The boundedness of commutators on locally compact Vilenkin groups

Let G be a locally compact Vilenkin group. In this paper, the authors investigate the boundedness of commutators of singluar integral operator on Triebel-Lizorkin spaces on G. Furthermore, the boundedness on the Herz-type Triebel-Lizorkin spaces are also studied.


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], [7], [12]) is that the commutators [b, T ] of singular integral operators are bounded on some L p (R n )(1 < p < ∞) if and only if b ∈ BM O, where [b, T ] is defined by [b, T ]f (x) = b(x)T f(x) − T (bf )(x).Later, Janson in [6] gave that [b, T ] is bounded from L p (R n ) to L q (R n ) when 1 < p < q < ∞ if and only if b ∈ Lip β and β = n 1 p − 1 q .In 1995, M. Paluszycński extended and generalized their results (see [10]).He proved that [b, T ] is bounded from L p (R n ) to Ḟ β,∞ p (R n ) is equivalent to b ∈ Lip β ,where 1 < p < ∞, 0 < β < 1. Motivated by their works, we consider the cases on Vilenkin groups.Moreover, Xu and Yang ( [13]) introduced Herz-type Triebel-Lizorkin spaces on R n .In this paper, we continue studying the properties of commutator on Herz-type Triebel-Lizorkin spaces on Vilenkin groups.
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 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. Set ϕ n = m n χ Gn − m n+1 χ Gn+1 , where χ Gn is the characteristic function of G n .Definition 1.1.Let 0 ≤ α < ∞, 0 < p, q < ∞, the homogeneous Herz spaces Kα,p q (G) are defined by Kα,p q (G) = {f : f is a measurable function on G with f Kα,p q (G) < ∞}, where Before defining the Herz-type Triebel-Lizorkin spaces on G, we give the notes of a second space of test functions and distributions.We refer to [4] for details.Let and define the convergence in Z(G) to be like in S(G).Let Z (G) be the space of linear functionals on Z(G) with convergence in Z (G) defined as in S (G) and denotes the set of constant distributions in S (G).
with the usual modification if q = ∞.
Here supp b denote the support of function b(x).
By the conclusion in [9], the functions f in Kα,p q (G) have the following characterization.

Proposition 1.1 ([9]). Let
with the infimum taken over all decompositions of f as above.
In order to study the boundedness of commutators, we also need to define Lipschitz spaces.For x 0 ∈ G, set , with the f Ij being the average of f over I j and the usual modification in the case of q = ∞.
Our main results are the following.
taking the supremum over all I j such that x ∈ I j , and then consider the Kα,p q norm of both sides, it is obviously to see that Theorem 1.2 is a consequence of the following theorem.
In section 2, we'll give the proof of Theorem 1.1.And Theorem 1.3 will be proved in section 3.

Boundedness of the commutator on the Triebel-Lizorkin spaces
Let T be a singular integral operator, i. e.
Before we start to show our theorems, we first give some equivalent norms of Triebel-Lizorkin spaces.In [1], A. Seeger gave the characterization of Triebel-Lizorkin spaces Ḟ p,q α on R n via means of maximal functions, which is introduced by Devore and Sharpley in [5] and Christ in [2].Paralleled to the situation in [1], using the maximal operator, we can obtain the following lemma. Write and Proof.According to the definition, the average over I j is dominated as Replacing x + h by y, A is easily controlled by To deal with the second term, using the differential theorem of integral, we have Since 0 < β < 1, k ≤ j and the domination of (2.1), we have where M is maximal operator.Consequently, Now we start to prove Theorem 1.1.
Proof of Theorem 1.1.
We give the estimations for L 1 , L 2 and L 3 respectively.By the definition of the Lipschitz space, the L 1 is easily to be dominated by the maximal operator, that is To estimate L 2 , choosing r > 1, we have Consequently, Then, using Hölder inequality and L r -boundedness of singular operator, we get Now we turn to deal with L 3 .By the hypothesis on the kernel K, we have It is easy to see where |b I k − b Ij |'s have the following estimates Taking account of all the estimates for L 1 , L 2 and L 3 together, we obtain Taking the supremum over all I j such that x ∈ I j , and the L p -norm on both sides, and using Lemma 2.1 we conclude that Thus, Theorem 1.1 is proved.

Boundedness of some sublinear operators on Herz space
In this section, we will consider the boundedness of some sublinear operators on Herz space.
Proof of Theorem 1.3.We first prove the case when r = 1.Let f ∈ Kα,p q (G).Then by Proposition 1.1, we have where , and b j 's are (α, q)-block supported in G j , j ∈ Z.Let χ n = χ Gn\Gn+1 .Since T is a sublinear operator, we have In the case of 0 p ≤ 1, using L q -boundedness of T , we get In the case of 1 < p ≤ ∞, by Hölder's inequality and the L q boundedness of T , we obtain where p is the conjugate index of p. Now we turn to estimate L 2 .For j Similar to the estimation of L 1 , we also consider two cases.
Case A: 0 < p ≤ 1.In this case, Using the Hölder's inequality, Using all the estimations above, we obtain When r > 1, similar to the proof of r = 1, we can decompose T f Kα,p q into two parts which are dominated by (L 1 + L 2 ) 1/p .The estimate for L 1 is easily obtained as in the proof of r = 1.We only consider to estimate L 2 now.By the hypothesis (1.1), we have So, if 0 < p ≤ 1, then Combining the estimations of L 1 with L 2 , we can deduce that This finishes the proof of Theorem 1.3.