Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Nondoubling Measures

where B(x, r) is the open ball centered at some point x ∈ R 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 ofCalderón-Zygmundoperators with nondoubling measures and many classical results have been proved still valid if the underlyingmeasure 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 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 > α; 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 wemean a (2, 2)-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


Introduction
We will work on the -dimensional Euclidean space R  with a nonnegative Radon measure  which only satisfies the following growth condition that there exists a consist  0 such that  ( (, )) ≤  0   , where (, ) is the open ball centered at some point  ∈ R  and having radius .The measure  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][2][3][4][5][6][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  ⊂ R  we mean a closed cube whose sides are paralleled to the axes and we denote its side length by () and its center by   .Let  and  be positive constants such that  > 1 and  >   ; for a cube , we say that  is (, )-doubling if () ≤ (), where  denotes the cube concentric with , having side length ().In what follows, for definiteness, if  and  are not specified, by a doubling cube we mean a (2, 2 +1 )-doubling cube.Especially, for any give cube , we denote by Q the smallest doubling cube in the family {2  } ∈N .For two cubes  ⊂ , set where  , is the first positive integer  such that (2  ) ≥ ().
Definition 1.Let  > 1 be some fixed constant, we say that a function  ∈ where the supremum is taken over all cubes concentered at some point of supp  and  Q() is the mean value of  on Q; namely, The minimal constant  in (3) and ( 4) is defined to be RBMO() norm of  and is defined by ‖‖ * .
In [1] Tolsa introduced the space RBMO() and showed that the definition of RBMO() is independent of the choice of numbers  > 1.In the sequel we will choose  = 2.
To state our results, we need to recall some necessary notation and definitions.Let (⋅, ⋅) be a locally integrable function on {R  × R  \ (, ) :  = }.Assume that there exists a constant  > 0 such that for ,  ∈ R  with  ̸ = , and for any , , and for any ,   ∈ R  .The Marcinkiewicz integral M associated with the above kernel  and the measure  as in ( 1) is defined by In what follows, we will always assume that M is bounded on  2 ().For  ∈ N and   ∈ RBMO(),  = 1, 2, . . ., , we formally define the multilinear commutator M ⃗  by where  not only satisfies the size condition (6) but also satisfies a strong Hömander type condition: for all ,   ∈ R  and For   = ,  = 1, 2, . . ., , we denote by    .
Remark 2. With the assumption that M is bounded on  2 (), Hu et al. [8] established that the integral M as in (8) with kernel  satisfying ( 6) and ( 7) is bounded from  1 () to  1,∞ (), from  1 () to  1 (), and from  ∞ () to RBLO(), respectively, where the function space RBLO() is the subset of function space RBMO().At the same time, the authors obtain the boundedness of the commutator M , , respectively, from   () to itself for  ∈ (1, ∞] and from the space  log () to  1,∞ (), where the kernel  satisfies ( 6) and (10).Such type of multilinear commutators when  is a nondoubling measure was introduced by Zhang [ 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.
Then, we denote We say that  ∈ with where the infimum is taken over all the possible decomposition of  in ( ⃗ , , , )-atomic blocks.From the definition, we have the following properties that for any and for any ,  ∈ N and and for any For some more details about this Hardy-type space, see [4] and its references.It is valuable to point out that, if  = 0, the space  1, ⃗ ,, () is just the Hardy space  1 () introduced by Tolsa in [1] with equivalent norm, which was proved by the authors in [10].
Let us state our main results as follows.
Throughout this paper,  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.
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  log () to  1,∞ () using a different method.