BLAISE PASCAL
Bechir Amri & Mohamed Sifi
Riesz transforms for Dunkl transform Volume 19, no1 (2012), p. 247-262.
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Riesz transforms for Dunkl transform
Bechir Amri Mohamed Sifi
Abstract
In this paper we obtain theLp-boundedness of Riesz transforms for the Dunkl transform for all 1< p <∞.
Transformées de Riesz associés à la transformée de Dunkl
Résumé
Dans cet article, nous étudions la bornitude des transformées de Riesz associées à la transformée de Dunkl sur les espacesLp, 1< p <∞.
1. Introduction
On the Euclidean space RN, N ≥ 1, the ordinary Riesz transform Rj, j = 1, ..., N is defined as the multiplier operator
R\j(f)(ξ) =−i ξj
kξkf(ξ).b (1.1)
It can also be defined by the principal value of the singular integral Rj(f)(x) =d0lim
ε→0
Z
kx−yk>ε
xj−yj
kx−ykf(y)dy
where d0 = 2N2 Γ(N+12 )
√π . It follows from the general theory of singular integrals that Riesz transforms are bounded onLp(RN, dx) for all 1< p <
∞. What is done in this paper is to extend this result to the context of Dunkl theory where a similar operator is already defined.
Dunkl theory generalizes classical Fourier analysis on RN. It started twenty years ago with Dunkl’s seminal work [3] and was further developed by several mathematicians. See for instance the surveys [5, 6, 7, 9] and the
Keywords:Dunkl transforms, Riesz Transforms, Singular integrals.
Math. classification:17B22, 32A55, 43A32, 42A45.
references cited therein. The study of theLp-boundedness of Riesz trans- forms for Dunkl transform onRN goes back to the work of S. Thangavelu and Y. Xu [10] where they established boundedness result only in a very special case of N = 1. It has been noted in [10] that the difficulty arises in the application of the classical Lp- theory of Caldéron-Zygmund, since Riesz transforms are singular integral operators. In this paper we describe how this theory can be adapted in Dunkl setting and gives anLp-result for Riesz transforms for all 1< p <∞. More precisely, through the fundamen- tal result of M. Rösler [6] for the Dunkl translation of radial functions, we reformulate a Hörmander type condition for singular integral operators.
The Riesz kernel is given by acting Dunkl operator on Dunkl translation of radial function.
This paper is organized as follows. In Section 2 we present some defi- nitions and fundamental results from Dunkl’s analysis. The Section 3 is devoted to proving Lp-boundedness of Riesz transforms. As applications, we will prove a generalized Riesz and Sobolev inequalities. Throughout this paper C denotes a constant which can vary from line to line.
Acknowledgments. The authors are partially supported by the project DGRST 04/UR/15-02 and the cooperation programs PHC Utique / CMCU 07G 1501 and 10G 1503.
2. Preliminaries
In this section we collect notations and definitions and recall some basic facts. We refer to [5, 3, 6, 7, 9].
Let G⊂ O(RN) be a finite reflection group associated to a reduced root system R and k : R → [0,+∞) be a G-invariant function (called multiplicity function). Let R+ be a positive root subsystem. We shall assume that R is normalized in the sense that kαk2 = hα, αi = 2 for all α ∈R, whereh, iis the standard Euclidean scalar product on RN.
The Dunkl operators Tξ, ξ ∈ RN are the following k–deformations of directional derivatives ∂ξ by difference operators :
Tξf(x) =∂ξf(x) + X
α∈R+
k(α)hα, ξif(x)−f(σα. x)
hα, xi , x∈RN
whereσα denotes the reflection with respect to the hyperplane orthogonal toα. For the standard basis vectors ofRN, we simply write Tj =Tej.
The operators ∂ξ and Tξ are intertwined by a Laplace–type operator Vkf(x) =
Z
RN
f(y)dµx(y),
associated to a family of compactly supported probability measures {µx|x∈RN}.
Specifically, µx is supported in the the convex hull co(G.x).
