International Journal of Analysis and Applications
ISSN 2291-8639
Volume 9, Number 2 (2015), 142-150
http://www.etamaths.com

CHARACTERIZATION OF (δ,γ)-DINI-LIPSCHITZ FUNCTIONS

IN TERMS OF THEIR JACOBI-DUNKL TRANSFORMS

A. BELKHADIR1,∗, A. ABOUELAZ2 AND R. DAHER3

Abstract. In this paper, we are going to define a generalized Dini-Lipschitz

class and give a characterization for functions belonging to by means of an
asymptotic estimating growth of the norm of their Jacobi-Dunkl transforms.

1. Introduction and Preliminaries

Younis theorem 5.2 [3] characterizes the set of functions in L2(R) satisfying the
Dini-Lipschitz condition by means of an asymptotic estimating growth of the norm
of their Fourier transforms. i.e.

Theorem 1.1. [3] Let δ ∈ (0, 1) , γ > 0 and f ∈ L2(R) . Then the following
conditions are equivalents:

(1) ‖f(t + h) −f(t)‖L2(R) = O
(
hδ(log 1

h
)−γ
)

, as h → 0 ;

(2)

∫
|λ|≥r

|f̂(λ)|2dλ = O
(
r−2δ(log r)−2γ

)
, as r → +∞ .

where f̂ is the Fourier transform of f .

In the following, Let α,β and ρ denote 3 reals such that α ≥ β ≥ −1
2

, α 6=
−1

2
and ρ = α + β + 1,

Aα,β(x) = 2
ρ(sinh |x|)2α+1(cosh |x|)2β+1.

In [1] we have established a characterization of functions f in L2(R,Aα,β(x)dx)
satisfying a certain Lipschitz condition, namely the equivalence between the two
following conditions:

(1) ||∆hf|| = ||τhf + τ−hf − 2f||L2(R,Aα,β(x)dx) = O(h
δ) , as h → 0;

(2)

∫
|λ|≥r

|Fα,β(f)(λ)|2dσ(λ) = O(r−2δ) , as r →∞ .

where Fα,β(f) stands for the Jacobi-Dunkl transform of f, and τh is the related
generalized translation operator .

This result has been generalized in [2] by using the higher powers: Λrα,β and

∆khf = ∆h(∆
k−1
h f), r ∈ N, k ∈ N

∗ . In this way, we are going in section 3 to define
a generalized Dini-Lipschitz class DLip[2, (δ,γ),k,r], δ ∈ (0, 1), γ > 0, and give a
characterization for functions belonging to by means of an asymptotic estimating

2010 Mathematics Subject Classification. 47B36, 46E35.
Key words and phrases. Younis theorem; Generalized Jacobi-Dunkl translation; Jacobi-Dunkl

transform; Dini-Lipschitz class; Sobolev spaces.

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.

142



CHARACTERIZATION OF (δ,γ)-DINI-LIPSCHITZ FUNCTIONS 143

growth of the norm of their Jacobi-Dunkl transforms, i.e. we show the equivalence
of the two following conditions:

(1) f ∈ DLip[2, (δ,γ),k,r] ;

(2)

∫ ∞
s

λ2r |Fα,β(f)(λ)|
2
dσα,β(λ) = O

(
s−2δ(log s)−2γ

)
, as s → +∞ .

In the following section we recapitulate some results related to the harmonic
analysis associated with the Jacobi-Dunkl operator Λα,β (see [4, 5, 6, 7, 9]).

2. Notations and Preliminaries

Notations:

• dσα,β(λ) =
|λ|

8π
√
λ2 −ρ2|Cα,β(

√
λ2 −ρ2)|

IR\]−ρ,ρ[(λ)dλ

where, Cα,β(µ) =
2ρ−iµΓ(α + 1)Γ(iµ)

Γ( 1
2
(ρ + iµ))Γ( 1

2
(α−β + 1 + iµ))

, µ ∈ C\ (iN) .

and IΩ is the characteristic function of Ω .
• Lp(Aα,β) (resp. Lp(σα,β) ,p ∈]0, +∞[ , the space of measurable functions

g on R such that

||g||Lp(Aα,β) =
(∫

R
|g(t)|pAα,β(t)dt

)1/p
< +∞ .

