# Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

Authors: Guanghui Lu; Shuangping Tao
Journal: Journal of Function Spaces
 (2015)

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## Abstract

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 commutator Mb generated by the Marcinkiewicz integral M and Lipschitz function b. The authors prove that Mb is bounded from the Lebesgue spaces Lp(μ) to weak Lebesgue spaces Lq(μ) for 1≤p<n/β, from the Lebesgue spaces Lp(μ) to the spaces RBMO(μ) for p=n/β, and from the Lebesgue spaces Lp(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.


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## Body

## 1. 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-Paleyg-function. It is firstly introduced by Marcinkiewicz on R and it is conjectured that it is bounded on Lp([0,2π]) for any p∈(1,∞) (see [1]). In 1958, Stein gave the higher-dimensional Marcinkiewicz integral (see [2]). Suppose that Ω is homogeneous of degree zero on Rn, for n≥2, and has mean value zero on the unit sphere Sn-1; Marcinkiewicz MΩ is defined by (1)MΩfx=∫0∞∫x-y≤tΩx-yx-yn-1fydμy2dtt31/2,x∈Rn.In [2], Stein proved that if Ω∈Lipα(Sn-1) for some α∈(0,1], then M was bounded on Lp(Rn) for any p∈(1,2] and also bounded from L1(Rn) to L1,∞(Rn). In 1990, Torchinsky and Wang established Lp(Rn)(1<p<∞) boundedness for the commutator generated by MΩ and BMO function (see [3]). In 2007, Mo and Lu obtained boundedness of the commutator generated by MΩ and Lipβ function in [4]. For more results about this operator, we refer the reader to see [5–8].Letμ be a nonnegative Radon measure on Rn which satisfies the polynomial growth condition; that is, there exist positive constants C0 and d∈(0,n] such that, for all x∈Rn and r∈(0,∞), μ(B(x,2r))≤C0rd, where B(x,r)={y∈Rn:x-y<r}. The analysis associated with nondoubling measures μ is proved to play a striking role in solving the long-standing open Painlevé problem by Tolsa in [9]. Obviously, the nondoubling measure μ with the polynomial growth condition may not satisfy the well-known 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 [10–14]). The Marcinkiewicz integral operators and commutators have also been discussed widely on the spaces with nondoubling measure (see [15–17]). 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 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(μ) 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,d,μ) is said to be upper doubling, if μ is Borel measure on χ and there exist a dominating function λ:X×(0,∞)→(0,∞) and a positive constant Cλ such that for each x∈X:r→λ(x,r) is nondecreasing and, for all x∈X and r∈(0,∞),(2)μBx,r≤λx,r≤Cλλx,r2.Remark 2.
(1) Obviously, a space of homogeneous type is a special case of upper doubling spaces, where one can take the dominating functionλ(x,r)=μ(B(x,r)). Moreover, let μ be a nonnegative Radon measure on Rn which only satisfies the polynomial growth condition. By taking λ(x,r)=C0rd, we see that (Rn,·,μ) is also an upper doubling measure space.
(2) It was proved in [20] that there exists another dominating function λ~ related to λ satisfying the property that there exists a positive constant Cλ~ such that λ~≤λ,Cλ~≤Cλ and, for all x,y∈χ with d(x,y)≤λ, (3)λ~x,r≤Cλ~λ~y,r.Based on this, we always assume that (X,d,μ) is a nonhomogeneous metric measure spaces with the dominating function λ 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,d) is said to be geometrically doubling, if there exists some N0∈N=1,2,… such that, for any ball B(x,r)⊂X, there exists a finite ball covering {B(xi,r/2)}i of B(x,r) such that the cardinality of this covering is at most N0.Remark 4.
Let(X,d) be a metric space. Hytönen showed that the following statements are mutually equivalent (see [18]):(1)
(X,d) is geometrically doubling.(2)
For anyϵ∈(0,1) and ball B(x,r)⊂X, there exists a finite ball covering {B(xi,ϵr)}i of B(x,r) such that the cardinality of this covering is at most N0ϵ-n. Here and in what follows, N0 is as Definition 3 and n=log2N0.(3)
For everyϵ∈(0,1), any ball B(x,r)⊂X can contain at most N0ϵ-n centers {xi}i of disjoint balls with radius ϵr.(4)
There existsM∈N such that any ball B(x,r)⊂X can contain at most M centers {xi}i of disjoint balls {B(xi,r/4)}i=1M.Now we recall the notion of the coefficientKB,S introduced by Hytönen (see [18]), which is analogous to the quantity KQ,R introduced by Tolsa (see [13, 14]).Definition 5.
