Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Nondoubling Measures

Abstract

Let 𝜇 be a nonnegative Radon measure on ℝ^𝑑 satisfying 𝜇(𝐵(𝑥,𝑟))≤𝐶₀𝑟^𝑛 for all 𝑥∈ℝ^𝑑,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).

1. Introduction

We will work on the 𝑑-dimensional Euclidean space ℝ^𝑑 with a nonnegative Radon measure 𝜇 which only satisfies the following growth condition that there exists a consist 𝐶₀ such that 𝜇(𝐵(𝑥,𝑟))≤𝐶₀𝑟^𝑛, (1) where 𝐵(𝑥,𝑟) is the open ball centered at some point 𝑥∈ℝ^𝑑 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 cube 𝑄⊂ℝ^𝑑 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 𝑥_𝑄. Let 𝛼 and 𝛽 be positive constants such that 𝛼>1 and 𝛽>𝛼^𝑛; 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 (2,2^𝑑+1)-doubling cube. Especially, for any give cube 𝑄, we denote by 𝑄 the smallest doubling cube in the family {2^𝑘𝑄}_𝑘∈ℕ. For two cubes 𝑄⊂𝑅, set 𝐾_𝑄,𝑅=1+^{𝑁𝑄,𝑅}_𝑘=1𝜇2^𝑘𝑄𝑙2^𝑘𝑄^𝑛, (2) where 𝑁_𝑄,𝑅 is the first positive integer 𝑘 such that 𝑙(2^𝑘𝑄)≥𝑙(𝑅).

Definition 1. Let 𝜌>1 be some fixed constant, we say that a function 𝑏∈𝐿¹_loc(𝜇) belongs to the space RBMO(𝜇) if there is a constant 𝐵>0 such that sup_𝑄1𝜇𝜌𝑄_𝑄𝑏(𝑥)𝑄−𝑚(𝑏)d𝜇(𝑥)≤𝐵, (3) and if 𝑄₁⊂𝑄₂ are doubling cubes, 𝑚𝑄₁(𝑏)𝑄−𝑚₂(𝑏)≤𝐵𝐾_𝑄1,𝑄2, (4) 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, 𝑚𝑄(𝑏)=1𝜇𝜌𝑄_𝑄𝑚𝑄(𝑏)d𝜇(𝑥). (5) 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. Let 𝐾(⋅,⋅) be a locally integrable function on {ℝ^𝑑×ℝ^𝑑⧵(𝑥,𝑦)∶𝑥=𝑦}. Assume that there exists a constant 𝐶>0 such that for 𝑥,𝑦∈ℝ^𝑑 with 𝑥≠𝑦, 𝐾𝑥,𝑦≤𝐶𝑥−𝑦^{−(𝑛−1)}, (6) and for any 𝑥, 𝑦, and 𝑦^′∈ℝ^𝑑, with |𝑥−𝑦|≥2|𝑦−𝑦^′|, _{𝑥−𝑦≥2𝑦−𝑦′}𝐾𝑥,𝑦−𝐾𝑥,𝑦^′+𝐾𝑦,𝑥−𝐾𝑦^′×1,𝑥(𝑥)𝑥−𝑦d𝜇≤𝐶, (7) for any 𝑦,𝑦^′∈ℝ^𝑑. The Marcinkiewicz integral ℳ associated with the above kernel 𝐾 and the measure 𝜇 as in (1) is defined by ℳ𝑓(𝑥)=^∞_{0𝑥−𝑦≤𝑡}𝐾𝑦𝑦𝑥,𝑦×𝑓d𝜇²d𝑡𝑡^31/2,𝑥∈ℝ^𝑑. (8) In what follows, we will always assume that ℳ is bounded on 𝐿²(𝜇). For 𝑚∈ℕ and 𝑏_𝑖∈RBMO(𝜇), 𝑖=1,2,…,𝑚, we formally define the multilinear commutator ℳ_𝑏 by ℳ_𝑏𝑓(𝑥)=^∞₀^𝑚_{𝑥−𝑦≤𝑡𝑙=1}𝑏_𝑙𝑥−𝑏_𝑙𝑦×𝐾𝑥,𝑦𝑓𝑦d𝜇𝑦²d𝑡𝑡^31/2, (9) where 𝐾 not only satisfies the size condition (6) but also satisfies a strong Hömander type condition: for all 𝑦,𝑦^′∈ℝ^𝑑 and 𝑟≥|𝑦−𝑦^′|^∞_𝑙=1𝑙^𝑚_{2𝑙𝑟<𝑥−𝑦≤2𝑙+1𝑟}𝐾𝑥,𝑦−𝐾𝑥,𝑦^′+𝐾𝑦,𝑥−𝐾𝑦^′1,𝑥(𝑥)𝑥−𝑦d𝜇≤𝐶. (10) For 𝑏_𝑖=𝑏, 𝑖=1,2,…,𝑚, we denote by 𝑀_𝑏^𝑚.