For every λ∈CN, the simultaneous eigenfunction problem, Tξf =hλ, ξif, ξ∈RN
has a unique solutionf(x) =Ek(λ, x) such thatEk(λ,0) = 1, which is given by
Ek(λ, x) = Vk(ehλ, .i)(x) = Z
RN
ehλ,yidµx(y), x∈RN. Furthermore λ7→Ek(λ, x) extends to a holomorphic function on CN.
Let mk be the measure onRN, given by dmk(x) = Y
α∈R+
|hα, xi|2k(α)dx.
Forf ∈L1(mk) (the Lebesgue space with respect to the measuremk) the Dunkl transform is defined by
Fk(f)(ξ) = 1 ck
Z
RN
f(x)Ek(−i ξ, x)dmk(x), ck = Z
RN
e−|x|
2
2 dmk(x).
This new transform shares many analogous properties of the Fourier trans- form.
(i) The Dunkl transform is a topological automorphism of S(RN) (Schwartz space).
(ii) (Plancherel Theorem) The Dunkl transform extends to an isomet- ric automorphism of L2(mk).
(iii) (Inversion formula) For everyf∈L1(mk) such thatFkf∈L1(mk), we have
f(x) =Fk2f(−x), x∈RN.
(iv) For allξ ∈RN and f ∈ S(RN)
Fk(Tξ(f))(x) =< iξ, x >Fk(f)(x), x∈RN. (2.1) Letx∈RN, the Dunkl translation operatorτxis defined onL2(mk) by, Fk(τx(f))(y) =Ek(ix, y)Fkf(y), y∈RN. (2.2) If f is a continuous radial function in L2(mk) withf(y) =fe(kyk), then
τx(f)(y) = Z
RN
fe qkxk2+kyk2+ 2< y, η >dµx(η). (2.3) This formula is first proved by M. Rösler [6] for f ∈ S(RN) and recently is extended to continuous functions by F. and H. Dai Wang [2].
We collect below some useful facts : (i) For all x, y∈RN,
τx(f)(y) =τy(f)(x). (2.4) (ii) For allx, ξ∈RN and f ∈ S(RN),
Tξτx(f) =τxTξ(f). (2.5) (iii) For all x∈RN and f, g∈L2(mk),
Z
RN
τx(f)(−y)g(y)dmk(y) = Z
RN
f(y)τxg(−y)dmk(y). (2.6) (iv) For allx∈RN and 1≤p≤2, the operator τx can be extended to
all radial functions f inLp(mk) and the following holds
kτx(f)kp,k≤ kfkp,k. (2.7) k.kp,k is the usual norm of Lp(mk).
3. Riesz transforms for the Dunkl transform.
In Dunkl setting the Riesz transforms (see [10]) are the operators Rj, j = 1...N defined onL2(mk) by
Rj(f)(x) =dklim
ε→0
Z
|y|>ε
τx(f)(−y) yj
kykpkdmk(y), x∈RN
where
dk = 2pk
−1 2
Γ(p2k)
√π ; pk = 2γk+N + 1 and γk= X
α∈R+
k(α).
It has been proved by S. Thangavelu and Y. Xu [10], thatRjis a multiplier operator given by
Fk(Rj(f))(ξ) =−i ξj
kξkFk(f)(ξ), f ∈ S(RN), ξ∈RN, (3.1) The authors state that if N = 1 and 2γk∈Nthe operatorRj is bounded onLp(mk), 1< p <∞. In [1] this result is improved by removing 2γk ∈N, where Riesz transform is called Hilbert transform. If γk= 0 (k= 0), this operator coincides with the usual Riesz transform Rj given by (1.1). Our interest is to prove the boundedness of this operator forN ≥2 andk≥0.
