# Boundedness of Multidimensional Dunkl-Hausdorff Operators

Authors: Radouan Daher; Faouaz Saadi
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In the present paper, we introduce the multidimensional Dunkl-Hausdorff operatorℋκ and we give simple sufficient conditions so that these operators be bounded on the weighted lebesgue spaces Lκpℝn and in the Hardy space Hκ1ℝn associated with the Dunkl operators. We also determine the Dunkl-Hausdorff operator ℋκ∗ that is adjoint to ℋκ.


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## Body

## 1. Introduction

The Hausdorff operator is one of the most important operators in harmonic analysis, and it is used to solve certain classical problems in analysis. In the one-dimensional case, Hausdorff operators on the real line were introduced in [1] and studied on the Hardy space in [2]. The natural generalization in several dimensions was introduced and studied in [3–5]. The reader can see a recent survey article [6] by Liflyand which contains the main results on Hausdorff operators in various settings and bibliography until 2013.Dunkl theory generalizes classical Fourier analysis onℝn. It started twenty years ago with Dunkl’s seminal work in [7] and was further developed by several mathematicians. In the frame of extending these results to the context of Dunkl theory, we have introduced and studied recently in [8] the Dunkl-Hausdorff operators in the one-dimensional case on the spaces Lκpℝ and Hκ1ℝ. Now, we are going to study the multidimensional Dunkl-Hausdorff operators in the spirit of those in [3, 5]. We can only establish a complete result for the finite reflection group G≃ℤ2n with the associated measure μκ given for every x=x1,…,xn∈ℝn by(1)dμκx=hκ2xdx,with hκ the ℤ2n-invariant function is defined by(2)hκx=∏j=1nxjκj=∏j=1nhκjxj,where κ1,…,κn are nonnegative real numbers.The paper is organised as follows. In Section2, we collect some definitions and results related to Dunkl’s analysis. In particular, we list the properties of the Dunkl transform and Hardy space associated with the Dunkl operators that will be relevant for the sequel. Section 3 is devoted to proving our main results.


## 2. Preliminaries

This section is devoted to the preliminaries and background. For further survey about this theory, the reader may refer to [9–11].


### 2.1. Notation

We denote byLκpℝn the space Lpℝn;dμκ and we use the shorter notation ⋅κ,p instead of ⋅Lκpℝn. For p∈1,+∞, the space Lκpℝn is of course the space of measurable functions on ℝn such that(3)fκ,p=∫ℝnfypdμκy1/p<+∞,if1≤p<+∞,fκ,∞=esssupy∈ℝnfy<+∞,otherwise.