(resp. ||g||Lp(σα,β) =
(∫

R
|g(λ)|pdσα,β(λ)

)1/p
< +∞) .

• D(R) the space of C∞-functions on R with compact support.
• S(R) the usual Schwartz space of C∞-functions on R rapidly decreasing

together with their derivatives, equipped with the topology of semi-norms
Lm,n , (m,n) ∈ N2 , where

Lm,n(f) = sup
x∈R,0≤k≤n

[
(1 + x2)m

∣∣∣∣ dkdxk f(x)
∣∣∣∣
]
< +∞.

• S1(R) = {(cosh t)−2ρf; f ∈S(R)} .
The topology of this space is given by the semi-norms L1m,n , (m,n) ∈ N2 ,
where

L1m,n(f) = sup
x∈R,0≤k≤n

[
(cosh t)−2ρ(1 + x2)m

∣∣∣∣ dkdxk f(x)
∣∣∣∣
]
< +∞.

•
(
S1(R)

)′
the topological dual of S1(R) .

Now, we introduce the Jacobi-Dunkl Transform and its basic properties:

The Jacobi-Dunkl function with parameters (α,β) , α ≥ β ≥ −1
2
,α 6= −1

2
, is

defined by :

(1) ∀x ∈ R, ψ(α,β)λ (x) =

{
ϕ

(α,β)
µ (x) −

i

λ

d

dx
ϕ

(α,β)
µ (x) , if λ ∈ C\{0};

1 , if λ = 0.



144 BELKHADIR, ABOUELAZ AND DAHER

with λ2 = µ2 + ρ2, ρ = α + β + 1 and ϕ
(α,β)
µ is the Jacobi function given by:

(2) ϕ(α,β)µ (x) = F

(
ρ + iµ

2
,
ρ− iµ

2
; α + 1,−(sinh x)2

)
,

where F is the Gaussian hypergeometric function given by

F(a,b,c,z) =

∞∑
m=0

(a)m(b)m
(c)mm!

zm , |z| < 1,

a,b,z ∈ C and c /∈−N;
(a)0 = 1, (a)m = a(a + 1)...(a + m− 1) . (see [4, 10, 11]).
ψ

(α,β)
λ is the unique C

∞-solution on R of the differentiel-difference equation

(3)

{
Λα,βu = iλu , λ ∈ C;
u(0) = 1.

where Λα,β is the Jacobi-Dunkl operator given by:

Λα,βu(x) =
du

dx
(x) +

A′α,β(x)

Aα,β(x)
×
u(x) −u(−x)

2
; i.e.

Λα,βu(x) =
du

dx
(x) + [(2α + 1) coth x + (2β + 1) tanh x] ×

u(x) −u(−x)
2

.

The function ψ
(α,β)
λ can be written in the form below (See [5]),

(4) ψ
(α,β)
λ (x) = ϕ

(α,β)
µ (x) + i

λ

4(α + 1)
sinh(2x)ϕ(α+1,β+1)µ (x) , ∀x ∈ R ,

where λ2 = µ2 + ρ2 , ρ = α + β + 1.

The Jacobi-Dunkl transform of a function f ∈ L1(Aα,β) is defined by :

(5) Fα,β(f)(λ) =
∫
R
f(x)ψ

(α,β)
−λ (x)Aα,β(x)dx, ∀λ ∈ R ;

The inverse Jacobi-Dunkl transform of a function h ∈ L1(σα,β) is:

(6) F−1α,β(h)(t) =
∫
R
h(λ)ψ

(α,β)
λ (t)dσα,β(λ) .

