# Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Nondoubling Measures

Authors: Songbai Wang; Yinsheng Jiang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Letμ be a nonnegative Radon measure on ℝd satisfying μ(B(x,r))≤C0rn for all x∈ℝd,r>0, and some fixed C0>0 and n∈(0,d]. The authors obtain some estimates for the multilinear commutators generated by Marcinkiewicz integral and RBMO(μ) functions on certain Hardy-type spaces, whereRBMO(μ) space was introduced by Tolsa in (Mathematische Annalen, 2001).


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## Body

## 1. Introduction

We will work on thed-dimensional Euclidean space ℝd with a nonnegative Radon measure μ which only satisfies the following growth condition that there exists a consist C0 such that
(1)μ(B(x,r))≤C0rn,
where B(x,r) is the open ball centered at some point x∈ℝd and having radius r. The measure μ in (1) is not assumed to satisfy the doubling condition which is a key assumption in the analysis on spaces of homogeneous type. In recent years, considerable attention has been paid to the study of function spaces and the boundedness of Calderón-Zygmund operators with nondoubling measures and many classical results have been proved still valid if the underlying measure is substituted by a nondoubling Radon measure as in (1); see [1–7] and their references. It is worth pointing out that the analysis with nondoubling measures plays an essential role in solving the long-standing Painlevé open problem; see [1].By a cubeQ⊂ℝd we mean a closed cube whose sides are paralleled to the axes and we denote its side length by l(Q) and its center by xQ. Let α and β be positive constants such that α>1 and β>αn; for a cube Q, we say that Q is (α,β)-doubling if μ(αQ)≤βμ(Q), where αQ denotes the cube concentric with Q, having side length αl(Q). In what follows, for definiteness, if α and β are not specified, by a doubling cube we mean a (2,2d+1)-doubling cube. Especially, for any give cube Q, we denote by Q~ the smallest doubling cube in the family {2kQ}k∈ℕ. For two cubes Q⊂R, set
(2)KQ,R=1+∑k=1NQ,Rμ(2kQ)[l(2kQ)]n,
where NQ,R is the first positive integer k such that l(2kQ)≥l(R).Definition 1.
Letρ>1 be some fixed constant, we say that a function b∈Lloc1(μ) belongs to the space RBMO(μ) if there is a constant B>0 such that
(3)supQ1μ(ρQ)∫Q|b(x)-mQ~(b)|dμ(x)≤B,
and if Q1⊂Q2 are doubling cubes,
(4)|mQ~1(b)-mQ~2(b)|≤BKQ1,Q2,
where the supremum is taken over all cubes concentered at some point of suppμ and mQ~(b) is the mean value of b on Q~; namely,
(5)mQ~(b)=1μ(ρQ)∫QmQ~(b)dμ(x).
The minimal constant B in (3) and (4) is defined to be RBMO(μ) norm of b and is defined by ∥b∥*.
In [1] Tolsa introduced the space RBMO(μ) and showed that the definition of RBMO(μ) is independent of the choice of numbers ρ>1. In the sequel we will choose ρ=2.
To state our results, we need to recall some necessary notation and definitions. LetK(·,·) be a locally integrable function on {ℝd×ℝd∖(x,y):x=y}. Assume that there exists a constant C>0 such that for x,y∈ℝd with x≠y,
(6)|K(x,y)|≤C|x-y|-(n-1),
and for any x, y, and y′∈ℝd, with |x-y|≥2|y-y′|,
(7)∫|x-y|≥2|y-y′|[|K(x,y)-K(x,y′)|+|K(y,x)-K(y′,x)|]×1|x-y|dμ(x)≤C,
for any y,y′∈ℝd. The Marcinkiewicz integral ℳ associated with the above kernel K and the measure μ as in (1) is defined by
(8)ℳ(f)(x)=(×f(y)dμ(y)∫|x-y|≤tK(x,y)|2∫0∞|∫|x-y|≤tK(x,y)×f(y)dμ(y)∫|x-y|≤tK(x,y)|2dtt3)1/2,x∈ℝd.
