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Extremal Functions and Calderon’s Formulas for the Riemann-Liouville Two-Wavelet Transform

Ahmed Chana∗, Abdellatif Akhlidj
Laboratory of Fundamental and Applied Mathematics, Department of Mathematics and Informatics, Facultyof Sciences Ain Chock, University of Hassan II, B.P 5366 Maarif, Casablanca, Moroccomaths.chana@gmail.com, akhlidj@hotmail.fr

∗Correspondence: maths.chana@gmail.com
Abstract. The Riemann-Liouville operator has been extensively investigated and his witnessed aremarkable development in numerous fields of harmonic analysis. Knowing the fact of the study ofthe time-frequency analysis are both theoritically interesting and pratically useful, we investigatedseveral problems for this subject on the setting of the Riemann-Liouville wavelet transform. Firstly,we introduce the notion of Riemann-Liouville two-wavelet and we present generalized version ofParseval’s, Plancherel’s, inversion and Calderon’s reproducing formulas. Next, using the theory ofreproducing kernels, we give best estimates and an integral representation of the extremal functionsrelated to the Riemann-Liouville wavelet transform on weighted Sobolev spaces.

1. Introduction
The mean operator is defined for a continuous function on R2, even with respect to the firstvariable by

R0(f )(x, t) =
1

2π

∫ 2π
0

f (x sin θ, t + x cos θ)dθ.Which means that R0(f )(x, t) is the mean value of f on the circle centered at (0, t) and radius
x . The operators R0 play an mportant role and has many applications, for example, in imageprocessing of so-called synthetic aperture radar (SAR) data see [10,11], or in the linearized inversescattering problem in acoustics see [5, 7].In [3], the authors have generalized R0 and its dual tR0 by introducing the so-called Riemann-Liouville operator defined on the space of continuous functions on R2, even with the respect to the

Received: 17 Aug 2024.Key words and phrases. Riemann-Liouville operator, Wavelet transform, Time-Frequency Analysis, Extremal Func-tions, Weighted Sobolev spaces.

1

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https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 2first variable by
Rα(f )(x, t) :=


α
π

∫ 1
−1
∫ 1
−1 f

(
xs
√

1− y2, t + xy
) (

1− y2
)α− 1

2
(

1− s2
)α−1

dyds if α > 0,

1
π

∫ 1
−1 f

(
r
√

1− y2, t + xy
)

dt√
1−t2 if α = 0,(1.1)Many harmonic analysis results related to the Riemann-Liouville operator (1.1) have been estab-lished see [1–4] and the references therein. The wavelet transform has a long story which stared in1984 with Mortel, a French petroleum engineer in connection with his study of seismic traces, themathematical foundations were given by Grossman and Mortel in [8, 9]. Mortel defined a waveletas a collection of functions constructed by using translation and dilatation of a single function

ψ ∈ L2(R) called the mother wavelet by:
ψb,a(t) =

1√
a
ψ(
t − b
a

),

where a > 0 is called the scaling parameter, wich measure the degree of compression and b ∈ Ris a translation parameter wich determines the time location of the wavelet. The theory of wavelethave applications in several research area as signal theory, time frequency analysis, geophysicsand medicine see [6, 9].A lot of attention has been given to various generalization of the classical Fourier transform,this paper focuses on the generalized Fourier transform associated with the Riemann-Liouvilleoperator (1.1) called the Riemann-Liouville transform, more precisely we consider a system ofpartial differential operator ∆1 and ∆2 defined by
∆1 :=

∂

∂x
; α ≥ 0, t > 0,

∆2 :=
∂2

∂t2
+

2α+ 1

t

∂

∂t
−
∂2

∂x2
, α ≥ 0.

From [3], the authors gives the connection between the eigenfunctions of this system denoted by
ϕµ,λ with (µ, λ) ∈ C2 and the Riemann-Liouville operator (1.1) as follows:

ϕµ,λ(x, t) = Rα(cos(µ.) exp(−iλ·))(x, t). (1.2)
Wavelet analysis has attracted attention for its ability to analyse rapidly changing transient signals,any application using the Fourier like transform can be formulated using wavelets to provide moretime and frequency information.The reason for the extension from one wavelet to two-wavelet comes from the extra degree offlexibility in signal analysis and imaging when the localization operators are used as time-varingfilters. This paper is an attempt to fill this gap by extending one wavelet to two wavelets in theRiemann-Liouville setting. The remainder of this paper is arranged as follows, in section 2 werecall the main results concerning the harmonic analysis associated with the Riemann-Liouvilletransform, in section 3 we introduce the notion of Riemann-Liouville two-wavelet and we give