To do this, we invoke the theory of singular integrals. Our basic is the following,
Theorem 3.1. LetKbe a measurable function onRN×RN\{(x, g.x); x∈ RN, g∈G} and S be a bounded operator from L2(mk) into itself, associ- ated with a kernel K in the sense that
S(f)(x) = Z
RN
K(x, y)f(y)dmk(y), (3.2) for all compactly supported function f in L2(mk) and for a.e x ∈ RN satisfying g.x /∈supp(f), for allg∈G. If K satisfies
Z
ming∈Gkg.x−yk>2ky−y0k
|K(x, y)− K(x, y0)|dmk(x)≤C, y, y0 ∈RN, (3.3) then S extends to a bounded operator from Lp(mk) into itself for all1 <
p≤2.
Proof. We first note that (RN, mk) is a space of homogenous type, that is, there is a fixed constant C >0 such that
mk(B(x,2r))≤Cmk(B(x, r)), ∀x∈RN, r >0 (3.4) where B(x, r) is the closed ball of radius r centered at x (see [8], Ch 1).
Then we can adapt to our context the classical technic which consist to show that S is weak type (1,1) and conclude by Marcinkiewicz interpola- tion theorem.
In fact, the Calderón-Zygmund decomposition says that for all f ∈ L1(mk)∩L2(mk) and λ >0, there exist a decomposition of f, f =h+b with b =Pjbj and a sequence of balls (B(yj, rj))j =(Bj)j such that for some constantC, depending only on the multiplicity function k
(i) khk∞≤Cλ;
(ii) supp(bj)⊂Bj; (iii)
Z
Bj
bj(x)dmk(x) = 0;
(iv) kbjk1,k ≤C λ mk(Bj);
(v) X
j
mk(Bj)≤C kfk1,k
λ .
The proof consists in showing the following inequality hold for w=hand w=b :
ρλ(S(w)) =mk
{x∈RN; |S(w)(x)|> λ
2}≤C kfk1,k
λ . (3.5)
By using the L2-boundedness of S we get ρλ(S(h))≤ 4
λ2 Z
RN
|S(h)(x)|2dmk(x)≤ C λ2
Z
RN
|h(x)|2dmk(x). (3.6) From (i) and (v),
Z
∪Bj
|h(x)|2dmk(x)≤Cλ2µk(∪Bj)≤Cλkfk1,k. (3.7) Since on (∪Bj)c,f(x) =h(x), then
Z
(∪Bj)c
|h(x)|2dmk(x)≤Cλkf|1,k. (3.8) From (3.6), (3.7) and (3.8), the inequality (3.5) is satisfied for h.
Next we turn to the inequality (3.5) for the functionb. Consider Bj∗ =B(yj,2rj ); and Q∗j = [
g∈G
g.B∗j.
Then
ρλ(S(b))≤mk
[
j
Q∗j+mk{x∈ [
j
Q∗jc;|S(b)(x)|> λ 2}.
Now by (3.4) and (v) mk [
j
Q∗j≤ |G|X
j
mk(Bj∗)≤CX
j
mk(Bj)≤C kfk1,k λ .
Furthermore if x /∈Q∗j, we have
ming∈Gkg.x−yjk>2ky−yjk, y∈Bj. Thus, from (3.2),(iii) ,(ii), (3.3), (iv)and (v)
Z
(∪Q∗j)c
|S(b)(x)|dmk(x)
≤X
j
Z
(Q∗j)c
|S(bj)(x)|dmk(x)
=X
j
Z
(Q∗j)c
Z
RN
K(x, y)bj(y)dmk(y)dmk(x)
=X
j
Z
(Q∗j)c
Z
RN
bj(y)K(x, y)− K(x, yj)dmk(y)dmk(x)
≤X
j
Z
RN
|bj(y)|
Z
(Q∗j)c
|K(x, y)− K(x, yj)|dmk(x)dmk(y)
≤X
j
Z
RN
|bj(y)|
Z
g∈Gminkg.x−yjk>2ky−yjk|K(x, y)− K(x, yj)|dmk(x)dmk(y)
≤CX
j
kbjk1,k
≤C kfk1,k. Therefore,
mk{x∈ [
j
Q∗jc;|S(b)(x)|> λ 2}
≤ 2 λ
Z
(∪Q∗j)c
|S(b)(x)|dmk(x)≤C kfk1,k λ .