Fα,β is a topological isomorphism from S1(R) onto S(R) , and extends uniquely to
a unitary isomorphism from L2(Aα,β) onto L

2(σα,β) . The Plancherel formula is
given by

(7) ‖f‖L2(Aα,β) = ‖Fα,β(f)‖L2(σα,β) .

For f ∈S1(R) we have the following inversion formula

(8) f(x) =

∫
R
Fα,β(f)(λ)ψ

(α,β)
λ (x)dσα,β(λ) , ∀x ∈ R ,

and the relation

(9) Fα,β(Λα,βf)(λ) = iλFα,β(f)(λ) .



CHARACTERIZATION OF (δ,γ)-DINI-LIPSCHITZ FUNCTIONS 145

Let f ∈ L2(Aα,β) . For all x ∈ R the operator of Jacobi-Dunkl translation τx is
defined by:

(10) τxf(y) =

∫
R
f(z)dνα,βx,y (z) , ∀ y ∈ R .

where να,βx,y , x,y ∈ R are the signed measures given by

(11) dνα,βx,y (z) =




Kα,β(x,y,z)Aα,β(z)dz , if x,y ∈ R∗;
δx , if y = 0;
δy , if x = 0.

Here, δx is the Dirac measure at x. And

Kα,β(x,y,z) = Mα,β(sinh(|x|) sinh(|y|) sinh(|z|))−2αIIx,y ×
∫π

0
ρθ(x,y,z)

×(gθ(x,y,z))
α−β−1
+ sin

2β θdθ.

Ix,y = [−|x|− |y|,−||x|− |y||] ∪ [||x| + |y||, |x| + |y|] ,
ρθ(x,y,z) = 1 −σθx,y,z + σ

θ
z,x,y + σ

θ
z,y,x

σθx,y,z =




cosh(x) + cosh(y) − cosh(z) cos(θ)
sinh(x) sinh(y)

, if xy 6= 0;

0 , if xy = 0.

for all x,y,z ∈ R , θ ∈ [0,π].

gθ(x,y,z) = 1 − cosh2 x− cosh2 y − cosh2 z + 2 cosh x cosh y cosh z cos θ .

t+ =

{
t , if t > 0;
0 , if t ≤ 0.

and

Mα,β =




2−2ρΓ(α + 1)
√
πΓ(α−β)Γ(β + 1

2
)

, if α > β;

0 , if α = β.

We have

(12) Fα,β(τhf)(λ) = ψ
α,β
λ (h).Fα,β(f)(λ) ; h,λ ∈ R .

Let g ∈ L2(σα,β) . Then the distribution Tgσα,β defined by

(13) 〈Tgσα,β,ϕ〉 =
∫
R
g(λ)ϕ(λ)dσα,β(λ) , ϕ ∈D(R) ,

belongs to S′(R) .
Let f ∈ L2(Aα,β) . Then the distribution TfAα,β defined by

(14) 〈TfAα,β,ϕ〉 =
∫
R
f(x)ϕ(x)Aα,β(x)dx, ϕ ∈S1(R) ,

belongs to
(
S1(R)

)′
.

Via the correspondance f 7→ TfAα,β , we identify L
2(Aα,β) as a subspace of(

S1(R)
)′

.



146 BELKHADIR, ABOUELAZ AND DAHER

The jacobi-dunkl transform of a distribution T ∈
(
S1(R)

)′
is defined by:

(15) 〈Fα,β(T),ϕ〉 = 〈T,F−1α,β(ϕ̌)〉 , ϕ ∈S(R) ,

where ϕ̌ is given by ϕ̌(x) = ϕ(−x) .
It is clear that Fα,β(T) ∈S′(R) .

The jacobi-dunkl transform of a distribution defined by f ∈ L2(Aα,β) is given
by the distribution TFα,β(f)σα,β ; i.e.

(16) Fα,β(TfAα,β ) = TFα,β(f)σα,β .

