Multilinear fractional integral operators on non-homogeneous metric measure spaces

In this paper, the boundedness in Lebesgue spaces for multilinear fractional integral operators and commutators generated by multilinear fractional integrals with an RBMO(μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{RBMO}(\mu)$\end{document} function on non-homogeneous metric measure spaces is obtained.


Introduction and preliminaries
It is well known that a space of homogeneous type is the space, which satisfies the assumption of the doubling measure condition, i.e. there exists a constant C > 0 such that µ(B(x, 2r)) ≤ Cµ(B(x, r)) for all x ∈ suppµ and r > 0. However non-doubling measure is a nonnegative measure µ only satisfies the polynomial growth condition, i.e., for all x ∈ X and r > 0, there exists a constant C > 0 and k ∈ (0, n] such that, µ(B(x, r)) ≤ C 0 r k , (1.1) where B(x, r) = {y ∈ X : |y − x| < r}.This brings rapid development in harmonic analysis (see [2,7,10,11,22,24,25,27]).As an important application, it is to solve the long-standing open Painlevé's problem (see [24]).
In [13], Hytönen pointed out that the doubling measure is not the special case of the non-doubling measures.To overcome this difficulty, a kind of metric measure space (X, d, µ), that satisfies the geometrically doubling and the upper doubling measure conditions (see Definition 1.1 and 1.2) is introduced by Hytönen in [13], which is called the non-homogeneous metric measure space.The highlight of this kind of space is that it includes both space of homogeneous type and metric spaces with polynomial growth measures as special cases.From then on, a lot of results paralleled to homogeneous spaces and non-doubling measure spaces are obtained (see [1,4,5,6,13,14,15,16,18,19,20,21] and the references therein).For example, Hytönen et al. [16] and Bui and Duong [1] independently introduced the atomic Hardy space H 1 (µ) and obtained that the dual space of H 1 (µ) is RBMO(µ).Bui and Duong [1] also proved that Calderón-Zygmund operator and commutators of Calderón-Zygmund operator with RBMO function are bounded in L p (µ) for 1 < p < ∞.Later, Lin and Yang [18] introduced the space RBLO(µ) and proved the maximal Calderón-Zygmund operator is bounded from L ∞ (µ) into RBLO(µ).Recently, some equivalent characterizations was established by Liu et al. [21] for the boundedness of Carderon-Zygmund operators on L p (µ) for 1 < p < ∞.Fu et al. [5,6] established the boundedness of multilinear commutators of Calderon-Zygmund operators and commutators of generalized fractional integrals with RBMO(µ).Fu et al. [4] partially established the theory of the Hardy space H p with p ∈ (0, 1] on (X, d, µ).The readers can refer to the survey [30] and the monograph [31] for more developments on harmonic analysis in non-homogeneous metric measure spaces.
At the other hand, the theory on multilinear integral operators has been studied by some researchers.Coifman and Meyers [3] firstly established the theory of bilinear Calderón-Zygmund operators.Later, Gorafakos and Torres [8,9] established the boundedness of multilinear singular integral on the product Lebesgue spaces and Hardy spaces.Xu [28,29] established the properties of multilinear singular integrals and commutators on non-doubling measure spaces (R n , µ).The bounedeness of multilinear fractional integral and commutators on non-doubling measure spaces (R n , µ) was proved by Lian and Wu in [17].In non-homogeneous metric measure spaces, Hu et al. [12] established the weighted norm inequalities for multilinear Calderón-Zygmund operators.The boundedness of commutators of multilinear singular integrals on Lebesgue spaces was obtained by Xie et al. in [26].
In this paper, multilinear fractional integral operator and commutators generated by multilinear fractional integral with RBMO(µ) function on non-homog eneous metric spaces are introduced.And it is proved that multilinear farctional integral operators and commutators are bounded in Lebesgue spaces on nonhomogeneous metric spaces, provided that factional integral is bounded from L r into L s for all r ∈ (1, 1/β) and 1/s = 1/r − β with 0 < β < 1.The results in this paper include the corresponding results on both the homogeneous spaces and (R n , µ) with non-doubling measure spaces.
We first recall some notations and definitions.
Definition 1.1.[13] A metric spaces (X, d) is called geometrically doubling if there exists some N 0 ∈ N such that, for any ball B(x, r) ⊂ X, there exists a finite ball covering {B(x i , r/2)} i of B(x, r) such that the cardinality of this covering is at most N 0 .
[13] A metric measure space (X, d, µ) is said to be upper doubling if µ is a Borel measure on X and there exists a function λ : X × (0, +∞) → (0, +∞) and a constant C λ > 0 such that for each x ∈ X, r −→ (x, r) is nondecreasing, and for all x ∈ X, r > 0, Remark 1.3.(i) A space of homogeneous type is a special case of upper doubling spaces, where one can take λ(x, r) ≡ µ(B(x, r)).On the other hand, a metric space (X, d, µ) satisfying the polynomial growth condition (1.1)(in particular, (X, d, µ) ≡ (R n , | • |, µ) with µ satisfying (1.1) for some k ∈ (0, n])) is also an upper doubling measure space if we take λ(x, r) ≡ Cr k .(ii) Let(X, d, µ) be an upper doubling space and λ be a function on X × (0, +∞) as in Definition 1.2.In [13], it was showed that there exists another function λ such that for all x, y ∈ X with d(x, y) ≤ r, λ(x, r) ≤ C λ(y, r).
(1.3) Thus, in this paper, we always suppose that λ satisfies (1.3) and λ(x, ar) ≥ a m λ(x, r) for all x ∈ X and a, r > 0.
Definition 1.5.[6] Let 0 ≤ γ < 1.For any two balls B ⊂ Q, set N B,Q be the smallest integer satisfying 6 N B,Q r B ≥ r Q , then one defines . (1.4) For γ = 0, we simply write . The multilinear fractional integral on nonhomogeneous metric measure spaces is defined as follows.
(ii) There exists 0 < δ ≤ 1 such that proved that Cd(x, x ′ ) ≤ max 1≤j≤m d(x, y j ) and for each j, proved that Cd(y j , y ′ j ) ≤ max 1≤j≤m d(x, y j ).
A multilinear operators I α,m is called the multilinear fractional integral operator with the above kernel K satisfying (1.5), (1.6) ( For m = 1, we simply write I α,1 by I α , which is the generalized fractional integral operator introduced by [6].(ii') There exists 0 < δ ≤ 1 such that and for any two doubling ball The minimal constant C appearing in (1.9) and (1.10) is defined to be the RBMO(µ) norm of f and denoted by ||b|| * . For Definition 1.9.A kind of commutators generated by multilinear fractional integral operators In particular, when m = 2, it is easy to see that ] and [b 2 , I α,2 ] are defined as follows respectively.
. Without loss of generality, we only consider the case of m = 2. Now let us state the main results. where Remark 1.12.For ||µ|| < ∞, by Lemma 2.1 in Section 2, Theorem 1.11 also holds if one assumes that X G(f 1 , f 2 )(x)dµ(x) = 0 with the operator G be replaced by This paper is organized as follows.Theorem 1.10 and Theorem 1.11 are proved in Section 2. In Section 3, some applications are stated.Throughout this paper, C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line.