Remark 2. With the assumption that ℳ 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 𝐿^1,∞(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 ℳ_𝑏,𝑚, respectively, from 𝐿^𝑝(𝜇) to itself for 𝑝∈(1,∞] and from the space 𝐿log𝐿(𝜇) to 𝐿^1,∞(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 𝐿^𝑝-boundedness has been established.
In this sequel, for 1≤𝑖≤𝑚, we denote by 𝐶^𝑚_𝑖 the family of all finite subsets 𝜎={𝜎(1),…,𝜎(𝑖)} of {1,2,…,𝑚} with 𝑖 different elements. For any 𝜎∈𝐶^𝑚_𝑖, the complementary sequence 𝜎^′ is given by 𝜎^′={1,2,…,𝑚}⧵𝜎. Let 𝑏=(𝑏₁,𝑏₂,…,𝑏_𝑚) be a finite family of locally integrable functions. For all 1≤𝑖≤𝑚 and 𝜎={𝜎(1),…,𝜎(𝑖)}∈𝐶^𝑚_𝑖, we define 𝑏(𝑥)−𝑏𝑦_𝜎=𝑏_𝜎(1)(𝑥)−𝑏_𝜎(1)𝑦⋯𝑏_𝜎(𝑖)(𝑥)−𝑏_𝜎(𝑖)𝑦;𝑏(𝑥)−𝑚_𝑄(𝑏)_𝜎=𝑏_𝜎(1)(𝑥)−𝑚_𝑄𝑏_𝜎(1)⋯𝑏_𝜎(𝑖)(𝑥)−𝑚_𝑄𝑏_𝜎(𝑖),𝑚_𝑅(𝑏)−𝑚_𝑄(𝑏)_𝜎=𝑚_𝑅𝑏_𝜎(1)−𝑚_𝑄𝑏_𝜎(1)⋯𝑚_𝑅𝑏_𝜎(𝑖)−𝑚_𝑄𝑏_𝜎(𝑖), (11) where 𝑄 and 𝑅 are cubes in ℝ^𝑑 and 𝑥, 𝑦∈ℝ^𝑑. With this notation, we write 𝑏_𝜎∗=𝑏_𝜎(1)∗⋯𝑏_{𝜎(𝑖)∗}. (12) If 𝜎={1,2,…,𝑚}, we simply write 𝑏_∗=𝑏_1∗⋯𝑏_𝑚∗. (13)

