EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 3, 2017, 544-551
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Generalization of Dunkl Dini Lipschitz Functions

Salah El Ouadih1,∗, Radouan Daher1

1 Department of Mathematics, Faculty of Sciences Aı̈n Chock, University Hassan II
Morocco

Abstract. Using a generalized spherical mean operator, we obtain a generalization of Younis’s
Theorem 5.2 in [12] for the Dunkl transform for functions satisfying the d-Dunkl Dini Lipschitz
condition in the space Lp(Rd, wl(x)dx), 1 < p ≤ 2, where wl is a weight function invariant under
the action of an associated reflection group.

2010 Mathematics Subject Classifications: 42B37

Key Words and Phrases: Dunkl transform, Dunkl kernel, generalized spherical mean operator

1. Introduction and Preliminaries

Younis’s Theorem 5.2 [12] characterized the set of functions in L2(R) satisfying the
Dini Lipschitz condition by means of an asymptotic estimate growth of the norm of their
Fourier transforms, namely we have

Theorem 1. [12] Let f ∈ L2(R). Then the following are equivalents

(i) ‖f(x+ h)− f(x)‖2 = O
(

hη

(log 1
h
)δ

)
, as h→ 0, 0 < η < 1, δ ≥ 0

(ii)

∫
|λ|≥s

|f̂(λ)|2dλ = O

(
s−2η

(log s)2δ

)
, as s→∞,

where f̂ stands for the Fourier transform of f .

In this paper, we obtain a generalization of Theorem 1.1 for the Dunkl transform on Rd
in the space Lp(Rd, wl(x)dx), 1 < p ≤ 2. For this purpose, we use a generalized spherical
mean operator.

We consider the Dunkl operators Dj , 1 ≤ j ≤ d, on Rd which are the differential- dif-
ference operators introduced by Dunkl in [3]. These operators are very important in pure
mathematics and in physics. The theory of Dunkl operators provides generalizations of
various multivariable analytic structures, among others we cite the exponential function,

∗Corresponding author.

Email addresses: salahwadih@gmail.com (S. El Ouadih), rjdaher024@gmail.com (R. Daher)

http://www.ejpam.com 544 c© 2017 EJPAM All rights reserved.



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 545

the Fourier transform and the translation operator. For more details about these opera-
tors see [6, 5]. The Dunkl Kernel El has been introduced by Dunkl in [4]. This Kernel is
used to define the Dunkl transform.
Let R be a root system in Rd, W the corresponding reflection group, R+ a positive sub-
system of R ( see [6, 5, 1, 8, 9]) and l a non-negative and W-invariant function defined on
R. The Dunkl operator is defined for f ∈ C1(Rd) by

Djf(x) =
∂f

∂xj
(x) +

∑
α∈R+

l(α)αj
f(x)− f(σα(x))

< α, x >
, x ∈ Rd(1 ≤ j ≤ d).

Here <,> is the usual Euclidean scalar product on Rd with the associated norm |.| and
σα the reflection with respect to the hyperplane Hα orthogonal to α, and αj =< α, ej >,
(e1, e2, ..., ed) being the canonical basis of Rd.
We consider the weight function

wl(x) =
∏
ζ∈R+

| < ζ, x > |2l(α), x ∈ Rd,

where wl is W-invariant and homogeneous of degree 2γ where

γ = γ(R) =
∑
ζ∈R+

l(ζ) ≥ 0.

The Dunkl kernel El on Rd × Rd has been introduced by C. F. Dunkl in [4]. For y ∈ Rd,
the function x 7→ El(x, y) is the unique solution on Rd of the following initial problem{

Dju(x, y) = yju(x, y) si 1 ≤ j ≤ d
u(0, y) = 0 for all y ∈ Rd

El is called the Dunkl kernel.

Lemma 1. [6] Let z, w ∈ Cd and λ ∈ C
1. El(z, 0) = 1, El(z, w) = El(w, z), El(λz,w) = El(z, λw).
2. For all ν = (ν1, ..., νd) ∈ Nd,x ∈ Rd,z ∈ Cd, we have

|∂νzEl(x; z)| ≤ |x||ν|exp(|x||Rez|,

where

∂νz =
∂|ν|

∂ν1z1 ....∂
ν2
zd

, |ν| = ν1 + ....+ νd.

In particular |∂νzEl(ix; z)| ≤ |x||ν| for all x, z ∈ Rd.

We denote by Lpl (R
d) = Lp(Rd, wl(x)dx), 1 < p ≤ 2, the space of measurable functions

on Rd with the norm

‖f‖p,l =

(∫
Rd
|f(x)|pwl(x)dx

) 1
p

<∞.