### 2.2. Dunkl Transform

Lete1,…,en be the standard basis of ℝn. We denote by σj (for each j, from 1 to n) the reflection with respect to the hyperplane perpendicular to ej; that is, to say, for every x=x1,…,xn∈ℝn(4)σjx=x−2x,ejej2ej=x1,…,xj−1,−xj,xj+1,…,xn.Of course⋅,⋅ is the usual inner product on ℝn×ℝn and ⋅ is the associated norm. Let G be the finite reflection group generated by σj:j=1,…,n; so, G is isomorphic to ℤ2n. Let κ1,κ2,…,κn be nonnegative real numbers.Associated with these objects are the Dunkl operatorsDk (for k=1,…,n) that have been introduced in [7] by Dunkl. They are given for x∈ℝn by(5)Dkfx=∂kfx+∑j=1nκjfx−fσjxx,ejek,ej=∂kfx+κkfx−fσkxxk,where ∂k denotes the usual partial derivative. A fundamental property of these differential-difference operators is their commutativity, that is, to say, DkDl=DlDk.Closely related to them is the so-called intertwining operatorVκ (the subscript means that the operator depends on the parameter κj, except in the rank-one case, where the subscript is then a single parameter), which is the unique linear isomorphism of ⊕m≥0Pm such that(6)VκPm=Pm,Vκ1=1,DkVκ=Vκ∂kfork=1,…,n,Vκfx=∫−1,1nfx1t1,…,xntn∏j=1nMκj1+tj1−tj2κj−1dt,with Mκj=Γκj+1/2/ΓκjΓ1/2 (where Γ is the well-known Gamma function).In order to define the Dunkl transform, we also need to introduce the Dunkl kernelEκ that is given for x∈ℂn by(7)Eκ⋅,xy=Vκe⋅,xy,y∈ℝn,and it satisfies the following basic properties: Eκix,y≤1 for x,y∈ℝn. Considering the definition of Eκ together with the explicit formula for Vκ gives us(8)Eκx,y=∏j=1nEκjxj,yj.In the rank-one case,Eκ is explicitly known. More precisely, it is given for both x and y in ℂ by(9)Eκx,y=jκ−1/2ixy+xy2κ+1jκ+1/2ixy,where jκ is the normalized Bessel function of the first kind and of order κ.We are now in a position to introduce the Dunkl transform that is taken with respect to the measureμκ defined by (1). For f∈Lk1Rn, the Dunkl transform of f, denoted by ℱκf, is given by(10)ℱκfx=cκ∫ℝnfyEκx,−iydμκy,x∈ℝn,where cκ is the following constant:(11)cκ−1=∫ℝne−x2/2dμκx=∏j=1ncκj−1.Ifκ1=⋯=κn=0, then Vκ=id and the Dunkl transform coincides with the Euclidean Fourier transform.Letx∈ℝn. The Dunkl translation operator τxκ is given for f∈Lk2Rn by(12)ℱκτxκfy=Eκix,yℱκfy,y∈ℝn.Proposition 1.
Letx∈ℝn. The operator τxκ extends to Lκpℝn for p∈1,+∞ and for f∈Lκpℝn, and we have(13)τxκfκ,p≤Cfκ,p,where C is independent of x and f.The Dunkl convolution operator is defined for bothf and g in Lk2Rn by(14)fκ∗gx=cκ∫ℝnfyτxκg−ydμκy,x∈ℝn.Proposition 2.
Assume thatp−1+q−1=1+r−1 with p,q,r∈1,+∞. Then, the map f,g↦fκ∗g defined on Lk2Rn×Lk2Rn extends to a continuous map from Lκpℝn×Lκqℝn to Lκrℝn, and we have(15)fκ∗gκ,r≤Cfκ,pgκ,q,where C is independent of f and g.We finally note that the Dunkl convolution satisfies the propertiesfκ∗g=gκ∗f and ℱκfκ∗g=ℱκf⋅ℱκg.The Dunkl Laplacian:(16)ℒfx=∑j=1nDj2fx=∑j=1n∂∂xj2fx+2kjxj∂∂xjfx−kjxj2fx−fσjx.


### 2.3. Hardy Space

We define the Riesz transforms in the Dunkl setting, putting(17)ℛj=Dj−ℒ−1/2.The operatorsℛj can be expressed as the Dunkl multiplier operators, namely,(18)ℛjfx=Dj−ℒ−1/2fx=Dj∫ℝn1ξEx,iξℱfξdμξ=∫ℝniξjξEx,iξℱfξdμξ.The Hardy type spaceHκ1ℝn for Dunkl analysis is defined to be(19)Hκ1ℝn≔f∈Lκ1ℝn;f1,κ+∑j=1nℛjf1,κ<∞,endowed with the norm(20)fHκ1≔f1,κ+∑j=1nℛjf1,κ.Anμκ-atom is a measurable function  a:ℝn​⟶ℂ  such that(i)
a is supported in a ball B(ii)
aκ,∞≲μκB−1(iii)
∫ℝnaxdμκx=0By definition, the atomic Hardy spaceHatom1 consists of all functions f∈Lk1Rn that can be written as f=∑ℓλℓaℓ, where the aℓ’s are atoms and ∑ℓλℓ<+∞, and the norm is given by(21)fHatom1=inf∑ℓλℓ,where the infimum is taken over all atomic decompositions of f.The following result was proved in [12].Proposition 3.
The spacesHκ1 and Hatom1 coincide and the norms fHκ1 and fHatom1 are equivalent.