Proof of Main Results
Proof of Theorem 1.10.
and for any two doubling ball Lemma 2.3.
[10] |m Proof.As L ∞ (µ) with compact support is dense in L p (µ) for 1 < p < ∞, we only consider f 1 , f 2 ∈ L ∞ (µ) with compact support.Also, by Corollary 3.11 in [4], without loss of generality, we assume b 1 , b 2 ∈ L ∞ (µ).As Theorem 9.1 in [25], in order to obtain (2.2), it suffices to show that holds for any x ∈ B, and for any ball B ⊂ Q with x ∈ B, where Q is a doubling ball.For any ball B, denote )), and For E 2 , let 1 < s such that 1 s + 1 r = 1, by Hölder's inequality, one deduces For E 3 , in a similar way we can obtain It follows from Hölder's inequality and Theorem 1.10 that For E 42 , using (i) of Definition 1.5, Lemma 2.2, Lemma 2.3, Hölder's inequality and the condition of λ(x, ar) ≥ a m λ(x, r), we have dµ(y 2 ) Similarly, we get For E 44 , by (ii) of Definition 1.5, Lemma 2.2, Lemma 2.3, Hölder's inequality and the properties of λ, we obtain Taking the mean over z 0 ∈ B, it deduces (2.9)So (2.5) can be obtain from (2.8) to (2.9).
Next we prove (2.6).Consider two balls B ⊂ Q with x ∈ B, where B is an arbitrary ball and Q is a doubling ball.Let N = N B,Q + 1, then we yield Using the method to estimate E 44 , we get Let us estimate F 2 .At first, we calculate Hence To estimate F 21 , we write By Hölder's inequality, we have Then we obtain . Using the fact that Q is a doubling balls, Lemma 3.1 and Hölder's inequality, we yield We can also obtain For H 5 , since z ∈ Q, by (i) of Definition 1.5, Lemma 2.2, Lemma 2.3, Hölder's inequality and the properties of λ and Q is a doubling ball, we deduce In the similar way to estimate m Q (H 5 ), it follows that From (2.1) in Lemma 2.2, we deduce F 22 and F 23 also have similar estimate of F 21 , therefore, p 2 ,(5) f 2 (x) .
From F 3 to F 6 , using the similar method to estimate I 4 , we conclude Here we omit the details.Thus Lemma 2.4 is proved.
is bounded from L p (µ) into L q (µ).Theorem 3.2.Under the same assumption as that of Lemma 3.1, the conclusions of Theorem 1.10 and Theorems 1.11 hold true, if I α therein is replaced by T α .

Remark 1 . 7 .
Because max 1≤j≤m d(x, y j ) ≤ m j=1 d(x, y j ) ≤ m max 1≤j≤m d(x, y j ), (ii) in Definition 1.5 is equivalent to (ii') in the following statement.