For any two ballsB⊂S, define (4)KB,S=1+∫2S∖B1λcB,dx,cBdμx,where above and in that follows, for a ball B=B(cB,rB) and ρ>0, ρB=B(cB,ρrB).cB is the center of ball B.Remark 6.
The following discrete version,K~B,S, of KB,S 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 KQ,R introduced by Tolsa [12] in the setting of nondoubling measures. For any two balls B⊂S, let K~B,S be defined by (5)K~B,S≔1+∑k=1NB,Sμ6kBλcB,6krB,where rB and rS, respectively, denote the radius of the balls B and S and NB,S denote the smallest integer satisfying 6NB,SrB≥rS. Obviously, KB,S≤CK~B,S. As was pointed by Bui and Duong in [24], in general, it is not true that KB,S~CK~B,S.In [25], Zhou and Wang gave the notion of Lipschitz function as follows.Definition 7.
Givenβ∈(0,1], the function f:X→C satisfies a Lipschitz condition of order β provided (6)fLipβ=supx≠y,x,y∈X⁡fx-fyλx,dx,yβ/n<∞,and we claim that f∈Lipβ.Remark 8.
Lipschitz condition can also be defined by1.3′fLipβ=supx≠y,x,y∈X⁡fx-fyλy,dx,yβ/n<∞.See [25].Letα,β∈(0,∞). A ball B⊂X is called (α,β)-doubling if μ(αB)≤βμ(B). It was proved in [18] that if a metric measure space (X,d,μ) is upper doubling and β>Cλlog2α=αν, then, for every ball B⊂X, there exists some j∈Z+ such that αjβ is (α,β)-doubling. Moreover, let (X,d) be geometrically doubling and β>αn with n=log2N0 and μ is Borel measure on X which is finite on bounded sets. In [18] Hytönen also showed that for μ-almost every x∈X there exist arbitrarily small (α,β)-doubling balls centered at x. Furthermore, the radii of their balls may be chosen to be of the form α-jr for j∈N and any preassigned number r∈(0,∞). Throughout this paper, for any α∈(1,∞) and ball B, B~ denotes the smallest (α,βα)-doubling ball of the form αjB with j∈Z+, where (7)βα=α3max⁡n,ν+30n+30ν.In what follows, by a doubling ball we mean a (6,β6)-doubling ball and B~6 is simply denoted by B~.Now we give the definition of Marcinkiewicz integral (see [21]).LetK(x,y) be a μ-locally integrable function on X×X∖{(x,x):x∈X}. Assume that there exists a positive constant C such that, for any x,y∈X with x≠y, (8)Kx,y≤Cdx,yλx,dx,yand, for any x,y,y′∈X, (9)∫dx,y≥2dy,y′Kx,y-Kx,y′+Ky,x-Ky′,x1dx,ydμx≤C.The Marcinkiewicz integralM(f) associated with the above kernel K(x,y) is defined by (10)Mfx=∫0∞∫dx,y≤tKx,yfydμy2dtt31/2,x∈X.Definition 9.
Letb∈Lipβ(μ) and K(x,y) satisfy (8) and (9). The commutator of Marcinkiewicz Mb(f) is formally defined by (11)Mbfx=∫0∞∫dx,y≤tKx,ybx-byfydμy2dtt31/2,x∈X.Obviously, by takingλ(x,r)≔Crn, we see that, in the classical Euclidean space Rn, if (12)Kx,y=Ωx-yx-yn-1with Ω homogeneous of degree zero and Ω∈Lipα(Sn-1) for some α∈(0,1], then K satisfies (8) and (9) and M as in (10) is just the Marcinkiewicz integral MΩ introduced by Stein in [2].In 2007, Hu et al. introduced a Hörmander-type condition in [8],  defined as follows: (13)supl>0,y,y′∈Rdy-y′≤l⁡∑k=1∞k∫2kl<x-y≤2k+1lKx,y-Kx,y′+Ky,x-Ky′,x1x-ydμx≤C.According to this, we will consider the following condition to replace (13).Definition 10.