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 3a generalized version of Parseval’s, Plancherel’s, inversion and Calderon’s reproducing formulasrelated to this transform, the last section is devoted to give an integral representation and bestestimates of extremal functions related to the Riemann-Liouville wavelet transform on weightedSobolev spaces.
2. Harmonic Analysis Associated with the Riemann-Liouville Operator

In this section we set some notations and we recall some results in harmonic analysis relatedto the Riemann-Liouville operator (1.1), for more details we refer the reader to [1–4, 15]. In thefollowing we denote by• K :=]0,+∞[×R equipped with the weighted Lebesgue measure µα given by
dµα(x, t) :=

x2α+1

2αΓ(α+ 1)
√

2π
dx ⊗ dt, , α ≥ 0,

where Γ is the Gamma function.• Lpα(K), 1 ≤ p ≤ ∞, the space of measurable functions on K, satisfying
‖f ‖p,µα :=


(∫
K |f (x, t)|pdµα(x, t)

)1/p
<∞, if p ∈ [1,+∞[,

ess sup(x,t)∈K |f (x, t)| <∞, if p = +∞.

• K̂ := [0,+∞[×R ∪ {(i s, y); (s, y) ∈ [0,+∞[×R; s 6 |y |}.• BK̂ the σ-algebra defined on K̂ by
BK̂ =

{
θ−1(B), B ∈ B([0,+∞[×R)},

where θ is the bijective function given by
θ(s, y) =

(√
s2 + y2, y

)
.

• dγα the measure defined on BK̂ by
∀A ∈ BK̂; γα(A) = µα(θ(A)).

and for all non-negative measurable function on K̂ we have∫
K̂
g(µ, λ)dγα(µ, λ) =

1

2αΓ(α+ 1)
√

2π

(∫ +∞
0

∫
R
g(µ, λ)

(
µ2 + λ2

)α
µdµdλ

+

∫
R

∫ |λ|
0

g(iµ, λ)
(
λ2 − µ2

)α
µdµdλ

)
.

(2.1)
• Lpα(K̂) with p ∈ [1,+∞] the space of measurable functions on K̂ satisfying

‖g‖p,γα :=


(∫
K̂ |g(λ,m)|pdγα(λ,m)

) 1
p <∞ if p ∈ [1,+∞[,ess sup(λ,m)∈K̂ |g(λ,m)| <∞, if p = +∞.

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 42.1. The eigenfunctions of the partial Differential operators ∆1 and ∆2. For (µ, λ) ∈ K̂ weconsider the following Cauchy problem
(S) :


∆1(u)(x, t) = −λ2u(x, t),

∆2(u)(x, t) = −µ2u(x, t)

u(0, 0) = 1; ∂u∂x (0, t) = 0.From [3], the Cauchy problem (S) admits a unique solution ϕµ,λ given by:
ϕµ,λ(x, t) = jα

(
x
√
µ2 + λ2

)
exp(−iλt), (2.2)

where jα is the spherical Bessel function of index α see [16] for more information about the Besselfunctions. The function ϕµ,λ is infinitely differentiable on R2, even with respect to each variableand we have the following important result:
sup

(x,t)∈R2

∣∣ϕµ,λ(x, t)
∣∣ = 1. for (µ, λ) ∈ K̂. (2.3)

2.2. The Riemann-Liouville transform.

Definition 2.1. The generalized Fourier transform Fα associated with the Riemann-Liouville op-erator (1.1) is defined on L1α(K) by
Fα(f )(µ, λ) =

∫
K
ϕµ,λ(x, t)f (x, t)dµα(x, t), for (µ, λ) ∈ K̂.

Some basic properties of this transform are as follows, for the proofs one can see [2–4].
Proposition 2.1.(1) For every f ∈ L1α(K), we have

‖Fα(f )‖∞,γα ≤ ‖f ‖1,µα . (2.4)
(2)(Inversion formula) For f ∈ (L1α ∩ L2α) (K) such that Fα(f ) ∈ L1α(K̂) we have

f (x, t) =

∫
K̂
ϕµ,λ(x, t)Fα(f )(µ, λ)dγα(µ, λ), a.e (x, t) ∈ K. (2.5)

(3) (Parseval formula) For all f , g ∈ L2α(K) we have∫
K
f (x, t)g(x, t)dµα(x, t) =

∫
K̂

Fα(f )(µ, λ)Fα(g)(µ, λ)dγα(µ, λ), (2.6)
in particular we have

‖f ‖2,µα = ‖Fα(f )‖2,γα . (2.7)(4) (Plancherel’s theorem) The Reimann-Liouville transform Fα can be extended to an isometricisomorphism from L2α(K) into L2α(K̂).