## 2.1. Notation

We denote byLκpℝn the space Lpℝn;dμκ and we use the shorter notation ⋅κ,p instead of ⋅Lκpℝn. For p∈1,+∞, the space Lκpℝn is of course the space of measurable functions on ℝn such that(3)fκ,p=∫ℝnfypdμκy1/p<+∞,if1≤p<+∞,fκ,∞=esssupy∈ℝnfy<+∞,otherwise.


## 2.2. Dunkl Transform

Lete1,…,en be the standard basis of ℝn. We denote by σj (for each j, from 1 to n) the reflection with respect to the hyperplane perpendicular to ej; that is, to say, for every x=x1,…,xn∈ℝn(4)σjx=x−2x,ejej2ej=x1,…,xj−1,−xj,xj+1,…,xn.Of course⋅,⋅ is the usual inner product on ℝn×ℝn and ⋅ is the associated norm. Let G be the finite reflection group generated by σj:j=1,…,n; so, G is isomorphic to ℤ2n. Let κ1,κ2,…,κn be nonnegative real numbers.Associated with these objects are the Dunkl operatorsDk (for k=1,…,n) that have been introduced in [7] by Dunkl. They are given for x∈ℝn by(5)Dkfx=∂kfx+∑j=1nκjfx−fσjxx,ejek,ej=∂kfx+κkfx−fσkxxk,where ∂k denotes the usual partial derivative. A fundamental property of these differential-difference operators is their commutativity, that is, to say, DkDl=DlDk.Closely related to them is the so-called intertwining operatorVκ (the subscript means that the operator depends on the parameter κj, except in the rank-one case, where the subscript is then a single parameter), which is the unique linear isomorphism of ⊕m≥0Pm such that(6)VκPm=Pm,Vκ1=1,DkVκ=Vκ∂kfork=1,…,n,Vκfx=∫−1,1nfx1t1,…,xntn∏j=1nMκj1+tj1−tj2κj−1dt,with Mκj=Γκj+1/2/ΓκjΓ1/2 (where Γ is the well-known Gamma function).In order to define the Dunkl transform, we also need to introduce the Dunkl kernelEκ that is given for x∈ℂn by(7)Eκ⋅,xy=Vκe⋅,xy,y∈ℝn,and it satisfies the following basic properties: Eκix,y≤1 for x,y∈ℝn. Considering the definition of Eκ together with the explicit formula for Vκ gives us(8)Eκx,y=∏j=1nEκjxj,yj.In the rank-one case,Eκ is explicitly known. More precisely, it is given for both x and y in ℂ by(9)Eκx,y=jκ−1/2ixy+xy2κ+1jκ+1/2ixy,where jκ is the normalized Bessel function of the first kind and of order κ.We are now in a position to introduce the Dunkl transform that is taken with respect to the measureμκ defined by (1). For f∈Lk1Rn, the Dunkl transform of f, denoted by ℱκf, is given by(10)ℱκfx=cκ∫ℝnfyEκx,−iydμκy,x∈ℝn,where cκ is the following constant:(11)cκ−1=∫ℝne−x2/2dμκx=∏j=1ncκj−1.Ifκ1=⋯=κn=0, then Vκ=id and the Dunkl transform coincides with the Euclidean Fourier transform.Letx∈ℝn. The Dunkl translation operator τxκ is given for f∈Lk2Rn by(12)ℱκτxκfy=Eκix,yℱκfy,y∈ℝn.Proposition 1.
Letx∈ℝn. The operator τxκ extends to Lκpℝn for p∈1,+∞ and for f∈Lκpℝn, and we have(13)τxκfκ,p≤Cfκ,p,where C is independent of x and f.The Dunkl convolution operator is defined for bothf and g in Lk2Rn by(14)fκ∗gx=cκ∫ℝnfyτxκg−ydμκy,x∈ℝn.Proposition 2.
Assume thatp−1+q−1=1+r−1 with p,q,r∈1,+∞. Then, the map f,g↦fκ∗g defined on Lk2Rn×Lk2Rn extends to a continuous map from Lκpℝn×Lκqℝn to Lκrℝn, and we have(15)fκ∗gκ,r≤Cfκ,pgκ,q,where C is independent of f and g.We finally note that the Dunkl convolution satisfies the propertiesfκ∗g=gκ∗f and ℱκfκ∗g=ℱκf⋅ℱκg.The Dunkl Laplacian:(16)ℒfx=∑j=1nDj2fx=∑j=1n∂∂xj2fx+2kjxj∂∂xjfx−kjxj2fx−fσjx.


