The boundedness of commutators on locally compact Vilenkin groups

c 2005, Scientific Horizon 

JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 3, Number 2 (2005), 209-222

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The boundedness of commutators on locally compact Vilenkin groups Canqin Tang, Qingguo Li and Bolin Ma

∗

(Communicated by Hans Triebel ) 2000 Mathematics Subject Classification. 43A70, 43A75. Keywords and phrases. Vilenkin group.

Commutator, Herz-type Triebel-Lizorkin space,

Abstract. 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.

1. 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 Lp (Rn )(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] n q n gave that [b, T ] is bounded from Lp (R  ) to L (R ) when 1 < p < q < ∞ 1 1 if and only if b ∈ Lip β and β = n − . In 1995, M. Paluszyc´ nski p q ∗

supported by NSF Grand 10371004

210 Boundedness of commutators on locally compact Vilenkin groups extended and generalized their results (see [10]). He proved that [b, T ] is bounded from Lp (Rn ) to F˙pβ,∞ (Rn ) 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 Rn . 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 {Gn }∞ n=−∞ such that (a) ∞ G = G and ∩ G = 0; (b) sup{order(G /Gn+1 ) : n ∈ Z} = ∪∞ n n n n=−∞ n=−∞ B < ∞. Choose Haar measure dx on G so that |G0 | = 1, where |A| denotes the measure of a measurable subset A of G. Let |Gn | = (mn )−1 for each n ∈ Z. Since 2mn ≤ mn+1 ≤ Bmn for each n ∈ Z, it follows that ∞  (mn )−α ≤ c(mk )−α n=k

and

k 

(mn )α ≤ c(mk )α

n=−∞

for any α > 0, k ∈ Z, where c is a constant independent of k. For each n ∈ Z we choose elements zl,n ∈ G(l ∈ Z+ ) so that the subsets Gl,n := zl,n + Gn of G satisfy Gk,n ∩ Gl,n = φ if k = l and ∪∞ l=0 Gl,n = G; moreover, we choose z0,n such that G0,n = Gn . We now define the function d : G × G → R by d(x, y) = 0 if x − y = 0 and d(x, y) = (mn )−1 if x − y ∈ Gn \Gn+1 . 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| = (mn )−1 if and only if x ∈ Gn \Gn+1 . Let S(G) be the space of test functions and S  (G) be the distribution space on G. Set ϕn = mn χGn − mn+1 χGn+1 , where χGn is the characteristic function of Gn . Definition 1.1. Let 0 ≤ α < ∞, 0 < p, q < ∞, the homogeneous Herz spaces K˙ qα,p (G) are defined by K˙ qα,p (G) = {f : f is a measurable function on G with f K˙ qα,p(G) < ∞}, where ⎫1/p ⎧ ∞ ⎬ ⎨  f K˙ qα,p(G) = m−αp f χGl \Gl+1 pLq (G) . l ⎭ ⎩ j=−∞

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]

C. Tang, Q. Li and B. Ma

211

for details. Let



ˆ Z(G) = ψ ∈ S(G) : ψ(0) = ψ(t)dt = 0 , G

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). Definition 1.2. Let α ∈ R, 0 < p, q ≤ ∞. Then ⎧  ⎫ 1/q     ∞ ⎨ ⎬   αq q  1, similar to the proof of r = 1, we can decompose T f K˙ qα,p into two parts which are dominated by (L1 + L2 )1/p . The estimate for L1 is easily obtained as in the proof of r = 1. We only consider to estimate L2 now. By the hypothesis (1.1), we have ⎛  qr ⎞1/q  r |b (y)| j dy dx⎠ (T bj )χn q ≤ C ⎝ Gn \Gn+1 Gj |x − y| 1/r  1

≤ Cmnr 1

≤ Cmnr So, if 0 < p ≤ 1, then ∞ 

L2 ≤ C

n=−∞ ∞ 

≤C ≤C

j=−∞ ∞ 

− q1

Gj − q1

α+ q1 − r1

mj

⎛ ⎝ m−αp n  |λj |p

|bj (y)|r dy

∞ 

.

|λj |p



j=n+1

mn mj

( r1 − 1q )p

⎞ ⎠ mαp j

( 1 − 1 −α)p  j−1   mn r q mj n=−∞

|λj |p < ∞ ,

j=−∞

1 1 − . r q If 1 < p ≤ ∞, then

where 0 < α