Some estimates for commutators of Littlewood-Paley g-functions

Abstract: The aim of this paper is to establish the boundedness of commutator [ ] b g , ̇r generated by Littlewood-Paley g-functions ġr and ∈ ( ) b μ RBMO on non-homogeneous metric measure space. Under assumption that λ satisfies ε-weak reverse doubling condition, the author proves that [ ] b g , ̇r is bounded from Lebesgue spaces ( ) L μ p into Lebesgue spaces ( ) L μ p for ∈ ( ∞) p 1, and also bounded from spaces ( ) L μ 1


Introduction
Although the metric measure spaces equipped with the polynomial growth conditions (see [1][2][3][4][5]) and spaces of homogeneous type in the sense of Coifman and Weiss [6,7] are two important classes of function spaces in harmonic analysis, there exist no relations between the non-doubling measure spaces and spaces of homogeneous type. To solve this problem, Hytönen [8] first introduced the non-homogeneous metric measure spaces satisfying the so-called geometrically doubling and upper doubling conditions. From then on, many papers focus on the properties of function spaces and operators over non-homogeneous metric measure spaces. For example, Cao and Zhou [9] obtained the definition of Morrey space on non-homogeneous metric measure space and also proved that Hardy-Littlewood maximal operator, Calderón-Zygmund operator and fractional integral are bounded on Morrey space. Fu and Zhao [10] proved that generalized homogeneous Littlewood-Paley g-function is bounded from atomic Hardy space ( ) H μ atb 1 into space ( ) L μ 1 and also bounded from space ( ) μ RBMO into space ( ) μ RBLO . For more development of harmonic analysis on non-homogeneous metric measure space the readers can see [11][12][13][14][15][16][17][18][19] and references therein.
Let ( ) d μ , , be a non-homogeneous metric measure space in the sense of Hytönen [8]. In this setting, the author proves that the and on space ( ) μ RBMO (see [10]). In 2021, Lu and Tao [11] proved that the g˙r is bounded on Lipschitz space ( ) μ Lip β and on generalized Morrey space .


Before stating the main results of this paper, we first recall some necessary notions. The following definitions of geometrically doubling and upper doubling conditions are from [8].
Moreover, Hytönen et al. [12] showed that there exists another dominating function Here and in what follows, we always assume that λ satisfies (1.2).
Although the doubling measure condition is not assumed uniformly for all balls on ( ) d μ , , , Hytönen [8] showed that there exist many balls satisfying the ( ) α β , -doubling condition, i.e., let α, where ( ) c B represents the center of ball B. For more properties of the K B S , , we can see [12, Lemma 2.1]. The following regularized bounded mean oscillation (RBMO) space was from [8].
if there exist a positive constant C and, for any ball B, a number f B such that and, for any two balls B and S such that ⊂ B S,

5)
The infimum of the positive constants C satisfying both (1.4) and (1.5) is defined to be the ( ) μ RBMO norm of f and denoted by ‖ ‖ ( ) f μ RBMO . Moreover, Hytönen [8] also showed that the space ( ) μ RBMO is independent of the choice of the constant ∈ ( ∞) ρ 1, .
In 2015, Tan and Li [13] gave an approximation of the identity ≔ { } ∈ S S k k associated with ( ( + )) on ( ) d μ , , , which are integral operators associated with kernels ( ) S x y , k on × satisfying the following conditions: , then there exists a non-negative constant C such that and Q x k , represents a fixed doubling ball center as x of generation k.
for all ∈ x y , , and D k be the corresponding integral operator associated with the kernel Then, the generalized homogeneous Littlewood-Paley g-function g˙r is defined by , f o r a n y .
,˙r , which is generated by b and g˙r as in (1.6), is defined by where ∈ x and ∈ ( ) The following definition of generalized Morrey space is from [14].
is defined by (1) From [14], Lu and Tao showed that the norm ‖ ‖ ( ) f M μ q p is independent of the choice of parameter k for > k 1.
, which was introduced by Cao and Zhou [9], that is, let > k 1 and < ≤ < ∞ q p 1 , then We now recall the following ε-weak reverse doubling condition introduced in [16].
, depending only on a and , such that, for all ∈ x , and, moreover, The main theorems of this paper are stated as follows: . Suppose that λ satisfies the ε-weak reverse doubling condition defined as in Definition 1.8. Then, there exists a constant > C 0 such that, for all ∈ ( ) . Suppose that λ satisfies the ε-weak reverse doubling condition defined as in Definition 1.8. Then, there exists a constant > C 0 such that, for all ∈ ( ) f L μ . Suppose that λ satisfies the weak reverse doubling condition defined as in Definition 1.8. Then, there exists a constant > C 0 such that, for all ∈ ( ) Finally, we make some conventions on notation. Throughout the paper, C represents a positive constant, which is independent of the main parameters. For any subset E of , we use χ E to denote its characteristic function. Given any ∈ ( ∞)

Preliminaries
To prove the main theorems of this paper, we should recall some necessary results in this section (see [10,14,16,17] The following maximal operators, respectively, defined for all and ∈ x , and ∈ x , we first claim that and, for all balls B, S satisfying ⊂ B S and ∋ B x, To prove    Together with the aforementioned estimates, to get (3.2), we still need to estimate the difference     With an argument similar to that used in the estimate of (3.6), it is easy to see that   Conflict of interest: Author states no conflict of interest.