We identify the tempered distribution given by Fα,β(f) and the function Fα,β(f) .
Let T ∈

(
S1(R)

)′
and consider the distribution Λα,βT defined by

(17) 〈Λα,β(T),ϕ〉 = −〈T, Λα,β(ϕ)〉 , for all ϕ ∈S1(R) .

(Note that S1(R) is unvariant under Λα,β) .
By using (9) it is easy to see that

(18) Fα,β(Λα,β(T)) = iλFα,β(T) .

For f ∈ L2(Aα,β) , we define the finite differences of first and higher order as
follows:

∆1hf = ∆hf = τhf + τ−hf − 2f = (τh + τ−h − 2E)f;
∆khf = ∆h(∆

k−1
h )f = (τh + τ−h − 2E)

kf , k = 2, 3, ...;

where E is the unit operator in L2(Aα,β) .

Lemma 2.1. The following inequalities are valids for Jacobi functions ϕα,βµ (h)

(1) |ϕ(α,β)µ (h)| ≤ 1 ;
(2) |1 −ϕ(α,β)µ (h)| ≤ h2λ2; where λ2 = µ2 + ρ2 .

Proof. (See [12], Lemmas 3.1-3.2) �

For α ≥ −1
2

, we introduce the Bessel normalized function of the first kind defined
by

jα(z) = Γ(α + 1)

∞∑
n=0

(−1)n(z
2
)2n

n!Γ(n + α + 1)
, z ∈ C.

We see that lim
z→0

jα(z) − 1
z2

6= 0 , by consequence, there exists c1 > 0 and η > 0
satisfying

(19) |z| ≤ η ⇒ |jα(z) − 1| ≥ c1|z|2 .

Lemma 2.2. Let α ≥ β ≥ −1
2

, α 6= −1
2

. Then for |υ| ≤ ρ , there exists a
positive constant c2 such that

|1 −ϕ(α,β)µ+iυ(t)| ≥ c2|1 − jα(µt)| .

Proof. (See [8], Lemma 9) �



CHARACTERIZATION OF (δ,γ)-DINI-LIPSCHITZ FUNCTIONS 147

3. Main Results

We denote by W
2,k
α,β , k ∈ N

∗ , the Sobolev space constructed by the operator
Λα,β ; i.e.

(20) W
2,k
α,β =

{
f ∈ L2(Aα,β); Λ

j
α,βf ∈ L

2(Aα,β), j = 0, 1, 2, ...,k
}

;

where, Λ0α,βf = f, Λ
1
α,βf = Λα,βf , Λ

r
α,βf = Λα,β(Λ

r−1
α,β f), r = 2, 3, ...

Lemma 3.1. Let f ∈ W 2,kα,β , k ∈ N
∗ . Then∥∥∆khΛrα,βf∥∥2L2(Aα,β) = 22k

∫
R
λ2r|1 −ϕµ(h)|2k|Fα,β(f)(λ)|2dσα,β(λ) ,

where r = 0, 1, ...,k.

Proof. Using the eveness of ϕµ and formula (4) we get

Fα,β(τhf + τ−hf − 2f)(λ = (ψ
(α,β)
λ (h) + ψ

(α,β)
λ (−h) − 2).Fα,β(f)(λ)

= 2(ϕ(α,β)µ (h) − 1).Fα,β(f)(λ).
and

(21) Fα,β(∆khf)(λ) = 2
k(ϕ(α,β)µ (h) − 1)

k.Fα,β(f)(λ).
Furthermore, we obtain by the formula (18)

(22) Fα,β(Λrα,βf)(λ) = (iλ)
rFα,β(f)(λ) .

Using the formulas (21) and (22) we get

Fα,β(∆khΛ
r
α,βf)(λ) = 2

k(iλ)r.(ϕ(α,β)µ (h) − 1)
k.Fα,β(f)(λ).