In what follows, we will always assume that ℳ is bounded on L2(μ). For m∈ℕ and bi∈RBMO(μ), i=1,2,…,m, we formally define the multilinear commutator ℳb→ by
(9)ℳb→(f)(x)=(×f(y)dμ(y)∫|x-y|≤tK(x,y)|2∫0∞|∫|x-y|≤t∏l=1m[bl(x-bl(y))]×K(x,y)f(y)dμ(y)∫|x-y|≤t∏l=1m[bl(x-bl(y))]|2dtt3)1/2,
where K not only satisfies the size condition (6) but also satisfies a strong Hömander type condition: for all y,y′∈ℝd and r≥|y-y′|(10)∑l=1∞lm∫2lr<|x-y|≤2l+1r[|K(x,y)-K(x,y′)|+|K(y,x)-K(y′,x)|]1|x-y|dμ(x)≤C.
For bi=b, i=1,2,…,m, we denote by Mbm.Remark 2.
With the assumption thatℳ is bounded on L2(μ), Hu et al. [8] established that the integral ℳ as in (8) with kernel K satisfying (6) and (7) is bounded from L1(μ) to L1,∞(μ), from H1(μ) to L1(μ), and from L∞(μ) to RBLO(μ), respectively, where the function space RBLO(μ) is the subset of function space RBMO(μ). At the same time, the authors obtain the boundedness of the commutator ℳb,m, respectively, from Lp(μ) to itself for p∈(1,∞] and from the space LlogL(μ) to L1,∞(μ), where the kernel K satisfies (6) and (10). Such type of multilinear commutators when μ is a nondoubling measure was introduced by Zhang [9] and its Lp-boundedness has been established.
In this sequel, for1≤i≤m, we denote by Cim the family of all finite subsets σ={σ(1),…,σ(i)} of {1,2,…,m} with i different elements. For any σ∈Cim, the complementary sequence σ′ is given by σ′={1,2,…,m}∖σ. Let b→=(b1,b2,…,bm) be a finite family of locally integrable functions. For all 1≤i≤m and σ={σ(1),…,σ(i)}∈Cim, we define
(11)[b(x)-b(y)]σ=[bσ(1)(x)-bσ(1)(y)]⋯[bσ(i)(x)-bσ(i)(y)];[b(x)-mQ(b)]σ=[bσ(1)(x)-mQ(bσ(1))]⋯[bσ(i)(x)-mQ(bσ(i))],[mR(b)-mQ(b)]σ=[mR(bσ(1))-mQ(bσ(1))]⋯[mR(bσ(i))-mQ(bσ(i))],
where Q and R are cubes in ℝd and x, y∈ℝd. With this notation, we write
(12)∥b→σ∥*=∥bσ(1)∥*⋯∥bσ(i)∥*.
If σ={1,2,…,m}, we simply write
(13)∥b→∥*=∥b1∥*⋯∥bm∥*.In this paper, we will deal with the multilinear commutators for Marcinkiewicz integrals on the Hardy-type spaces with nondoubling measures. Now we recall the definition of the atomic block Hardy spaces.Definition 3.
Letρ>1, 1<ρ≤∞, γ,τ∈ℕ. Suppose bl∈RBMO(μ) for l=1,2,…,τ. A function h∈Lloc(μ) is called a (b→,p,τ,γ)-atomic block if (a)
there exists some cubeR such that supp(h)⊂R;(b)
∫ℝd‍h(y)bσ(y)dμ(y)=0 for all 0≤l≤τ and σ∈Clτ;(c)
fori=1,2, there is function ai supported on cubes Qi⊂R and numbers ri∈ℝ such that h=r1a1+r2a2 and
(14)∥ai∥Lp(μ)≤[μ(ρQi)]1/(p-1)KQi,R-γ.Then, we denote
(15)|h|Hb→,τ,γ1,p(μ)=r1+r2.
