Boundedness of Marcinkiewicz Integrals on RBMO Spaces over Nonhomogeneous Metric Measure Spaces

We all know that the Littlewood-Paley g-function has been playing an important role in harmonic analysis and the Marcinkiewicz integral is an essential Littlewood-Paley g-function. As an analogy to Littlewood-Paley g-function without going into the interior of the unit disk, in 1938,Marcinkiewicz introduced the integral on one dimensional Euclidean space R, which now is called the Marcinkiewicz integral, and conjectured that it is bounded on L([0, 2π]) for p ∈ (1,∞) (see [1]). In 1944, Zygmund proved theMarcinkiewicz conjecture by using the complex variable method in [2]. Particularly, in 1958, Stein introduced the higher dimensional Marcinkiewicz integral (see [3]). Let Ω be homogeneous of degree zero in R for n ≥ 2 and integrable and have mean value zero on the unit sphere S. The higher dimensional Marcinkiewicz integralMΩ is defined by


Introduction
We all know that the Littlewood-Paley -function has been playing an important role in harmonic analysis and the Marcinkiewicz integral is an essential Littlewood-Paley -function.As an analogy to Littlewood-Paley -function without going into the interior of the unit disk, in 1938, Marcinkiewicz introduced the integral on one dimensional Euclidean space R, which now is called the Marcinkiewicz integral, and conjectured that it is bounded on   ([0, 2]) for  ∈ (1, ∞) (see [1]).In 1944, Zygmund proved the Marcinkiewicz conjecture by using the complex variable method in [2].Particularly, in 1958, Stein introduced the higher dimensional Marcinkiewicz integral (see [3]).Let Ω be homogeneous of degree zero in R  for  ≥ 2 and integrable and have mean value zero on the unit sphere S −1 .The higher dimensional Marcinkiewicz integral M Ω is defined by ,  ∈ R  . ( Recently, many papers focus on the boundedness of this operator on various function spaces.We refer the reader to see [4,5]. Many results from real analysis and harmonic analysis on the classical Euclidean spaces have been extended to the space of homogeneous type by Coifman and Weiss in [6].Recall that a metric space (X, ) equipped with a Borel measure  is called a space of homogeneous type, if (X, , ) satisfies the following doubling measure condition that there exists a positive constant   such that, for all balls (, ) := { ∈ X : (, ) < } with  ∈ X and  ∈ (0, ∞),  ( (, 2)) ≤    ( (, )) . ( Meanwhile, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid for nondoubling measures.In particular, let  be a nonnegative Radon measure on R  which only satisfies the polynomial growth condition that there exists a positive constant  0 and  ∈ (0, ] such that, for all  ∈ R  and  ∈ (0, ∞), where (, ) := { ∈ R  : | − | < }.Such a measure  need not satisfy the doubling condition (2 However, in 2010, Hytönen pointed out that the measures satisfying the polynomial growth condition are different from the doubling measures in [10].Hytönen introduced a new class of metric measure spaces which satisfy the upper doubling condition and the geometrically doubling condition (resp., see Definitions 1 and 3 below).This new class of metric measure space is called the nonhomogeneous metric measure space, which includes both spaces of homogeneous type and metric spaces with the measures satisfying (3) as special cases.We refer the reader to the monograph [11] for several recent developments on harmonic analysis in this setting.
In this paper, we mainly discuss the boundedness of Marcinkiewicz integrals M on RBMO() with nonhomogeneous metric measure spaces.Now we recall some notations and definitions as follows.
Definition 1.A metric measure space (X, , ) is said to be upper doubling, if  is Borel measure on X and there exists 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 (, ) =   , we see that (R  , | ⋅ |, ) is also an upper doubling measure space.
(2) It was proved that there exists a 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, in this paper, we always assume that the dominating function  also satisfies (5).
Definition 3. A metric space (X, ) is said to be geometrically doubling, if there exist 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 .
(2) For any  ∈ (0, 1) and ball (, ) ⊂ X, there exists a finite ball covering {(  , )}  of (, ) such that the cardinality of this covering is at most  − .Here and in what follows,  0 is as in Definition 3 and  = log 2  0 .
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 [12]), which is more close to the quantity  , introduced by Tolsa [13] 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  , is the smallest integer satisfying 6  ,   ≥   .Obviously,  , ≤  K, .That was pointed by Bui and Duong in [12]; in general, it is not true that  , ∼  K, .
It was proved in [10] 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,  >   with  = log 2  0 , and  is Borel measure on X which is finite on bounded sets.In [10] Hytönen also showed that, for -almost every  ∈ X, there exist arbitrarily small (, )-doubling balls centered at .Furthermore, the radius of these balls may be chosen to be 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 If  = 6, we denote the ball B simply by B. Let (, ) be a -locally integrable function on X × X \ {(, ) :  ∈ X}.Assume that there exists a positive constant  such that, for any ,  ∈ X with  ̸ = , and, for any , , The Marcinkiewicz integral M() associated with the above kernel (, ) is defined by Obviously, by taking (, ) =   , 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 ( 5) and ( 8).In 2014, Lin and Yang [14] established the equivalent boundedness of Marcinkiewicz integral M with kernel (, ) satisfying ( 9) and (10).In this note, we make some modification for the kernel.Besides satisfying the regular condition (9), (, ) also satisfies that, for any ,   and  ∈ X with (, ) ≥ 2(,   ), there exists 0 <  ≤ and, for any  ∈ X and any two positive real numbers ,  with  < , When the kernel satisfies condition (13), it also satisfies (10).
Throughout this paper, we denote by  a positive constant which is independent of the main parameters involved, but it may be different from line to line.For any  ∈ [1, ∞], we denote by   its conjugate index; namely, 1/ + 1/  = 1.

Main Result and Its Proof
We give the main result as follows.
In order to prove the theorem, we need the following two lemmas.The following useful properties of  , were proved in [10,16]; see also [17].
(iii) For any  ∈ [1, ∞), there exists a positive constant C, depending on , such that, for all balls ,  , B ≤ C.
(v) There exists a positive constant c such that, for all balls  ⊂  ⊂ ,  , ≤ c , ; moreover, if  and  are concentric, then  , ≤  , .
Moreover, the minimal constant  as above is equivalent to ‖‖ RBMO() .
Proof of Theorem 11.For  ∈ RBMO(), we have that M() is finite -almost everywhere and decompose for any two balls  ⊂ .
By the vanishing ( 14), we have By applying the Minkowski inequality and vanishing condition (14), we have Then it is easy to get that, for any ,  ∈ X, which follows that (28) Applying Hölder's inequality, [10, Corollary 6.3], and  2 ()-boundedness of M, we deduce that
From this and the Minkowski inequality, it follows that From (31), (32), and (38), we deduce that On the other hand, By applying a similar argument of ( 29 If  ∈ RBMO() such that M() is infinite on a set of positive measures, it is easy to prove that we take a ball  such that namely, M()() ∉  loc ().So, M()() ∉ RBMO().We complete the proof of Theorem 11.
1 loc () is said to be the space RBMO(), if there exists a positive constant  and a number   for any ball  such that, for all balls , and, for balls  ⊂ ,       −       ≤  , .