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 52.3. Generalized translation operator Associated with the Riemann-Liouville operator.

Definition 2.2. The translation operator associated with Riemann-Liouville transform is defined on
Lpα(K), for all (x, t), (y , s) ∈ K, by

τ
(x,t)
α (f )(y , s) =

Γ(α+ 1)√
πΓ(α+ 1/2)

∫ π

0

f
(√

x2 + y2 + 2xy cos θ, t + s
)

sin2α θdθ.

The following proposition summarizes some properties of the Riemann-Liouville translation operatorsee [2–4].
Proposition 2.2. For all (x, t), (y , s) ∈ K, f ∈ Lpα(K) we have:(1) ∫

K
τ
(x,t)
α (f )(y , s)dµα(y , s) =

∫
K
f (y , s)dµα(y , s). (2.8)

(2) for f ∈ Lpα(K) with p ∈ [1; +∞] τ
(x,t)
α (f ) ∈ Lpα(K) and we have∥∥∥τ (x,t)α (f )

∥∥∥
p,µα
≤ ‖f ‖p,µα . (2.9)

(3) For f ∈ L1α(K), τ
(x,−t)
α (f ) ∈ L1α(K) and we have

Fα

(
τ
(x,−t)
α (f )

)
(µ, λ) = ϕµ,λ(x, t)Fα(f )(µ, λ), ∀(µ, λ) ∈ K̂. (2.10)

By using the generalized translation, we define the generalized convolution product of f , g by
(f ∗α g) (x, t) =

∫
K
τ
(x,−t)
α (f̌ )(y , s)g(y , s)dµα(y , s).

where f̌ (y , s) = f (y ,−s).With this convolution product (K, ∗α) is a hypergroup in the sense of Jewett [13].We have the following results for the proofs, we refer the reader to [2–4].
Proposition 2.3.(1)(Young’s inequality) for all p, q, r ∈ [1; +∞] such that: 1p + 1

q = 1 + 1
r and for all f ∈ Lpα(K), g ∈

Lqα(K) the function f ∗α g belongs to the space Lrα(K) and we have
‖f ∗α g‖r,µα ≤ ‖f ‖p,µα‖g‖q,µα (2.11)

(2) For f , g ∈ L2α(K) the function f ∗α g belongs to L2α(K) if and only if the function Fα(f )Fα(g)belongs to L2α(K̂) and in this case we have
Fα (f ∗α g) = Fα(f )Fα(g). (2.12)

(3) For f , g ∈ L2α(K) then we have∫
K
|f ∗α g(x, t)|2 dµα(x, t) =

∫
K̂
|Fα(f )(µ, λ)|2 |Fα(g)(µ, λ)|2 dγα(µ, λ), (2.13)

where both integrals are simultaneously finite or infinite.

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 63. Calderón’s Reproducing Formula for the Riemann-Liouville Two-wavelet Transform
Using the harmonic analysis associated with the Riemann-Liouville transform, the main purposeof this section is to define the wavelet transform associated with the Riemann-Liouville operatorand to give generalized Parseval’s, Plancherel’s, inversion and Calderon’s reproducing formulasrelated to this transform which generalizes all the results proved in [4].

Notation: we denote by•Lpα(R+ ×K),1 ≤ p ≤ +∞ the space of measurable functions on R+ ×Ksatisfying
‖f ‖p,θα :=


(∫ +∞
0

∫
K |f (a, x, t)|pdθα(a, x, t)

) 1
p
<∞, if p ∈ [1,+∞[,

ess sup |f (a, x, t)
(a,x,t)∈R+×K

| <∞, if p = +∞..where θα is the measure defined on R+ ×K by
dθα(a, x, t) := a2α+2da ⊗ dµα(x, t).