## 2.3. Hardy Space

We define the Riesz transforms in the Dunkl setting, putting(17)ℛj=Dj−ℒ−1/2.The operatorsℛj can be expressed as the Dunkl multiplier operators, namely,(18)ℛjfx=Dj−ℒ−1/2fx=Dj∫ℝn1ξEx,iξℱfξdμξ=∫ℝniξjξEx,iξℱfξdμξ.The Hardy type spaceHκ1ℝn for Dunkl analysis is defined to be(19)Hκ1ℝn≔f∈Lκ1ℝn;f1,κ+∑j=1nℛjf1,κ<∞,endowed with the norm(20)fHκ1≔f1,κ+∑j=1nℛjf1,κ.Anμκ-atom is a measurable function  a:ℝn​⟶ℂ  such that(i)
a is supported in a ball B(ii)
aκ,∞≲μκB−1(iii)
∫ℝnaxdμκx=0By definition, the atomic Hardy spaceHatom1 consists of all functions f∈Lk1Rn that can be written as f=∑ℓλℓaℓ, where the aℓ’s are atoms and ∑ℓλℓ<+∞, and the norm is given by(21)fHatom1=inf∑ℓλℓ,where the infimum is taken over all atomic decompositions of f.The following result was proved in [12].Proposition 3.
The spacesHκ1 and Hatom1 coincide and the norms fHκ1 and fHatom1 are equivalent.


## 3. Hausdorf Operator

We define the multivariate Dunkl-Hausdorff operatorℋκ=ℋκΦ,A acting on Borel measurable functions f:ℝn⟶ℂ by(22)ℋκfx≔∫ℝnΦsfAsxdμκs.


### 3.1. Boundedness ofℋ on the Spaces Lκpℝn

Let1≤p≤∞ and denoted by p∗, the exponent conjugate to p; that is, let 1/p+1/p∗=1 with the agreement that 1/∞≔0.For a nonsingular diagonal matrixA=niaga1,…,an, we denote by vA=∏j=1najs2κj.Theorem 1.
Iff∈Lκpℝn for some 1≤p≤∞, As=diaga1s,…,ans is a nonsingular diagonal matrix, and if(23)Kp,κ≔∫ℝnΦsvAs−1detAs−11/pdμκs<∞,then(24)ℋfp,κ≤Kp,κfp,κ.Clearly, the constantKp,κ depends not only on p and κ but also on Φ and A, but does not depend on f.Proof.
First, assume that1<p<∞. Making use of Fubini’s theorem and Minkowski’s inequality in the general form and integrating by substitution yields the following:(25)ℋfp,κ=∫Rn∫ℝnΦsfAsxdμκspdμκx1/p≤∫Rn∫ℝnΦsfAsxpnμκx1/pdμκs=∫RnΦs∫ℝnfvpdetA−1s∏j=1najs2κjdμκv1/pdμκs=fp,κ∫RnΦsvAs−1detAs−1−1/pdμκs,≕Kp,κfp,κ,which proves (24) in case 1<p<∞.
In casep=1, the abovementioned argument works without using Minkowski’s inequality. The case p=∞ is trivial.


#### 3.1.1. Examples

(1)
Ifn=1 and Φs=φ1/s/s, then
(26)
ℋ
κ
f
x
=
∫
0
∞
φ
s
s
2
κ
+
1
f
x
s
d
s
=
∫
0
∞
φ
s
s
2
α
+
2
f
x
s
d
s
,
α
=
κ
−
1
2
,
which is the same definition given in [8].
(2)
By choosingΦs1,s2=χ0,12s1,s2 and As1,s2=diags1,s2, we get the bivariate n=2 Dunkl-Cesàro operator Cκ:
(27)
C
κ
x
1
,
x
2
=
∬
0,1
2
f
s
1
x
1
,
s
2
x
2
d
μ
κ
1
s
1
d
μ
κ
2
s
2
=
1
x
1
2
κ
1
+
1
x
2
2
κ
2
+
1
∫
0
x
1
∫
0
x
2
f
t
1
,
t
2
d
μ
κ
1
t
1
d
μ
κ
2
t
2
,
x
1
x
2
≠
0.


