MULTILINEAR COMMUTATORS FOR FRACTIONAL INTEGRALS IN NON-HOMOGENEOUS SPACES

Abstract Under the assumption that μ is a non-doubling measure on R, the authors obtain the (L, L)-boundedness and the weak type endpoint estimate for the multilinear commutators generated by fractional integrals with RBMO(μ) functions of Tolsa or with Oscexp Lr (μ) functions for r ≥ 1, where Oscexp Lr (μ) is a space of Orlicz type satisfying that Oscexp Lr (μ) = RBMO(μ) if r = 1 and Oscexp Lr (μ) ⊂ RBMO(μ) if r > 1.

The purpose of this paper is to prove the (L p , L q )-boundedness and the weak type estimate for any multilinear commutator generated by the fractional integral I α related to a measure µ as in (1.1) for 0 < α < n with any RBMO(µ) function of Tolsa in [20] or with any Osc exp L r (µ) function for r ≥ 1, motivated by [16], [3] for commutators on R d in the case that µ is the d-dimensional Lebesgue measure there.Here, for 0 < α < n and all x ∈ supp (µ), The behavior of such fractional integrals on a metric space was recently studied by García-Cuerva and Gatto in [4].Chen and Sawyer [2] showed that the first order commutator generated by I α and RBMO(µ) function enjoys the same (L p , L q ) mapping properties as in the case that µ is the Lebesgue measure.However, it seems that the argument used in [2] does not apply to the multilinear commutator here, we will employ some ideas used in [2] and some new ideas different from ones used in [2].
Before stating our results, let us introduce some notation and recall some definitions.Throughout this paper, we only consider closed cubes with sides parallel to coordinate axes.Let α > 1 and β > α n .We say that a cube Q is a (α, β)-doubling cube if µ(αQ) ≤ βµ(Q), where αQ denotes the cube with the same center as Q and l(αQ) = 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 given cube Q, we denote by Q the smallest doubling cube which contains Q and has the same center as Q.
, where N Q,R is the smallest positive integer k such that l(2 k Q) ≥ l(R).
If γ = 0, then we denote K Q,R by K Q,R .Tolsa in [20] first introduced the concept of K Q,R and gave its several useful properties.Chen and Sawyer in [2] introduced K (γ) Q,R and established some properties on K (γ) Q,R similar to those on K Q,R .
Using K Q,R , Tolsa in [20] introduced the space RBMO(µ) with the non-doubling measure µ, which is proved to be a good substitute of the classical space BMO in this case.Definition 1.1.Let ρ > 1 be some fixed constant.We say that a function f ∈ L 1 loc (µ) is in RBMO(µ) if there exists some constant C > 0 such that for any cube Q centered at some point of supp (µ), for any two doubling cube Q ⊂ R, where m Q (f ) denotes the mean of f over cube Q.The minimal constant C as above is the RBMO(µ) norm of f and is denoted by f * .
It has been shown in [20] that the definition of RBMO(µ) is independent of chosen constant ρ.In this paper, we need to choose different ρ in the proof of Theorem 1.1 and Theorem 1.2 below, respectively.
For any m ∈ N, 0 < α < n and b i ∈ RBMO(µ), i = 1, 2, . . ., m, the multilinear commutator, When m = 1, the operator I α; b1,...,bm is just the commutator [b 1 , I α ], which is a variant on the non-doubling measure of the classical commutator studied by Chanillo in [1].In [16], Pérez and Trujillo-González studied the boundedness of multilinear operators of this type generated by Calderón-Zygmund operators with BMO(R d ) functions or with Osc exp L r functions for r ≥ 1 in the case that µ is the d-dimensional Lebesgue measure.An extensive study of multilinear operators of this type can also be founded in [9].In [7], the authors obtained the corresponding results of multilinear operators generated by Calderón-Zygmund operators with RBMO(µ) functions or with Osc exp L r (µ) functions for r ≥ 1 in the case that µ is a non doubling measure.The multilinear commutator I α; b1,...,bm can be regarded as a natural variant of these multilinear operators in [9], [16], [7].
Chen and Sawyer in [2] proved that I α; b1 as in (1.2) is bounded from L p (µ) to L q (µ) provided that 1 < p < n/α and 1/q = 1/p − α/n.For m ≥ 2, a conclusion similar to that for the case m = 1 in [2] can be obtained as follows.
To consider the endpoint case of Theorem 1.1, we introduce the following function space, which is a variant with a non-doubling measure of the space Osc exp L r in [16].
The minimal constant C 1 satisfying (i) and (ii) is the Osc exp L r (µ) norm of f and is denoted by f Osc exp L r (µ) .
Obviously, for any r ≥ 1, Osc exp L r (µ) ⊂ RBMO(µ).Moreover, from John-Nirenberg's inequality in [20] (see also Lemma 3.1 below), it follows that Osc expL(µ) = RBMO(µ).We remark that it was pointed by Pérez and Trujillo-González in [16] that if µ is the d-dimensional Lebesgue measure in R d , the counterpart in [16] of the space Osc exp L r (µ) when r > 1 is a proper subspace of the classical space BMO(R d ).However, it is still unknown if the space Osc exp L r (µ) is a proper subspace of the space RBMO(µ) when µ is a non-doubling measure?
In what follows, C denotes a constant that is independent of main parameters involved but whose value may differ from line to line.For any index p ∈ [1, ∞], we denote by p its conjugate index, namely, 1/p + 1/p = 1.For A ∼ B, we mean that there is a constant C > 0 such that For a µ-locally integrable function f and a cube Q, we define and Then the generalized Hölder inequality holds for µ-locally integrable functions f and b i , i = 1, 2, . . ., m, and any cube Q, provided that r 1 , r 2 , . . ., r m ≥ 1, 1/r = 1/r 1 + • • • + 1/r m .If µ is the d-dimensional Lebesgue measure, (1.3) was proved in [16].It is easy to see that the proof in [16] still works in non-homogeneous spaces.