We say that f∈Hb→,τ,γ1,p(μ), if there exist (b→,p,τ,γ)-atomic blocks {hj}j∈ℕ such that
(16)f=∑j=1∞hj,
with ∑j=1∞‍|hj|Hb→,τ,γ1,p(μ)<∞. The Hb→,τ,γ1,p(μ) norm of f is defined by
(17)∥f∥Hb→,τ,γ1,p(μ)=inf{∑j|bj|Hb→,τ,γ1,p(μ)},
where the infimum is taken over all the possible decomposition of f in (b→,p,τ,γ)-atomic blocks.
From the definition, we have the following properties that for anyτ∈ℕ, 1<p≤∞ and γ1,γ2∈ℕ with 1≤γ1<γ2,
(18)Hb→,τ,γ21,p(μ)⊂Hb→,τ,γ11,p(μ)⊂H1(μ),
and for any τ,γ∈ℕ and 1<p1<p2≤∞,
(19)Hb→,τ,γ1,∞(μ)⊂Hb→,τ,γ1,p2(μ)⊂Hb→,τ,γ1,p1(μ)⊂H1(μ),
and for any k1,k2,γ∈ℕ, 1<p≤∞ and 0<k1<k2<τ,
(20)Hb→,k2,γ1,p(μ)⊂Hb→,k1,γ1,p(μ)⊂H1(μ).
For some more details about this Hardy-type space, see [4] and its references. It is valuable to point out that, if τ=0, the space Hb→,τ,γ1,p(μ) is just the Hardy space H1(μ) introduced by Tolsa in [1] with equivalent norm, which was proved by the authors in [10].Let us state our main results as follows.Theorem 4.
Let1<p≤∞, m∈ℕ, and bl∈RBMO(μ), for l=1,2,…,m, and let ℳb→ be as in (9) with K satisfying (6) and (10). Then ℳb→ is bounded from Hb→,m-1,m+11,p(μ) to weak L1(μ).Theorem 5.
Let1<p≤∞, m∈ℕ, bl∈RBMO(μ), for l=1,2,…,m, and ℳb→ be as in (9) with K satisfying (6) and (10). Then ℳb→ is bounded from Hb→,m,m+11,p(μ) to L1(μ).Remark 6.
Form=1, Theorem 4 is just Theorem 3.6 in [8], so we extend the result significantly.Throughout this paper,C denotes a constant that is independent of the main parameters involved but whose value may differ from line to line. We use the constant with subscripts to indicate its dependence on the parameters in the subscripts.


## 2. Proof of Theorems

To prove our Theorems, we need the following lemma; see [1].Lemma 7.
Letm∈ℕ and bl∈RBMO(μ), for l=1,2,…,m, ρ>1 and 1≤p<∞. Then there exists a constant C>0 such that for any cube Q,
(21){1μ(2Q)∫Q∏l=1m|bl(x)-mQ~(bl)|pdμ(x)}1/p≤C∏l-1m∥bl∥*.Proof of Theorem4.
For each fixedf∈Hb→,m-1,m+11,p(μ), one has the decomposition
(22)f=∑jhj,
where hj,s are (b→,p,m-1,m+1)-atomic blocks as in Definition 3, such that
(23)∑j|hj|Hb→,m-1,m+11,p(μ)≤2∥f∥Hb→,m-1,m+11,p(μ).
Let Rj be a cube such that supp(hj)⊂Rj; for each fixed j, decompose hj as
(24)hj(x)=rj(1)aj(1)(x)+rj(2)aj(2)(x),
where for i=1,2,rj(i)∈ℝ and
(25)|hj|Hb→,m-1,m+11,p(μ)=rj(1)+rj(2);aj(i) is supported on some cube Qj(i) such that Qj(i)⊂Rj and it satisfies the following condition:
(26)∥aj(i)∥Hb→,m-1,m+11,p(μ)≤μ(4Qj(i))1/p-1[KQj(i),Rj]-m-1.