Definition 3.1. Let ψ1, ψ2 ∈ L2α(K), the pair (ψ1, ψ2) is said to be a Rieman-Liouville two-waveleton K if for almost all (µ, λ) ∈ K̂ we have
0 < Cψ1,ψ2 :=

∫ ∞
0

Fα(ψ1)

(
µ

a
,
λ

a

)
Fα(ψ2)

(
µ

a
,
λ

a

)
da

a
< +∞. (3.1)

Remark 3.1. Its clear that if ψ = ψ1 = ψ2, we have
Cψ1,ψ2 = Cψ :=

∫ ∞
0

∣∣∣∣Fα(ψ)

(
µ

a
,
λ

a

)∣∣∣∣2 daa < +∞, (3.2)
in this case we say that ψ is a Riemann-Liouville wavelet in L2α(K).Let a > 0, we define the dilatation operator Da of a measurable function ψ on C2 by

Da(ψ)(x, t) = aα+3/2ψ(ax, at), (x, t) ∈ C2.

the dilatation operator Da satisfies the following properties• For all ψ ∈ Lpα(K) we have Da(ψ) ∈ Lpα(K) and
‖Da(ψ)‖p,µα = a(

1
2
− 1
p
)(2α+3)‖ψ‖p,µα . (3.3)

• For all ψ ∈ L2(K) we have
Fα(Da(ψ))(µ, λ) =

1

aα+3/2
Fα(ψ)

(
µ

a
,
λ

a

)
. (3.4)

Let ψ be a Riemann-Liouville wavelet on K in Lp(K) with 1 ≤ p ≤ ∞, for all
a > 0,(x, t) ∈ K we define the function

ψa,x,t(y , s) = τ
(x,−t)
α (Da(ψ))(y , s). (3.5)

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 7By using the relations (2.9) and (3.3) we find that ψa,x,t ∈ Lpα(K) and
‖ψa,x,t‖p,µα ≤ a

( 1
2
− 1
p
)(2α+3)‖ψ‖p,µα .. (3.6)

Definition 3.2. ( [4]) Let ψ be a Riemann-Liouville wavelet on K in L2α(K) the continuous wavelettransform Sαψ associated with the Riemann-Liouville operator is defined for a function f ∈ L2α(K)and (a, x, t) ∈ R+ ×K by
Sαψ(f )(a, x, t) :=

∫
K
f (y , s)ψa,x,t(y , s)dµα(y , s). (3.7)

Remark 3.2. The Riemann-Liouville wavelet transform (3.7) can be written as
Sαψ(f )(a, x, t) = (Da(ψ̌) ∗α f )(x, t) = 〈f , ψa,x,t〉α. (3.8)

The following result gives the relation between the Riemann-Liouville transform Fα and theRiemann-Liouville wavelet transform Sαψ .
Proposition 3.1. Let ψ be a Riemann-Liouville wavelet for all f ∈ L2(K) we have

Fα

[
Sαψ(f )(a, .)

]
(µ, λ) =

1

aα+3/2
Fα (f ) (µ, λ)Fα(ψ)

(
µ

a
,
λ

a

)
. (3.9)

Proof. Is a consequence of the convolution theorem (2.12) and the relations (3.4),(3.8). �

The following theorem generalizes the Parseval’s formula for the Riemann-Liouville wavelettransform Sαψ(f ) proved in [4].
Theorem 3.1. Let (ψ1, ψ2) be a Rieman-Liouville two-wavelet on K for all f , g ∈ L2(K) we have∫ +∞

0

∫
K
Sαψ1(f )(a, x, t)Sαψ2(g)(a, x, t)dθα(a, x, t) = Cψ1,ψ2

∫
K
f (y , s)g(y , s)dµα(y , s), (3.10)

where Cψ1,ψ2 is the constant given by the relation (3.1).
Proof. By using the relations (2.5),(2.12),(3.4), (3.8) and Fubini’s theorem we get∫ +∞
0

∫
K
Sαψ1(f )(a, x, t)Sαψ1(g)(a, x, t)dθα(a, x, t)

=

∫ +∞
0

[

∫
K

(Da(ψ̌1) ∗α f )(x, t)(Da(ψ̌2) ∗α g)(x, t)dµα(x, t)]a2α+2da

=

∫ +∞
0

[

∫
K̂

Fα(Da(ψ̌1))(µ, λ)Fα(f )(µ, λ)Fα(Da(ψ̌2))(µ, λ)Fα(g)(µ, λ)dγα(λ,m)]a2α+2da

= Cψ1,ψ2

∫
K̂

Fα(f )(µ, λ)Fα(g)(µ, λ)dγα(µ, λ),

by using Parseval’s formula for the Riemann-Liouville transform (2.5) we find the disered result. �
In the following we establish an inversion formula for the Riemann-Liouville two-wavelet trans-form.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.20 8