Proof of Theorem 1.1
Before we begin to prove Theorem 1.1, we state one equivalent norm for the space RBMO(µ) and some lemmas which play important roles in the proof.
Let ρ > 1.For any given function b ∈ RBMO(µ), there exists some constant C 2 > 0 and a collection of numbers , where the infimum is taken over all C 2 > 0 as above.Then there is a constant C > 0 such that for all b ∈ RBMO(µ), (2.1) see [20].
Proof of Theorem 1.1:For simplicity, we only consider the case of m = 2.If m ≥ 2, we can deduce the conclusion of the theorem by induction on m.
We leave the details to the reader; see also the proof of Theorem 1.2 below.For all r ∈ (1, n/α), we will prove the following sharp maximal function estimate Then choose r such that 1 < r < p < n/α and 1/q = 1/p − α/n.By (2.3), (2.2), Lemma 2.1, Lemma 2.2 and Lemma 2.3, we obtain which is the desired conclusion.
for any cube Q, and for all cubes Q ⊂ R. For any cube Q, we let .
To establish (2.3), it suffices to verify that holds for any cube Q and any x ∈ Q, and for any cubes Q ⊂ R with x ∈ Q, where Q is an arbitrary cube and R is a doubling cube.By a method similar to that in [2], from (2.4) and (2.5), it is then easy to deduce the sharp maximal function estimate (2.3).
To obtain the estimate (2.4) for any fixed Q and x ∈ Q, write The Hölder inequality and Corollary 3.5 in [20] yield that for 1 < r < n/α, and (2.7) An estimate similar to that for I 2 tells us that To estimate I 4 , let Let s = √ r and 1/ν = 1/s − α/n.From the Hölder inequality, (2.1) and Lemma 2.1, it follows that r,(9/8) f (x). (2.9) To estimate F, by the Hölder inequality, we first have that for where we used the estimate for j = 1, 2. From (2.10) and the choice of h Q , it follows that Therefore, r,(9/8) f (x).
Combining (2.6), (2.7), (2.8) and (2.12) yields (2.4).We now check (2.5) for chosen {h Q } Q as above.Consider two cubes Similar to the estimate for F, we easily obtain We estimate F 2 by decomposing it into From the fact that R is a doubling cube, it is easy to deduce that We further estimate M 1 by writing From the fact that R is a doubling cube, it follows that An estimate similar to that for E and the fact that R is a doubling cube tell us that Noting that l(2 N Q) ∼ l(R) and 2 N Q ⊃ R, by the Hölder inequality and (2.1), for y ∈ R, we have r,(9/8) f (x).
All the estimates above lead to (2.15) Similarly, we can prove (2.16) Combining the estimates (2.14), (2.15) and (2.16) leads to (2.17) Let us now estimate L 3 by first decomposing where we used the Hölder inequality and (2.1).Taking the mean over Q, we then obtain r,(9/8) f (x).

