Xianming HOUHuoxiong WU

School of Mathematical Sciences,Xiamen University,Xiamen 361005,China

E-mail:houxianming37@163.com;huoxwu@xmu.edu.cn

Abstract In this paper,we establish the following limiting weak-type behaviors of Littlewood-Paley g-function g?:for nonnegative function f ∈ L1(Rn),

Key words limiting behaviors;weak-type bounds;Littlewood-Paley g-functions;Marcinkiewicz integrals;Dini conditions

1 Introduction

It is well known that the Littlewood-Paley operators,such as the Littlewood-Paley gfunction,the area integral,-function and the Marcinkiewicz integral,play very important roles in harmonic analysis,complex analysis and PDEs.Readers may consult[1,2,5,7,9,13–23]and among a large number of references for their recent development and applications.

Recently,the limiting weak-type behaviors for singular or fractional integrals and maximal operators have been established,which are closely related to the best constants of the weak-type endpoint bounds for such operators,see[3,4,8,10,11]et al.This article aims to explore the limiting weak-type behaviors of the Littlewood-Paley funtions,including the Littlewood-Paley g-functions and Marcinkiewicz integrals.Before stating our main results,we fi rst recall a few de fi nitions,notations and relevant results.

Let ?(x) ∈ L1(Rn)satisfy Z

Then the generalized Littlewood-Paley g-function g?is de fi ned by

where ?r(x)=r?n?(x/r).Suppose that ?(x) ∈ L1(Rn)satis fi es(1.1)and the following conditions

and

where C and σ>0 are both independent of x and h.It follows from[12,Theorem 5.1.2]that g?is of type(p,p)(1

On the other hand,suppose that Sn?1denotes the units sphere on the Euclidean space Rnand 0≤α

De fi ne the Marcinkiewicz integral and its fractional version μ?,αwith homogeneous kernel ?as follows

where ? satis fi es the vanishing condition

when α =0 and we denoteμ?,0= μ?.Meanwhile,the singular and fractional integral operator T?,αde fi ned by

Similarly,we denote T?,0by T?.

For 0< α

n?α(Sn?1),note that T?,αis of weak type(1,n/(n ? α))and|μ?,α(f)(x)| ≤ T|?|,α(|f|)(x),we know that μ?,αis of weak type(1,n/(n ? α)).For α =0,it follows from[6,Theorem 1]thatμ?is bounded from L1(Rn)to L1,∞(Rn)provided? ∈ LlogL(Sn?1).

For any ξ∈ Sn?1,0< δ

Janakiraman[11]established the following limiting result

which gives a lower bound for the weak type constant of T?.Corresponding conclusions for Hardy-Littlewood maximal function was also established in[11].Recently,Ding and Lai[3,4]extended the above results to T?,αand the maximal operators M?,αwith homogeneous kernels? satisfying the following-Dini condition(which is weaker than(1.7),see[3])

Inspired by the above results,the main purpose of this paper is to study the limiting weak-type behaviors of g?and μ?,α.Our arguments can deal with the following more general cases.

Let V be an absolutely continuous measure with respect to Lebesgue measure on Rnwith V(Rn)<∞,and Vtdenote the dilation of V:Vt(E)=V(E/t)for t>0.De fi ne

This paper is organized as follows.In Section 2,we give the proof of Theorem 1.1 and a further remark.The proof of Theorem 1.3 will be given in Section 3.

Throughout this paper,the letter C,sometimes with additional parameters,will stand for positive constants,not necessarily the same one at each occurrence,but independent of the essential variables.

2 Proof of Theorem 1.1

This section is devoted to proving Theorem 1.1.To do this,we need to fi rst establish the following several auxiliary lemmas.

Note that for|x|>?εt,|y|<εtand 0<ε<1,we have

Then

And in view of(1.3)and(1.4),we have

where we make the change of variable u:=x/r in the second inequality and the fourth inequality follows from the fact|u?y/r|>|u|/(1+3√t).Combining the above estimates of II1and II2,

we get

This combining(2.2)and(2.6)leads to

Letting t→ 0+,and then letting ε→ 0+,we get

Proof of Theorem 1.1It is not difficult to see that

where in the second equality we make the change of variable r′:=r/t.Hence,

It follows from Lemma 2.2 that

This fi nishes the proof of Theorem 1.1.?

Remark 2.3We remark that Φ(x):=is a homogeneous function of degree?n,and conclusion(2)in Theorem 1.3 is stronger than conclusion(1).

Indeed,supposed that conclusion(2)in Theorem 1.1 holds.Letbe as before,and assume V(Rn)=1.It is easy to check that

which implies that conclusion(1)in Theorem 1.1 holds.

3 Proof of Theorem 1.3

In this section,we will give the proof of Theorem 1.3.At fi rst,we recall and establish two auxiliary lemmas.

Lemma 3.1(see[3]) Let 0≤ α 0,

Lemma 3.2Let V be an absolutely continuous measure with respect to Lebesgue measure on Rnand V(Rn)<∞.Let 0≤α

ProofThe proof of Lemma 3.2 is similar to that of Lemma 2.2.Employing the notations εt,?εt,ε,dV1t(x),dV2t(x)as in the proof of Lemma 2.2,for any given λ>0,set

This together with Minkowski’s inequality and the mean value theorem implies