By the Plancherel formula (7), we have the result. �

Definition 3.2. Let δ ∈ (0, 1),γ > 0 and k ∈ N∗ . A function f ∈ W 2,kα,β is said
to be in the (δ,γ)-Dini-Lipschitz class, denoted by DLip[2, (δ,γ),k,r] , if∥∥∆khΛrα,βf∥∥L2(Aα,β) = O

(
hδ(log

1

h
)−γ
)
, as h −→ 0,

where r = 0, 1, ...,k.

Theorem 3.3. Let f ∈ W 2,kα,β , k ∈ N
∗ . Then the following are equivalents:

(1) f ∈ DLip[2, (δ,γ),k,r] ;

(2)

∫ ∞
s

λ2r |Fα,β(f)(λ)|
2
dσα,β(λ) = O

(
s−2δ(log s)−2γ

)
, as s → +∞ .

Proof. (1) ⇒ (2): Assume that f ∈ DLip[2, (δ,γ),k,r] ; then∥∥∆khΛrα,βf∥∥L2(Aα,β) = O
(
hδ(log

1

h
)−γ
)

as h −→ 0.

by lemma 3.1, we have

∫
R
λ2r|1 −ϕµ(h)|2k|Fα,β(f)(λ)|2dσα,β(λ) = O

(
h2δ(log

1

h
)−2γ

)



148 BELKHADIR, ABOUELAZ AND DAHER

If |λ| ∈ [ η
2h
, η
h

] then |µh| ≤ η (recall that λ2 = µ2 + ρ2). We get by (19): |jα(µh)−

1| ≥ c1µ2h2. From |λ| ≥
η

2h
we have, µ2h2 ≥

η2

4
− ρ2h2; then we can find

a positive constant c3 = c3(η,α,β) such that µ
2h2 ≥ c3 (take h < 1) ; thus,

|jα(µh) − 1| ≥ c1c3. This inequality and lemma 2.2 implys that: |1 −ϕ
(α,β)
µ (h)| ≥

c1c2c3 = C.

Hence, 1 ≤
1

C2
|1 −ϕ(α,β)µ (h)|2. Then,

∫
η
2h
≤|λ|≤η

h

λ2r|Fα,β(f)(λ)|2dσα,β(λ) ≤
1

C2k

∫
η
2h
≤|λ|≤η

h

λ2r|1 −ϕ(α,β)µ (h)|
2k

×|Fα,β(f)(λ)|2dσα,β(λ)

≤
1

C2k

∫
R
λ2r|1 −ϕ(α,β)µ (h)|

2k|Fα,β(f)(λ)|2dσα,β(λ)

= O

(
h2δ(log

1

h
)−2γ

)
.

Then we have,

∫
s≤|λ|≤2s

λ2r|Fα,β(f)(λ)|2dσα,β(λ) = O
(
s−2δ(log s)−2γ

)
, as s → +∞.

Or equivalently

∫
s≤|λ|≤2s

λ2r|Fα,β(f)(λ)|2dσα,β(λ) ≤ K1O
(
s−2δ(log s)−2γ

)
, as s → +∞,

where K1 is some positive constant . It follows that,

∫
|λ|≥s

λ2r|Fα,β(f)(λ)|2dσα,β(λ) =
∞∑
i=0

∫
2is≤|λ|≤2i+1s

λ2r|Fα,β(f)(λ)|2dσα,β(λ)

≤ K1
∞∑
i=0

(2is)−2δ(log 2is)−2γ

≤ K1

(
∞∑
i=0

(2i)−2δ

)(
s−2δ(log s)−2γ

)
≤ K

(
s−2δ(log s)−2γ

)
.

where K =
K1

1 − 2−2δ
. This proves that:

∫
|λ|≥s

λ2r|Fα,β(f)(λ)|2dσα,β(λ) = O
(
s−2δ(log s)−2γ

)
, as s → +∞.

