International Journal of Analysis and Applications
ISSN 2291-8639
Volume 9, Number 1 (2015), 39-44
http://www.etamaths.com

DUNKL LIPSCHITZ FUNCTIONS FOR THE GENERALIZED

FOURIER-DUNKL TRANSFORM IN THE SPACE L2α,n

R. DAHER, S. EL OUADIH∗ AND M. EL HAMMA

Abstract. In this paper, using a generalized translation operator, we prove

the estimates for the generalized Fourier-Dunkl transform in the space L2α,n
on certain classes of functions.

1. Introduction and Preliminaries

In [5], E. C. Titchmarsh’s characterizes the set of functions in L2(R) satisfying
the Cauchy-Lipschitz condition by means of an asymptotic estimate growth of the
norm of their Fourier transform, namely we have
Theorem 1.1 Let δ ∈ (0, 1) and assume that f ∈ L2(R). Then the following are
equivalents
(i) ‖f(t + h) −f(t)‖ = O(hδ), as h → 0,

(ii)

∫
|λ|≥r

|f̂(λ)|2dλ = O(r−2δ) as r →∞,

where f̂ stands for the Fourier transform of f.

In this paper, we consider a first-order singular differential-difference operator Λ on
R which generalizes the Dunkl operator Λα, We prove an analog of Theorem 1.1 in
the generalized Fourier-Dunkl transform associated to Λ in L2α,n . For this purpose,
we use a generalized translation operator. We point out that similar results have
been established in the context of non compact rank one Riemannian symetric s-
paces [4].
In this section, we develop some results from harmonic analysis related to the
differential-difference operator Λ. Further details can be found in [1] and [6]. In all
what follows assume where α > −1/2 and n a non-negative integer.
Consider the first-order singular differential-difference operator on R

Λf(x) = f′(x) +

(
α +

1

2

)
f(x) −f(−x)

x
− 2n

f(−x)
x

.

For n = 0, we regain the differential-difference operator

Λαf(x) = f
′(x) +

(
α +

1

2

)
f(x) −f(−x)

x
,

2010 Mathematics Subject Classification. 46L08.
Key words and phrases. differential-difference operator; generalized Fourier-Dunkl transform;

generalized translation operator.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

39



40 DAHER, OUADIH AND HAMMA

which is referred to as the Dunkl operator of index α + 1/2 associated with the re-
flection group Z2 on R. Such operators have been introduced by Dunkl (see [3], [7])
in connection with a generalization of the classical theory of spherical harmonics.
Let M be the map defined by

Mf(x) = x2nf(x), n = 0, 1, ...

Let Lpα,n, 1 ≤ p < ∞, be the class of measurable functions f on R for which

‖f‖p,α,n = ‖M−1f‖p,α+2n < ∞,

where

‖f‖p,α =
(∫

R
|f(x)|p|x|2α+1dx

)1/p
.

If p = 2, then we have L2α,n = L
2(R, |x|2α+1).

The one-dimensional Dunkl kernel is defined by

eα(z) = jα(iz) +
z

2(α + 1)
jα+1(iz),z ∈ C,(1)

where

jα(z) = Γ(α + 1)

∞∑
m=0

(−1)m(z/2)2m

m!Γ(m + α + 1)
,z ∈ C,(2)

is the normalized spherical Bessel function of index α. It is well-known that the
functions eα(λ.), λ ∈ C, are solutions of the differential-difference equation

Λαu = λu,u(0) = 1.

Lemma 1.2 For x ∈ R the following inequalities are fulfilled
i) |jα(x)| ≤ 1,
ii) |1 − jα(x)| ≤ |x|,
iii) |1 − jα(x)| ≥ c with |x| ≥ 1, where c > 0 is a certain constant which depends
only on α.
Proof. Similarly as the proof of Lemma 2.9 in [2].
For λ ∈ C, and x ∈ R, put

ϕλ(x) = x
2neα+2n(iλx),

where eα+2n is the Dunkl kernel of index α + 2n given by (1).
Proposition 1.3
i) ϕλ satisfies the differential equation

Λϕλ = iλϕλ.

ii) For all λ ∈ C, and x ∈ R

|ϕλ(x)| ≤ |x|2ne|Imλ||x|.

