given 0 < k < 1. Let Q and Q be the cubes in R satisfying Q ⊂ Q. If Q ∈ O{a},k(or O{b},k); then Q ∈ O{a},k(or O{b},k). Lemma 5 can be obtained from Lemma 3.3 in [4] by translating the cubes in it. 4. The Main Results Now we come to the main results of the paper. Theorem 6. Let 0 < k < 1 and 1 ≤ p < ∞. If w ∈ A {a,b},loc, then the double points local maximal operator M{a,b},k,loc is bounded on L(w) when p > 1 and bounded from L(w) into L 1,∞ (w). Moreover, the corresponding constant depends only on the double points local k0 − Ap constant of w for some k0, 0 < k0 < 1. Proof. DenoteM{a,b},k,loc byMk,loc for short.The definition of G 󸀠 a,b gives that Mk,locf(x) = Mk,loc(fχG󸀠 a,b )(x) for x ∈ Ga,b. First, we consider the case p > 1 ∫ R 󵄨󵄨󵄨󵄨Mk,locf (x) 󵄨󵄨󵄨󵄨 p w (x) dx = ∑ Ga,b∈Ga,b ∫ Ga,b 󵄨󵄨󵄨󵄨Mk,locf (x) 󵄨󵄨󵄨󵄨 p w (x) dx = ∑ Ga,b∈Ga,b ∫ Ga,b 󵄨󵄨󵄨󵄨󵄨 Mk,loc (fχG󸀠 a,b ) (x) 󵄨󵄨󵄨󵄨󵄨 p w (x) dx ≤ ∑ Ga,b∈Ga,b ∫ Ga,b 󵄨󵄨󵄨󵄨󵄨 M(fχG󸀠 a,b ) (x) 󵄨󵄨󵄨󵄨󵄨 p w (x) dx. (14) SinceM is sublinear and G a,b ⊂ (G 󸀠 a ∩ G 󸀠 b), we have ∫ R 󵄨󵄨󵄨󵄨Mk,locf (x) 󵄨󵄨󵄨󵄨 p w (x) dx ≤ 2 p ∑ Ga,b∈Ga,b (∫ Ga,b 󵄨󵄨󵄨󵄨 M (fχG󸀠 a ) (x) 󵄨󵄨󵄨󵄨 p w (x) dx +∫ Ga,b 󵄨󵄨󵄨󵄨 M (fχG󸀠 b ) (x) 󵄨󵄨󵄨󵄨 p w (x) dx) 4 Abstract and Applied Analysis = 2 p (∫ R 󵄨󵄨󵄨󵄨 M (fχG󸀠 a ) (x) 󵄨󵄨󵄨󵄨 | p


Introduction
The Hardy-Littlewood maximal operator  is defined on the class of the locally integrable functions on R  by  () = sup where the supremum is taken over all cubes  ⊂ R  containing , 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 variable   spaces.Gallardo [3] characterized the pairs of weights (, ) for which the Hardy-Littlewood maximal operator  satisfies a weak type integral inequality.
For given ,  ∈ R  and  > 0, denote by O {,}, () the family of all cubes centered at  with radius  ≤  min{‖ − ‖, ‖ − ‖}, and define Note that for every  ∈ O {,}, , 0 <  < where the supremum is taken over all cubes  ∈ O {,}, containing .It generalizes the local maximal operator  ,loc defined in [4] for locally integrable functions on R  \ {0} by The purpose of this paper is to investigate the double points local Hardy-Littlewood maximal operator.The paper was organized as follows.
for 0 < ,  < 1. Section 3 includes some lemmas that need to prove the main result Theorem 6 given in Section 4. We prove that for a weight  satisfying the double points local   condition, the operator  {,},,loc is bounded on   (), 1 <  < ∞ and bounded from  1 () to  1,∞ ().
That  is a weight on R  means that  is a nonnegative locally integrable function and finite almost everywhere.  () and  1,∞ () denote the class of weighed   space and weighed weak  1 space, with the norm

Double Points Local 𝐴 𝑝 Weights
For a weight  on R  , we say that  ∈   for 1 <  < ∞ if there is a constant  > 0 such that sup here and below 1/ + 1/  = 1.We say that  ∈ where the constant  is independent of  (cf.[6]).
The class of the local   ,loc weights is defined in [4] by considering the cubes in O  and the corresponding constant is denoted as ‖‖   ,loc .Lin and Stempak proved that for  ∈   ,loc , the local Hardy-Littlewood maximal operator is bounded from   () to   () when  > 1 and from  1 () to  1,∞ ().
In order to prove the coincidence of the classes   {,},,loc for 0 <  < 1, we need the following lemma.
The proofs of Lemma 2 and Proposition 3 are analogous to the proofs of Lemma 2.1 and Proposition 2.2 in [4], respectively, so we omit them here.
By Proposition 3, the class   {,},,loc is independent of the choice of  ∈ (0, 1) and we will denote it by   {,},loc .

Preparatory Lemmas
For a given 0 <  < 1,  ∈ R  ,  = ( 1 ,  2 , . . .,   ), considering a grid of R  \{} based on the sequence {± − } ∈Z , which results in the collection G  of all the rectangles ), but at least one of the intervals must be of the first type ) If one of the intervals is of the second type, we say that   is of the first category, otherwise of the second category.
For   ∈ G  , define    to be the union of the cubes in O {}, satisfying  ∩   ̸ = 0.If   is of the first category, we define    to be the least cube containing    with the largest distance from the point  along each axis.If   is of the second category, we define   , to be the least cube containing   , =    ∩  ,  with the largest distance from the point  along each axis, where for every  = 1, 2, . . ., , According to the definitions above, if   is of the first category, we have Lemma 5 can be obtained from Lemma 3.3 in [4] by translating the cubes in it.
The proof of Theorem 7 is analogous to that of Theorem 4.2 in [4], and we omit it here.