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. Let 𝜌>1, 1<𝜌≤∞, 𝛾,𝜏∈ℕ. Suppose 𝑏_𝑙∈RBMO(𝜇) for 𝑙=1,2,…,𝜏. A function ℎ∈𝐿_loc(𝜇) is called a (𝑏,𝑝,𝜏,𝛾)-atomic block if (a)there exists some cube 𝑅 such that supp(ℎ)⊂𝑅;(b)∫_ℝ𝑑ℎ(𝑦)𝑏_𝜎(𝑦)d𝜇(𝑦)=0 for all 0≤𝑙≤𝜏 and 𝜎∈𝐶^𝜏_𝑙;(c)for 𝑖=1,2, there is function 𝑎_𝑖 supported on cubes 𝑄_𝑖⊂𝑅 and numbers 𝑟_𝑖∈ℝ such that ℎ=𝑟₁𝑎₁+𝑟₂𝑎₂ and 𝑎_{𝑖𝐿𝑝(𝜇)}≤𝜇𝜌𝑄_𝑖^1/(𝑝−1)𝐾_𝑄^−𝛾_𝑖,𝑅. (14)Then, we denote |ℎ|_𝐻^1,𝑝_{𝑏,𝜏,𝛾(𝜇)}=𝑟₁+𝑟₂. (15) We say that 𝑓∈𝐻^1,𝑝_{𝑏,𝜏,𝛾}(𝜇), if there exist (𝑏,𝑝,𝜏,𝛾)-atomic blocks {ℎ_𝑗}_𝑗∈ℕ such that 𝑓=^∞_𝑗=1ℎ_𝑗, (16) with ∑^∞_𝑗=1|ℎ_𝑗|_𝐻^1,𝑝_{𝑏,𝜏,𝛾(𝜇)}<∞. The 𝐻^1,𝑝_{𝑏,𝜏,𝛾}(𝜇) norm of 𝑓 is defined by 𝑓_𝐻^1,𝑝_{𝑏,𝜏,𝛾(𝜇)}=inf_𝑗𝑏_{𝑗𝐻1,𝑝𝑏,𝜏,𝛾(𝜇)}, (17) 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 any 𝜏∈ℕ, 1<𝑝≤∞ and 𝛾₁,𝛾₂∈ℕ with 1≤𝛾₁<𝛾₂, 𝐻^1,𝑝_{𝑏,𝜏,𝛾2}𝜇⊂𝐻^1,𝑝_{𝑏,𝜏,𝛾1}𝜇⊂𝐻¹𝜇, (18) and for any 𝜏,𝛾∈ℕ and 1<𝑝₁<𝑝₂≤∞, 𝐻^1,∞_{𝑏,𝜏,𝛾}𝜇⊂𝐻^1,𝑝2_{𝑏,𝜏,𝛾}𝜇⊂𝐻^1,𝑝1_{𝑏,𝜏,𝛾}𝜇⊂𝐻¹𝜇, (19) and for any 𝑘₁,𝑘₂,𝛾∈ℕ, 1<𝑝≤∞ and 0<𝑘₁<𝑘₂<𝜏, 𝐻^1,𝑝_{𝑏,𝑘2,𝛾}𝜇⊂𝐻^1,𝑝_{𝑏,𝑘1,𝛾}𝜇⊂𝐻¹𝜇. (20) 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 𝐻^1,𝑝_{𝑏,𝜏,𝛾}(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. Let 1<𝑝≤∞, 𝑚∈ℕ, and 𝑏_𝑙∈𝑅𝐵𝑀𝑂(𝜇), for 𝑙=1,2,…,𝑚, and let ℳ_𝑏 be as in (9) with 𝐾 satisfying (6) and (10). Then ℳ_𝑏 is bounded from 𝐻^1,𝑝_{𝑏,𝑚−1,𝑚+1}(𝜇) to weak 𝐿¹(𝜇).

Theorem 5. Let 1<𝑝≤∞, 𝑚∈ℕ, 𝑏_𝑙∈𝑅𝐵𝑀𝑂(𝜇), for 𝑙=1,2,…,𝑚, and ℳ_𝑏 be as in (9) with 𝐾 satisfying (6) and (10). Then ℳ_𝑏 is bounded from 𝐻^1,𝑝_{𝑏,𝑚,𝑚+1}(𝜇) to 𝐿¹(𝜇).

Remark 6. For 𝑚=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. Let 𝑚∈ℕ and 𝑏_𝑙∈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, 1𝜇(2𝑄)_𝑄^𝑚_𝑙=1𝑏_𝑙(𝑥)𝑄−𝑚𝑏_𝑙^𝑝d𝜇(𝑥)^1/𝑝≤𝐶^𝑚_𝑙−1𝑏_𝑙∗. (21)