The generalized Fourier-Dunkl transform we call the integral transform

FΛf(λ) =
∫
R
f(x)ϕ−λ(x)|x|2α+1dx,λ ∈ R,f ∈ L1α,n.



DUNKL LIPSCHITZ FUNCTIONS 41

Let f ∈ L1α,n such that FΛ(f) ∈ L1α+2n = L1(R, |x|2α+4n+1dx). Then the inverse
generalized Fourier-Dunkl transform is given by the formula

f(x) =

∫
R
FΛf(λ)ϕλ(x)dµα+2n(λ),

where

dµα+2n(λ) = aα+2n|λ|2α+4n+1dλ, aα =
1

22α+2(Γ(α + 1))2
.

Proposition 1.4
i) For every f ∈ L2α,n,

FΛ(Λf)(λ) = iλFΛ(f)(λ).

ii) For every f ∈ L1α,n ∩L2α,n we have the Plancherel formula∫
R
|f(x)|2|x|2α+1dx =

∫
R
|FΛf(λ)|2dµα+2n(λ).

iii) The generalized Fourier-Dunkl transform FΛ extends uniquely to an isometric
isomorphism from L2α,n onto L

2(R,µα+2n).

The generalized translation operators τx, x ∈ R, tied to Λ are defined by

τxf(y) =
(xy)2n

2

∫ 1
−1

f(
√
x2 + y2 − 2xyt)

(x2 + y2 − 2xyt)n

(
1 +

x−y√
x2 + y2 − 2xyt

)
A(t)dt

+
(xy)2n

2

∫ 1
−1

f(−
√
x2 + y2 − 2xyt)

(x2 + y2 − 2xyt)n

(
1 −

x−y√
x2 + y2 − 2xyt

)
A(t)dt,

where

A(t) =
Γ(α + 2n + 1)

√
πΓ(α + 2n + 1/2)

(1 + t)(1 − t2)α+2n−1/2.

Proposition 1.5 Let x ∈ R and f ∈ L2α,n. Then τxf ∈ L2α,n and

‖τxf‖2,α,n ≤ 2x2n‖f‖2,α,n.

Furthermore,

FΛ(τxf)(λ) = x2neα+2n(iλx)FΛ(f)(λ).(3)

2. Main Results

In this section we give the main result of this paper. We need first to define
(ψ,δ,β)-generalized Dunkl Lipschitz class.
Definition 2.1. Let δ > 1 and β > 0. A function f ∈ L2α,n is said to be in the
(ψ,δ,β)-generalized Dunkl Lipschitz class, denoted by DLip(ψ,δ,β), if

‖τhf(x) + τ−hf(x) − 2h2nf(x)‖2,α,n = O(hδ+2nψ(hβ)) as h → 0,

where
(a) ψ is a continuous increasing function on [0,∞),



42 DAHER, OUADIH AND HAMMA

(b) ψ(0) = 0 , ψ(ts) = ψ(t)ψ(s) for all t,s ∈ [0,∞),
(c) and ∫ 1/h

0

s1−2δψ(s−2β)ds = O(h2δ−2ψ(h2β)), h → 0.

Theorem 2.2. Let f ∈ L2α,n. Then the following are equivalents

(a) f ∈ DLip(ψ,δ,β)

(b)

∫
|λ|≥r

|FΛf(λ)|2dµα+2n(λ) = O(r−2δψ(r−2β)), as r →∞.

Proof. (a) ⇒ (b). Let f ∈ DLip(ψ,δ,β). Then we have

‖τhf(x) + τ−hf(x) − 2h2nf(x)‖2,α,n = O(hδ+2nψ(hβ)) as h → 0.