This achieves the proof of (3.5) for b.
Now, we will give an integral representation for the Riesz transformRj. For this end, we put for x, y∈RN and η∈co(G.x)
A(x, y, η) = q
kxk2+kyk2−2< y, η >= q
ky−ηk2+kxk2− kηk2. It is easy to check that
ming∈Gkg.x−yk ≤A(x, y, η)≤max
g∈G kg.x−yk. (3.9) The following inequalities are clear
∂A`
∂yr
(x, y, η)≤CA`−1(x, y, η),
∂2A`
∂yr∂ys
(x, y, η)≤CA`−2(x, y, η) (3.10) and for α∈R+,
∂A`
∂yr(x, σα.y, η)≤CA`−1(x, σα.y, η),
∂2A`
∂yr∂ys(x, σα.y, η)≤CA`−2(x, σα.y, η), (3.11) where r, s= 1, ...N and `∈R.
Let us set K(1)j (x, y) =
Z
RN
ηj−yj
Apk(x, y, η)dµx(η) Kj(α)(x, y) = 1
< y, α >
Z
RN
h 1
Apk−2(x, y, η) − 1
Apk−2(x, σα.y, η)
idµx(η), Kj(x, y) = dknK(1)j (x, y) + X
α∈R+
k(α)αj
pk−2K(α)j (x, y)o, where α∈R+.
Proposition 3.2. If f ∈L2(mk) with compact support, then for all x ∈ RN satisfying g.x /∈supp(f) for all g∈G, we have
Rj(f)(x) = Z
RN
Kj(x, y)f(y)dmk(y).
Proof. Let f ∈ L2(mk) be a compact supported function and x ∈ RN, such that g.x /∈supp(f) for all g ∈G. For 0< ε <min
g∈G min
y∈supp(f)
|g.x−y|
and n∈N, we consider ϕen,ε aC∞-function onR, such that:
• ϕen,ε is odd .
• ϕen,ε is supported in{t∈R; ε≤ |t| ≤n+ 1}.
• ϕen,ε= 1 in{t∈R; ε+n1 ≤t≤n}.
• |ϕen,ε| ≤1.
Let
φen,ε(t) = Z t
−∞
ϕen,ε(u)
|u|pk−1 du and φn,ε(y) =φen,ε(kyk),
for t∈R and y ∈RN. Clearly,φn,ε is a C∞ radial function supported in the ballB(0, n+ 1) and
n→+∞lim ϕen,ε(kyk) = 1, ∀y∈RN, kyk> ε.
The dominated convergence theorem, (2.5) and (2.6) yield Z
kyk>ετx(f)(−y) yj
kykpkdmk(y)
= lim
n→∞
Z
RN
τx(f)(−y) yj
kykpkϕen,ε(kyk)dmk(y)
= lim
n→∞
Z
RN
τx(f)(−y)Tj(φn,ε)(y)dmk(y)
= lim
n→∞
Z
RN
f(y)Tjτx(φn,ε)(−y)dmk(y).
Now we have Tjτx(φn,ε)(−y)
= Z
RN
(ηj−yj)ϕen,ε(A(x, y, η)) Apk(x, y, η) dµx(η)
+ X
α∈R+
k(α)αj
Z
RN
φen,ε(A(x, σα.y, η))−φen,ε(A(x, y, η))
< y, α > dµx(η),
where from (3.9)
ε < A(x, y, η) ; ε < A(x, σα.y, η), y∈supp(f), η ∈co(G.x).
Then with the aid of dominated convergence theorem
n→∞lim Tjτx(φn,ε)(−y) = 1
dkKj(x, y), and
dk
Z
kyk≥ετx(f)(−y) yj
kykpkdmk(y) = Z
RN
Kj(x, y)f(y)dmk(y).
Letting ε→0, it follows that Rj(f)(x) =
Z
RN
Kj(x, y)f(y)dmk(y),
which proves the result.
Now, we are able to state our main result.
Theorem 3.3. The Riesz transformRj,j = 1...N, is a bounded operator from Lp(mk) into itself, for all 1< p <∞.
