Two Weight Characterization of New Maximal Operators

For the last twenty years, there has been a great deal of interest in the theory of two weight. In the present paper, we investigate the two weight norm inequalities for fractional new maximal operator on the Lebesgue space. More specifically, we obtain that the sufficient and necessary conditions for strong and weak type two weight norm inequalities for a new fractional maximal operators by introducing a class of new two weight functions. In the discussion of strong type two weight norm inequalities, we make full use of the properties of dyadic cubes and truncation operators, and utilize the space decomposition technique which space is decomposed into disjoint unions. In contrast, weak type two weight norm inequalities are more complex. We have the aid of some good properties of Ap weight functions and ingeniously use the characteristic function. What should be stressed is that the new two weight functions we introduced contains the classical two weights and our results generalize known results before. In this paper, it is worth noting that w(x)dx may not be a doubling measure if our new weight functions .. Since | | 1, our new weight functions are including the classical Muckenhoupt weights.


Introduction
In 1972, Muckenhopt [1] established the weight theory when studying the Lebesgue boundedness of classical Hardy-Littlewood maximal operators. Subsequently, he further obtained the two weight boundedness of Hardy-Littlewood maximal operators and founded that the two weight condition is a necessary and sufficient condition for the two weight weak boundedness of Hardy-Littlewood maximal operators. In addition, he discussed strong boundedness. Muckenhopt and Wheeden [2] founded that the two weight condition is only a necessary but not sufficient condition for Hardy-Littlewood maximal operator and Hilbert transform to have two weight strong boundedness, which is essentially different from the one weight case. Therefore, as a generalization of the one weight case, it is more difficult to discuss the boundedness of operators with two weights than with one weight. In 1982, great progress was made in the two weighted results. Sawyer [3] obtained the necessary and sufficient conditions for the two weight , of Hardy-Littlewood maximal operators to be bounded from to with 1 ∞. In 2000, Cruz-Uribe [4] gave a new proof of this result, and we can see a lot of work related to this topic in this paper, for example, see [5][6][7][8].
Next, some necessary definitions and notations are given. In this paper, , denotes the cube centered at and of the sidelength . Similarly, given , and 0, we will write for the λ-dilate cube, which is the cube with the same center and with sidelength . denote constants independent of parameters and may take different values in different places. Let , ! " denote the characteristic function of . | | denote the Lebesgue measure of . # denote the conjugate index of , i.e. where, Then we say that ∈ A . The classical Hardy-Littlewood maximal operator M is defined by Obviously, D ≤ B C D ≤ BD , 6. 7. ∈ and B C D is lower semi-continuous.
denote the classical Muckenhoupt weights; see [10]. It  pseudo-difffferential operator is a standard Calderón-Zygmund operator; see also /12 − 151. Applying the new maximal functions, Pan [16] obtained the strong type and weak end-point estimates for certain classes of multilinear operators and their iterated commutators with new BMO functions.
In this paper, the fractional form of a new maximal operator considered by Tang in [9] is introduced: given 0 ≤ 4 < 1, the fractional new maximal function B C 2 D is defined as where the supremum is taken over all the cube containing .
Next, we introduce a new class of two weight functions, which includes the classical two weights in [3]. (ii) When p = 1, for a. e. x ∈ Q, Then we say that , ∈ ,h 2 . It is easy to know that We end this section with the outline of this paper. Section 2 contains Theorem 2.1.1, Theorem 2.2.1 and the proofs of them. we extend the corresponding strong and weak results to the fractional new maximal operators. In Section 3, we give a conclusion.