Let1≤s<∞,0<ϵ<1. The kernel K(x,y) is said to satisfy a Hörmander-type condition if there exist cs>1 and Cs>0 such that, for any x∈X and l>csd(x,0),(14)supl>0,y,y′∈Xdy,y′≤l⁡∑k=1∞2kϵλx,2kl1λx,2kl∫2kl<dx,y≤2k+1lKx,y-Kx,y′+Ky,x-Ky′,x1dx,ysdμx1/s≤Cs.We denote byHs the class of kernels satisfying this condition. It is obvious that these classes are nested: (15)Hs2⊂Hs1⊂H1,1<s1<s2<∞.Now we recall the notion ofRBMO(μ) (see [18]) as follows.Definition 11.
Letρ∈(1,∞). A function f∈Lloc1(μ) is said to be in the RBMO(μ) if there exist a positive constant C and a complex number fB for any balls B,(16)1μρB∫Bfy-fBdμy≤C,and that for any balls B⊂S,(17)fB-fS≤CKB,S.And fRBMO(μ) is defined to be the infimum of the positive constants C in the above inequalities. From [18], it follows that the definition RBMO(μ) is independent of the choice of ρ∈(1,∞).We begin with recalling some useful properties ofKB,S in Definition 5 (see [18]).Lemma 12.
(1) For all ballsB⊂R⊂S, it holds true that KB,R≤KB,S.
(2) For anyρ∈[1,∞), there exists a positive constant C, depending on ρ, such that, for all balls B⊂S with rS≤ρrB,KB,S≤C.
(3) For anyα∈(1,∞), there exists a positive constant C~, depending on α, such that, for all balls B,KB,B~α≤C~.
(4) There exists a positive constantc such that, for all balls B⊂R⊂S,KB,S≤KB,R+cKR,S. In particular, if B and R are concentric, then c=1.
(5) There exists a positive constantC~ such that, for all balls B⊂R⊂S,KB,R≤c~KB,S; moreover, if B and R are concentric, then KR,S≤KB,S.The following characterizations ofLipβ(μ)(0<β≤1) from [25] play a key role in the proofs of theorems.Lemma 13.
For a functionf∈Lloc1(μ), the following conditions (i), (ii), and (iii) are equivalent.
(
i
) There exist some constant C1 and a collection of numbers of fB, one for each ball B, such that these two properties hold: For any ball B with radius r(18)1μ2B∫Bfx-fBdμx≤C1λxB,rBβ/nand, for any ball R such that B⊂R and rR≤2r, (19)fB-fR≤C1λxB,rBβ/n.
(
i
i
) There is a constant C2 such that (20)fx-fy≤C2λx,dx,yβ/n,for μ-almost every x and y in the support of μ.
(
i
i
i
) For any given p, 1≤p≤∞, there is a constant C(p), such that, for every ball B of radius r, we have (21)1μB∫Bfx-mBfpdμx1/p≤CpλxB,rBβ/n,where mB=(1/μB)∫B‍f(y)dμ(y) and also for any ball R such that B⊂R and rR≤2r(22)mBf-mRf≤CpλxB,rBβ/n.
In addition, the quantitiesinf⁡C1, inf⁡C2, and inf⁡C(p) with a fixed p are equivalent and denoted by bLipβ(μ).Remark 14.
For0<β≤1, (20) is equivalent to (23)fB-fR≤C1′KB,Sλx,rβ/n,for any two balls B⊂R with rB≤2rR (see [25]).The organization of this paper is as follows. In Sections2 and 3, we study the commutator Mb in the case of b∈Lipβ(μ) and establish that Mb is bounded from the Lebesgue spaces Lp(μ) to the Lebesgue spaces Lq(μ) for 1≤p<n/β, from the Lebesgue spaces Lp(μ) to the spaces RBMO(μ) for p=n/β, and from the Lebesgue spaces Lp(μ) to the Lipschitz spaces Lip(β-n/p)(μ) for n/β<p≤∞. In Section 4, we establish the boundedness of commutator of the Marcinkiewicz integral Mb from the Morrey space Mqp(μ) to the Morrey spaces Mts(μ), from the Morrey spaces Mqp(μ) to the spaces RBMO(μ), and from the Morrey spaces Mqp(μ) to the Lipschitz spaces Lip(β-n/p)(μ). Finally, we establish the boundedness of Mb in Hfin1,∞,0(μ) for 1/q=1-β/n.Throughout this paper, we use the constantC with subscripts to indicate its dependence on the parameters. For a μ-measurable set E, χE denotes its characteristic function. For any p∈[1,∞], we denote by p′ its conjugate index; namely, 1/p+1/p′=1.