With the aid of the formulate that for x,y∈ℝd,
(27)∏l=1m[bl(x)-bl(y)]=∑l=0m∑σ∈Clm[b(x)-mR~j(b)]σ[mRj~(b)-b(y)σ′],
we can write
(28)ℳ(f)(x)≤ℳ(∑j∏l=1m‍[mRj~(bl)-bl]hj)(x)+∑l=1m∑σ∈Clm|[b(x)-mRj~(b)]σ|×ℳ([mRj~(b)-b]σ′)(x)=F1(x)+F2(x).
With weak (L1,L1)-boundedness of ℳ, Hölder’s inequality, Lemma 7, and (26), it states that
(29)μ({x∈ℝd:|F1(x)|>λ})≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm|[mQj(i)~(b)-mRj~(b)]σ|×∫Qj(i)|[b(y)-mQj(i)(b)]σ′||aj(i)(x)|dμ(y)≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm[KQj(i),Rj]l×∥b→σ∥*∥aj(i)∥Lp(μ)∥b→σ′∥*μ(2Qj(i))1/p′≤Cλ∥b→∥*∑j∑i=12|rj(i)|,
where we have used the face derived from [1],
(30)|mQj(i)~(b)-mRj~(b)|≤CKQj(i),Rj∥b∥*.
Now we turn to estimate the term F2(x). Write
(31)μ({x∈ℝd:|F2(x)|>λ})≤Cλ∑j∑l=1m∑σ∈Clm∫2Rj|[b(x)-mRj~(b)]σ|×ℳ([mRj~(b)-b]σ′hj)(x)dμ(x)+Cλ∑j∑l=1m∑σ∈Clm∫ℝd∖2Rj|[b(x)-mRj~(b)]σ|×ℳ([mRj~(b)-b]σ′hj)(x)dμ(x)=G1+G2.
To estimateG1, further decompose
(32)G1≤Cλ∑j∑i=12|rj(i)|×∑l=0m∑σ∈Clm∑s=0l∑η∈Csl|[mQj(i)~(b)-mRj~(b)]η|×∫2Qj(i)|[b(x)-mQj(i)(b)]η′|×ℳ([mRj~(b)-b]σ′aj(i))(x)dμ(x)+Cλ∑j∑i=12|rj(i)|×∑l=0m∑σ∈Clm∫2Rj∖2Qj(i)|[b(x)-mRj~(b)]σ|×ℳ([mRj~(b)-b]σ′aj(i))(x)dμ(x)=G11+G12.
Choose 1<p1<p<∞; the Hölder inequality, Lp(μ)-boundedness of ℳ (see [9]), Lemma 7, (26), and (30) yield that
(33)G11≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm∥b→σ∥*×[KQj(i),Rj]lμ(4Qj(i))1/p1′×∥ℳ([mRj~(b)-b]σ′aj(i))∥Lp1(μ)≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm∥b→σ∥*[KQj(i),Rj]lμ(4Qj(i))(1/p1′)×(∫Qj(i)|[b(x)-mRj~(b)]σ′|pp1/(p-p1)dμ(x))(p-p1)/pp1×∥aj(i)∥Lp(μ)≤Cλ∑j∑i=12|rj(i)|∥b→∥*∥aj(i)∥Lp(μ)×μ(4Qj(i))((1-1)/p)[KQj(i),Rj]m≤Cλ∥b→∥*∑j∑i=12|rj(i)|.