Proof. Clearly, from (3.1) and Plancherel’s theorem Rj is bounded from L2(mk) into itself, with adjoint operatorR∗j =−Rj. Thus, via duality it’s enough to consider the range 1< p≤2 and apply Theorem 3.1. In view of Proposition 3.2 it only remains to show that Kj satisfies condition (3.3).
Let y, y0∈RN,y6=y0 andx∈RN, such that
ming∈Gkg.x−yk>2ky−y0k. (3.12) By mean value theorem,
|K(1)j (x, y)− K(1)j (x, y0)| =
N
X
i=0
(yi−(y0)i) Z 1
0
∂K(1)j
∂yi (x, yt)dt
=
N
X
i=0
(yi−(y0)i) Z 1
0
Z
RN
δi,j
Apk(x, yt, η) +pk((yt)i−ηi)(ηj−(yt)j)
Apk+2(x, yt, η)
dµx(η)
≤ Cky−y0k Z 1
0
Z
RN
1
Apk(x, yt, η) dµx(η)dt.
where yt=y0+t(y−y0) and δi,j is the Kronecker symbol.
In view of (3.9) and (3.12), we obtain
ky−y0k< A(x, yt, η), η ∈co(G.x).
Therefore,
K(1)j (x, y)− K(1)j (x, y0)
≤ Cky−y0k Z 1
0
Z
RN
1
ky−y0k2+A2(x, yt, η)
pk 2
dµx(η)dt.
≤ Cky−y0k Z 1
0
τx(ψ)(yt)dt where ψ is the function defined by
ψ(z) = 1
(ky−y0k2+kzk2)pk2 , z∈RN. Using Fubini’s theorem, (2.4) and (2.7), we get
Z
ming∈Gkg.x−yk>2ky−y0k
|K(1)j (x, y)− K(1)j (x, y0)|dmk(x)
≤ Cky−y0k Z 1
0
Z
RN
τ−yt(ψ)(x)dmk(x)dt
≤ C|y−y0| Z
RN
ψ(z)dmk(z) =C Z
RN
du (1 +u2)pk2
=C0.
This established the condition (3.3) for Kj(1).
To deal with Kj(α), α ∈ R+, we put for x, y ∈ RN , η ∈ co(G.x) and t∈[0,1]
U(x, y, η) = A2pk−4(x, y, η),
Vα(x, y, η) = Apk−2Apαk−2(Apk−2+Apαk−2), hα,t(y) = y+t(σα.y−y) =y−t < y, α > α, By mean value theorem we have
K(α)j (x, y) = Z
RN
1
< y, α >
U(x, σα.y, η)−U(x, y, η) Vα(x, y, η) dµx(η)
= −
Z
RN
Z 1 0
∂αU(x, hα,t(y), η)
Vα(x, y, η) dt dµx(η) and
K(α)j (x, y)− Kj(α)(x, y0)
= Z
RN
Z 1
0
Z 1
0
∂y−y0
∂αU(x, hα,t(.), η) Vα(x, ., η)
(yθ)dθ dt dµx(η). (3.13) Here the derivations are taken with respect to the variabley.
To simplify, let us denote by
A=A(x, yθ, η); Aα =A(x, σα.yθ, η) Then using (3.10) and the fact
kη−hα,t(yθ)k ≤max(kη−yθk,kη−σα(yθ)k), we obtain
∂U
∂yr(x, hα,t(yθ), η) ≤ CA2pk−5+A2pαk−5
∂2U
∂yr∂ys
(x, hα,t(yθ), η) ≤ CA2pk−6+A2pαk−6, r, s= 1, ..., N.
This gives us the following estimates
∂αU(x, hα,t(yθ), η)≤CA2pk−5+A2pαk−5, (3.14)
∂y−y0
∂αU(x, hα,t(.), η)(yθ)≤Cky−y0kA2pk−6+A2pαk−6. (3.15) By (3.10) and (3.11), we also have
∂Vα
∂yr((x, yθ, η))≤CApk−3Apαk−3(Apk−2+Apαk−2)(A+Aα).