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 546

The Dunkl transform is defined for f ∈ L1
l (Rd) = L1(Rd, wl(x)dx) by

F(f)(ξ) = f̂(ξ) = c−1l

∫
Rd
f(x)El(−iξ, x)wl(x)dx,

where the constant cl is given by

cl =

∫
Rd
e−
|z|2
2 wl(z)dz.

The Dunkl transform shares several properties with its counterpart in the classical case,
we mention here in particular that Plancherel’s Theorem holds in L2

l (Rd), when both f

and f̂ are in L1
l (Rd), we have the inversion formula

f(x) =

∫
Rd
f̂(ξ)El(ix, ξ)wl(ξ)dξ, x ∈ Rd.

By Plancherel’s Theorem and the Marcinkiewicz interpolation theorem (see [10]), we get
for f ∈ Lpl (R

d) with 1 < p ≤ 2 and q such that 1
p + 1

q = 1,

‖F(f)‖q,l ≤ K‖f‖p,l, (1)

where K is a positive constant.
The generalized spherical mean value of f ∈ Lpl (R

d) is defined by

Mhf(x) =
1

dl

∫
Sd−1

τxf(hy)dµl(y), x ∈ Rd, h > 0.

where τx Dunkl translation operator (see [9, 11]), µ be the normalized surface measure on
the unit sphere Sd−1 in Rd and set dµl(y) = wl(y)dµ(y), µl is a W-invariant measure on
Sd−1 and dl = µl(Sd−1).
We see that Mhf ∈ Lpl (R

d) whenever f ∈ Lpl (R
d) and

‖Mhf‖p,l ≤ ‖f‖p,l.

for all h > 0.
For β ≥ −12 , we introduce the Bessel normalized function of the first kind jβ defined by

jβ(z) = Γ(β + 1)

∞∑
n=0

(−1)n(z/2)2n

n!Γ(n+ β + 1)
, z ∈ C. (2)

Lemma 2. (Analog of lemma 2.9 in [2]) The following inequality is true

|1− jβ(x)| ≥ c,

with |x| ≥ 1, where c > 0 is a certain constant which depend only on β.



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 547

Moreover, from (1) we see that

lim
z→0

jγ+ d
2
−1(z)− 1

z2
6= 0. (3)

Lemma 3. [7] Let f ∈ Lpl (R
d). Then

M̂hf(ξ) = jγ+ d
2
−1(h|ξ|)f̂(ξ).

The first and higher order finite differences of f (x) are defined as follows

Zhf(x) = (Mh − I)f(x),

where I is the identity operator Lpl (R
d).

Zkhf(x) = Zh(Zk−1h f(x)) = (Mh − I)kf(x) =

k∑
i=0

(−1)k−i(ki )M
i
hf(x),

where M0
hf(x) = f(x), M i

hf(x) = Mh(M i−1
h f(x)), i = 1, 2, .. and k = 1, 2, ...

From Lemma 3, we obtain

Ẑkhf(ξ) = (jγ+ d
2
−1(h|ξ|)− 1)kf̂(ξ).

By (1), we have∫
Rd
|1− jγ+ d

2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ ≤ Kq‖Zkhf(x)‖qp,l, (4)

where 1
p + 1

q = 1.

2. Dunkl Dini Lipschitz Condition

Definition 1. Let f ∈ Lpl (R
d), and define

‖Zkhf(x)‖p,l ≤ C
hη

(log 1
h)δ

, δ ≥ 0,

i.e.,

‖Zkhf(x)‖p,l = O

(
hη

(log 1
h)δ

)
,

for all x in Rd and for all sufficiently small h,C being a positive constant. Then we say
that f satisfies a d-Dunkl Dini Lipschitz of order η, or f belongs to Lip(η, δ).



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 548

Definition 2. If however

‖Zkhf(x)‖p,l
hη

(log 1
h
)δ

→ 0, as h→ 0,

i.e.,

‖Zkhf(x)‖p,l = O

(
hη

(log 1
h)δ

)
, as h→ 0, δ ≥ 0,

then f is said to be belong to the little d-Dunkl Dini Lipschitz class lip(η, δ).

Remark. It follows immediately from these definitions that

lip(η, δ) ⊂ Lip(η, δ).

Theorem 2. Let η > 1. If f ∈ Lip(η, δ), then f ∈ lip(1, δ).

Proof. For x ∈ Rd , h small and f ∈ Lip(η, δ) we have

‖Zkhf(x)‖p,l ≤ C
hη

(log 1
h)δ

.

Then

(log
1

h
)δ‖Zkhf(x)‖p,l ≤ Chη.