## 2. Boundedness ofMb in Lebesgue Spaces

In this section, we investigate the boundedness of commutatorMb as in (11) in the Lebesgue spaces. The main results are listed as follows.Theorem 15.
Letb∈Lipβ(μ),0<β≤1,1≤p<n/β, and 1/q=1/p-β/n. Suppose that K(x,y) satisfies (8) and (9), M is bounded on L2(μ), and Mb is defined as (11). Then there exists a positive constant C such that, for all λ>0, one has (24)μx∈X:Mbfx>λ≤CfL1μλn/n-β.Proof.
By Minkowski inequality and the kernel condition, we deduce that(25)Mbfx≤∫XKx,ybx-byfy∫dx,y∞dtt31/2dμy≤C∫Xdx,yλx,dx,ybx-byfy1dx,ydμy≤CbLipβμIβfx,where Iβ denotes the fractional integral operator defined by (26)Iβfx=∫Xfyλx,dx,yβ-n/ndμy,0<β<n.By applying (2) and Theorem  1.1 in [26], it is easy to get that there exists a positive constant C>0, such that, for all η>0, one has (27)μx∈X:Iβfx>η≤CfL1μηn/n-β.This finishes the proof of Theorem 15.By applying Marcinkiewicz interpolation theorem, it is easy to get the following result.Corollary 16.
Letb∈Lipβ(μ),0<β≤1,1<p<n/β, and 1/q=1/p-β/n. Suppose that K(x,y) satisfies (8) and (9), M is bounded on L2(μ), and Mb is defined as (11). Then there exists a positive constant C such that, for all λ>0,(28)MbfLqμ≤CbLipβμfLpμ.


## 3. Boundedness ofMb in Lipschitz Spaces

In this section, we investigate the boundedness of commutatorMb as in (11) in the Lipschitz space.Theorem 17.
Let0<ϵ<1,0<β<min⁡1/2,ϵ,n/β<p<∞. Suppose that b∈Lipβ(μ) and K(x,y) satisfies (8) and Hp′ condition. Then Mb is bounded from the Lebesgue spaces Lp(μ) to the Lipschitz spaces Lip(β-n/p)(μ); namely, there exists a positive constant C, such that (29)MbfLipβ-n/pμ≤CbLipβμfLpμ.Proof.
For any ballsR and S in X such that R⊂S satisfies rS≤2rR, let (30)aR=mRMbfχX∖2R,aS=mSMbfχX∖2S.It is easy to see that aR and aS are real numbers. By Lemma 13, we need to show that there exists a constant C>0 such that(31)1μ2R∫RMbfx-aRdμx≤CbLipβμfLpμλcR,rRβ/n-1/p,(32)aR-aS≤CbLipβμfLpμλcR,rRβ/n-1/p.Firstly, let us estimate (31). For a fixed ball R and x∈R, decompose f=f1+f2, where f1=fχ2R and f2=f-f1. Write that (33)1μ2R∫RMbfx-aRdμx≤1μ2R∫RMbf1xdμx+1μ2R∫RMbf2x-aRdμx=I1+I2.Thus, for 1<p1<n/β<p and 1/q1=1/p1-β/n, by Hölder inequality and Lemma 13, we can get (34)I1≤1μ2R∫RMbf1xq1dμx1/q1μR1-1/q1≤CbLipβμμ2R∫2Rfxp1dμx1/p1μR1-1/q1≤CbLipβμμ2R∫2Rfxpdμx1/pμ2R1/p1-1/pμR1-1/q1≤CbLipβμfLpμλcR,rRβ/n-1/p.
Secondly, let us proveI2. For any x,y∈R, one has (35)I2=1μ2R∫R1μR∫RMbf2xdμy-1μR∫RMbfχX∖2Rydμydμx≤1μR∫2R∫RMbf2x-Mbf2ydμxdμy.