Proof of Theorem 4. For each fixed 𝑓∈𝐻^1,𝑝_{𝑏,𝑚−1,𝑚+1}(𝜇), one has the decomposition 𝑓=_𝑗ℎ_𝑗, (22) where ℎ^,_𝑗𝑠 are (𝑏,𝑝,𝑚−1,𝑚+1)-atomic blocks as in Definition 3, such that _𝑗ℎ_{𝑗𝐻1,𝑝𝑏,𝑚−1,𝑚+1(𝜇)}≤2𝑓_𝐻^1,𝑝_{𝑏,𝑚−1,𝑚+1(𝜇)}. (23) Let 𝑅_𝑗 be a cube such that supp(ℎ_𝑗)⊂𝑅_𝑗; for each fixed 𝑗, decompose ℎ_𝑗 as ℎ_𝑗(𝑥)=𝑟_𝑗⁽¹⁾𝑎_𝑗⁽¹⁾(𝑥)+𝑟_𝑗⁽²⁾𝑎_𝑗⁽²⁾(𝑥), (24) where for 𝑖=1,2,𝑟_𝑗^(𝑖)∈ℝ and ℎ_{𝑗𝐻1,𝑝𝑏,𝑚−1,𝑚+1(𝜇)}=𝑟_𝑗⁽¹⁾+𝑟_𝑗⁽²⁾; (25)𝑎_𝑗^(𝑖) is supported on some cube 𝑄_𝑗^(𝑖) such that 𝑄_𝑗^(𝑖)⊂𝑅_𝑗 and it satisfies the following condition: 𝑎_𝑗^(𝑖)_{𝐻1,𝑝𝑏,𝑚−1,𝑚+1(𝜇)}≤𝜇4𝑄_𝑗^{(𝑖)1/𝑝−1}𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^−𝑚−1. (26) With the aid of the formulate that for 𝑥,𝑦∈ℝ^𝑑, ^𝑚_𝑙=1𝑏_𝑙(𝑥)−𝑏_𝑙𝑦=^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝑏(𝑥)−𝑚𝑅_𝑗(𝑏)_𝜎𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′, (27) we can write ℳ𝑓(𝑥)≤ℳ_𝑗^𝑚_𝑙=1𝑚𝑅_𝑗𝑏_𝑙−𝑏_𝑙ℎ_𝑗(𝑥)+^𝑚_{𝑙=1𝜎∈𝐶𝑚𝑙}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎𝑚𝑅×ℳ_𝑗(𝑏)−𝑏_𝜎′(𝑥)=𝐹₁(𝑥)+𝐹₂(𝑥). (28) With weak (𝐿¹,𝐿¹)-boundedness of ℳ, Hölder’s inequality, Lemma 7, and (26), it states that 𝜇𝑥∈ℝ^𝑑∶𝐹₁(𝑥≤𝐶)>𝜆𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝑚𝑄_𝑗^(𝑖)(𝑏)−𝑚𝑅_𝑗(𝑏)_𝜎×_{𝑄𝑗(𝑖)}𝑏𝑦−𝑚_𝑄𝑗^(𝑖)(𝑏)_𝜎′𝑎_𝑗^(𝑖)(𝑥𝑦≤𝐶)d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑙×𝑏_𝜎∗𝑎_𝑗^(𝑖)_{𝐿𝑝(𝜇)}𝑏_𝜎′∗𝜇2𝑄_𝑗^{(𝑖)1/𝑝′}≤𝐶𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖), (29) where we have used the face derived from [1], 𝑚𝑄_𝑗^(𝑖)(𝑏)−𝑚𝑅_𝑗(𝑏)≤𝐶𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗‖𝑏‖_∗. (30) Now we turn to estimate the term 𝐹₂(𝑥). Write 𝜇𝑥∈ℝ^𝑑∶𝐹₂(𝑥≤𝐶)>𝜆𝜆_𝑗^𝑚_{𝑙=1𝜎∈𝐶𝑚𝑙2𝑅𝑗}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎𝑚𝑅×ℳ_𝑗(𝑏)−𝑏_𝜎′ℎ_𝑗(𝑥)(𝑥)+𝐶d𝜇𝜆_𝑗^𝑚_{𝑙=1𝜎∈𝐶𝑚𝑙ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎𝑚𝑅×ℳ_𝑗(𝑏)−𝑏_𝜎′ℎ_𝑗(𝑥)d𝜇(𝑥)=𝐺₁+𝐺₂. (31)
To estimate 𝐺₁, further decompose 𝐺₁≤𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)×^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}^𝑙_{𝑠=0𝜂∈𝐶𝑙𝑠}𝑚𝑄_𝑗^(𝑖)(𝑏)−𝑚𝑅_𝑗(𝑏)_𝜂×_{2𝑄𝑗(𝑖)}𝑏(𝑥)−𝑚_𝑄𝑗^(𝑖)(𝑏)_𝜂′𝑚𝑅×ℳ_𝑗(𝑏)−𝑏_𝜎′𝑎_𝑗^(𝑖)(𝑥)(𝑥)+𝐶d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)×^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙2𝑅𝑗⧵2𝑄𝑗(𝑖)}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎𝑚𝑅×ℳ_𝑗(𝑏)−𝑏_𝜎′𝑎_𝑗^(𝑖)(𝑥)d𝜇(𝑥)=𝐺₁₁+𝐺₁₂. (32) Choose 