### 3.2. The Adjoint Operator toℋκ

Consider the Hausdorff operatorℋκΦ;A for which condition (23) is satisfied for some 1≤p≤∞. By Theorem 1, the operator ℋκ is bounded on Lkpℝn. For Ψs=ΦsvAs−1detAs−1 we claim that the operator ℋκ∗=ℋκΨs;A−1 is bounded on the conjugate space Lkp∗ℝn. Indeed, by assumption, we have(28)Kp∗,κℋκ∗=∫ℝnΦsvAs−1detAs−1vAsdetAs1/p∗dμκs=∫ℝnΦsvAs−1detAs−11/pnμκs=Kp,κℋκ.This implies the case1<p≤∞ in the following corollary, while the case p=1 is trivial.Corollary 1.
If condition (23) is satisfied for some 1≤p≤∞, then the operator ℋκ∗ is bounded on Lkp∗ℝn:(29)ℋκ∗fp∗,κ≤Kp,κfp∗,κ.We also claim that thisℋκ∗ is the adjoint operator to ℋκ in the sense that we attempt to make clear in the following.Letℋκ=ℋκΦ;A be a Hausdorff operator, for which condition (23) is satisfied for some 1≤p≤∞ and let ℋκ∗=:ℋκΨs;A−1. If f∈Lkpℝn and g∈Lkp∗ℝn, then(30)∫ℝngxℋκfxdμκx=∫ℝnfxℋκ∗gxdμκx.Observe thatℋκ∗ is also a Hausdorff operator according to our definition.Proof.
We have seen above that by virtue of Theorem1 and Corollary 1 both integrals in (30) exist as Lebesgue integrals. So, we may apply Fubini’s theorem twice as follows:(31)∫ℝngxℋκfxdμκx=∫ℝn∫ℝnΦsfAsxdμαsgxdμκx=∫ℝnΦs∫ℝnfAsxgxdμκxdμκs=∫ℝnΦs∫ℝnfvgA−1svdetA−1s∏j=1najs2κjdμκvdμκs=∫ℝnfv∫ℝnΦsvAs−1detAs−1gA−1svdμκsdμκv=∫ℝnfvℋκ∗gvdμκv,which is (30).


### 3.3. Commuting Relations with Dunkl Transform

In the following theorem, we answer the claim indicated in the title of this section.Theorem 2.
(i)
If condition (23) is satisfied for p=1, then
(32)
ℱ
κ
ℋ
κ
f
u
=
ℋ
κ
∗
ℱ
κ
u
.
(ii)
If condition (23) is satisfied for p=∞, then(33)ℱκℋκfu=ℋκ∗ℱκu.Proof.
(i)
Thus, we may apply Fubini’s theorem twice in order to obtain that(34)cκ−1ℱκℋκfu=∫ℝnℋκfxEκu,−ixdμκx=∫ℝn∫ℝnΦsfAsxdμκsEκu,−ixdμκx=∫ℝnΦs∫ℝnfAsxEκu,−ixdμκxdμκs=∫ℝnΦs∫ℝnfv∏j=1najs2κj+1−1Eκu,−iA−1svdμκvdμκs=∫ℝnΦs∏j=1najs2κj+1−1∫ℝnfvEκA−1su,−ivdμκvdμκs=∫ℝnΦsvAs−1detAs−1cκ−1ℱκfA−1sudμκs=cκ−1ℋκ∗ℱκu.
(ii)
The proof of (33) is analogous to that of (32).