Now let us estimate|Mb(f2)(x)-Mb(f2)(y)|. In order to do this, we write (36)D1x,y=∫0∞∫dx,z<t<dy,zKx,zbz-mRbf2zdμz2dtt31/2,D2x,y=∫0∞∫dy,z<t<dx,zKy,zbz-mRbf2zdμz2dtt31/2,D3x,y=∫0∞∫dx,z≤tdy,z≤tKx,z-Ky,zbz-mRbf2zdμz2dtt31/2.
Forx,y∈R, we have (37)Mbf2x-Mbf2y=∫0∞∫dx,z≤tKx,zbx-bzf2zdμz2dtt31/2-∫0∞∫dy,z≤tKy,zby-bzf2zdμz2dtt31/2≤∫0∞∫dx,z≤tKx,zbx-bzf2zdμz-∫dy,z≤tKy,zby-bzf2zdμz2dtt31/2≤∑i=13Dix,y+∫0∞∫dx,z≤tKx,zbx-mRbf2zdμz2dtt31/2+∫0∞∫dy,z≤tKy,zby-mRbf2zdμz2dtt31/2≤∑i=13Dix,y+H1+H2.
Noticing that, for anyx,y∈R, z∈X∖2R, it holds true that d(x,z)≥rR. By Hölder inequality and Lemma 13, we can deduce that (38)D1x,y≤C∫0∞∫dx,z<t<dy,zdx,zλx,dx,zbz-mRbf2zdμz2dtt31/2≤Cdx,y1/2∫X∖2Rbz-mRbλx,dx,zfz1dx,z1/2dμz≤C∑k=1∞1λcR,2k+1R∫2kRbz-mRbfzdμz≤CfLpμ∑k=1∞1λcR,2k+1R∫2k+1Rbz-mRbp′dμz1/p′≤CbLipβμfLpμ∑k=1∞1λx,2k+1rRμ2k+1R1-1/pλx,2k+1rRβ/n≤CbLipβμfLpμλcR,rRβ/n-1/p.
Asz∈X∖2R, we have (39)bz-mRb≤CbLipβμλcR,2krRβ/n.
By a similar argument as in the estimate ofD1(x,y), it follows that (40)D2x,y≤CbLipβμfLpμλcR,rRβ/n-1/p.
As0<β<min⁡{1/2,ϵ}, noting that K(x,y) satisfies the condition Hp′, we have by applying Minkowski inequality (41)D3x,y≤∫XKx,z-Ky,zbz-mRbf2z∫dy,z≤tdx,z≤tdtt31/2dμz≤CbLipβμ∑k=1∞λcR,2krRβ/n∫2k+1R∖2kRKx,z-Ky,zfzdy,zdμz≤CbLipβμfLpμ∑k=1∞λcR,2krRβ/n∫2k+1R∖2kRKx,z-Ky,z1dx,yq′dμz1/q′≤CbLipβμfLpμ∑k=1∞λcR,2krRβ/n-1/p2-ϵ2ϵλx,2krR1λcR,2krR∫2k+1R∖2kRKx,z-Ky,z1dx,yq′dμz1/q′≤CbLipβμfLpμλcR,rRβ/n-1/p.
Finally, let us estimateH1. For any x,y∈R,z∈X∖2R, we have(42)H1≤∫X∖2RKx,zbx-mRbfz∫dx,z∞dtt31/2dμz≤CbLipβμλx,rRβ/n∑k=1∞∫2k+1R∖2kRfzλx,dx,zdμz≤CbLipβμλx,rRβ/n∑k=1∞1λcR,2k+1rR∫2k+1R∖2kRfzdμz≤CbLipβμfLpμλcR,rRβ/n-1/p.
By a similar argument forH1, we can get that (43)H2≤CbLipβμfLpμλcR,rRβ/n-1/p.
Combining these estimates, we can conclude(44)I2≤CbLipβμfLpμλcR,rRβ/n-1/p.Thus, (31) is proved.
Now, let us proceed to show (32). For any balls R⊂S⊂X with x∈R, where R is an arbitrary ball and S is a doubling ball, denote NR,S+1 simply by N. Write (45)aR-aS≤mRMbfχX∖2NR-mSMbfχX∖2NR+mRMbfχ2NR∖2R+mSMbfχ2NR∖2S=E1+E2+E3.
