Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Let (X, d, μ) be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorM b generated by the Marcinkiewicz integralM and Lipschitz function b. The authors prove that Mb is bounded from the Lebesgue spaces L p (μ) to weak Lebesgue spaces L(μ) for 1 ≤ p < n/β, from the Lebesgue spaces L(μ) to the spaces RBMO(μ) for p = n/β, and from the Lebesgue spaces L(μ) to the Lipschitz spaces Lip(β−n/p)(μ) for n/β < p ≤ ∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.


Introduction
As we know, the Littlewood-Paley operators are playing an important role in harmonic analysis and PDE.The Marcinkiewicz integral is an essential Littlewood-Paley function.It is firstly introduced by Marcinkiewicz on R and it is conjectured that it is bounded on   ([0, 2]) for any  ∈ (1, ∞) (see [1]).In 1958, Stein gave the higherdimensional Marcinkiewicz integral (see [2]).Suppose that Ω is homogeneous of degree zero on R  , for  ≥ 2, and has mean value zero on the unit sphere S −1 ; Marcinkiewicz M Ω is defined by (1) In [2], Stein proved that if Ω ∈ Lip  (S −1 ) for some  ∈ (0, 1], then M was bounded on   (R  ) for any  ∈ (1,2] and also bounded from  1 (R  ) to  1,∞ (R  ).In 1990, Torchinsky and Wang established   (R  ) (1 <  < ∞) boundedness for the commutator generated by M Ω and BMO function (see [3]).
In 2007, Mo and Lu obtained boundedness of the commutator generated by M Ω and Lip  function in [4].For more results about this operator, we refer the reader to see [5][6][7][8].
Let  be a nonnegative Radon measure on R  which satisfies the polynomial growth condition; that is, there exist positive constants  0 and  ∈ (0, ] such that, for all  ∈ R  and  ∈ (0, ∞), ((, 2)) ≤  0   , where (, ) = { ∈ R  : | − | < }.The analysis associated with nondoubling measures  is proved to play a striking role in solving the long-standing open Painlevé problem by Tolsa in [9].Obviously, the nondoubling measure  with the polynomial growth condition may not satisfy the wellknown doubling condition, which is a key assumption in harmonic analysis on spaces of homogeneous type.Since then, many results from real analysis and harmonic analysis on the classical Euclidean spaces have been extended to the spaces with nondoubling measures satisfying the polynomial growth condition (see [10][11][12][13][14]).The Marcinkiewicz integral operators and commutators have also been discussed widely on the spaces with nondoubling measure (see [15][16][17]).In 2010, Hytönen introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions, which are called nonhomogeneous metric measure space in [18].In particular, in recent years, a lot of classical results have been proved to be still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen et al. (see [19,20]).For 2 Journal of Function Spaces example, Lin and Yang in [21] obtained the boundedness of Marcinkiewicz integral on nonhomogeneous metric measure spaces.In 2013, Cao and Zhou considered some operators on Morrey spaces over nonhomogeneous metric measure spaces in [22].
In this paper, we will give some estimates for the commutator of Marcinkiewicz integral on the Lebesgue spaces, Lipschitz spaces, RBMO() spaces, Morrey spaces, and Hardy spaces on nonhomogeneous metric measure spaces.
To state our main results, we first recall some necessary notions and remarks.The notion of upper doubling metric measure spaces was originally introduced by Hytönen [18] (see also [19]) as follows.
Definition 1.A metric measure space (X, , ) is said to be upper doubling, if  is Borel measure on  and there exist a dominating function  : X×(0, ∞) → (0, ∞) and a positive constant   such that for each  ∈ X :  → (, ) is nondecreasing and, for all  ∈ X and  ∈ (0, ∞), ) . ( Remark 2. (1) Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating function (, ) = ((, )).Moreover, let  be a nonnegative Radon measure on R  which only satisfies the polynomial growth condition.By taking (, ) =  0   , we see that (R  , | ⋅ |, ) is also an upper doubling measure space.
(2) It was proved in [20] that there exists another dominating function λ related to  satisfying the property that there exists a positive constant  λ such that λ ≤ ,  λ ≤   and, for all ,  ∈  with (, ) ≤ , λ (, ) ≤  λ λ (, ) . ( Based on this, we always assume that (X, , ) is a nonhomogeneous metric measure spaces with the dominating function  that satisfies (3).
The following notion of the geometrically doubling condition is well-known in analysis on metric spaces, which was firstly introduced by Coifman and Weiss in [23, pp. 66-67].Definition 3. A metric space (X, ) is said to be geometrically doubling, if there exists some  0 ∈ N = {1, 2, . ..} such that, for any ball (, ) ⊂ X, there exists a finite ball covering {(  , /2)}  of (, ) such that the cardinality of this covering is at most  0 .Remark 4. Let (X, ) be a metric space.Hytönen showed that the following statements are mutually equivalent (see [18]): (1) (X, ) is geometrically doubling.
Now we recall the notion of the coefficient  , introduced by Hytönen (see [18]), which is analogous to the quantity  , introduced by Tolsa (see [13,14]).Definition 5.For any two balls  ⊂ , define where above and in that follows, for a ball  = (  ,   ) and  > 0,  = (  ,   ).  is the center of ball .