The elementary inequality u+v
u`+v` ≤ 3
u`−1+v`−1, u, v >0, `≥1, leads to
∂y−y0Vα(x, yθ, η) Vα2(x, yθ, η)
≤ Cky−y0k Aα+A
Apk−1Apαk−1(Apk−2+Apαk−2)
≤ Cky−y0k 1
Apk−1Apαk−1(Apk−3+Apαk−3). (3.16) Now (3.14), (3.15) and (3.16) yield
∂y−y0
∂αU(x, hα,t(.), η) Vα(x, ., η)
(yθ)
≤ Cky−y0k A2pk−6+A2pαk−6 Apk−2Apαk−2(Apk−2+Apαk−2) + Cky−y0k A2pk−5+A2pαk−5
Apk−1Apαk−1(Apk−3+Apαk−3)
≤ Cky−y0k 1 A2Apαk−2
+ 1
Apk−2A2α
+ Cky−y0k 1 AApαk−1
+ 1
Apk−1Aα
≤ Cky−y0k 1 Apk + 1
Apαk
where in the last equality we have used the fact that 1
uv`−1 ≤ 1 u` + 1
v`, u, v >0 and `≥1.
Thus, in view of (3.13),
K(α)j (x, y)− Kj(α)(x, y0)
≤Cky−y0k Z 1
0
Z
RN
h 1
Apk(x, yθ, η) + 1 Apk(x, σαyθ, η)
i
dµx(η)dθ.
Then by same argument as for K(1)j we obtain Z
ming∈Gkg.x−yk>2ky−y0k
|K(2)j (x, y)− K(2)j (x, y0)|dmk(x)≤C, which established the condition (3.3) for the kernelK(α)j and furnishes the
proof.
As applications, we will prove a generalized Riesz and Sobolev inequal- ities
Corollary 3.4 (Generalized Riesz inequalities). For all 1< p <∞ there exists a constant Cp such that
||TrTs(f)||k,p≤Cp||∆kf||k,p, for all f ∈ S(RN), (3.17) where ∆k is the Dunkl laplacian: ∆kf =
N
X
r=1
Tr2(f) Proof. From (2.1) and (3.1) one can see that
TrTs(f) =RrRs(−∆k)(f), r, s= 1...N, f ∈ S(RN).
Then (3.17) is concluded by Theorem 3.3.
Corollary 3.5 (Generalized Sobolev inequality). For all 1 < p ≤ q <
2γ(k) +N with 1 q = 1
p− 1
2γ(k) +N, we have
kfkq,k ≤Cp,qk∇kfkp,k (3.18) for all f ∈ S(RN). Here ∇kf = (T1f, ..., TNf) and|∇kf|= (
N
X
r=1
|Trf|2)12.
Proof. For allf ∈ S(RN), we write Fk(f)(ξ) = 1
kξk
N
X
r=1
−iξr kξk
iξrFk(f)(ξ)
= 1
kξk
d
X
r=1
−iξr kξk
Fk(Trf)(ξ). This yields to the following identity
f =Ik1
N
X
j=1
Rj(Tjf), where
Ikβ(f)(x) = (dβk)−1 Z
RN
τyf(x)
kyk2γ(k)+N−βdmk(y), here
dβk = 2−γ(k)−N/2+β Γ(β2) Γ(γ(k) +N−β2 ).
Theorem 1.1 of [4] asserts that Ikβ a bounded operator from Lp(mk) to Lq(mk). Then (3.18) follows from Theorem 3.3.
References
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Bechir Amri
Department of Mathematics University of Tunis
Preparatory Institute of Engineer Studies of Tunis
1089 Montfleury, Tunis, Tunisia bechir.amri@ipeit.rnu.tn
Mohamed Sifi
Department of Mathematics University of Tunis El Manar Faculty of Sciences of Tunis
2092 Tunis El Manar, Tunis, Tunisia mohamed.sifi@fst.run.tn