### 3.4. Boundedness ofℋκ on the Dunkl Hardy Spaces Hκ1ℝn

Proceeding similarly to the proof of Theorem 1 in [13], we are going to extend the earlier version of H1 estimates, as the strongest and sharp can be found in [4].LetB2=maxx=1BxT, where ⋅ denotes the Euclidean norm. We will say that Φ∈Lκ,B2 if(35)ΦLκ,B2=∫ℝnΦxBx2Ndμκx<∞.Theorem 3.
The Hausdorff operatorℋκf is bounded on the real Hardy space Hκ1ℝn provided Φ∈Lκ,A−12ℝn, and(36)ℋfHκ1≤ΦLκ,A−12fHκ1.We shall need the following inequalities for volumes of the Euclidean balls (see [12]). Recall that  k1,…,kn≥0  and N=n+∑j=1n2kj. On ℝn, equipped with the Euclidean distance, the measure μκ has the following rescaling properties:(37)dμκλx=λNdμx,∀λ∈ℝ∗,(38)μκBλx,λr=λNμκBx,r,∀x∈ℝn,∀λ∈ℝ∗.Moreover,(39)μκBx,r≍rn∏j=1nxj+r2kj.Proof.
We have(40)ℋfHκ1=∑j=0nℛκ,jℋfLκ1ℝn=∑j=0n∫ℝnℛκ,jℋfxdμκx≤∑j=0n∫ℝnΦs∫ℝnℛκ,jfAsxdμκxdμκs≤∫ℝnΦs∑j=0nℛκ,jfAsxLκ1ℝndμκs=∫ℝnΦsfAsxHκ1dμκs.
Let(41)fAsx=∑kλkbkAsx,where bκ is an μκ-atom supported in the ball By,r.
We will show that multiplyingbkAsx by a constant depending on s we get an atomic decomposition of f itself, with no composition in the argument.
We have(42)∫ℝnbkAsxdμκx=∫ℝnbkv∏j=1nasj2κj+1−1dμκv=0.
Letls=minajs, we have(43)suppbkAs⋅⊂x∈ℝn;Asx−y,Asx−y≤r2,since(44)Asx−y,Asx−y=Asx−A−1sy,Asx−A−1sy≥l2sx−A−1sy,Asx−A−1sy,then(45)suppbkAs⋅⊂BA−1sx0,rls.
It follows from (38) and (39) that(46)μκBA−1sy,rls≍rlsn∏j=1nyjas+rls2kj≤rlsn∏j=1nyjls+rls2kj≍μκByls,rls=ls−NμκBy,r.
Therefore,lsNbkAs⋅ is an atom, and(47)fAs⋅Hκ1≤A−1s2NfHκ1.


## 3.1. Boundedness ofℋ on the Spaces Lκpℝn

Let1≤p≤∞ and denoted by p∗, the exponent conjugate to p; that is, let 1/p+1/p∗=1 with the agreement that 1/∞≔0.For a nonsingular diagonal matrixA=niaga1,…,an, we denote by vA=∏j=1najs2κj.Theorem 1.
Iff∈Lκpℝn for some 1≤p≤∞, As=diaga1s,…,ans is a nonsingular diagonal matrix, and if(23)Kp,κ≔∫ℝnΦsvAs−1detAs−11/pdμκs<∞,then(24)ℋfp,κ≤Kp,κfp,κ.Clearly, the constantKp,κ depends not only on p and κ but also on Φ and A, but does not depend on f.Proof.
First, assume that1<p<∞. Making use of Fubini’s theorem and Minkowski’s inequality in the general form and integrating by substitution yields the following:(25)ℋfp,κ=∫Rn∫ℝnΦsfAsxdμκspdμκx1/p≤∫Rn∫ℝnΦsfAsxpnμκx1/pdμκs=∫RnΦs∫ℝnfvpdetA−1s∏j=1najs2κjdμκv1/pdμκs=fp,κ∫RnΦsvAs−1detAs−1−1/pdμκs,≕Kp,κfp,κ,which proves (24) in case 1<p<∞.
In casep=1, the abovementioned argument works without using Minkowski’s inequality. The case p=∞ is trivial.