Remark 6.
The following discrete version, K, , of  , defined in Definition 5, was first introduced by Bui and Duong in nonhomogeneous metric measure spaces (see [24]), which is more close to the quantity  , introduced by Tolsa [12] in the setting of nondoubling measures.For any two balls  ⊂ , let K, be defined by where   and   , respectively, denote the radius of the balls  and  and  , denote the smallest integer satisfying 6  ,   ≥   .Obviously,  , ≤  K, .As was pointed by Bui and Duong in [24], in general, it is not true that  , ∼  K, .
Let ,  ∈ (0, ∞).A ball  ⊂ X is called (, )-doubling if () ≤ ().It was proved in [18] that if a metric measure space (X, , ) is upper doubling and  >  log 2   =  ] , then, for every ball  ⊂ X, there exists some  ∈ Z + such that    is (, )-doubling.Moreover, let (X, ) be geometrically doubling and  >   with  = log 2  0 and  is Borel measure on X which is finite on bounded sets.In [18] Hytönen also showed that for -almost every  ∈ X there exist arbitrarily small (, )-doubling balls centered at .Furthermore, the radii of their balls may be chosen to be of the form  −  for  ∈ N and any preassigned number  ∈ (0, ∞).Throughout this paper, for any  ∈ (1, ∞) and ball , B denotes the smallest (,   )-doubling ball of the form    with  ∈ Z + , where In what follows, by a doubling ball we mean a (6,  6 )-doubling ball and B6 is simply denoted by B. Now we give the definition of Marcinkiewicz integral (see [21]).
The Marcinkiewicz integral M() associated with the above kernel (, ) is defined by Obviously, by taking (, ) fl   , we see that, in the classical Euclidean space R  , if with Ω homogeneous of degree zero and Ω ∈ Lip  (S −1 ) for some  ∈ (0, 1], then  satisfies ( 8) and ( 9) and M as in ( 10) is just the Marcinkiewicz integral M Ω introduced by Stein in [2].
In 2007, Hu et al. introduced a Hörmander-type condition in [8], defined as follows: According to this, we will consider the following condition to replace (13).
The kernel (, ) is said to satisfy a Hörmander-type condition if there exist   > 1 and   > 0 such that, for any  ∈ X and ℓ >   (, 0), We denote by H  the class of kernels satisfying this condition.It is obvious that these classes are nested: Now we recall the notion of RBMO() (see [18]) as follows.
Definition 11.Let  ∈ (1, ∞).A function  ∈  1 loc () is said to be in the RBMO() if there exist a positive constant  and a complex number   for any balls , and that for any balls  ⊂ ,       −       ≤  , .
And ‖‖ RBMO() is defined to be the infimum of the positive constants  in the above inequalities.From [18], it follows that the definition RBMO() is independent of the choice of  ∈ (1, ∞).
We begin with recalling some useful properties of  , in Definition 5 (see [18]).
The following characterizations of Lip  () (0 <  ≤ 1) from [25] play a key role in the proofs of theorems.

