Commutators of θ-type generalized fractional integrals on non-homogeneous spaces

The aim of this paper is to establish the boundednes of the commutator [b,Tα]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[b,T_{\alpha }]$\end{document} generated by θ-type generalized fractional integral Tα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\alpha }$\end{document} and b∈RBMO˜(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b\in \widetilde{\mathrm{RBMO}}(\mu )$\end{document} over a non-homogeneous metric measure space. Under the assumption that the dominating function λ satisfies the ϵ-weak reverse doubling condition, the author proves that the commutator [b,Tα]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[b,T_{\alpha }]$\end{document} is bounded from the Lebesgue space Lp(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}(\mu )$\end{document} into the space Lq(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}(\mu )$\end{document} for 1q=1p−α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{q}=\frac{1}{p}-\alpha $\end{document} and α∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha \in (0,1)$\end{document}, and bounded from the atomic Hardy space H˜b1(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\widetilde{H}^{1}_{b}(\mu )$\end{document} with discrete coefficient into the space L11−α,∞(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\frac{1}{1-\alpha },\infty }(\mu )$\end{document}. Furthermore, the boundedness of the commutator [b,Tα]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[b,T_{\alpha }]$\end{document} on a generalized Morrey space and a Morrey space is also got.


Introduction
The theories of the function spaces and the singular integral operators play an important role in the fields of the harmonic analysis and PDE. In particular, during the past 20 to 25 years, many authors have paid much attention to the space of homogeneous type in the sense of Coifman and Weiss [4,5] and the metric measure space endowed with nondoubling measure; for example, see [8,15,17,18,[20][21][22] and the references therein.
However, the non-doubling measure may not satisfy the well-known doubling condition being a key assumption on spaces of homogeneous type. To solve this problem, Hytönen in [9] introduced a new class of metric measure spaces satisfying the so-called upper doubling and the geometrically doubling conditions (see Definitions 1.1 and 1.3 below, respectively). For convenience, the new metric measure space is now called a non-homogeneous metric measure space. Since then, many papers about the different kinds of function spaces and singular integral operators on non-homogeneous metric measure space have been widely focused on; for example, see [2, 7, 11-14, 16, 19] and the references therein. Furthermore, see the monograph [23] to find the more development on harmonic analysis in this new context.
Let (X , d, μ) be a non-homogeneous metric measure space in the sense of Hytönen [9]. In this setting, Lin et al. [12] proved that the commutator T b := bT -Tb generated by the Calderón-Zygmund operator T and the function b ∈ RBMO(μ) is bounded from the atomic Hardy space H 1 (μ) with discrete coefficient into the space L 1,∞ (μ), and bounded from Lebesgue space L p (μ) into the space L p (μ) for p ∈ (1, ∞). Moreover, Ri and Zhang in [16] proved that the commutators of θ -type Calderón-Zygmund operators with RBMO functions is bounded from the L ∞ (μ) into the space RBMO(μ), and bounded from the Hardy space H 1 (μ) into the L 1 (μ). In [7], Fu et al. established some equivalent characterizations for the boundedness of the generalized fractional integrals over (X , d, μ), moreover, the boundedness of the multilinear commutators of generalized fractional integrals with RBMO(μ) functions on Orlicz spaces is obtained. Motivated by these results, in this paper, we will mainly establish the boundedness of the commutator [b, T α ] generated by θ -type generalized fractional integral T α and b ∈ RBMO(μ) on the Lebesgue space, atomic Hardy space with discrete coefficient, Morrey space and generalized Morrey space.
Before presenting the organization of this paper, we need to recall some necessary notions. The following definitions of the upper doubling and geometrically doubling conditions are from [9]. Definition 1.1 ([9]) A metric measure space (X , d, μ) is said to be upper doubling, if μ is Borel measure on X and there exist a dominating function λ : X × (0, ∞) → (0, ∞) and a positive constant C λ depending only on λ such that, for each x ∈ X , r → λ(x, r) is non-decreasing and, for all x ∈ X and r ∈ (0, ∞), (1. We now recall the definition of the discrete coefficient K (ρ) B,S originally introduced by Bui and Duong (see [1]), which is closer to the quantity introduced by Tolsa in [20]. Definition 1.5 ([1]) For any ρ ∈ (1, ∞) and any two balls B ⊂ S ⊂ X , let , (1.3) where N (ρ) B,S represents the smallest integer satisfying ρ N (ρ) B,S r B ≥ r S , and c B and r B are the center and radius of ball B, respectively.