1<𝑝₁<𝑝<∞; the Hölder inequality, 𝐿^𝑝(𝜇)-boundedness of ℳ (see [9]), Lemma 7, (26), and (30) yield that 𝐺₁₁≤𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝑏_𝜎∗×𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑙𝜇4𝑄_𝑗^{(𝑖)1/𝑝′1}×ℳ𝑚𝑅_𝑗(𝑏)−𝑏_𝜎′𝑎_𝑗^(𝑖)_{𝐿𝑝1(𝜇)}≤𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝑏_𝜎∗𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑙𝜇4𝑄_𝑗^{(𝑖)(1/𝑝′1)}×_{𝑄𝑗(𝑖)}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎′^{𝑝𝑝1/(𝑝−𝑝1)}d𝜇(𝑥)^{(𝑝−𝑝1)/𝑝𝑝1}×𝑎_𝑗^(𝑖)_{𝐿𝑝(𝜇)}≤𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑏_∗𝑎_𝑗^(𝑖)_{𝐿𝑝(𝜇)}×𝜇4𝑄_𝑗^{(𝑖)((1−1)/𝑝)}𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑚≤𝐶𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖). (33) On the other hand, supp(𝑎_𝑗^(𝑖))⊂𝑄_𝑗^(𝑖)⊂𝑅_𝑗, 𝑥∈2𝑅_𝑗⧵2𝑄_𝑗^(𝑖) and 𝑦∈𝑄_𝑗^(𝑖); it is easy to get that |𝑥−𝑦| is equivalent to |𝑥−𝑥_𝑄^(𝑖)_𝑗| and |𝑥−𝑦|≥(1/8)𝑙(𝑄_𝑗^(𝑖)). By (6), the Hölder inequality, the Minkowski inequality, Lemma 7, (26) and (30), one has 𝐺₁₂=𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙2𝑅𝑗⧵2𝑄𝑗(𝑖)}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×^∞_{0𝑥−𝑦≤𝑡}𝐾𝑚𝑅𝑥,𝑦_𝑗(𝑏)−𝑏𝑦_𝜎′×𝑎_𝑗^(𝑖)𝑦d𝜇𝑦²d𝑡𝑡^31/2(𝑥)≤𝐶d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙2𝑅𝑗⧵2𝑄𝑗(𝑖)}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×_{𝑄𝑗(𝑖)}1𝑥−𝑦^𝑛−11𝑙𝑄_𝑗^(𝑖)𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′×𝑎_𝑗^(𝑖)𝑦𝑦(𝑥)≤𝐶d𝜇d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)×^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}^{𝑁𝑗𝑗2𝑄(𝑖),2𝑅}_{𝑘=12𝑅𝑗⧵2𝑄𝑗(𝑖)}𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎𝑙𝑄_𝑗^(𝑖)𝑥−𝑥_𝑄𝑗^{(𝑖)𝑛−1}d𝜇(𝑥)×_{𝑄𝑗(𝑖)}𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′×𝑎_𝑗^(𝑖)𝑦𝑦≤𝐶d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑎_𝑗^(𝑖)_(𝜇)𝜇2𝑄_𝑗^{(𝑖)1/𝑝′}𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗×^𝑚_𝑙=1𝐾_{2𝑘+1𝑄𝑗}^(𝑖)_,𝑅𝑗^𝑙𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑚−𝑙≤𝐶𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖), (34) where we have used the fact that for 1≤𝑘≤𝑁_2𝑄𝑗^(𝑖)_,2𝑅𝑗, 𝑚2^𝑘+1𝑄_𝑗^(𝑖)(𝑏)−𝑚𝑅_𝑗(𝑏)≤𝐶𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗‖𝑏‖_∗. (35)