Therefore
(log 1

h)δ

h
‖Zkhf(x)‖p,l ≤ Chη−1,

which tends to zero with h→ 0. Thus

(log 1
h)δ

h
‖Zkhf(x)‖p,l → 0, h→ 0.

Then f ∈ lip(1, δ).

Theorem 3. If η < ν, then Lip(η, 0) ⊃ Lip(ν, 0) and lip(η, 0) ⊃ lip(ν, 0).

Proof. We have 0 ≤ h ≤ 1 and η < ν, then hν ≤ hη.
Then the proof of the theorem is immediate.

3. New Results on Dunkl Dini Lipschitz Class

Theorem 4. Let η > 2k. If f belong to the d-Dunkl Dini Lipschitz class, i.e.,

f ∈ Lip(η, δ), η > 2k, δ ≥ 0.

Then f is equal to the null function in Rd.



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 549

Proof. Assume that f ∈ Lip(η, δ). Then

‖Zkhf(x)‖p,l ≤ C
hη

(log 1
h)δ

.

From (4), we have∫
Rd
|1− jγ+ d

2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ ≤ KqCq
hqη

(log 1
h)qδ

.

Then ∫
Rd
|1− jγ+ d

2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ

h2qk
≤ KqCq

hqη−2qk

(log 1
h)qδ

,

Since η > 2k we have

lim
h→0

hqη−2qk

(log 1
h)qδ

= 0.

Thus

lim
h→0

∫
Rd

(
|1− jγ+ d

2
−1(h|ξ|)|

|ξ|2h2

)qk
|ξ|2qk|f̂(ξ)|qwl(ξ)dξ = 0.

and also from the formula (3) and Fatou’s theorem, we obtain∫
Rd
|ξ|2qk|f̂(ξ)|qwl(ξ)dξ = 0.

Hence |ξ|2kf̂(ξ) = 0 for all ξ ∈ Rd, then f(x) is the null function.

Analog of the Theorem 4, we obtain this theorem.

Theorem 5. Let f ∈ Lpl (R
d). If f belong to lip(2, 0), i.e.,

‖Zkhf(x)‖p,l = O(h2), as h→ 0.

Then f is equal to null function in Rd.

Now, we give another the main result of this paper analog of Theorem 1.

Theorem 6. Let f ∈ Lpl (R
d). If f(x) belong to Lip(η, δ), then∫

|ξ|≥s
|f̂(ξ)|qwl(ξ)dξ = O

(
s−qη

(log s)qδ

)
, s→∞,

where 1
p + 1

q = 1.



S. El Ouadih, R. Daher / Eur. J. Pure Appl. Math, 10 (3) (2017), 544-551 550

Proof. Suppose that f ∈ Lip(η, δ). Then

‖Zkhf(x)‖p,l = O

(
hη

(log 1
h)δ

)
, h→ 0.

From (4), we have∫
Rd
|1− jγ+ d

2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ ≤ Kq‖Zkhf(x)‖qp,l.

If |ξ| ∈ [ 1h ,
2
h ] then h|ξ| ≥ 1 and Lemma 2 implies that

1 ≤ 1

cqk
|1− jγ+ d

2
−1(h|ξ|)|

qk.

Then ∫
1
h
≤|ξ|≤ 2

h

|f̂(ξ)|qwl(ξ)dξ ≤
1

cqk

∫
1
h
≤|ξ|≤ 2

h

|1− jγ+ d
2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ

≤ 1

cqk

∫
Rd
|1− jγ+ d

2
−1(h|ξ|)|

qk|f̂(ξ)|qwl(ξ)dξ

≤ Kq

cqk
‖Zkhf(x)‖qp,l

= O

(
hqη

(log 1
h)qδ

)
.

So we obtain ∫
s≤|ξ|≤2s

|f̂(ξ)|qwl(ξ)dξ ≤ C ′
s−qη

(log s)qδ
,

where C ′ is a positive constant. Now, we have∫
|ξ|≥s
|f̂(ξ)|qwl(ξ)dξ =

∞∑
i=0

∫ 2i+1s

2is
|f̂(ξ)|qwl(ξ)dξ

≤ C ′
(

s−qη

(log s)qδ
+

(2s)−qη

(log 2s)qδ
+

(4s)−qη

(log 4s)qδ
+ · · ·

)
≤ C ′

s−qη

(log s)qδ
(
1 + 2−qη + (2−qη)2 + (2−qη)3 + · · ·

)
≤ Kη

s−qη

(log s)qδ
,

where Kη = C ′(1− 2−qη)−1 since 2−qη < 1.
Consequently ∫

|ξ|≥s
|f̂(ξ)|qwl(ξ)dξ = O

(
s−qη

(log s)qδ

)
, as s→∞.



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