High School for Gifted Students,Hanoi National University of Education, 136 Xuan Thuy,Hanoi,Vietnam

E-mail:hunghaduy@gmail.com

Luong Dang KY?

Applied Analysis Research Group,Faculty of Mathematics and Statistics, Ton Duc Thang University,Ho Chi Minh City,Vietnam

E-mail:luongdangky@tdt.edu.vn

NEW WEIGHTED MULTILINEAR OPERATORS AND COMMUTATORS OF HARDY-CES`ARO TYPE?

Ha Duy HUNG

High School for Gifted Students,Hanoi National University of Education, 136 Xuan Thuy,Hanoi,Vietnam

E-mail:hunghaduy@gmail.com

Luong Dang KY?

Applied Analysis Research Group,Faculty of Mathematics and Statistics, Ton Duc Thang University,Ho Chi Minh City,Vietnam

E-mail:luongdangky@tdt.edu.vn

This paper deals with a general class of weighted multilinear Hardy-Ces`aro operators that acts on the product of Lebesgue spaces and central Morrey spaces.Their sharp bounds are also obtained.In addition,we obtain sufcient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Ces`aro operators(with symbols in central BMO spaces)are bounded on the product of central Morrey spaces.These results extends known results on multilinear Hardy operators.

Hardy-Ces`aro operators;Hardy’s inequality;multilinear Hardy operator;weig

hted Hardy-Littlewood averages

2010 MR Subject Classifcation42B35(46E30;42B15;42B30)

1 Introduction

The Hardy integral inequality and its variants play an important role in various branches of analysis such as approximation theory,diferential equations,theory of function spaces etc (see[4,10,18,20,29,30]and references therein).The classical Hardy operator,its variants and extensions were appeared in various papers(refer to[4,5,13-15,20,21,29-31]for surveys and historical details about these difererent aspects of the subject).On the other hand,the study of multilinear operators is not motivated by a mere quest to generalize the theory of linear operators but rather by their natural appearance in analysis.Coifman and Meyer in their pioneer work in the 1970s were one of the frst to adopt a multilinear point of view in their study of certain singular integral operators,such as the Calder′on commutators,paraproducts, and pseudodiferential operators.

Let ψ:[0,1]→[0,∞)be a measurable function.The weighted Hardy operator Uψis defned on all complex-valued measurable functions f on Rdas

Theorem 1.1(see[31])Let 1＜p＜∞and ψ:[0,1]→[0,∞)be a measurable function. Then,Uψis bounded on Lp(Rd)if and only if

Furthermore,

Theorem 1.1 implies immediately the following celebrated integral inequality,due to Hardy [20]

For further applications of Theorem 1.1,for examples the sharp bounds of classical Riemann-Liouville integral operator,see[13,14].Recently,Chuong and Hung[7]introduced a more general form of Uψo(hù)perator as follows.

Here ψ:[0,1]→[0,∞),s:[0,1]→R are measurable functions and f is a measurable complex valued function on Rd.The authors in[7]obtained sharp bounds of Uψ,son weighted Lebesgue and BMO spaces,where weights are of homogeneous type.They also obtained a characterization on weight functions so that the commutator of Uψ,s,with symbols in BMO,is bounded on Lebesgue spaces.Moreover,the authors in[7]proved the boundedness on Lebesgue spaces of the following operator

Very recently,the multilinear version of Uψo(hù)perators has been introduced by Fu,Gong, Lu and Yuan[14].They defned the weighted multilinear Hardy operator as

As showed in[14],when d=1 and

where α∈(0;m),then

Motivated from[7,13,14,17,21],this paper aims to study the boundedness of a more general multilinear operator of Hardy type as follows.

A mutilinear version of Hψ,scan be defned as

where R+is the set of all positive real numbers.

2 Notations and Defnitions

The weighted BMO space BMO(ω)is defned as the set of all functions f,which are of bounded mean oscillation with weight ω,that is,

The case ω≡1 of(2.2)corresponds to the class of functions of bounded mean oscillation of John and Nirenberg[22].We obverse that L∞(Rd)?BMO(ω).

Next we recall the defnition of Morrey spaces.It is well-known that Morrey spaces are useful to study the local behavior of solutions to second-order elliptic partial diferential equations and the boundedness of Hardy-Littlewood maximal operator,the fractional integral operators, singular integral operators(see[1,6,24]).We notice that the weighted Morrey spaces were frst introduced by Komori and Shirai[24],where they used them to study the boundedness of some important classical operators in harmonic analysis like as Hardy-Littlewood maximal operator,Calder′on-Zygmund operators.

