Boundedness Characterization of Maximal Commutators on Orlicz Spaces in the Dunkl Setting

Vagif S.Guliyev,Yagub Y.Mammadov and Fatma A.Muslumova

1 Department of Mathematics,Dumlupinar University,Kutahya,Turkey

2 Institute of Mathematics and Mechanics of NASA,AZ 1141 Baku,Azerbaijan

3 Department of Informatics,Nakhchivan State University,Nakhchivan,Azerbaijan

4 Nakhchivan Teacher-Training Institute,Nakhchivan,Azerbaijan.

Abstract.On the real line,the Dunkl operators are differential-difference operators associated with the reflection group Z2 on R,and on the Rd the Dunkl operatorsare the differential-difference operators associated with the reflection group on Rd.In this paper,in the setting R we show that b ∈BMO(R,dmν)if and only if the maximal commutator Mb,ν is bounded on Orlicz spaces LΦ(R,dmν).Also in the setting Rdwe show thatif and only if the maximal commutator Mb,k is bounded on Orlicz spaces

Key words:Maximal operator,Orlicz space,Dunkl operator,commutator,BMO.

1 Introduction

Norm inequalities for several classical operators of harmonic analysis have been widely studied in the context of Orlicz spaces. It is well known that many of such operators fail to have continuity properties when they act between certain Lebesgue spaces and,in some situations,the Orlicz spaces appear as adequate substitutes.For example,the Hardy-Littlewood maximal operator is bounded onLpfor 1＜p ＜∞,but not onL1,but using Orlicz spaces,we can investigate the boundedness of the maximal operator nearp=1,see[13]and[4]for more precise statements.

LetTbe the classical singular integral operator,thecommutator[b,T]generated byTand a suitable functionbis given by

A well-known result due to Coifman,et al.[3](see also[11])states thatb∈BMO(Rn)if and only if the commutator[b,T]is bounded onLp(Rn)for 1＜p＜∞.

Maximal commutator of Hardy-Littlewood maximal operatorMwith a locally integrable functionbis defined by

where the supremum is taken over all ballsB ?Rncontainingx. We refer to[2]for a detailed investigation of the operatorsMband the commutator of the maximal operator[b,M]and references therein. For the boundedness of these operators in Orlicz spaceLΦ(Rn)see for instance[5,7].

In[9],Dunkl introduced a family of first order differential-difference operators which play the role of the usual partial differentiation for the reflection group structure.For a real parameterν ≥?1/2,we consider theDunkl operator,associated with the reflection group Z2on R:

Note that

The organization of this paper is as follows.In Sections 2 and 3,we give some preliminaries in the Dunkl setting,respectively,on R and Rd.We then present the boundedness of maximal commutators associated with Dunkl operators in Orlicz spacesLΦ(R,dmν)in Section 4 and the boundedness of maximal commutators associated with Dunkl operators in Orlicz spacesin Section 5.

Finally,we make some conventions on notation.Bywe mean thatA≤CBwith some positive constantCindependent of appropriate quantities.Ifand,we writeA≈Band say thatAandBare equivalent.

2 Preliminaries in the Dunkl setting on R

Letν＞?1/2 be a fixed number andmνbe theweighted Lebesgue measureon R given by

For anyx∈R andr＞0,letB(x,r):={y∈R:|y|∈]max{0,|x|?r},|x|+r[}.ThenB(0,r)=]?r,r[and

Themaximal operator Mνassociated with Dunkl operator on the real line is given by

and themaximal commutator Mb,νassociated with Dunkl operator on the real line and with a locally integrable functionis defined by

For a functionbdefined on R,we let,for anyx∈R,

andObviously,for anyx∈R,b+(x)?b?(x)=b(x).The following relations between[b,Mν]andMb,νare valid:

Letbbe any non-negative locally integrable function.Then

holds for all

Ifbis any locally integrable function on R,then

holds for all(see,for example,[2]).Recall also that Orlicz space was first introduced by Orlicz in[15,16]as a generalizations of Lebesgue spacesLp.Since then this space has been one of important functional frames in the mathematical analysis,and especially in real and harmonic analysis.Orlicz space is also an appropriate substitute forL1space when the spaceL1does not work.

To introduce the notion of Orlicz spaces in the Dunkl setting on R,we first recall the definition of Young functions.

Definition 2.1.A functionΦ:[0,∞)→[0,∞]is called aYoung functionifΦis convex,leftcontinuous,limr→+0Φ(r)=Φ(0)=0andlimr→∞Φ(r)=∞.