Now we turn to estimate 𝐺₂: 𝐺₂≤𝐶𝜆_𝑗^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×^{|𝑥−𝑥𝑅𝑗√|+𝑑𝑙(𝑅𝑗)}_{0𝑥−𝑦≤𝑡}𝐾×𝑚𝑅𝑥,𝑦_𝑗(𝑏)−𝑏𝑦_𝜎′×ℎ_𝑗𝑦d𝜇𝑦²d𝑡𝑡^31/2(𝑥)+𝐶d𝜇𝜆_𝑗^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×^∞_{𝑥−𝑥𝑅𝑗+√𝑅𝑑𝑙𝑗𝑥−𝑦≤𝑡}𝐾×𝑚𝑅𝑥,𝑦_𝑗(𝑏)−𝑏𝑦_𝜎′×ℎ_𝑗𝑦d𝜇𝑦²d𝑡𝑡^31/2d𝜇(𝑥)=𝐺₂₁+𝐺₂₂. (36)
Note that for 1≤𝑘<∞, 𝑚2^𝑘𝑅_𝑗(𝑏)−𝑚𝑅_𝑗(𝑏)≤𝐶𝑘‖𝑏‖_∗. (37)
By (6), the Minkowski inequality, Lemma 7, the Hölder inequality, (26), and (37), we obtain that 𝐺₂₁≤𝐶𝜆_𝑗^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×_𝑅𝑗𝐾𝑚𝑅𝑥,𝑦_𝑗(𝑏)−𝑏𝑦_𝜎′ℎ_𝑗𝑦×^{|𝑥−𝑥𝑅𝑗√|+𝑑𝑙(𝑅𝑗)}_𝑥−𝑦d𝑡𝑡^31/2𝑦(𝑥)≤𝐶d𝜇d𝜇𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)×^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}^∞_{𝑘=12𝑘+1𝑅𝑗⧵2𝑘𝑅𝑗}𝑙𝑅_𝑗𝑥−𝑥_𝑅𝑗^𝑛+1/2×𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎(𝑥)×d𝜇_𝑄𝑖𝑗𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′^𝑝′d𝜇𝑦^1/𝑝′×𝑎_𝑗^(𝑖)𝑦_{𝐿𝑝(𝜇)}≤𝐶𝜆_𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}^∞_𝑘=1𝑙𝑅_𝑗𝑙2^𝑘𝑅_𝑗^𝑛+1/2×𝜇2^𝑘+2𝑅_𝑗𝐾_𝑅𝑗,2^𝑘+1_𝑅𝑗𝑏_𝜎∗×𝐾_𝑄𝑗^(𝑖)_,𝑅𝑗^𝑚−𝑙𝑏_𝜎′∗×𝑎_𝑗^(𝑖)𝑦_{𝐿𝑝(𝜇)}𝜇2𝑄_𝑗^{(𝑖)1/𝑝′}≤𝐶𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖). (38) 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 _{ℝ𝑑⧵2𝑅𝑗}𝑏(𝑥)𝑅−𝑚_𝑗𝜎𝐾𝑥,𝑦−𝐾𝑥,𝑥_𝑅𝑗×1𝑥−𝑥_𝑅𝑗d𝜇(𝑥)≤𝐶𝑏_𝜎∗. (39) Then, we can get the estimate of 𝐺₂₂: 𝐺₂₂≤𝐶𝜆_𝑗^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×_ℝ𝑑𝐾𝑥,𝑦−𝐾𝑥,𝑥_𝑅𝑗×𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′×ℎ_𝑗𝑦𝑦×1d𝜇𝑥−𝑥_𝑅𝑗+√𝑑𝑙𝑅_𝑗(𝑥)≤𝐶d𝜇𝜆_𝑗^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙𝑅𝑗ℝ}^𝑑_⧵2𝑅𝑗𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎×1𝑥−𝑥_𝑅𝑗×𝐾𝑥,𝑦−𝐾𝑥,𝑥_𝑅𝑗×𝑏(𝑥)𝑅−𝑚_𝑗(𝑏)_𝜎′×ℎ_𝑗(𝑥)𝑦≤𝐶d𝜇d𝜇𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖)𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}𝑏_𝜎∗𝑎_𝑗^(𝑖)_{𝐿𝑝(𝜇)}×_{𝑄𝑗(𝑖)}𝑚𝑅_𝑗(𝑏)−𝑏𝑦_𝜎′^𝑝′d𝜇𝑦^1/𝑝′≤𝐶𝜆𝑏_∗𝑗²_𝑖=1𝑟_𝑗^(𝑖). (40) Combining the estimates for 𝐺₂₁ and 𝐺₂₂ then yields the desired estimate for 𝐺₂, which together with the estimate for 𝐺₁ indicates that 𝜇𝑥∈ℝ^𝑑∶ℳ𝑓(𝑥)>𝜆≤𝐶𝜆⁻¹𝑓_𝐻^1,𝑝_{𝑏,𝑚−1,𝑚+1(𝜇)}. (41)
This finishes the proof of Theorem 4.