The spaces of bounded central mean oscillation CMOq?Rd?appear naturally when considering the dual spaces of the homogeneous Herz type Hardy spaces and were introduced by Lu and Yang(see[27]).The relationships between central BMO spaces and Morrey spaces were studied by Alvarez,Guzm′an-Partida and Lakey[2].Furthermore,they introduced λ-central BMO spaces and central Morrey spaces as follows.

Defnition 2.4Let α be a real number.Let Wαbe the set of all functions ω on Rd, which are measurable,ω(x)＞0 for almost everywhere x∈Rd,0＜RSdω(y)dσ(y)＜∞,and are absolutely homogeneous of degree α,that is ω(tx)=|t|αω(x),for all t∈R{0},x∈Rd.

3 S Sh p aa rc p es Boundedness ofon the Product of Weighted Lebesgue

In this section,we prove sharp bound for Hardy-Ces`arooperators on the product of weighted Lebesgue spaces and weighted central Morrey spaces under certain conditions of the associated weights.Before stating our results we make some conventions on the notation.

For m exponents 1≤pj＜∞,j=1,···,m and α1,···,αm＞-d we will often write p for the number given by

and

If ωk∈Wαk,k=1,···,m,we will set

It is obvious that ω∈Wα.We say that(ω1,···,ωm)satisfes the W-→αcondition if

Notice that the weights ω as defned in(3.2)were used in[19]to obtain multilinear extrapolation results.On the other hand,Condition(3.3)holds for power weights.In fact,(3.3) becomes to equality for ωk(x)=|x|αk,k=1,···,m.

First we will show the following result which can be viewed as an extension of Theorem 1.1 to the multilinear case.

Furthermore,

In order to prove the converse of the theorem,we frst need the following lemma.

Lemma 3.2Let w∈Wα,α＞-d and ε＞0.Then the function

for each k=1,···,m.For each x∈Rdwhich|x|≥1,let

This implies that

Notice that

Thus,letting ε→0+and by Lebesgue’s dominated convergence theorem,we obtain

Combine(3.6)and(3.7),we obtain the result. ?

Analogous to the proof of Theorem 3.1,one can prove the following result.

(i)If in addition

and

(ii)Conversely,if

Let ωk≡1,we have that αk=0 thus(3.10)becomes equality and(3.12)holds only when λ1p1=···=λmpm.Hence,we obtain Theorem 2.1 in[14].

This means that

Thus,

This leads us to

Therefore,

(i)If in addition

and

(ii)Conversely,if

4 Commutators of Weighted Multilinear Hardy-Ces`aro Operator

We use some analogous tools to study a second set of problems related now to multilinear versions of the commutators of Coifman,Rochberg and Weiss[9].The boundedness of commutators of weighted Hardy operators on the Lebesgue spaces,with symbols in BMO space, have been studied in[13]by Fu,Liu and Lu.Their main idea is to control the commutators of weighted Hardy operators by the weighted Hardy-Littlewood maximal operators.However this method is not easy to extend to deal with the multilinear case,since the maximal operators which are suitable to control commutator in this case are less known.In[14],the authors give an idea to replace Lebesgue spaces with central Morrey spaces and symbols are taken in central BMO spaces.By this method,we can avoid to use multilinear maximal operators.

In what follows,we set

We shall consider weight functions ωk∈Wαk,k=1,···,m,such that

We note here that D is fnite is not enough to imply C is fnite(see[13,14]).But it is easy to see that,if we assume in addition that for each k=1,···,m such that|sk(t)|≥c＞1 for all t∈[0,1]nor|sk(t)|≤c＜1 for all t∈[0,1]n,then C is fnite if and only if D is fnite.Thus, Theorem 4.2 implies immediately that

Now we will give the proof of Theorem 4.2.

For any xi,yi,zi,ti∈C with i=1,2,we have the following elementary inequality

Similarly to the estimate of I1,we have that

Now we give the estimate for I3.Applying H¨older’s inequality,we get that

Notice that[0,1]nis the union of pairwise disjoint subsets S?1,?2,where

Thus we obtain that

Then Minkowski’s inequality and H¨olders inequality show us that

From the estimates of I1and I3,we deduce that

It can be deduced from the estimates of I1,I2,I3,I4that

Combining the estimates of I1,I2,I3,I4,I5and I6gives

This proves(i).

Lemma 4.4([7,Lemma 2.3])If ω∈Wαand has doubling property,then log|x|∈BMO(ω).

Also we have

Let B=B(0,R)be any ball of Rd,then

Taking the supremum over R＞0,we get

which ends the proof of Theorem 4.2.

AcknowledgementsThe authors would like to thank Prof.Nguyen Minh Chuong for many helpful suggestions and discussions.

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?Received August 7,2014;revised April 13,2015.The frst author is supported by Vietnam National Foundation for Science and Technology Development(101.02-2014.51).

?Corresponding author:Luong Dang KY.