By a similar argument forI2, we have (46)E1≤CbLipβμfLpμλcR,rRβ/n-1/p.As y∈S,z∈2NR∖2R, it follows that (47)Mbfχ2NR∖2Sy≤C∫2NR∖2SKy,zby-bz∫dy,z∞dtt31/2dμz≤C∫2NR∖2Sfzλy,dy,zby-bzdμz≤CbLipβμfLpμλcR,rSβ/n-1μ2NR1-1/p≤CbLipβμfLpμλcR,rRβ/n-1/p.Therefore, we obtain that (48)E3≤CbLipβμfLpμλcR,rRβ/n-1/p.With a similar argument to that in the estimate of E3, we can get (49)E2≤CbLipβμfLpμλcR,rRβ/n-1/p.Thus, (32) is proved and this finishes the proof of Theorem 17.Theorem 18.
Letb∈Lipβ(μ),0<β≤1, and p=n/β. Suppose that K(x,y) satisfies (9) and Hp′ condition, M is bounded on L2(μ), and Mb is defined as in (11). Then there exists a positive constant C such that, for all bounded functions f with compact support, one has(50)MbfRBMOμ≤CbLipβμfLn/βμ.Proof.
With a slight change in the proof of Theorem17, by applying Lemma 13, it is not difficult to get the proof of Theorem 18. We omit the details here.


## 4. Boundedness ofMb in Morrey Spaces

In this section, we investigate the boundedness for the commutatorMb defined as (11) in the Morrey space Mpq(μ). Before stating our main results, we need to recall the definition of the Morrey space.Definition 19.
Letk>1 and 1≤q≤p<∞:(51)Mqpμ=f∈Llocq:fMqpμ<∞,where (52)fMqpμ=supB⁡μkB1/p-1/q∫Bfxqdμx1/q.It is easy to see thatLp(μ)=Mpp(μ) and Mq1p⊂Mq2p for 1≤q2≤q1≤p. 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 k (see [22]).Suppose thatQ={B⊂X:Bis(6,β6)-doubling ball}. Let (53)Mqpd=f∈Llocq:fMqpd=supB∈Q⁡μB1/p-1/q∫Bfqdμ1/q<∞.In [22], it was proved that (54)Mqpd≈Mqpμ,as1≤q<p<∞,β6>6nqp/p-q.Theorem 20.
Letb∈Lipβ(μ)(0<β≤1),1<q≤p<n/β,1<t≤s<n/β, and 1/s=1/p-α/n,t/s=p/q. Suppose that K(x,y) satisfies (8) and (9), M is bounded on L2(μ), and Mb is defined as in (11). Then Mb is bounded from the space Mqp(μ) to the space Mts(μ); namely, there exists a positive constant C such that (55)MbfMtsμ≤CbLipβμfMqpμ.Proof.
By Minkowski inequality and the kernel conditions, we can get(56)Mbfx≤∫XKx,ybx-byfy∫dx,y∞dtt31/2dμy≤CbLipβμ∫Xfyλx,dx,y1-β/ndμy≤CbLipβμIβfx,from the following fact proved in [22]: (57)IβfMtsμ≤CfMqpμ,so we can easily get the proof of Theorem 20.The following theorem is adapted from [28].Theorem 21.
Letb∈Lipβ(μ)(0<β≤1),1≤q<p, and p>n/β. Suppose that K(x,y) satisfies (9) and Hp′ condition, M is bounded on L2(μ), and Mb is defined as in (11). Then there exists a positive constant C such that, for all f∈Mqp(μ),(58)MbfLipβ-n/pμ≤CbLipβμfMqpμ.Theorem 22.
Letb∈Lipβ(μ),0<β≤1, and p=n/β. Suppose that K(x,y) satisfies (9) and Hn/(n-β) condition, M is bounded on L2(μ), and Mb is defined as in (11). Then there exists a positive constant C such that, for all f with compact support,(59)MbfRBMOμ≤CbLipβμfLn/βμ.Theorem 23.
Letb∈Lipβ(μ)(0<β≤1),1≤q<p=n/β. Suppose that K(x,y) satisfies (9) and Hp′ condition, M is bounded on L2(μ), and Mb is defined as in (11). Then there exists a positive constant C such that, for all f∈Mqp(μ),(60)MbfRBMOμ≤CbLipβμfMqpμ.Remark 24.
SinceMpp(μ)=Lp(μ), 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 p=q=n/β. 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.


## 5. Boundedness ofMb in Hardy Spaces

This section is devoted to the behavior ofMb in Hardy spaces. In order to define the Hardy space H1(μ), we need to recall the grand maximal operator Mϕ introduced in [14].Definition 25.