Theorem 3.2. Let (ψ1, ψ2) be a Riemann-Liouville two-wavelet such that Cψ1,ψ2 6= 0 for all
f ∈ L1α(K) such that Fα(f ) ∈ L1α(K̂) ∩ L∞α (K̂) we have

f (·) =
1

Cψ1,ψ2

∫ ∞
0

(∫
K
Sαψ1(f )(a, x, t)ψ2,a,x,t(·)dµα(x, t)

)
a2α+2da.

Proof. Let f , g ∈ L2α(K), by using the relation (3.10), Fubini’s theorem we find that
∫
K
f (y , s)g(y , s)dµα(y , s) =

1

Cψ1,ψ2

∫ +∞
0

∫
K
Sαψ1(f )(a, x, t)Sαψ2(g)(a, x, t)dθα(a, x, t)

=
1

Cψ1,ψ2

∫
K

[

∫ ∞
0

(∫
K
Sαψ1(f )(a, x, t)ψ2,a,x,t(y , s)dµα(x, t)

)
a2α+2da]g(y , s)dµα(y , s)which gives the result. �

The rest of this subsection is devoted to give a Calderón’s reproducing formula for the Reimann-Liouville two-wavelet (ψ1, ψ2) under the following condition
Cψ1,ψ2 6= 0 and Fα(Da(ψ1)),Fα(Da(ψ2)) ∈ L∞α (K̂), (3.11)

Proposition 3.2. For 0 < ε < δ <∞, we put
Gε,δ(x, t) :=

1

Cψ1,ψ2

∫ δ

ε

(
Da(ψ̌2) ∗α Da(ψ1)

)
(x, t)a2α+2da

and
Kε,δ(λ,m) :=

1

Cψ1,ψ2

∫ δ

ε

Fα(ψ1)

(
µ

a
,
λ

a

)
Fα(ψ2)

(
µ

a
,
λ

a

)
da

a
.Under the condition (3.11) we have

Gε,δ ∈ L2α(K), Kε,δ ∈ L1α(K̂) ∩ L∞α (K̂)

and
Fα(Gε,δ)(λ,m) = Kε,δ(λ,m) (3.12)

Proof. By using Hölder’s inequality for the measure a2α+2da we obtain
‖Gε,δ‖22,µα ≤

δ2α+3 − ε2α+3

C2ψ1,ψ2

∫ δ

ε

(∫
K

∣∣∣(Da(ψ̌2) ∗α Da(ψ1)
)

(x, t)
∣∣∣2 dµα(x, t)

)
a2α+2da,

By using the relations (2.13) and (3.4) we find that
‖Gε,δ‖22,µα ≤

δ2α+3 − ε2α+3

C2ψ1,ψ2
‖Fα(Da(ψ2))‖2∞,γα‖ψ1‖

2
2,µα

∫ δ

ε

da

a
<∞.

Which prove that Gε,δ ∈ L2α(K), the result Kε,δ ∈ L1α(K̂) ∩ L∞α (K̂) can be easily checked, on theother hand by using the relations (2.5), (2.10), (3.4) and Fubini’s theorem we find that
Gε,δ(x, t) =

∫
K̂
ϕµ,λ(x, t)Kε,δ(µ, λ)dγα(µ, λ),

inversion formula (2.5) gives the relation (3.12). �

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 9We can now state the main result of this section
Theorem 3.3. (First Calderón’s reproducing formula)Let (ψ1, ψ2) be a Reimann-Liouville two-wavelet satisfying the condition (3.11) and let 0 < ε <

δ <∞ then for all f ∈ L2α(K), the function fε,δ given by
fε,δ(x, t) =

1

Cψ1,ψ2

∫ δ

ε

(∫
K
Sαψ1(f )(a, y , s)ψ2,a,x,t(y , s)dµα(y , s)

)
a2α+2da,

belongs to L2α(K) and satisfies
lim

ε→0,δ→∞
‖fε,δ − f ‖2,µα = 0. (3.13)

Proof. It is easy to see that
fε,δ = f ∗α Gε,δthen by using the relations (2.7) and (3.12) we find that

‖fε,δ − f ‖22,µα =

∫
K̂
|Fα(f )(µ, λ)|2(1−Kε,δ(µ, λ))2dγα(µ, λ),

the relation (3.13) follows from the admissibility condition (3.1) and the dominated convergencetheorem. �

4. Extremal Functions Associated with the Riemann-Liouville Wavelet Transform
By using the theory of reproducing kernels [18,19], the main purpose of this section is to studythe extremal functions associated with the Riemann-Liouville wavelet transform and to give anintegral representation and best estimate of these functions on weighted Sobolev spaces.