From the convexity and Φ(0)=0 it follows that any Young function is increasing.If there existss ∈(0,∞)such that Φ(s)=∞,then Φ(r)=∞for allr ≥s.The set of Young functions such that

is denoted byY.If Φ∈Y,then Φ is absolutely continuous on every closed interval in[0,∞)and bijective from[0,∞)to itself.

For a Young function Φ and 0≤s≤∞,let

If Φ∈Y,then Φ?1is the usual inverse function of Φ.It is well known that

A Young function Φ is said to satisfy the Δ2-condition,denoted also as Φ∈Δ2,if

for someC＞1.If Φ∈Δ2,then Φ∈Y.A Young function Φ is said to satisfy the?2-condition,denoted also by Φ∈?2,if

for someC＞1.In what follows,for any subsetEof R,we useχEto denote itscharacteristic function.

Definition 2.2.(Orlicz Space).For a Young functionΦ,the set

is called theOrlicz space. IfΦ(r):=rp for all r ∈[0,∞),1≤p ＜∞,then LΦ(R,dmν)=Lp(R,dmν).IfΦ(r):=0for all r ∈[0,1]andΦ(r):=∞for all r ∈(1,∞),then LΦ(R,dmν)=L∞(R,dmν). Thespace(R,dmν)is defined as the set of all functions f such that f χB ∈LΦ(R,dmν)for all balls B?R.

LΦ(R,dmν)is a Banach space with respect to the norm

For a measurable functionfon R andt＞0,let

Definition 2.3.Theweak Orlicz space

is defined by the norm

We note that

The following analogue of the H?lder inequality is well known(see,e.g.,[17]).

Theorem 2.1.Let the functions f and g be measurable onR.For a Young functionΦand its complementary functionthe following inequality is valid

By elementary calculations we have the following property.

Lemma 2.1.LetΦbe a Young function and B be a ball inR.Then

By Theorem 2.1,Lemma 2.1 and(2.2)we obtain the following estimate.

Lemma 2.2.For a Young functionΦand for the ball B the following inequality is valid:

The known boundedness statement forMνin Orlicz spaces on spaces of homogeneous type runs as follows.

Theorem 2.2.([6])LetΦbe any Young function.Then the maximal operator Mν is bounded from LΦ(R,dmν)to WLΦ(R,dmν)and forΦ∈?2bounded in LΦ(R,dmν).

3 Preliminaries in the Dunkl setting on Rd

We consider Rdwith the Euclidean scalar productand its associated norm‖x‖:=for anyx ∈Rd. For anyv ∈Rd{0}letσvbe the reflection in the hyperplaneHv ?Rdorthogonal tov:

A finite setR?Rd{0}is called aroot system,ifσvR=Rfor allv∈R.We assume that it is normalized by‖v‖2=2 for allv∈R.

Throughout this paper,we assume thatkv ≥0 for allv ∈Rand we denote byhkthe weight function on Rdgiven by

The functionhkisG-invariant and homogeneous of degreeγk,where

Closely related to them is the so-called intertwining operatorVκ(the subscript means that the operator depends on the parametersκi,except in the rank-one case where the subscript is then a single parameter). Theintertwining operator Vκis the unique linear isomorphism of⊕n≥0Pnsuch that

withPnbeing the subspace of homogeneous polynomials of degreenindvariables.The explicit formula ofVkis not known in general(see[19]). For the groupandfor allx∈Rd,it is an integral transform

LetB(x,r):={y ∈Rd:|x?y|＜r}denote the ball in Rdthat centered inx ∈Rdand having radiusr＞0.Then having

where

Sd?1is the unit sphere on Rdwith the normalized surface measuredσ.

Themaximal operator Mkassociated with the Dunkl operator on Rdis given by

and themaximal commutator Mb,kassociated with the Dunkl operator on Rdand with a locally integrable functionis defined by

In what follows,for any subsetEof Rd,we useχEto denote itscharacteristic function.Now,we introduce the notion of Orlicz spaces in the Dunkl setting on Rdas follows.

Definition 3.1.(Orlicz Space).For a Young functionΦ,the set

is called theOrlicz space.IfΦ(r):=rp for all r∈[0,∞),1≤p＜∞,thenIfΦ(r):=0for all r∈[0,1]andΦ(r):=∞for all r∈(1,∞),then

Thespacefor all balls B?Rd.

is a Banach space with respect to the norm

For a measurable functionfon Rdandt＞0,let

Definition 3.2.The weak Orlicz space

is defined by the norm

We note that

The following analogue of the H?lder inequality is well known(see,e.g.,[17]).

Theorem 3.1.Let the functions f and g be measurable onRd.For a Young functionΦand its complementary functionthe following inequality is valid

By elementary calculations we have the following property.