On the other hand, supp(aj(i))⊂Qj(i)⊂Rj, x∈2Rj∖2Qj(i) and y∈Qj(i); it is easy to get that |x-y| is equivalent to |x-xQj(i)| and |x-y|≥(1/8)l(Qj(i)). By (6), the Hölder inequality, the Minkowski inequality, Lemma 7, (26) and (30), one has
(34)G12=Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm‍∫2Rj∖2Qj(i)|[b(x)-mRj~(b)]σ|×(∫0∞|∫|x-y|≤tK(x,y)[mRj~(b)-b(y)]σ′×aj(i)(y)dμ(y)∫K(x,y)|2dtt3)1/2dμ(x)≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm‍∫2Rj∖2Qj(i)|[b(x)-mRj~(b)]σ|×∫Qj(i)‍1|x-y|n-11l(Qj(i))|[mRj~(b)-b(y)]σ′|×|aj(i)(y)|dμ(y)dμ(x)≤Cλ∑j∑i=12|rj(i)|×∑l=0m∑σ∈Clm∑k=1N2Qj(i),2Rj‍∫2Rj∖2Qj(i)|[b(x)-mRj~(b)]σ|l(Qj(i))|x-xQj(i)|n-1dμ(x)×∫Qj(i)|[mRj~(b)-b(y)]σ′|×|aj(i)(y)|dμ(y)≤Cλ∑j∑i=12|rj(i)|∥aj(i)∥(μ)μ(2Qj(i))1/p′KQj(i),Rj×∑l=1m[K2k+1Qj(i),Rj]l[KQj(i),Rj]m-l≤Cλ∥b→∥*∑j∑i=12|rj(i)|,
where we have used the fact that for 1≤k≤N2Qj(i),2Rj,
(35)|m2k+1Qj(i)~(b)-mRj~(b)|≤CKQj(i),Rj∥b∥*.
Now we turn to estimateG2:
(36)G2≤Cλ∑j‍∑l=0m‍∑σ∈Clm‍∫ℝd∖2Rj‍|[b(x)-mRj~(b)]σ|×(∫0|x-xRj|+dl(Rj)|∫|x-y|≤tK(x,y)×[mRj~(b)-b(y)]σ′×hj(y)dμ(y)∫|x-y|≤tK(x,y)|2dtt3∫0|x-xRj|+dl(Rj)|∫|x-y|≤tK(x,y))1/2dμ(x)+Cλ∑j∑l=0m∑σ∈Clm∫ℝd∖2Rj|[b(x)-mRj~(b)]σ|×(∫|x-xRj|+dl(Rj)∞|∫|x-y|≤tK(x,y)×[mRj~(b)-b(y)]σ′×hj(y)dμ(y)∫|x-y|≤tK(x,y)|2dtt3∫|x-xRj|+dl(Rj)∞|∫|x-y|≤tK(x,y))1/2dμ(x)=G21+G22.
Note that for1≤k<∞,
(37)|m2kRj~(b)-mRj~(b)|≤Ck∥b∥*.
By (6), the Minkowski inequality, Lemma 7, the Hölder inequality, (26), and (37), we obtain that
(38)G21≤Cλ∑j∑l=0m∑σ∈Clm∫ℝd∖2Rj|[b(x)-mRj~(b)]σ|×∫Rj‍|K(x,y)[mRj~(b)-b(y)]σ′hj(y)|×(∫|x-y||x-xRj|+dl(Rj)dtt3)1/2dμ(y)dμ(x)≤Cλ∑j∑i=12|rj(i)|×∑l=0m∑σ∈Clm∑k=1∞∫2k+1Rj∖2kRjl(Rj)|x-xRj|n+1/2×|[b(x)-mRj~(b)]σ|dμ(x)×(∫Qji|[mRj~(b)-b(y)]σ′|p′dμ(y))1/p′×∥aj(i)(y)∥Lp(μ)≤Cλ∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm∑k=1∞l(Rj)[l(2kRj)]n+1/2×μ(2k+2Rj)[KRj,2k+1Rj]∥b→σ∥*×[KQj(i),Rj]m-l∥b→σ′∥*×∥aj(i)(y)∥Lp(μ)μ(2Qj(i))1/p′≤Cλ∥b→∥*∑j∑i=12|rj(i)|.