4.1. Sobolev type spaces Associated with the Riemann-Liouville Transform. Let s > 0, we definethe Sobolev spaces associated with the Riemann-Liouville transform as
Hsα(K) :=

{
f ∈ L2α(K)/

(
1 + µ2 + 2λ2

)s/2
Fα(f ) ∈ L2α(K̂)

}
.

The space Hsα(K) provided with the inner product
〈f , g〉Hsα :=

∫
K̂

(
1 + µ2 + 2λ2

)s
Fα(f )(µ, λ)Fα(g)(µ, λ)dγα(µ, λ), (4.1)

and the norm
‖f ‖2Hsα := 〈f , f 〉Hsα =

∫
K̂

(
1 + µ2 + 2λ2

)s |Fα(f )(µ, λ)|2dγα(µ, λ), (4.2)
is a Hilbert space.
Definition 4.1. Let ψ be a Riemann-Liouville wavelet on K in L2α(K), we introduce the innerproduct in the Hilbert space Hsα(K) for any fixed β > 0 by

〈f , g〉Hsψ,β := β〈f , g〉Hsα + 〈Sαψ(f ), Sαψ(g)〉θα , (4.3)

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 10the norm associated to this inner product is defined by
‖f ‖2Hsψ,β := β‖f ‖2Hsα + ‖Sαψ1(f )‖22,θα . (4.4)

We have the following result
Proposition 4.1. Let s > 2α+3

2 , ψ be aRiemann-Liouville wavelet on K in L2α(K) and β > 0 thenwe have
f ∈ Hsψ,β(K)⇒ Fα(f ) ∈ L1α(K̂) (4.5)

Proof. Let f ∈ Hsψ,β(K), by using the relations (2.9), (3.9), (4.2) and (4.4) we find that
‖f ‖2Hsψ,β =

∫
K̂

[
β
(

1 + µ2 + 2λ2
)s

+ Cψ

]
|Fα(f )(µ, λ)|2dγα(µ, λ) (4.6)

by using Hölder’s inequality, the relation (2.1) and the fact that s > 2α+3
3 we find that

‖Fα(f )‖1,γα ≤ ‖f ‖Hsψ,β

(∫
K̂

dγα(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ

) 1
2

<∞

wich give the result. �

Theorem 4.1. Let s > 2α+3
2 , ψ be a Riemann-Liouville wavelet on K in L2α(K) and β > 0 then thespace (Hsψ,β(K), 〈, 〉Hsψ,β) is a reproducing kernel Hilbert space with kernel function given by

Kψ,β[(x, t), (y , z)] =

∫
K̂

ϕµ,−λ(x, t)ϕµ,λ(y , z)

β (1 + µ2 + 2λ2)s + Cψ
dγα(µ, λ) (4.7)

that is for every (y , z) ∈ K,(1) the function (x, t)→ Kr,h[(x, t), (y , z)] ∈ Hsψ,β(K).(2) For every f ∈ Hsψ,β(K) and (y , z) ∈ K we have
f (y , z) =

〈
f ,Kψ,β[·, (y , z)]

〉
Hsψ,β

.

Proof. Let (y , z) ∈ K, by using the fact that s > 2α+3
2 and the relation (2.3) we find that thefunction

(µ, λ)→
ϕµ,λ(y , z)

β (1 + µ2 + 2λ2)s + Cψbelongs to L1α(K̂)∩L2α(K̂), by using Plancherel’s theorem for the Riemann-Liouville transform thereexist a unique function in L2α(K), wich we denote by Kψ,β[·, (y , z)] such that
Fα

(
Kψ,β[·, (y , z)]

)
=

ϕµ,λ(y , z)

r [1 + λ2 (1 +m2)]s + Ch
, (4.8)

by using the relation (2.5) we find that
Kψ,β[(x, t), (y , z)] =

∫
K̂

ϕµ,−λ(x, t)ϕµ,λ(y , z)