Lemma 13. For a function 𝑓 ∈ 𝐿 1 𝑙𝑜𝑐 (𝜇), the following conditions (i), (ii), and (iii) are equivalent.
() There exist some constant  1 and a collection of numbers of   , one for each ball , such that these two properties hold: For any ball  with radius for any two balls  ⊂  with   ≤ 2  (see [25]).
The organization of this paper is as follows.In Sections 2 and 3, we study the commutator M  in the case of  ∈ Lip  () and establish that M  is bounded from the Lebesgue spaces   () to the Lebesgue spaces   () for 1 ≤  < /, from the Lebesgue spaces   () to the spaces RBMO() for  = /, and from the Lebesgue spaces   () to the Lipschitz spaces Lip (−/) () for / <  ≤ ∞.In Section 4, we establish the boundedness of commutator of the Marcinkiewicz integral M  from the Morrey space    () to the Morrey spaces    (), from the Morrey spaces    () to the spaces RBMO(), and from the Morrey spaces    () to the Lipschitz spaces Lip (−/) ().Finally, we establish the boundedness of M  in  1,∞,0 fin () for 1/ = 1 − /.Throughout this paper, we use the constant  with subscripts to indicate its dependence on the parameters.For a -measurable set ,   denotes its characteristic function.For any  ∈ [1, ∞], we denote by   its conjugate index; namely, 1/ + 1/  = 1.

Boundedness of M 𝑏 in Lebesgue Spaces
In this section, we investigate the boundedness of commutator M  as in (11) in the Lebesgue spaces.The main results are listed as follows.
Theorem 15.Let  ∈   (), 0 <  ≤ 1, 1 ≤  < /, and 1/ = 1/ − /.Suppose that (, ) satisfies ( 8) and ( 9), M is bounded on  2 (), and M  is defined as (11).Then there exists a positive constant  such that, for all  > 0, one has Proof.By Minkowski inequality and the kernel condition, we deduce that where   denotes the fractional integral operator defined by By applying (2) and Theorem 1.1 in [26], it is easy to get that there exists a positive constant  > 0, such that, for all  > 0, one has
This finishes the proof of Theorem 15.
By applying Marcinkiewicz interpolation theorem, it is easy to get the following result.

Boundedness of M 𝑏 in Lipschitz Spaces
In this section, we investigate the boundedness of commutator M  as in (11) in the Lipschitz space.
Suppose that (, ) satisfies ( 9) and H   condition, M is bounded on  2 (), and M  is defined as in (11).Then there exists a positive constant  such that, for all bounded functions  with compact support, one has     M  ()    () ≤  ‖‖   ()          / () .
Proof.With a slight change in the proof of Theorem 17, by applying Lemma 13, it is not difficult to get the proof of Theorem 18.We omit the details here.

Boundedness of M 𝑏 in Morrey Spaces
In this section, we investigate the boundedness for the commutator M  defined as (11) in the Morrey space    ().Before stating our main results, we need to recall the definition of the Morrey space.Definition 19.Let  > 1 and 1 ≤  ≤  < ∞: where It is easy to see that   () =    () and    1 ⊂    2 for 1 ≤  2 ≤  1 ≤ .If the underlying spaces are replaced by the nonhomogeneous spaces of Tolsa or Euclidean spaces, the definition of Morrey spaces can be seen in [27].Cao and Zhou proved that the Morrey space is independent of the choice of  (see [22]).
The following theorem is adapted from [28].
Based on Theorem 1.2 in [14], we give the following definition of the Hardy space  1 () from [12].
Then there exists a positive constant  such that, for all  ∈    (),     M  ()    () ≤  ‖‖   ()            () .(60) Remark 24.Since    () =   (), the proof of Theorem 21 is similar to Theorem 17. Theorem 22 can be immediately deduced as a conclusion of Theorem 21 in the case of  =  = /.By applying Lemma 13, with a slight change in the proof of Theorem 21, it is not difficult to show Theorem 23.Thus, we omit the proofs of Theorems 21-23.
, Zhou and Wang gave the notion of Lipschitz function as follows.