(2) ⇒ (1) : Suppose now that∫
|λ|≥s

λ2r|Fα,β(f)(λ)|2dσα,β(λ) = O
(
s−2δ(log s)−2γ

)
, as s → +∞.

we have to show that:



CHARACTERIZATION OF (δ,γ)-DINI-LIPSCHITZ FUNCTIONS 149

∫
R
λ2r|1 −ϕ(α,β)µ (h)|

2k|Fα,β(f)(λ)|2dσα,β(λ) = O
(
h2δ(log

1

h
)−2γ

)
, as h → 0.

Write: ∫
R
λ2r|1 −ϕ(α,β)µ (h)|

2k|Fα,β(f)(λ)|2dσα,β(λ) = I1 + I2,

where:

I1 =

∫
|λ|≤1

h

λ2r|1 −ϕ(α,β)µ (h)|
2k|Fα,β(f)(λ)|2dσα,β(λ) ;

I2 =

∫
|λ|> 1

h

λ2r|1 −ϕ(α,β)µ (h)|
2k|Fα,β(f)(λ)|2dσα,β(λ).

Estimate I1 and I2 . From (1) of lemma 2.1 we can write,

I2 ≤ 4k
∫
|λ|> 1

h

λ2r|Fα,β(f)(λ)|2dσα,β(λ) , (s =
1

h
)

= O

(
h2δ(log

1

h
)−2γ

)
.

Using the inequalities (1) and (2) of lemma 2.1 we get

I1 =

∫
|λ|≤1

h

λ2r|1 −ϕ(α,β)µ (h)|
2k|Fα,β(f)(λ)|2dσα,β(λ)

≤ 22k−1
∫
|λ|≤1

h

λ2r|1 −ϕ(α,β)µ (h)|.|Fα,β(f)(λ)|
2dσα,β(λ)

≤ 22k−1h2
∫
|λ|≤1

h

λ2r.λ2|Fα,β(f)(λ)|2dσα,β(λ).

Consider the function

ψ(s) =

∫ ∞
s

λ2r|Fα,β(f)(λ)|2dσα,β(λ).

Since ψ(s) = O
(
s−2δ(log s)−2γ

)
, an integration by parts gives:

22k−1h2
∫ 1

h

0

λ2r.λ2|Fα,β(f)(λ)|2dσα,β(λ) = 22k−1h2
∫ 1

h

0

(
−s2ψ′(s)

)
ds

= 22k−1h2

(
−

1

h2
ψ(

1

h
) + 2

∫ 1
h

0

sψ(s)ds

)

≤ 22k−1h2
∫ 1

h

0

sψ(s)ds

≤ C1.h2
∫ 1

h

0

s1−2δ(log s)−2γds

≤ C2.h2δ(log
1

h
)−2γ.



150 BELKHADIR, ABOUELAZ AND DAHER

Hence,

I1 = O

(
h2δ(log

1

h
)−2γ

)
.

Finally we get

I1 + I2 = O

(
h2δ(log

1

h
)−2γ

)
+ O

(
h2δ(log

1

h
)−2γ

)
= O

(
h2δ(log

1

h
)−2γ

)
Which completes the proof of the theorem.

�

Corollary 3.4. Let f ∈ W 2,kα,β such that f ∈ DLip[2, (δ,γ),k,r] . Then∫ ∞
s

|Fα,β(f)(λ)|
2
dσα,β(λ) = O

(
s−2(δ+r)(log s)−2γ

)
, as s → +∞.

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α,β

(R), International Journal of Analysis and Applications,
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[2] A. Belkhadir and A. Abouelaz, Generalization of Titchmarsh’s theorem for the Jacobi-Dunkl
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[3] M. S. Younis, Fourier Transforms of Dini-Lipschitz functions, International Journal of Math-

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Department of Mathematics and Informatics, Faculty of Science Ain Chock, Univer-
sity of Hassan II, Casablanca, Morocco

∗Corresponding author