From formulas (1), (2) and (3), we have the generalized Fourier-Dunkl transform
of τhf(x) + τ−hf(x) − 2h2nf(x) is 2h2n(jα+2n(λh) − 1)FΛf(λ).
By Plancherel equality, we obtain

‖τhf(x)+τ−hf(x)−2h2nf(x)‖22,α,n = 4h
4n

∫ +∞
−∞

|jα+2n(λh)−1|2|FΛf(λ)|2dµα+2n(λ).

If |λ| ∈ [ 1
h
, 2
h

], then |λh| ≥ 1 and (iii) of Lemma 1.2 implies that

1 ≤
1

c2
|jα+2n(λh) − 1|2.

Then∫
1
h
≤|λ|≤2

h

|FΛf(λ)|2dµα+2n(λ) ≤
1

c2

∫
1
h
≤|λ|≤2

h

|jα+2n(λh) − 1|2|FΛf(λ)|2dµα+2n(λ)

≤
1

c2

∫ +∞
−∞

|jα+2n(λh) − 1|2|FΛf(λ)|2dµα+2n(λ)

≤
h−4n

4c2
‖τhf(x) + τ−hf(x) − 2h2nf(x)‖22,α,n

= O(h2δψ(h2β)).

We obtain ∫
r≤|λ|≤2r

|FΛf(λ)|2dµα+2n(λ) ≤ Cr−2δψ(r−2β), r →∞,

where C is a positive constant. Now,∫
|λ|≥r

|FΛf(λ)|2dµα+2n(λ) =
∞∑
i=0

∫
2ir≤|λ|≤2i+1r

|FΛf(λ)|2dµα+2n(λ)

≤ Cr−2δψ(r−2β)
∞∑
i=0

(2−2δψ(2−2β))i

≤ CCδ,βr−2δψ(r−2β),

where Cδ,β = (1 − 2−2δψ(2−2β)))−1 since 2−2δψ(2−2β) < 1.
Consequently∫

|λ|≥r
|FΛf(λ)|2dµα+2n(λ) = O(r−2δψ(r−2β)), as r →∞.



DUNKL LIPSCHITZ FUNCTIONS 43

(b) ⇒ (a). Suppose now that∫
|λ|≥r

|FΛf(λ)|2dµα+2n(λ) = O(r−2δψ(r−2β)), as r →∞.

and write

‖τhf(x) + τ−hf(x) − 2h2nf(x)‖22,α,n = 4h
4n(I1 + I2),

where

I1 =

∫
|λ|< 1

h

|jα+2n(λh) − 1|2|FΛf(λ)|2dµα+2n(λ),

and

I2 =

∫
|λ|≥1

h

|jα+2n(λh) − 1|2|FΛf(λ)|2dµα+2n(λ).

Firstly, we use the formulas |jα+2n(λh)| ≤ 1 and

I2 ≤ 4
∫
|λ|≥1

h

|FΛf(λ)|2dµα+2n(λ) = O(h2δψ(h2β)), as h → 0.

Set

φ(x) =

∫ +∞
x

|FΛf(λ)|2dµα+2n(λ).

Integrating by parts we obtain∫ x
0

λ2|FΛf(λ)|2dµα+2n(λ) =
∫ x

0

−λ2φ′(λ)dλ = −x2φ(x) + 2
∫ x

0

λφ(λ)dλ

≤ C1
∫ x

0

λ1−2δψ(λ−2β)dλ = O(x2−2δψ(x−2β)),

where C1 is a positive constant.
We use the formula (ii) of Lemma 1.2∫ +∞
−∞

|jα+2n(λh) − 1|2|FΛf(λ)|2dµα+2n(λ) = O

(
h2
∫
|λ|< 1

h

λ2|FΛf(λ)|2dµα+2n(λ)

)
+ O(h2δψ(h2β))

= O(h2h−2+2δψ(h2β)) + O(h2δψ(h2β))

= O(h2δψ(h2β)),

and this ends the proof.�

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44 DAHER, OUADIH AND HAMMA

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Department of Mathematics, Faculty of Sciences Äın Chock, University Hassan II,
Casablanca, Morocco

∗Corresponding author