Littlewood-Paley算子交换子在Herz型Hardy空间上的弱型估计

Weak Type Estimates for Commutators of Littlewood-Paley Operators on Herz-Type Spaces

QU Meng, SHU Li-sheng

【摘 要】Abstract: We consider the weak type endpoint estimate for the commutators generated by the Littlewood-Paley oerators and BMO functions. We prove that these commutators are bounded from the Herz-type Hardy spaces to homogeneous weak Herz-type spaces 【期刊名称】安徽师范大学学报（自然科学版） 【年(卷),期】2012(035)001 【总页数】4

【关键词】Key words: Herz-type Hardy space; commutator; Littlewood-Paley operator

【文献来源】https://www.zhangqiaokeyan.com/academic-journal-cn_journal-anhui-normal-university-natural-science_thesis/0201247714592.html

1 Introduction and the main result

Let ε>0, fix a function ψ satisfies the following properties： (b) |ψ(x)|≤C(1+|x|)-(n+ε)，

(c) |ψ(x+y)-ψ(x)|≤C|y|ε(1+|x|)-(n+1+ε) when 2|y|<|x|.

The Littlewood-Paley operator (associated with ψ) is defined by

where ψt(x)=t-nψ(x/t) for t>0. Let b∈BMO(Rn), the commutator generated by b and gψ is defined by

This commutator was introduced in [1], where the Lp boundedness of

this operator was shown.

In 2003, Liu, Lu and Xu proved that gψ,b is bounded both from to Lp(Rn) and from to Lp,∞(Rn) for where and denote and atom Hardy space and an atom weak Hardy space related to b respectively(see [4]).

On the other hand, the Herz type spaces and Herz type Hardy spaces became a very active subject in harmonic analysis, since many new properties and applications have been found in the last decade (see[7]). In 2004, Zhang [9] considered gψ,b on Herz-type Hardy spaces. She proved that gψ,b is bounded from to where 0

Received date:2011-05-27

Foundation item: Partially by National Natural Science Foundation Funds(10701010), Key University Science Research Project of Anhui Province(KJ2011A138) and Ph.D. Science Research Fund of Anhui Normal University.

Author's brief: Qu Meng(1977-), male, born in Zongyang County, Anhui Province, doctor.

Before stating our main result, we first recall some notations (see[2,5,6,7]). For k∈Z, let Bk={x∈Rn:|x|≤2k}, Ak=Bk\\Bk-1={x∈Rn:2k-1<|x|≤2k} and χk=χAk (the characteristic function of the set Ak).

Definition 1.1 Let 0

k∈Z

and

f

be

a

measurable

function

on

Rn,

let

mk(λ,f)=|{x∈Ak:|f(x)|>λ}|.

Definition 1.2 Let 0

Definition 1.3 Let 10;

2. A function a(x) on Rn is called central (α,q,b)-atom, if it satisfies (i),(ii),(iii) and (iv)

Definition 1.4 Let 0

are central (α,q)-atoms supported on B(0,2j) and

More, where the infimum take over all representations of f. Now let us state the main result of this note.

Theorem 1.1 If 0

Remark 1.1 Note that ? Comparing Theorem 1.1 with the result in [9], we know that for the endpoint weak type estimate, both the domain and the range of commutator gψ,b are replaced by larger spaces.

2 Proof of Theorem 1.1