B,S was given in [9] and [10] as follows. That is, for any two balls B ⊂ S ⊂ X , define . (1.4) In general, K B,S and K (ρ) B,S are not equivalent, but, if we take (X , d, μ) = (R d , | · |, μ) and λ(x, r) := Cr d as in (1.1), it is not difficult to find that with implicit equivalent positive constants independent of the balls B and S; see [12] for more details. In addition, by (1.1) and a change of variables, it is easy to obtain the other form of the K (ρ) B,S , that is, . (1.6) Next, we recall the following definition of the fractional coefficient K α B,S given in [7]. 1-α where N B,S is the smallest integer satisfying 6 N B,S r B ≥ r S . Remark 1.8 If we take α ≡ 0 in Definition 1.7, then the fractional coefficient K α B,S is just the K B,S introduced by Bui and Duong in [1]. Moreover, the reader can see [7,Lemma 3.4] to find the other properties of the coefficient K B,S .
Although the measure doubling condition is not assumed uniformly for all balls on (X , d, μ), Hytönen has showed that there are many balls having (η, β)-doubling property. Namely, for η, β > 1, a ball B ⊂ X is said to be (η, β)-doubling if μ(ηB) ≤ βμ(B). Meanwhile, Hytönen [9] proved that if a metric measure space (X , d, μ) is upper doubling and β > C log 2 η λ =: η ν , then, for each ball B ⊂ X , there exists some j ∈ Z + such that η j B is (η, β)doubling. Moreover, let (X , d) be a geometrically doubling, β > η n with n := log 2 N 0 and μ Borel measure on X being finite on bounded sets. Hytönen also showed that there exist arbitrarily small (η, β)-doubling balls centered at x for μ-a.e x ∈ X . Furthermore, the radius of there balls may be chosen to be of the form η -j r for j ∈ N and any preassigned number r ∈ (0, ∞). Throughout this paper, for any η ∈ (1, ∞) and ball B, the smallest (η, β η )doubling ball of the form η j B with j ∈ Z + is denoted byB η , where β η := max η 3n , η 3ν + 30 n + 30 ν . (1.7) In this paper, if there is no special explanation, we always set η = 6 and simply denote B 6 by B.
The definition of θ -type generalized fractional integral is as follows.
Let L ∞ b (μ) be the space of all L ∞ (μ) functions with bounded support. A linear operator T α is called a θ -type generalized fractional integral with kernel K α satisfying (1.9) and (1.10) if, for all f ∈ L ∞ b (μ) and x / ∈ supp(f ), We now recall the notation of the space RBMO(μ) given in [6]. . A function f ∈ L 1 loc (μ) is said to be in the space RBMO ρ,γ (μ) if there exist a positive constant C and, for any ball B ⊂ X , a number f B such that and, for any two balls B and S such that B ⊂ S, The infimum of the positive constant C satisfying both (1.12) and (1.13) is defined to be the RBMO ρ,γ (μ) norm of f and denoted by f RBMO ρ,γ (μ) .
Given a function b ∈ RBMO(μ), the commutator [b, T α ] associated with the θ -type generalized fractional integral T α is, respectively, defined by (1.14) The following notion of the -weak reverse doubling condition is from [7]; also see [13].
, depending only on a and X , such that, for all x ∈ X , and, moreover, The organization of this paper is as follows. In Sect. 2, we mainly recall some necessary lemmas being used in the proof of the main theorems. In Sect. 3, we will prove that the commutator [b, T α ] generated by the θ -type fractional integral operator T α and b ∈ RBMO(μ) is bounded from the Lebesgue space L p (μ) into the space L q (μ), where 1 q = 1 pα with α ∈ (0, 1) and p ∈ (1, 1 α ). In Sect. 4, via decomposition of the atomic, the boundedness of the commutator [b, T α ] from the atomic Hardy space T α ] on the Morrey space and the generalized Morrey space is also presented in Sects. 5 and 6, respectively.
Finally, we make some conventions on notation. Throughout the paper, C represents for a positive constant that is independent of the main parameters involved, but may be different from line to line. For a μ-measurable set E, χ E denotes its characteristic function. For any p ∈ [1, ∞], we denote by p its conjugate index, that is, 1 p + 1 p = 1. In addition, for any f ∈ L 1 loc (μ) and any measurable set E of X , m B (f ) represents the mean value of the function f over ball B, namely, m B (f ) :