For G22, by the generalization of the Hölder inequality, John-Nirenberg’s inequality, (10), and a similar statement in [4, pages 12-13], we have
(39)∫ℝd∖2Rj|[b(x)-mRj~]σ||K(x,y)-K(x,xRj)|×1|x-xRj|dμ(x)≤C∥b→σ∥*.
Then, we can get the estimate of G22:
(40)G22≤Cλ∑j∑l=0m∑σ∈Clm∫ℝd∖2Rj|[b(x)-mRj~(b)]σ|×|∫ℝd{K(x,y)-K(x,xRj)}×[mRj~(b)-b(y)]σ′×hj(y)dμ(y)∫ℝd{K(x,y)-K(x,xRj)}|×1|x-xRj|+dl(Rj)dμ(x)≤Cλ∑j∑l=0m∑σ∈Clm∫Rj∫ℝd∖2Rj|[b(x)-mRj~(b)]σ|×1|x-xRj|×|K(x,y)-K(x,xRj)|×|[b(x)-mRj~(b)]σ′|×|hj|dμ(x)dμ(y)≤Cλ∥b→∥*∑j∑i=12|rj(i)|∑l=0m∑σ∈Clm∥b→σ∥*∥aj(i)∥Lp(μ)×{∫Qj(i)|[mRj~(b)-b(y)]σ′|p′dμ(y)}1/p′≤Cλ∥b→∥*∑j∑i=12|rj(i)|.
Combining the estimates for G21 and G22 then yields the desired estimate for G2, which together with the estimate for G1 indicates that
(41)μ({x∈ℝd:ℳ(f)(x)>λ})≤Cλ-1∥f∥Hb→,m-1,m+11,p(μ).
This finishes the proof of Theorem4.Proof of Theorem5.
By a standard argument, we only need to verify that for any(b→,p,m,m+1)-atomic block h as in Definition 3 with ρ=4,
(42)∫ℝd‍ℳb→(h)(x)dμ(x)≤C|h|Hb→,m,m+11,p(μ),
where C is a constant independent of h. Let all the notation be the same as in Definition 3. By our choices, ai,i=1,2, now they satisfy satisfies the following condition:
(43)∥ai∥Lp(μ)≤μ(4Qi)1/p-1[KQi,R]-m-1.
Write(44)∫ℝdℳb→(h)(x)dμ(x)≤∫2Rℳb→(h)(x)dμ(x)+∫ℝd∖2Rℳb→(h)(x)dμ(x)=W1+W2.
To estimate W1, we further decompose
(45)W1≤∑i=12|ri|∫2Qiℳb→(h)(x)dμ(x)+∑i=12|ri|∫2R∖2Qiℳb→(h)(x)dμ(x)=W11+W12.
The Hölder inequality, Lp-boundedness of ℳb→ (see [9]), and (43) tell us
(46)W11≤∑i=12|ri|∥ℳb→(ai)∥Lp(μ)μ(2Qi)1/p′≤C∥b→∥*∑i=12|ri|∥ai∥Lp(μ)μ(2Qi)1/p′≤C∥b→∥*∑i=12|ri|.
From (6), the Hölder inequality, Lemma 7, and (43), it follows that
(47)W12≤C∑i=12|ri|∑l=0m∑σ∈Clm∑k=1N2Qi2R∫2k+1Qi∖2kQi|[b(x)-mQi~]σ|×∫Qi|[mQi~(b)-b(y)]σ′||ai(y)|×1|x-y|n-11l(Qi)dμ(y)≤C∑i=12|ri|∑l=0m∑σ∈Clm{∫Qi|ai(y)|pdμ(y)}1/p×{∫Qi|[mQi~(b)-b(y)]σ′|p′dμ(y)}1/p′×∑k=1N2Qi,2R1l(Qi)l(2kQi)n-1×∫2k+1Qi∑s=0l∑η∈Csl|[b(x)-m2k+1Qi~(b)]η×[m2k+1Qi~(b)-mQi~(b)]η′|dμ(x)≤C∥b→∥*∑i=12|ri|∥ai∥Lp(μ)μ(2Qi)1/p′[KQi,R]m+1≤C∥b→∥*∑i=12|ri|.