Proof of Theorem 5. 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, _ℝ𝑑ℳ_𝑏(ℎ)(𝑥)d𝜇(𝑥)≤𝐶|ℎ|_𝐻^1,𝑝_{𝑏,𝑚,𝑚+1(𝜇)}, (42) where 𝐶 is a constant independent of ℎ. Let all the notation be the same as in Definition 3. By our choices, 𝑎_𝑖,𝑖=1,2, now they satisfy satisfies the following condition: 𝑎_{𝑖𝐿𝑝(𝜇)}≤𝜇4𝑄_𝑖^1/𝑝−1𝐾_{𝑄𝑖,𝑅}^−𝑚−1. (43)
Write _ℝ𝑑ℳ_𝑏(ℎ𝑥)(𝑥)≤)(d𝜇_2𝑅ℳ_𝑏(ℎ𝑥)(𝑥)+)(d𝜇_{ℝ𝑑⧵2𝑅}ℳ_𝑏(ℎ)(𝑥)d𝜇(𝑥)=𝑊₁+𝑊₂. (44) To estimate 𝑊₁, we further decompose 𝑊₁≤²_𝑖=1𝑟_{𝑖2𝑄𝑖}ℳ_𝑏(ℎ𝑥)(𝑥)+)(d𝜇_2𝑖=1𝑟_{𝑖2𝑅⧵2𝑄𝑖}ℳ_𝑏(ℎ)(𝑥)d𝜇(𝑥)=𝑊₁₁+𝑊₁₂. (45) The Hölder inequality, 𝐿^𝑝-boundedness of ℳ_𝑏 (see [9]), and (43) tell us 𝑊₁₁≤²_𝑖=1𝑟_𝑖ℳ_𝑏𝑎_{𝑖𝐿𝑝(𝜇)}𝜇2𝑄_𝑖^1/𝑝′≤𝐶𝑏_∗²_𝑖=1𝑟_𝑖𝑎_{𝑖𝐿𝑝(𝜇)}𝜇2𝑄_𝑖^1/𝑝′≤𝐶𝑏_∗²_𝑖=1𝑟_𝑖. (46) From (6), the Hölder inequality, Lemma 7, and (43), it follows that 𝑊₁₂≤𝐶²_𝑖=1𝑟_𝑖^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙}^{𝑁𝑖2𝑄2𝑅}_{𝑘=12𝑘+1𝑄𝑖⧵2𝑘𝑄𝑖}𝑏(𝑥)𝑄−𝑚_𝑖𝜎×_𝑄𝑖𝑚𝑄_𝑖(𝑏)−𝑏𝑦_𝜎′𝑎_𝑖𝑦×1𝑥−𝑦^𝑛−11𝑙𝑄_𝑖d𝜇𝑦≤𝐶²_𝑖=1𝑟_𝑖^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙𝑄𝑖}𝑎_𝑖𝑦^𝑝d𝜇𝑦^1/𝑝×_𝑄𝑖𝑚𝑄_𝑖(𝑏)−𝑏𝑦_𝜎′^𝑝′d𝜇𝑦^1/𝑝′×^{𝑁𝑖2𝑄,2𝑅}_𝑘=11𝑙𝑄_𝑖𝑙2^𝑘𝑄_𝑖^𝑛−1×_{2𝑘+1𝑄𝑖}^𝑙_{𝑠=0𝜂∈𝐶𝑙𝑠}𝑏(𝑥)2−𝑚_𝑘+1𝑄_𝑖(𝑏)_𝜂×𝑚2^𝑘+1𝑄_𝑖(𝑏)𝑄−𝑚_𝑖(𝑏)_𝜂′(𝑥)𝑏d𝜇≤𝐶_∗²_𝑖=1𝑟_𝑖𝑎_{𝑖𝐿𝑝(𝜇)}𝜇2𝑄_𝑖^1/𝑝′𝐾_{𝑄𝑖,𝑅}^𝑚+1≤𝐶𝑏_∗²_𝑖=1𝑟_𝑖. (47) Now we turn to estimate 𝑊₂. Involving the vanishing moment of ℎ, we can write 𝑊₂≤^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ𝑑⧵2𝑅}𝑏(𝑥)−𝑚𝑅(𝑏)_𝜎×ℳ_𝑏𝑚𝑅(𝑏)−𝑏_𝜎′ℎ(𝑥)d𝜇(𝑥)≤𝐶^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ𝑑⧵2𝑅}𝑏(𝑥)−𝑚𝑅(𝑏)_𝜎×^{|𝑥−𝑥𝑅√|+}₀^{𝑑𝑙(𝑅)}_{𝑥−𝑦≤𝑡}𝐾𝑥,𝑦𝑚𝑅(𝑏)−𝑏𝑦_𝜎′×ℎ_𝑗𝑦d𝜇𝑦²d𝑡𝑡^31/2d𝜇(𝑥)+𝐶^𝑚_{𝑙=0𝜎∈𝐶𝑚𝑙ℝ𝑑⧵2𝑅}𝑏(𝑥)−𝑚𝑅(𝑏)_𝜎×^∞_{𝑥−𝑥𝑅+√(𝑅)𝑑𝑙𝑥−𝑦≤𝑡}𝐾𝑥,𝑦𝑚𝑅(𝑏)−𝑏𝑦_𝜎′×ℎ_𝑗𝑦d𝜇𝑦²d𝑡𝑡^31/2d𝜇(𝑥). (48) An argument similar to estimate 𝐺₂ together with the Hölder inequality, Lemma 7, and (43) tell us that 𝑊₂≤𝐶𝑏_∗²_𝑖=1𝑟_𝑖. (49)
The estimates for 𝑊₁ and 𝑊₂ lead to (42). The proof of Theorem 5 is completed.