Strong Boundedness of New Maximal Operators
The purpose of this paper is to study the strong and weak type inequalities of fractional new maximal operators by introducing a new class of two weight functions containing the classical two weight, and obtain their two weight characterization in Lebesgue spaces.
The main result of this section is to obtain the two weight strong boundedness of fractional new maximal operators. (ii) For every D ∈ , The following lemma is needed to prove the theorem. Lemma 2.1.2. Let 0 ≤ 4 < 1, D ≥ 0 is a locally integrable function. If for every cube and some > 0 such that Then there exists a dyadic cube P such that ⊂ 3u and Proof. Take w ∈ x makes 2 y@A ≤ ℓ < 2 y , so there exists dyadic cubes u A , u V , … , u | , 1 ≤ } ≤ 2 whose generate by 2 y and intersect . Since ℓou~p = 2 y > • , by Vitali covering lemma, we get ⊂ 3u~ for every € . In addition, it is assumed that there is no dyadic cube u such that Then, This contradicts the hypothesis. Therefore, there exists a dyadic cube u, Next, we give the proof of the theorem. Proof of Theorem 2.1.1. Set T = A@ i . In (ii), take D = T! = , it is easy to see that (ii) implicature (i). Next, we prove that (i) implicature (ii). Firstly, without losing generality, Let D ∈ be a nonnegative bounded function with compact support. This ensures that B C 2 D is almost everywhere finite.
This estimate shows that for each fixed w, the dyadic cube u F y is bounded. Therefore, there exists such a succollection of disjoint dyadic cubes ‡u~y‰~ that for a € , each F y is contained in 3u~y . Therefore, … y ⊂ I 3u~ỹ . Next, we decompose the … y : The expected inequalities can be obtained by using the consistency of ž y and the control convergence theorem. Therefore, only proving that R y : , T → h ®, uniformly bounded is necessary. R y : H , T → H ®, is obvious, by Marcinewicz interpolation theorem, it is only to prove the uniform boundedness of R y : A , T → h/ ,H ®, . So we only need to prove that ™ €, w ∈ ®: R y ℎ €, w > • ≤ ; For this reason, fix ℎ ≥ 0 is a bounded function with compact support. Set ± ² = ™ €, w ∈ ®: R y ℎ €, w > • = ™ €, w ∈ ⋀˜: R y ℎ €, w > •.
Since ~y ⊂ 3u~y, we have The dydic cube in the family ‡u~y: €, w ∈ ± ² ‰ is bounded. In fact, if €, w ∈ ± ² , then |w| ≤ -, and for each w, the cube ‡u~y‰~ is bounded. This allows us to select the largest subset family ™u X • X for each €, w ∈ ± ² , u~y ⊂ u X . By ~y 's are pairwise disjoint and ~y ⊂ 3u~y, where we used the definition of two weight. Since the cube u X is selected from the maximal disjoint subcollection ‡u~y: €, w ∈ ± ² ‰ , there exists u X = u~y for each · , with €, w ∈ ± ² . In this case, R y ℎ €, w > , and since €, w ∈ ± ² , Notice that q/p ≥ 1 , the maxima of cube u X and their disjoints, we have where is independent of , so the proof is completed.

Weak Boundedness of New Maximal Operators
The main result of this section is to obtain the two weight weak boundedness of fractional new maximal operators. iii For every D ≥ 0 and cube , Proof of Theorem 2.2.1. The proof will be carried out in the following manner: ii ⇒ iii ⇒ i ⇒ iii ⇒ ii ii ⇒ iii Take D ≥ 0 and cube such that (3) . So it can be assumed that D ∈ A , by defining D y = D! = 5,y , then we have D y → D. In this case, the k limit of D y in (ii) is independent of each constant C, so (ii) about f holds. With these, we will prove that the inequality of expectation holds when D ≥ 0, D ∈ ∩ A . Set If ∈ ² , from the definition of and the relationship between maximal function and central maximal function, there exists F > 0 such that

Conclusion
This paper gives the sufficient and necessary conditions for strong and weak type two weight norm inequalities for a new fractional maximal operators by introducing a class of new two weight functions which include classical two weight functions on Lebesgue spaces and our main results will shed some new lights on boundedness of other operators and their commutators on Lebesgue spaces. In addition, we can also consider the two weight characterization of new maximal operators on Morrey spaces.