Considerf∈Lloc1(μ), defined by (61)Mϕfx=supφ~x⁡∫Xfφdμ,where the notation φ~x means that φ∈L1(μ)∩C1(X) and satisfies(1)
φL1(μ)≤1,(2)
0≤φ(y)≤1/λ(y,d(x,y)) for all y∈X,(3)
|φ′(y)|≤1/[λ(y,d(x,y))](n+1)/n for all y∈X.Based on Theorem  1.2 in [14], we give the following definition of the Hardy space H1(μ) from [12].Definition 26.
The Hardy spaceH1(μ) is the set of all functions f∈L1(μ) satisfying that ∫X‍fdμ=0 and Mϕf∈L1(μ). Moreover, the norm of H1(μ) is defined by (62)fH1μ=fL1μ+MϕμL1μ.Now we recall the definition of the atomic Hardy space from [20].Definition 27.
Letρ>1. A function h∈Lloc1(μ) is called an atomic block if(1)
there exists some ballB such that supp⁡(h)⊂B,(2)
∫X‍h(x)dμ(x)=0,(3)
fori=1,2, there are functions ai supported on cubes Bi⊂S and numbers λi∈R such that h=λ1a1+λ2a2 and (63)aiL∞μ≤μρBiKBi,S-1.
Then define(64)hHatb1,∞,0μ=λ1+λ2.
DefineHatb1,∞,0(μ) and Hfin1,∞,0(μ) as follows: (65)fHatb1,∞,0μ=inf⁡∑j∞hjHatb1,∞,0μ:f=∑j=1∞hj,hjj∈Nare1,∞,0-atoms,fHfin1,∞,0μ=inf⁡∑jkhjHatb1,∞,0μ:f=∑j=1khj,hjj=1kare1,∞,0-atoms,where the infimum is taken over all possible decompositions of f in atomic blocks and Hfin1,∞,0(μ) is the set of all finite linear combinations of (1,∞,0)-atoms.Remark 28.
It was proved in [12, 20] that, for each ρ>1, the atomic Hardy space Hatb1,∞,0(μ) is independent of the choice of ρ and the spaces H1(μ) and H1,∞,0(μ) coincide with equivalent norms.Then we state the main theorem which was adapted from [28] of this section.Theorem 29.
Let0<β≤1,b∈Lipβ(μ), and 1/q=1-β/n. Suppose that K(x,y) satisfies (9) and Hq condition. Then Mb is bounded from the Hardy spaces H1(μ) into the Lebesgue spaces Lq(μ); namely, there exists a positive constant C such that for all f∈Hfin1,∞,0(μ)(66)MbfLqμ≤CbLipβμfHfin1,∞,0μ.Proof.
Without loss of generality, we may assume thatρ=4 and f=∑‍h as a finite of atomic blocks defined in Definition 27. It is easy to see that we only need to prove the theorem for one atomic block h. Let B be a ball such that supp⁡(h)⊂B,∫X‍h(x)dμ(x)=0, and (67)hx=λ1a1x+λa2x,where λi, for i=1,2, is a real number, |hi|Hatb1,∞,0(μ)=λ1+λ2, and ai, for i=1,2, is a bounded function supported on some balls Bi⊂S and it satisfies (68)aiL∞μ≤μ4BiKBi,S-1.Write (69)MbhLqμ≤∫2SMbhxqdμx1/q+∫X∖2SMbhxqdμx1/q≤∫2SMbhxqdμx1/q+∫X∖2S∫0dx,xS+2rS∫dx,y≤tKx,ybx-byhydμy2dtt3q/2dμx1/q+∫X∖2S∫dx,xS+2rS∞∫dx,y≤tKx,ybx-byhydμy2dtt3q/2dμx1/q=I+II+III.By (67), we have (70)I≤λ1∫2SMba1xqdμx1/q+λ2∫2SMba2xqdμx1/q=I1+I2.To estimate I1, write (71)I1≤λ1∫6B1Mba1xqdμx1/q+λ1∫2S∖6B1Mba1xqdμx1/q=I11+I12.