Preliminaries
In this section, we will recall some necessary lemmas which is used in the proof of the main theorems in this paper. We first need to recall some properties of the discrete coefficient K B,S (see [12]).
We now recall the following characterizations of the space RBMO(μ) given in [12].
Also, we need to recall some results given in [1].

4)
and are bounded on L p (μ) and also bounded from Next, we recall the following lemma from [7].
and the supremum is taken over all balls B x.
Let α ∈ (0, 1). For all f ∈ L 1 loc (μ) and x ∈ X , the sharp maximal function M ,α of f is defined by We now recall the following lemma from [7].
Finally, we recall the following lemmas given in [7].
where C x is a positive constant, depending on x, and C a positive constant depending only on C λ , β 6 and α.

Boundedness of [b, T α ] on Lebesgue space
In this section, we will establish the boundedness of the commutator [b, T α ] generated by the θ -type generalized fractional integral T α and the space RBMO(μ) on the Lebesgue space L p (μ) for p ∈ (1, ∞). Moreover, the endpoint boundedness of the commutator [b, T α ] is also obtained.
We now state the main theorems of this section as follows.
pα, and K α satisfy (1.9) and (1.10). Suppose that T α is as in (1.11). Then there exists a constant C > 0, such that, for all f ∈ L p (μ), To prove the main theorem, we need to establish the following lemma about the θ -type generalized fractional integral T α . Lemma 3.2 Let α ∈ (0, 1), p ∈ (1, 1 α ) with 1 q = 1 pα, and K α satisfy (1.9). Then there exists a positive constant C, such that, for all f ∈ L p (μ), Proof For any x ∈ X , by applying (1.9) and (1.11), we can get where I α represents the fractional integral operator defined by Furthermore, by the (L p (μ), L q (μ))-boundedness of I α (see [7]), it is difficult to obtain Hence, the proof of Lemma 3.2 is completed. Now we give the proof of Theorem 3.1.
Proof of Theorem 3.1 In a slightly modified way similar to that used in the proof of Theorem 3.10 in [7], it is not difficult to prove that the case μ(X ) = ∞ holds. Thus, without loss of generality, we may assume μ(X ) = ∞. Assume that p ∈ (1, 1 α ). First, we claim that, for all r ∈ (1, ∞) and f ∈ L p (μ), Once (3.1) is obtained, taking 1 < r < p < 1 α , by applying Lemmas 2.4, 2.5 and 2.6, we can deduce that which is the desired consequence. Thus, we need to show (3.1). By the definition of the sharp maximal operator M ,α in (2.6), we should show that, for all x and balls B with B x, and, for all balls B, S and B x, where With a slightly modified argument similar to that used in the estimates (3.6), (3.7) and (3.8) in [7, Theorem 3.9], it is not difficult to see that (3.2) holds, too. However, to estimate (3.2), we still need to estimate the difference With an argument similar to that used in the estimate for I 1 in [7, Theorem 3.9], it is not difficult to get Hence, we only need to estimate A 2 . For any y 1 , y 2 ∈ B, by applying (1.8), (1.9), (1.16), the Hölder inequality and (2.3), we have where we have used the fact that Further, combining the estimates for A 1 and A 2 , we conclude that This, together with (3.6), (3.7) and (3.8)  Write Following the proof of [3, Theorem 1] and [1, Theorem 7.6], it is not difficult to get Finally, we turn to the estimate of D 3 . For any z ∈ B, by applying (1.9), Definition 1.7, the Hölder inequality, (2.3) and (2.5), we can conclude that Further, by applying the definition of m B (f ), we get Thus, combining the estimates D 1 , D 4 and D 5 , we complete the estimate for (3.3). If B is a doubling ball and x ∈ B, by (3.2), we have Similar to (3.3), for all doubling balls B ⊂ S with x ∈ B such that K α B,S ≤ P α , we have Thus, by applying Lemma 2.8, we know that, for all doubling B ⊂ S with x ∈ B, and, using (3.4), we can get Combining (3.5) and the above inequality, we can get (3.1). Hence, the proof of Theorem 3.1 is completed.