Then, we can get the following corollaries directly.

Corollary 8. Let 1<𝑝≤∞, 𝑚∈ℕ, and 𝑏∈𝑅𝐵𝑀𝑂(𝜇), and let ℳ_𝑏^𝑚 be as in (9) with 𝐾 satisfying (6) and (10). Then ℳ_𝑏^𝑚 is bounded from 𝐻_𝑏^1,𝑝_{𝑚,𝑚−1,𝑚+1}(𝜇) to weak 𝐿¹(𝜇).

Corollary 9. Let 1<𝑝≤∞,  𝑚∈ℕ, and 𝑏∈𝑅𝐵𝑀𝑂(𝜇), and let ℳ_𝑏^𝑚 be as in (9) with 𝐾 satisfying (6) and (10). Then ℳ_𝑏^𝑚 is bounded from 𝐻_𝑏^1,𝑝_{𝑚,𝑚,𝑚+1}(𝜇) to 𝐿¹(𝜇).

We remark that Corollary 9 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 𝐿^1,∞(𝜇) using a different method.

Theorem HL (see [12]). Let 1<𝑝₁,𝑞₁<∞,1<𝑟≤∞,1/𝑞₁=1/𝑝₁−1/𝑟, and 1/𝑞₀=1−1/𝑟, and let T be a sublinear operator that is bounded from 𝐿^𝑝1(𝜇) to weak 𝐿^𝑞1(𝜇) and is bounded from 𝐻¹(𝜇) to weak 𝐿^𝑞0(𝜇); then there is a positive constant 𝐶 such that for any 𝜆>0, measurable function 𝑓 with  ∫_ℝ𝑑|𝑓(𝑥)|𝑙𝑜𝑔(2+|𝑓(𝑥)|)𝑑𝜇(𝑥)<∞,𝜇𝑥∈ℝ^𝑑∶𝑇𝑓(𝑥)>𝜆≤𝐶_ℝ𝑑𝑓(𝑥)𝜆𝑓(𝑥𝑓log2+)^𝑞0₁⁻¹𝜆^𝑞0d𝜇(𝑥)^𝑞0. (50)

From the above Theorem  HL and Theorem 4, one has the following corollary.

Corollary 10. Let 1<𝑝≤∞,  𝑏∈𝑅𝐵𝑀𝑂(𝜇),   and 𝐾 satisfy (6) and (10). Then ℳ_𝑏 is bounded from 𝐿log𝐿(𝜇) to 𝐿^1,∞(𝜇); that is, 𝜇𝑥∈ℝ^𝑑∶ℳ_𝑏𝑓(𝑥)>𝜆≤𝐶_ℝ𝑑𝑓(𝑥)𝜆𝑓(𝑥log2+)𝜆d𝜇(𝑥). (51)

Conflict of Interests

The authors declare that they have no conflict of interests.