The boundedness of multilinear commutators on locally compact Vilenkin groups

Let G be a locally compact Vilenkin group. In this paper, the authors investigate the boundedness of multilinear commutators of fractional integral operator on Lebesgue spaces on G . Furthermore, the boundedness on Hardy spaces are also obtained in this paper.


Introduction
The commutators have been studied by many authors for a long time. A well known result which is discovered by Coifman, Rocherg and Weiss ( [3], [6], [9]) is that the commutators [b, T ] of singular integral operators are bounded on some L p (R n ) ( A natural generalization of the commutator T m b is given by where m ∈ N. It was shown in [1] that it is bounded on L p (ω)(1 < p < ∞) when ω ∈ A p . And in [2], the authors prove the L p (ω)(1 < p < ∞)boundedness for multilinear commutators. And similar results can be found in [10]. This is the motivation of considering the boundedness for multilinear commutators of fractional integral operator on locally compact Vilenkin group G.
In order to state our results more precisely we first introduce some notations and definitions.
Throughout this paper, G will denote a bounded locally compact Vilenkin group, that is, G is a locally compact Abelian group containing a strictly decreasing sequence of compact open subgroups for any α > 0, k ∈ Z, where c is a constant independent of k . For each n ∈ Z we choose elements z l,n ∈ G(l ∈ Z + ) so that the subsets G l,n := z l,n + G n of G satisfy G k,n ∩ G l,n = φ if k = l and ∪ ∞ l=0 G l,n = G; moreover, we choose z 0,n such that G 0,n = G n . We now define the function d : Then d is a metric on G and the topology on G generated by this metric is the same as the original topology on G. For x ∈ G, set |x| = d(x, 0). Then |x| = (m n ) −1 if and only if x ∈ G n \ G n+1 . Let S(G) be the space of test functions and S (G) be the distribution space on G. And χ Gn is the characteristic function of G n . C can be denote various constants.
We also recall the definition of space of bounded mean oscillation. For x 0 ∈ G, set I j = x 0 +G j , we say a locally integrable function b has bounded where the supremum is taken over all cosets I j ⊂ G (see [4]). Since the topological nature os G at any x ∈ G is the same as it is at 0, we choose x 0 = 0 in our article.

Main results and proofs
Define multilinear commutator of fractional integral operator I α as Given any positive integer m, for all 1 ≤ j ≤ m, we denote by C m j the family of all finite subsets σ = {σ(1), · · · , σ(j)} of {1, · · · , m} of j different elements. For any σ ∈ C m j , we associate the complementary sequence σ given by σ = {1, 2, · · · , m} \ σ , and suppose b i BMO = 1 for i = 1, 2, · · · , m. Let p is adjoint index of p. We have the following theorem.
To prove this theorem, we need the following lemmas.

Lemma 2.2 (see [5] and [8]). Let
Proof. We firstly consider the case of m = 1, that is, choose λ = b G k and 1 < q < /δ, using Hölder inequality and Jenson inequality, we can deduce that For III, we have Next, we turn to estimate IV. Let for any n ∈ Z, and Therefore, we have proved that for m = 1.
If m ≥ 2, let λ = (λ 1 , · · · , λ m ), we have where C m,j are constants depending on m and j . Now, for fixed x ∈ G, for any number c and G k x, since 0 < δ < 1, we have For the first two parts I and II , choose λ j = (b j ) G k , j = 1, 2 · · · , m, and 1 < q < δ/ , similar to the case of m = 1 , we can deduce that and By Jenson inequality, we have To estimate V , let similar to the case in m = 1 , it can follow that Combine with I, II, IV, V , we finish the proof of Lemma 2.3.
Proof of Theorem 2.1. We first take it for granted that M ([b, I α ]f ) ∈ L q (G) and we'll check this to the end of the proof.
We proceed by induction on m. For m = 1 , by lemma 2.1, 2.2 and 2.3, we have Suppose now that for m − 1 the theorem is true, and let us prove it for m. The same argument as used above and the induction hypothesis give . By the boundedness of maximal operator, we only need to prove [b, I α ]f ∈ L q (G). Suppose for any Using the boundedness of b i , we get For the second term, according to the boundedness of b i and x / ∈ G k , y ∈ G k , |x − y| ∼ |x|, we have which is finite by lemma 2.2.
For the general case, we will truncate the symbols b j as follows. Denote b N j = min{b j , N} , take into account the fact that f has compact support, we deduce that any product b N j1 · · · b N ji f converges in any L q (G)(q > 1) to b j1 · · · b ji f as N → ∞. By the above discussion, we have By Fatou's lemma, we conclude that the theorems holds for this general case. The theorem is proved.
Furthermore, we can discuss the boundedness on Hardy spaces. and where the infimum is taken over all the decomposition of f as above.

Theorem 2.2.
Let b is the same as Theorem 2.1. Let 1/2 < p ≤ 1, Proof. We only need to prove theorem for the (p, ∞, b)-atom. Suppose supp a ⊂ G k , then Choose p 1 , q 1 such that 1 < p 1 < q 1 < ∞ and 1/q = 1/p − α , then q < q 1 . By Hölder inequality and Theorem 2.1, we have Note that x ∈ G \ G k , and y ∈ G k , let λ j = (b j ) G k for any j , then Using the vanishing moment condition (iii), we get Similar to the proof of Theorem 2.1, we deduce that Here m ≥ 1, 1/2 < p ≤ 1 and 1/q = 1/p − α .