We choosep1 and q1 such that 1<p1<n/β,1<q<q1, and 1/q1=1/p1-n/β. By Hölder inequality, the fact that KB1,S≥1, and (Lp1(μ),Lq1(μ))-boundedness of Mb (see Theorem 15), we have (72)I11≤λ1∫6B1Mba1xq1dμx1/q1μ6B11/q-1/q1≤Cλ1bLipβμa1Lp1μμ6B11/q-1/q1≤Cλ1bLipβμ.
DenoteN2B1,2S simply by N1. Invoking the fact that a1L∞(μ)≤[μ(4B1)KB1,S]-1, therefore, we get (73)I12≤λ1∑k=1N1+1∫6k+1B1∖6kB1∫0∞∫dx,y≤tKx,ybx-bya1ydμy2dtt3q/2dμx1/q≤Cλ1∑k=1N1+1∫6k+1B1∖6kB1∫B1bx-byλx,dx,ya1ydμyqdμx1/q≤Cλ1bLipβμ∑k=1N1+1∫6k+1B1∖6kB1∫B11λx,dx,y1-β/na1ydμyqdμx1/q≤Cλ1bLipβμ∑k=1N1+1λx,6k+1rB1qβ/n-1μ6k+1B1a1L∞μqμB1q1/q≤Cλ1bLipβμKB1,S-q∑k=2N1+1μ6k+1B1λx,6k+1rB11/q≤Cλ1bLipβμ,and here we have used the fact that (74)∑k=2N1+1μ6k+1B1λx,6k+1rB1≤CKB1,S.So we have (75)I1≤Cλ1bLipβμ.
With an argument similar to that used in the proof ofI1, we have (76)I2≤Cλ1bLipβμ.
Combining the estimates forI1 and I2 yields I.
Fori=1,2,y∈Bi⊂S,x∈X∖2S, we have d(x,y)~d(x,xS)~d(x,xS)+2rS. By Minkowski inequality, we get (77)II≤∫X∖2S∫XKx,ybx-byhy∫dx,ydx,xS+2rSdtt31/2dμyqdμx1/q≤CbLipβμ∫S∫X∖2S1λx,dx,y1-β/nrS1/2dx,y1/2qdμx1/qhydμy≤CbLipβμ∫S∑k=1∞∫2k+1S∖2kS1λx,dx,y1-β/nrS1/2dx,y1/2qdμx1/qhydμy≤CbLipβμ∑k=1∞rS1/22krS1/21λx,2k+1rSμ2k+1S∫Shydμy≤C∑j=12λj.
For anyy∈S, we have t≥d(x,xS)+2rS≥d(x,xS)+d(y,xS)≥d(x,y). It follows that (78)III≤∫X∖2S∫SKx,ybx-byhy∫dx,xS+2rS∞dtt31/2dμyqdμx1/q≤C∫X∖2S∫SKx,ybx-mSbhydx,xS+2rSdμyqdμx1/q+C∫X∖2S∫SKx,ymSb-byhydx,xS+2rSdμyqdμx1/q=III1+III2.
ForIII1, by Minkowski inequality, we conclude that(79)III1≤C∫S∑k=1∞∫2k+1S∖2kSKx,y-Kx,xSbx-mSbdx,xS+2rSqdμx1/qhydμy≤CbLipβμ∫S∑k=1∞2-kϵ2kϵλcS,2k+1rS1/q1λcS,2k+1rS∫2k+1S∖2kSλx,dx,xSβ/nKx,y-Kx,xSdx,xSqdμx1/qhydμy≤CbLipβμ∑j=12λjaL1μ≤CbLipβμ∑j=12λj,and here we have used the fact that 1/q=1-β/n and 0<ϵ≤1.
Now we turn to estimateIII2. Noting that, for any y∈S,x∈X∖2S, we have d(x,y)~d(x,xS)+2rS and by Minkowski inequality,(80)III2≤C∫S∑k=1∞∫2k+1S∖2kSKx,ydx,ymSb-byqdμx1/qhydμy≤C∫S∑k=1∞∫2k+1S∖2kSmSb-byλx,dx,yqdμx1/qhydμy≤CbLipβμ∫S∑k=1∞λcS,rSβ/nλcS,2k+1rS-1μ2k+1S1/q∑j=12λjajydμy≤CbLipβμ∑j=12λj.So we have (81)III≤CbLipβμ∑j=12λj.
Combining the estimates forI,II, and III yields that (82)MhLqμ≤ChHatb1,∞,0μ.This finishes the proof of Theorem 29.



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*Source: 548165-2015-12-22.xml*