### 3.1.1. Examples

(1)
Ifn=1 and Φs=φ1/s/s, then
(26)
ℋ
κ
f
x
=
∫
0
∞
φ
s
s
2
κ
+
1
f
x
s
d
s
=
∫
0
∞
φ
s
s
2
α
+
2
f
x
s
d
s
,
α
=
κ
−
1
2
,
which is the same definition given in [8].
(2)
By choosingΦs1,s2=χ0,12s1,s2 and As1,s2=diags1,s2, we get the bivariate n=2 Dunkl-Cesàro operator Cκ:
(27)
C
κ
x
1
,
x
2
=
∬
0,1
2
f
s
1
x
1
,
s
2
x
2
d
μ
κ
1
s
1
d
μ
κ
2
s
2
=
1
x
1
2
κ
1
+
1
x
2
2
κ
2
+
1
∫
0
x
1
∫
0
x
2
f
t
1
,
t
2
d
μ
κ
1
t
1
d
μ
κ
2
t
2
,
x
1
x
2
≠
0.


## 3.1.1. Examples

(1)
Ifn=1 and Φs=φ1/s/s, then
(26)
ℋ
κ
f
x
=
∫
0
∞
φ
s
s
2
κ
+
1
f
x
s
d
s
=
∫
0
∞
φ
s
s
2
α
+
2
f
x
s
d
s
,
α
=
κ
−
1
2
,
which is the same definition given in [8].
(2)
By choosingΦs1,s2=χ0,12s1,s2 and As1,s2=diags1,s2, we get the bivariate n=2 Dunkl-Cesàro operator Cκ:
(27)
C
κ
x
1
,
x
2
=
∬
0,1
2
f
s
1
x
1
,
s
2
x
2
d
μ
κ
1
s
1
d
μ
κ
2
s
2
=
1
x
1
2
κ
1
+
1
x
2
2
κ
2
+
1
∫
0
x
1
∫
0
x
2
f
t
1
,
t
2
d
μ
κ
1
t
1
d
μ
κ
2
t
2
,
x
1
x
2
≠
0.


## 3.2. The Adjoint Operator toℋκ

Consider the Hausdorff operatorℋκΦ;A for which condition (23) is satisfied for some 1≤p≤∞. By Theorem 1, the operator ℋκ is bounded on Lkpℝn. For Ψs=ΦsvAs−1detAs−1 we claim that the operator ℋκ∗=ℋκΨs;A−1 is bounded on the conjugate space Lkp∗ℝn. Indeed, by assumption, we have(28)Kp∗,κℋκ∗=∫ℝnΦsvAs−1detAs−1vAsdetAs1/p∗dμκs=∫ℝnΦsvAs−1detAs−11/pnμκs=Kp,κℋκ.This implies the case1<p≤∞ in the following corollary, while the case p=1 is trivial.Corollary 1.
If condition (23) is satisfied for some 1≤p≤∞, then the operator ℋκ∗ is bounded on Lkp∗ℝn:(29)ℋκ∗fp∗,κ≤Kp,κfp∗,κ.We also claim that thisℋκ∗ is the adjoint operator to ℋκ in the sense that we attempt to make clear in the following.Letℋκ=ℋκΦ;A be a Hausdorff operator, for which condition (23) is satisfied for some 1≤p≤∞ and let ℋκ∗=:ℋκΨs;A−1. If f∈Lkpℝn and g∈Lkp∗ℝn, then(30)∫ℝngxℋκfxdμκx=∫ℝnfxℋκ∗gxdμκx.Observe thatℋκ∗ is also a Hausdorff operator according to our definition.Proof.
We have seen above that by virtue of Theorem1 and Corollary 1 both integrals in (30) exist as Lebesgue integrals. So, we may apply Fubini’s theorem twice as follows:(31)∫ℝngxℋκfxdμκx=∫ℝn∫ℝnΦsfAsxdμαsgxdμκx=∫ℝnΦs∫ℝnfAsxgxdμκxdμκs=∫ℝnΦs∫ℝnfvgA−1svdetA−1s∏j=1najs2κjdμκvdμκs=∫ℝnfv∫ℝnΦsvAs−1detAs−1gA−1svdμκsdμκv=∫ℝnfvℋκ∗gvdμκv,which is (30).