β (1 + µ2 + 2λ2)s + Cψ
dγα(µ, λ),

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 11furthermore by using the relations (2.3), (4.6) and (4.8) we find that∥∥Kψ,β[·, (y , z)]
∥∥2
Hsψ,β
≤
∫
K̂

dγα(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ
<∞.

wich proves that Kψ,β[·, (y , z)] ∈ Hsψ,β(K). Let f ∈ Hsψ,β(K), by using the relations the relations(3.10),(4.1),(4.3),( and (4.8) we find that〈
f ,Kψ,β[·, (y , z)]

〉
Hsr,h

=

∫
K̂
ϕµ,λ(y , z)Fα(f )(µ, λ)dγα(µ, λ),

inversion formula (2.5) gives the disered result. �

In the following we give the main result of this section.
Theorem 4.2. Let ψ be a Riemann-Liouville wavelet in L2α(K), s > 2α+3

2 ,
g ∈ L2α (R+ ×K) and β > 0 then the infimum

inf
f ∈Hsα(K)

{
β‖f ‖2Hsα +

∥∥Sαψ(f )− g
∥∥2
2,θα

} (4.9)
is attained by a unique function f ∗g,ψ,β given explicitly by

f ∗g,ψ,β(x, t) =

∫ +∞
0

∫
K
g(a, y , z)φψ,β (a, (y , z), (x, t)) dθα(a, y , z), (4.10)

where φψ,β is given by
φψ,β (a, (y , z), (x, t)) = a−

2α+3
2

∫
K̂

ϕµ,−λ(x, t)ϕµ,λ(y , z)Fα(ψ)
(
µ
a ,

λ
a

)
β (1 + µ2 + 2λ2)s + Cψ

dγα(µ, λ). (4.11)
Proof. The existence and unicity of the extremal function f ∗g,ψ,β solution of the problem (4.9) isassured in [18,19], moreover this solution is given by

f ∗g,ψ,β(x, t) =
〈
g, Sαψ

(
Kψ,β[·, (x, t)]

)〉
θα
, (4.12)

where Kψ,β is the kernel given by (4.8), by using the relations (2.6), (2.10), (3.4) and (4.8) we findthat
Sαψ
(
Kψ,β[·, (x, t)]

)
(a, y , z) = a−

2α+3
2

∫
K̂

ϕµ,λ(x, t)ϕµ,−λ(y , z)Fα(ψ)
(
µ
a ,

λ
a

)
β (1 + µ2 + 2λ2)s + Cψ

dγα(µ, λ), (4.13)
by using the relations (4.12) and (4.13) we find the desired result. �

We have the following results
Theorem 4.3. Let s > 2α+3

2 , ψ be a Riemann-Liouville wavelet in on K in L2α(K), , and g ∈
L2α (R+ ×K) , β > 0 then we have

(i) f ∗g,ψ,β(x, t) =

∫ +∞
0

∫
K̂

ϕµ,−λ(x, t)Fα(ψ)
(
µ
a ,

λ
a

)
Fα(g(a, .))(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ
a
2α+1
2 da ⊗ dγα(µ, λ).(4.14)

(i i) Fα(f ∗g,ψ,β)(µ, λ) =

∫ +∞
0

Fα(ψ)
(
µ
a ,

λ
a

)
Fα(g(a, .))(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ
a
2α+1
2 da (4.15)

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 12

(i i i) ‖f ∗g,ψ,β‖Hsα ≤
‖g‖2,θα√

2β
. (4.16)

Proof. (i) Is a consequence of (4.10), (4.11) and Fubini’s theorem.(ii) Is a consquence of Fubini’s theorem and the relations (2.5),(4.10) and (4.11).(iii)By using the relation (4.2) we find that
‖f ∗g,ψ,β‖2Hsα =

∫
K̂

(
1 + µ2 + 2λ2

)s |Fα(f ∗g,ψ,β)(µ, λ)|2dγα(µ, λ),

by using Holder’s inequality and the relations (3.2),(4.15) we find that
‖f ∗g,ψ,β‖2Hsα ≤

1

2β

∫
K̂

(∫ +∞
0

|Fα(g(a, .))(µ, λ)|2a2α+2da
)
dγα(µ, λ)

, By using Fubini’s theorem and Plancherel’s formula (2.7) we find that
‖f ∗g,ψ,β‖2Hsα ≤

1

2β
‖g‖22,θαwhich gives the result.