Boundedness of [b, T α ] on Hardy space with discrete coefficient
As is well known, the dual space of atomic Hardy space H 1,q,γ atb,ρ (μ), which was introduced by Fu et al. [6], is the space RBMO(μ) associated with the discrete coefficient K (ρ) B,S . But,in this section, we will consider the boundedness of the commutator [b, T α ] on the atomic Hardy space H 1,q,γ atb,b,ρ (μ) which is the subspace of the H 1,q,γ atb,ρ (μ). First, we give the definition of the atomic Hardy space H 1,q,γ atb,b,ρ (μ) being slightly modified in [15] and [6]. called a (b, q, γ , ρ)  A function f ∈ L 1 (μ) is said to belong to the atomic Hardy space H The H 1,q,γ atb,b,ρ (μ) norm of f is defined by where the infimum is taken over all the possible decompositions of f as above.
With an argument similar to that used in [6], it is not difficult to show that, for any q ∈ (1, ∞], the atomic Hardy space H 1,q,γ atb,b,ρ (μ) is independent of the choices of ρ and γ and that, for all q ∈ (1, ∞), the spaces H 1,q,γ atb,b,ρ (μ) and H 1,∞,γ atb,b,ρ (μ) coincide with equivalent norms. Thus, in this section, we denote the space H 1,q,γ atb,b,ρ (μ) by H 1  b (μ). The main theorems of this section are stated as follows.
Proof of Theorem 4.2. Without loss of generality, we may assume that ρ = 6 as in (4.1) and q 0 := 1 1-α for α ∈ (0, 1). For any function f ∈ H 1 b (μ), by Definition 4.1, we can get a decomposition Moreover, for each fixed i, we can further decompose the atomic block h i as where, for any j ∈ {1, 2}, κ i,j ∈ C, a i,j is a bounded function supported on some ball B i,j ⊂ S i satisfying and For F 1 i , by applying (4.2), write Choosing suitable p 1 , q 1 , such that 1 < p 1 < 1 α , 1 < q < q 1 and 1 q 1 = 1 p 1α. Applying the Hölder inequality, K (2) B i,j ,S i ≥ 1 and Theorem 3.1, we can deduce that (1.9), the Minkowski inequality, (2.3) and the fact that a i,j L ∞ (μ) ≤ [μ(2B i,j )] -1 [ K (2) B i,j ,S i ] -2 , we can deduce that Combining the estimates for F 1,1 i and F 1,2 i , we have Now we turn to F 2 i , by applying the vanishing condition of h i , the Hölder inequality, (1.10), (1.16) and y ∈ S i , we can get where we have used the following inequality (see [16]): Combining the above estimates for F 1 i and F 2 i , we can deduce that Thus, we complete the proof of Theorem 4.2.

Boundedness of [b, T α ] on Morrey space
In this section, we will mainly establish the boundedness of the commutator [b, T α ] generated by T α and the space RBMO(μ) on the Morrey space. Before giving the main result of this section, we first recall the definition of the Morrey space introduced by Cao and Zhou in [2].
By applying Lemma 3.2, we have For any x ∈ B, by applying (1.9), the Hölder inequality and Lemma 2.2, we can get