## 3.3. Commuting Relations with Dunkl Transform

In the following theorem, we answer the claim indicated in the title of this section.Theorem 2.
(i)
If condition (23) is satisfied for p=1, then
(32)
ℱ
κ
ℋ
κ
f
u
=
ℋ
κ
∗
ℱ
κ
u
.
(ii)
If condition (23) is satisfied for p=∞, then(33)ℱκℋκfu=ℋκ∗ℱκu.Proof.
(i)
Thus, we may apply Fubini’s theorem twice in order to obtain that(34)cκ−1ℱκℋκfu=∫ℝnℋκfxEκu,−ixdμκx=∫ℝn∫ℝnΦsfAsxdμκsEκu,−ixdμκx=∫ℝnΦs∫ℝnfAsxEκu,−ixdμκxdμκs=∫ℝnΦs∫ℝnfv∏j=1najs2κj+1−1Eκu,−iA−1svdμκvdμκs=∫ℝnΦs∏j=1najs2κj+1−1∫ℝnfvEκA−1su,−ivdμκvdμκs=∫ℝnΦsvAs−1detAs−1cκ−1ℱκfA−1sudμκs=cκ−1ℋκ∗ℱκu.
(ii)
The proof of (33) is analogous to that of (32).


## 3.4. Boundedness ofℋκ on the Dunkl Hardy Spaces Hκ1ℝn

Proceeding similarly to the proof of Theorem 1 in [13], we are going to extend the earlier version of H1 estimates, as the strongest and sharp can be found in [4].LetB2=maxx=1BxT, where ⋅ denotes the Euclidean norm. We will say that Φ∈Lκ,B2 if(35)ΦLκ,B2=∫ℝnΦxBx2Ndμκx<∞.Theorem 3.
The Hausdorff operatorℋκf is bounded on the real Hardy space Hκ1ℝn provided Φ∈Lκ,A−12ℝn, and(36)ℋfHκ1≤ΦLκ,A−12fHκ1.We shall need the following inequalities for volumes of the Euclidean balls (see [12]). Recall that  k1,…,kn≥0  and N=n+∑j=1n2kj. On ℝn, equipped with the Euclidean distance, the measure μκ has the following rescaling properties:(37)dμκλx=λNdμx,∀λ∈ℝ∗,(38)μκBλx,λr=λNμκBx,r,∀x∈ℝn,∀λ∈ℝ∗.Moreover,(39)μκBx,r≍rn∏j=1nxj+r2kj.Proof.
We have(40)ℋfHκ1=∑j=0nℛκ,jℋfLκ1ℝn=∑j=0n∫ℝnℛκ,jℋfxdμκx≤∑j=0n∫ℝnΦs∫ℝnℛκ,jfAsxdμκxdμκs≤∫ℝnΦs∑j=0nℛκ,jfAsxLκ1ℝndμκs=∫ℝnΦsfAsxHκ1dμκs.
Let(41)fAsx=∑kλkbkAsx,where bκ is an μκ-atom supported in the ball By,r.
We will show that multiplyingbkAsx by a constant depending on s we get an atomic decomposition of f itself, with no composition in the argument.
We have(42)∫ℝnbkAsxdμκx=∫ℝnbkv∏j=1nasj2κj+1−1dμκv=0.
Letls=minajs, we have(43)suppbkAs⋅⊂x∈ℝn;Asx−y,Asx−y≤r2,since(44)Asx−y,Asx−y=Asx−A−1sy,Asx−A−1sy≥l2sx−A−1sy,Asx−A−1sy,then(45)suppbkAs⋅⊂BA−1sx0,rls.
It follows from (38) and (39) that(46)μκBA−1sy,rls≍rlsn∏j=1nyjas+rls2kj≤rlsn∏j=1nyjls+rls2kj≍μκByls,rls=ls−NμκBy,r.
Therefore,lsNbkAs⋅ is an atom, and(47)fAs⋅Hκ1≤A−1s2NfHκ1.



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*Source: 1071746-2020-04-10.xml*