Proof of Theorem 1.2
We begin with some preliminary lemmas.
The following lemma on the Calderón-Zygmund decomposition in non-homogeneous spaces can be found in [22], [19].Lemma 3.2.Assume that µ satisfies (1.1).For any f ∈ L 1 (µ) and any λ > 0 (with λ > 2 d+1 f L 1 (µ) / µ if µ < ∞), we have (a) There exists a family of almost disjoint cubes {Q j } j (that is, (c) For each fixed j, let R j be a (6, 6 n+1 )-doubling cube concentric with Q j , with l(R j ) > 4l(Q j ) and set Then there exists a family of functions φ j with supp φ j ⊂ R j and with constant sign satisfying = 1 for j = 1, . . ., m, if the theorem is true when f L 1 (µ) = 1 and b j Osc exp L r j (µ) = 1 for j = 1, . . ., m, by (3.1) and the inequalities for any s > 0, t 1 , t 2 ≥ 0 and for t ≥ 0, we easily deduce that the theorem still holds for any bounded f with compact support and any b j ∈ Osc exp L r j (µ) for j = 1, . . ., m.
In what follows, we prove the theorem by two steps.
Step I: In this step, we prove Theorem 1.2 for m = 1.
For any fixed bounded and compact supported function f and any fixed λ > 2 d+1 f L 1 (µ) / µ , applying the Calderón-Zygmund decomposition (see Lemma 3.2) to f at level λ q0 , we obtain a sequence of cubes {Q j } j with bounded overlaps, that is, for all η > 2. ( (III) For each fixed j, let R j be the smallest (6, 6 n+1 )-doubling cube of the form Then there are a function φ j with supp φ j ⊂ R j and a constant C > 0 satisfying and It is easy to see that the conclusion of the theorem still holds if Let 1 < p 1 < n/α and 1/q 1 = 1/p 1 − α/n.Recall that and I α; b1 is bounded from L p1 (µ) to L q1 (µ) by Theorem 1.1 (see also [2]).This via the fact that g L ∞ (µ) ≤ Cλ q0 gives Noting that r = r 1 when m = 1 and therefore, the proof of the theorem can be reduced to proving that For each fixed j, set b ) and write The weak type (1, q 0 ) boundedness of I α (see Proposition 6.2 in [6]) tells us that It is obvious that R j is also (2, β d )-doubling and R j = R j .Thus, by Lemma 2.8 in [20], A trivial computation shows that K 6Qj ,Rj ≤ C (see also Lemma 2.1 (3) in [20]).This together with the estimate µ(2R j ) ≤ µ(6R j ) ≤ 6 n+1 µ(R j ) in turn implies that which is a desired estimate for V.
On the other hand, by the generalized Hölder inequality (1.3) and Lemma 3.1, we have From the fact that (see [16] and the related references there) and the inequality which is a desired estimate for U. Combining (3.5) and (3.6), we obtain that Now we turn our attention to I(x).Denote by x j the center of Q j .Let θ be a bounded function with θ L q 0 (µ) ≤ 1 and the support con- Invoking the condition (1.1), we have Therefore, which is a desired estimate for G. On the other hand, applying the Hölder inequality and the (L p1 , L q1 ) boundedness of I α (see Proposition 6.2 in [6]), we obtain where we have used the estimate that by recalling that R j is (2, β d )-doubling and R j = R j .
To estimate H, observe that for x ∈ 2R j \2Q j , Note that there is no (6, 6 n+1 )-doubling cube between Q j and R j .It follows that for each integer k with 0 We thus obtain that for each fixed j, Combining the estimates (3.8), (3.9) and (3.10) above yields and so The estimates (3.7) and (3.11) then yield (3.4), and we have completed the proof of the theorem for m = 1.
Step II: In this step, we prove Theorem 1.2 for all m ∈ N by induction on m.To this end, we assume that m ≥ 2 is an integer and that for any 1 ≤ i ≤ m − 1 and any subset σ = {σ(1), . . ., σ(i)} of {1, . . ., m}, Theorem 1.2 is true.We now verify that Theorem 1.2 is also true for m.
For each fixed f and λ > 0, we decompose f by the same way as in Step I; see (I), (II) and (III) in the proof of Step I.And let Q j , R j , φ j , w j , g, h be the same as in Step I. To prove the theorem in this case, it suffices to verity In particular, when σ = {1, . . ., m}, we denote I α; b1,...,bm simply by where Q is any cube in R d and y, z ∈ R d .With the aid of the formula for y, z ∈ R d , where if i = 0, then σ = {1, . . ., m} and σ = ∅, it is easy to see An argument similar to that for I(x) in Step I gives us that , and an argument similar to that for II(x) as in Step I yields .

Now we estimate I III
α; b σ h(x).For each fixed i with 1 ≤ i ≤ m − 1, the induction hypothesis now states that For A, we consider the following two cases.