Lemma 3.1.LetΦbe a Young function and B be a ball inRd.Then

By Theorem 3.1,Lemma 3.1 and(2.2)we obtain the following estimate.

Lemma 3.2.For a Young functionΦand for the ball B the following inequality is valid:

The known boundedness statement forMkin Orlicz spaces on spaces of homogeneous type runs as follows.

Theorem 3.2.([6])LetΦbe any Young function.Then the maximal operator Mk is bounded from to and forΦ∈?2

4 Boundedness of maximal commutators associated with Dunkl operators in Orlicz spaces LΦ(R,dmν)

In this section,we investigate the boundedness of the maximal commutatorMb,νand the commutator of the maximal operator,[b,Mν],in Orlicz spacesLΦ(R,dmν).

We recall the definition of the spaceBMO(R,dmν).

Definition 4.1.Suppose that.Let

where,for anyx∈R andr＞0,

Define

Modulo constants,the spaceBMO(R,dmν)is a Banach space with respect to the norm

We will need the following properties ofBMO-functions(see[11]):

where 1≤p＜∞and the positive equivalence constants are independent ofb,and

where the positive constantCis independent ofb,x,randt.

Next,we recall the notion of weights.Letwbe a locally integrable and positive function on(R,dmν).The functionwis called aMuckenhoupt A1(R,dmν)weightif there exists a positive constantCsuch that for any ballB

Lemma 4.1.([6,Chapter 1])Let ω ∈A1(R,dmν).Then the reverse H?lder inequality holds,that is,there exist q＞1and a positive constant C such that

for all balls B.

Lemma 4.2.LetΦbe a Young function withΦ∈Δ2,B be a ball inRand f ∈LΦ,ν(B).Then we have

for some1＜p＜∞,where the positive constant C is independent of f and B.Proof.The left-hand side inequality is just Lemma 2.2.

Next we prove the right-hand side inequality.Our idea is from[10].TakewithNote thatsince Φ∈Δ2,thereforeMνis bounded onfrom Theorem 2.2.Letand define a function

where

For everywiththe functionRghas the following properties:

for almost everyx∈R;

·Mν(Rg)(x)≤2QRg(x)for allx∈R,that is,Rgis a MuckenhouptA1(R,dmν)weight with theA1constant less than or equal to 2Q.

By Lemma 4.1,there exist positive constantsq ＞1 andCindependent ofgsuch that for all ballsB,

By Lemma 2.2,we obtain

Thus,we have

Since the Luxembourg-Nakano norm is equivalent to the Orlicz norm we obtain

Consequently,the right-hand side inequality follows withp=q′.

We have the following result from(4.1)and Lemma 4.2.

Lemma 4.3.Let b∈BMO(R,dmν)andΦbe a Young function withΦ∈Δ2.Then

where the positive equivalence constants are independent of b.

By Theorem 2.2 and Theorem 1.13 in[2]we obtain the following theorem.

Theorem 4.1.Let b ∈BMO(R,dmν)andΦ∈?2. Then the operator Mb,ν is bounded on LΦ(R,dmν),and the inequality

holds with the positive constant C0independent of f.

The following theorem is valid.

Theorem 4.2.Let b∈BMO(R,dmν)andΦbe a Young function.Then the conditionΦ∈?2is necessary for the boundedness of Mb,ν on LΦ(R,dmν).

Proof.Assume that(4.4)holds.For the particular symbolb(·):=log|·|∈BMO(R,dmν)andf:=χB(0,r)for allr＞0,(4.4)becomes

where,B:=B(0,r),ar:=mνB(0,r),u＞0 andv＞1.By Lemma 2.1 and(2.2),we have

On the other hand,ifx/∈B(0,r)thenB(0,r)?B(x,2|x|)because for anyy∈B(0,r)we have

Also for eachy∈B(0,r),we have

Therefore

Following the ideas of[14],forwithwe obtain

Since the Luxembourg-Nakano norm is equivalent to the Orlicz norm

(more precisely,it follows that

Hence,(4.5)implies that

foru＞0 or

for everyt＞0,and so~Φ satisfies the Δ2condition.

By Theorems 4.1 and 4.2 we have the following result.

Corollary 4.1.Let b ∈BMO(R,dmν)andΦ∈Y.Then the conditionΦ∈?2is necessary and sufficient for the boundedness of Mb,ν on LΦ(R,dmν).

Theorem 4.3.Φbe a Young function.The condition b∈BMO(R,dmν)is necessary for the boundedness of Mb,ν on LΦ(R,dmν).

Proof.Suppose thatMb,νis bounded fromLΦ(R,dmν)toLΦ(R,dmν).Choose any ballBin R;by(2.2),we have

Thus,b∈BMO(R,dmν).

