Ronghui LIU(劉榮輝) Feng LIU(劉鳳) Huoxiong WU(伍火熊),?

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

2.College of Mathematics and System Science,Shandong University of Science and Technology,Qingdao 266590,China E-mail:liuronghuimath@126.com;FLiu@sdust.edu.cn;huoxwu@xmu.edu.cn

Abstract This paper studies the mixed radial-angular integrability of parametric Marcinkiewicz integrals along“polynomial curves”.Under the assumption that the kernels satisfy certain rather weak size conditions on the unit sphere with radial roughness,the authors prove that such operators are bounded on the mixed radial-angular spaces.Meanwhile,corresponding vector-valued versions are also obtained.

Key words Marcinkiewicz integrals;rough kernels;mixed radial-angular spaces

1 Introduction

Let Sbe the unit sphere in R,n≥2,with normalized Lebesgue measure dσ=dσ(·).For any f∈L(R),0

Therefore,from the perspective of radial and angular integrability,Lebesgue norms can be interpreted as certain special norms with the same integrability in the radial and angular directions.Inspired by this version of things,we naturally consider the case of Lebesgue norms with different integrability in the radial and angular directions,namely,

It is worth pointing out that the spaces G(S)were introduced by Grafakos and Stefanov[14]in studying the L-boundedness of singular integrals with rough kernels,and it was shown in [14]that

On the other hand,as a special case of the Littlewood-Paley-Stein functons,Marcinkiewicz integrals play important roles in harmonic analysis.We refer the readers to [1,2,5,8,9,16,19,23,26–29]etc.for their development and applications.In this paper,we will focus on the following parametric Marcinkiewicz integrals along polynomial curves:

where P(t) is a real polynomial on R of degree N satisfying P(0)=0,? enjoys (1.2) and ρ is for a complex number with ρ=τ +i?(τ,? ∈R,τ >0),h ∈△(R) for 1 ≤γ ≤∞,where△(R) is the set of all measurable functions h defined on Rsatisfying

In light of the aforementioned facts concerning the Marcinkiewicz integral operators,two questions that arise naturally are:

In this paper,we shall give affirmative answers to the above questions.Our results can be formulated as follows:

Theorem 1.1

Let P(t) be a real polynomial on R of degree N satisfying P(0)=0.Suppose that ? satisfies(1.2)and ? ∈G(S)for some β >1.Then the following inequalities hold:

Theorem 1.2

Let P(t) be a real polynomial on R of degree N satisfying P(0)=0.Suppose that ? satisfies (1.2) and ? ∈WG(S) for some β >1,h ∈?(R) for some γ ∈(1,∞].Then the following inequalities hold:

Corollary 1.3

Let P(t)be a real polynomial on R of degree N and let it satisfy P(0)=0.Suppose that ? satisfies(1.2)and ? ∈∩G(S).Then the inequalities(1.4)–(1.6)hold provided that one of the following conditions holds:

Remark 1.4

It should be pointed out that the inclusion relation between WG(S)and G(S) is not clear for n >2 at present.Even so,the results of Theorem 1.2 are also a meaningful extension to those of Theorem 1.1 because of (1.3),even for h ≡1.

for 2 ≤p<2β/s,provided that ? ∈G(S) for any s>β,or for β

2.These results partly recover the corresponding results in [5].

The rest of this paper is organized as follows:in Section 2,we will establish a key lemma,which contains a general criterion on the weighted inequality of the convolution operators.By virtue of the general criterion,we only need to check the corresponding conditions to derive the weighted inequality of Marcinkiewicz integrals.Therefore,the boundedness of the parametric Marcinkiewicz integral operators on the mixed radial-angular spaces,vector-valued mixed radial-angular spaces will automatically hold by [20,Proposition 1].The proofs of theorems will be given in Section 3.Finally,as applications of our main results,we obtain the mixed radial-angular integrability for the corresponding parametric Marcinkiewicz integral operators related to the area integrals and the Littlewood-Paley gfunction in Section 4.We would like to remark that some ideas in the proofs of our main results are taken from [10,14,15,18,21].

2 Main Lemma and Key Proposition

In this section,we will establish a general criterion on the weighted boundedness of the convolution operators,which is a key to our proofs.On the basis of the general criterion,then,a key proposition related to Marcinkiewicz integral operators is also obtained accordingly.

By means of Lemma 2.1 we can prove the following weighted norm inequality on Marcinkiewicz integral operators,which plays a significant role in the proof of Theorem 1.2:

Proposition 2.2

Let P(t) be a real polynomial on R of degree N satisfying P(0)=0.Suppose that ? satisfies (1.2) and ? ∈WG(S) for some β >1,h ∈?(R) for some γ ∈(1,∞].Then the following inequalities hold:

To obtain the proofs of our theorems,we also need the following criterion about the boundedness of the operators on the mixed radial-angular spaces:

for all s>sand any nonnegative measurable function u on R,where the operator Mmaps any radial function to radial function,and satisfies

3 Proofs of Theorems

This section is devoted to the proofs of our main theorems.In fact,we only prove Theorem 1.2 because the conclusions of Theorems 1.1 can be similarly obtained by a slight modification.More precisely,suppose that ? ∈G(S) for some β >1,h ≡1 and make the following substitution in Theorem 1.2 :β →2β,this reduces it to Theorem 1.1.Therefore,we focus on the proof of Theorem 1.2.

First,we consider the condition (i).By (2.19),we have

4 Additional Results