�

corollary 4.1. Let s > 2α+3
2 , ψ be a Riemann-Liouville wavelet on K in L2α(K), ,and β > 0, for all

f ∈ Hsα(K) and g = Sαψ(f ), the extremal function f ∗Sαψ(f ),ψ,β satisfies the following properties
(i) Fα(f ∗Sαψ(f ),ψ,β

)(µ, λ) =
CψFα(f )(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ
. (4.17)

(i i) ‖f ∗Sαψ(f ),ψ,β‖Hsα ≤

√
Cψ
2β
‖f ‖2,µα (4.18)

Proof. (i) By using the relations (3.2), (3.9) and (4.15) we find the result.(ii) Is a consequence of (3.9) and (4.16). �

Theorem 4.4. (Second Calderon’s reproducing formula)Let s > 2α+3
2 , ψ be a Riemann-Liouville wavelet in on K in L2α(K), ,and β > 0, for all f ∈ Hsα(K)and g = Sαψ(f ), the extremal function f ∗Sαψ(f ),ψ,β satisfies

lim
β→0+

∥∥∥f ∗Sαψ(f ),ψ,β − f ∥∥∥Hsα = 0.

Moreover we have f ∗Sαψ(f ),ψ,β −→ f uniformly when β −→ 0+.
Proof. By using the relation (4.17) we find that

Fα(f ∗Sαψ(f ),ψ,β
− f )(µ, λ) =

−β
(

1 + µ2 + 2λ2
)s

Fα(f )(µ, λ)

β (1 + µ2 + 2λ2)s + Cψ
(4.19)

https://doi.org/10.28924/ada/ma.4.20


Eur. J. Math. Anal. 10.28924/ada/ma.4.20 13consequently we find that∥∥∥f ∗Sαψ(f ),ψ,β − f ∥∥∥Hsα =

∫
K̂

β2
(

1 + µ2 + 2λ2
)3s |Fα(f )(µ, λ)|2

β (1 + µ2 + 2λ2)s + Cψ
dγα(µ, λ)

by using the dominated convergence theorem and the fact that
β2
(

1 + µ2 + 2λ2
)3s |Fα(f )(µ, λ)|2

β (1 + µ2 + 2λ2)s + Cψ
≤
(

1 + µ2 + 2λ2
)s |Fα(f )(µ, λ)|2 ,

we deduce that
lim
β→0+

∥∥∥f ∗Sαψ(f ),ψ,β − f ∥∥∥Hsα = 0on the other hand by using inversion formula (2.5) and the relation (4.19) we find that
f ∗Sαψ(f ),ψ,β

(y , v)− f (y , v) =

∫
K̂

Fα(f ∗Sαψ(f ),ψ,β
− f )(µ, λ)ϕµ,λ(y , v)dγα(µ, λ),

=

∫
K̂

−β
(

1 + µ2 + 2λ2
)s

Fα(f )(λ)ϕµ,λ(y , v)

β (1 + µ2 + 2λ2)s + Cψ
dγα(µ, λ)

again by dominated convergence theorem and the fact that∣∣∣∣∣−β
(

1 + µ2 + 2λ2
)s

Fα(f )(λ)ϕµ,λ(y , s)

β (1 + µ2 + 2λ2)s + Cψ

∣∣∣∣∣ ≤ |Fα(f )(µ, λ)|

we deduce that
lim
β→0+

∥∥∥f ∗Sαψ(f ),ψ,β − f ∥∥∥∞,µα = 0

which proves that f ∗Sαψ(f ),ψ,β −→ f uniformly when β −→ 0+. �

Authors’ contribution
The authors contributed equally to this work.

References
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https://doi.org/10.1088/1742-6596/73/1/012019

	1. Introduction
	2. Harmonic Analysis Associated with the Riemann-Liouville Operator
	2.1. The eigenfunctions of the partial Differential operators 1 and 2
	2.2. The Riemann-Liouville transform 
	2.3. Generalized translation operator Associated with the Riemann-Liouville operator

	3.  Calderón's Reproducing Formula for the Riemann-Liouville Two-wavelet Transform
	4. Extremal Functions Associated with the Riemann-Liouville Wavelet Transform
	4.1. Sobolev type spaces Associated with the Riemann-Liouville Transform

	Authors' contribution
	References