By Theorems 4.1 and 4.3 we have the following result.

Corollary 4.2.LetΦbe a Young function withΦ∈?2.Then the condition b ∈BMO(R,dmν)is necessary and sufficient for the boundedness of Mb,ν on LΦ(R,dmν).

From(2.1)and Corollary4we deduce the following conclusion.

Corollary 4.3.LetΦbe a Young function withΦ∈?2.Then the conditions b+∈BMO(R,dmν)and b?∈L∞(R,dmν)are sufficient for the boundedness of[b,Mν]on LΦ(R,dmν).

5 Boundedness of maximal commutators associated with Dunkl operators in Orlicz spaces

In this section,we investigate the boundedness of the maximal commutatorMb,kand the commutator of the maximal operator,[b,Mk],in Orlicz spacesIndeed,these results and their proofs are similar to those presented in Section 4 with slight modifications.For the convenience of the reader,we give the details.

We recall the definition of the space

Definition 5.1.

where,for any x∈Rd and r＞0,

Define

Modulo constants,the spaceis a Banach space with respect to the norm‖·‖BMO(k).

We will need the following properties ofBMO-functions(see[11]):

where 1≤p＜∞and the positive equivalence constants are independent ofb,and

where the positive constantCis independent ofb,x,randt.

Next,we recall the notion of weights.Letwbe a locally integrable and positive function onThe functionwis called aif there exists a positive constantCsuch that for any ballB

Lemma 5.1.([6,Chapter 1])1and a positive constant C such that

for all balls B.

Lemma 5.2.LetΦbe a Young function withΦ∈Δ2,B be a ball inRd and.Then we have

for some1＜p＜∞,where the positive constant C is independent of f and B.

Proof.The left-hand side inequality is just Lemma 3.2.

Next we prove the right-hand side inequality.We use some ideas from[10].TakewithNote thatsince Φ∈Δ2,thereforeMkis bounded onfrom Theorem 3.2.Letand define a function

where

For everywiththe functionRghas the following properties:

for almost everyx∈Rd;

for allx ∈Rd,that is,Rgis a Muckenhoupt

weight with theA1constant less than or equal to 2Q.

By Lemma 4.1,there exist positive constantsq ＞1 andCindependent ofgsuch that for all ballsB,

By Lemma 3.2,we obtain

Thus,we have

Since the Luxembourg-Nakano norm is equivalent to the Orlicz norm we obtain

Consequently,the right-hand side inequality follows withp=q.

We have the following result from(5.1)and Lemma 4.2.

Lemma 5.3.Φbe a Young function withΦ∈Δ2.Then

where the positive equivalence constants are independent of b.

By Theorem 3.2 and Theorem 1.13 in[2]we get the following theorem.

Theorem 5.1.Φ∈?2.Then the operator Mb,k is bounded onand the inequality

holds with the positive constant C0independent of f.

The following theorem is valid.

Theorem 5.2.Φbe a Young function.Then the conditionΦ∈?2is necessary for the boundedness of Mb,k on

Proof.Assume that(5.4)holds.For the particular symbolandf:=χB(0,r)for allr＞0,(5.4)becomes

whereandv＞1.By Lemma 3.1 and(2.2),we have

On the other hand,ifx/∈B(0,r)thenB(0,r)?B(x,2|x|)because for anyy∈B(0,r)we have

Also for eachy∈B(0,r),we have

Therefore,

Following the ideas of[14],forwithwe obtain

Since the Luxembourg-Nakano norm is equivalent to the Orlicz norm

(more precisely,it follows that

Hence,(5.5)implies that

foru＞0 or

for everyt＞0,and sosatisfies the Δ2condition.

By Theorems 5.1 and 5.2 we have the following result.

Corollary 5.1.Φ∈Y.Then the conditionΦ∈?2is necessary and sufficient for the boundedness of Mb,k on

Proof.Suppose thatMb,kis bounded fromtoChoosing any ballBin Rd,by(2.2),we obtain

Thus,

By Theorems 5.1 and 5.3 we have the following result.

Corollary 5.2.LetΦbe a Young function withΦ∈?2. Then the condition b ∈BMO(Rd,,k on

From(2.1)and Corollary 5 we deduce the following conclusion.

Corollary 5.3.LetΦbe a Young function withΦ∈?2.Then the conditions b+∈BMO(Rd,[b,Mk]on LΦ(Rd,

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions.The work of the first named author is partially supported by the grant of 1st Azerbaijan-Russia Joint Grant Competition(Agreement number no. EIFBGM-4-RFTF-1/2017-21/01/1).