Now we turn to estimate W2. Involving the vanishing moment of h, we can write
(48)W2≤∑l=0m∑σ∈Clm∫ℝd∖2R|[b(x)-mR~(b)]σ|×ℳb→([mR~(b)-b]σ′h)(x)dμ(x)≤C∑l=0m∑σ∈Clm‍∫ℝd∖2R|[b(x)-mR~(b)]σ|×(hj(y)dμ(y)∫|x-y|≤tK(x,y)|2∫0|x-xR|+dl(R)|∫|x-y|≤tK(x,y)[mR~(b)-b(y)]σ′×hj(y)dμ(y)∫|x-y|≤tK(x,y)|2dtt3∫0|x-xR|+dl(R)|∫|x-y|≤tK(x,y))1/2dμ(x)+C∑l=0m∑σ∈Clm‍∫ℝd∖2R|[b(x)-mR~(b)]σ|×(hj(y)dμ(y)∫|x-y|≤tK(x,y)|2∫|x-xR|+dl(R)∞|∫|x-y|≤tK(x,y)[mR~(b)-b(y)]σ′×hj(y)dμ(y)∫|x-y|≤tK(x,y)|2dtt3∫|x-xR|+dl(R)∞|∫|x-y|≤tK(x,y))1/2dμ(x).
An argument similar to estimate G2 together with the Hölder inequality, Lemma 7, and (43) tell us that
(49)W2≤C∥b→∥*∑i=12|ri|.
The estimates forW1 and W2 lead to (42). The proof of Theorem 5 is completed.Then, we can get the following corollaries directly.Corollary 8.
Let1<p≤∞, m∈ℕ, and b∈RBMO(μ), and let ℳbm be as in (9) with K satisfying (6) and (10). Then ℳbm is bounded from Hbm,m-1,m+11,p(μ) to weak L1(μ).Corollary 9.
Let1<p≤∞,  m∈ℕ, and b∈RBMO(μ), and let ℳbm be as in (9) with K satisfying (6) and (10). Then ℳbm is bounded from Hbm,m,m+11,p(μ) to L1(μ).We remark that Corollary9 is a generalization version of Theorem 1 in [11].As a special example of Theorem 3.4 in [8], we conclude that the commutator version for Marcinkiewicz integral are bounded from LlogL(μ) to L1,∞(μ) using a different method.Theorem HL (see [12]). Let1<p1,q1<∞,1<r≤∞,1/q1=1/p1-1/r,and1/q0=1-1/r, and let T be a sublinear operator that is bounded fromLp1(μ)to weakLq1(μ)and is bounded fromH1(μ)to weakLq0(μ); then there is a positive constantCsuch that for anyλ>0, measurable functionfwith∫ℝd‍|f(x)|log(2+|f(x)|)dμ(x)<∞,(50)μ({x∈ℝd:|Tf(x)|>λ})≤C(∫ℝd‍|f(x)|λlog(2+|f(x)|∥f∥1q0-1λq0)dμ(x))q0.From the above Theorem  HL and Theorem4, one has the following corollary.Corollary 10.
Let1<p≤∞,  b∈RBMO(μ),   and K satisfy (6) and (10). Then ℳb is bounded from LlogL(μ) to L1,∞(μ); that is,
(51)μ({x∈ℝd:|ℳbf(x)|>λ})≤C∫ℝd|f(x)|λlog(2+|f(x)|λ)dμ(x).



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*Source: 394732-2013-12-11.xml*