




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 8doi: 10.28924/ada/ma.5.8
Uncertainty Principles and Extremal Functions for Bessel Multiplier Operators in Quantum

Calculus

Ahmed Chana∗, Abdellatif Akhlidj
Laboratory of Fundamental and Applied Mathematics, Department of Mathematics and Informatics, Faculty

of Sciences Ain Chock, University of Hassan II, B.P 5366 Maarif, Casablanca, Morocco
maths.chana@gmail.com, akhlidj@hotmail.fr
∗Correspondence: maths.chana@gmail.com

Abstract. Using the q-Jackson integral and some elements of the q-harmonic analysis associatedwith the q-Bessel operator for fixed 0 < q < 1, we introduce the q-Bessel multiplier operators and wegive some new results related to these operators as Plancherel’s, Calderón’s reproducing formulas andHeisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernelswe give best estimates and an integral representation of the extremal functions related to theseoperators on weighted Sobolev spaces.

1. Introduction
The q-theory, called also in some literature quantum calculus began to arise. Interest in thistheory is grown at an explosive note by both physicists and mathematicians due to a large numberof its application domains, for more information about quantum calculus one can see [20].Recently, many reasercher have been investigated the behavior of the q-theory to several alreadystudied for the Fourier analysis, for example sampling theorem [2], Paley-Wiener theorem [1],uncertainty principles [31], wavelet transform [15], wavelet packet [6], Ramanujan master theorem[16], Sobolev type spaces [27] and wave equation [29]. In their seminal papers, Hörmander’s andMikhlin’s [18,25] initiated the study of boundedness of the translation invariant operators on Rd . Thetranslation invariant operators on Rd characterized using the classical Euclidean Fourier transform

F(f ) therefore they also known as Fourier multipliers. Given a measurable function
m : Rd −→ C

its Fourier multiplier is the linear map Tm given for all λ ∈ Rd by the relation
F(Tm(f ))(λ) = m(λ)F(f )(λ) (1.1)

Received: 22 Nov 2024.
Key words and phrases. Quantum calculus; q-Bessel transform; Calderón’s reproducing formulas; Extremal functions;Heisenberg’s uncertainty principle; Approximation theory; Sobolev spaces.

1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 2The Hörmander-Mikhlin fundamental condition gives a criterion for Lp-boundedness for all 1 <

p <∞ of Fourier multiplier Tm in terms of derivatives of the symbol m, more precisely if∣∣∂γλm(λ)
∣∣ . |λ|−|γ| f or 0 ≤ |γ| ≤

[
d

2

]
+ 1. (1.2)

Then, Tm can be extended to a bounded linear operator from Lp(Rd) into itself .The condition (1.2) imposes m to be a bounded function, smooth over Rd\{0} satisfying certainlocal and asymptotic behavior. Locally, m admits a singularity at 0 with a mild control of deriva-tives around it up to order [d2 ] + 1. This singularity links to deep concepts in harmonic analysisand justifies the key role of Hörmander-Mikhlin theorem in Fourier multiplier Lp-theory, this con-dition defines a large class of Fourier multipliers including Riesz transforms and Littelwood-Paleypartitions of unity which are crucial in Fourier summability or Pseudo-differential operator.Theboundedness of Fourier multipliers is useful to solve problems in the area of mathematical analysisas Probability theory see [24], Stochastic processus see [5], and the study of nonlinear partialdifferential equations see [22]. For its importance many researcher extend the theory of Fouriermultiplier to different setting for example in the Dunkl-Weinstein setting [33], in the Laguerre-Bessel setting [8], in the q-Fourier setting [26, 31, 32] and the q-cosine Fourier setting [3]. Thegeneral theory of reproducing kernels is stared with Aronszajn’s in [4] in 1950, next the authorsin [23, 30] applied this theory to study Tikhonov regularization problem and they obtained ap-proximate solutions for bounded linear operator equations on Hilbert spaces with the viewpoint ofnumerical solutions by computers. This theory has gained considerable interest in various field ofmathematical sciences especially in Engineering and numerical experiments by using computerssee [30].This paper focuses on the generalized Fourier transform associated with the q-Bessel operatorcalled the q-Bessel transform introduced in [11], more precisely we define the following q-differentialoperator for 0 < q < 1 by
∆q,αf (x) =

f
(
q−1x

)
−
(

1 + q2α
)
f (x) + q2αf (qx)

x2
, ∀x 6= 0. (1.3)

The eigenfunctions of the operator (1.3) are related to the Hahn-Exton q-Bessel function jα(x ; q2)defined in [15]. The q-Bessel tranform Hq,α is defined on L1α(R+q ) by
Hq,α(f )(λ) =

∫ ∞
0

jα(λx ; q2)f (x)dµq,α(x), for λ ∈ R+q

where dµq,α is the measure on R+q given later. Let σ be a function in L2α(R+q ) and β ∈ R+q , theq-Bessel L2α-multiplier operators are defined for smooth function f on R+q as
Mq,σ,β(f )(x) := H−1q,α

(
σβHq,α(f )

)
(x) (1.4)

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 3where the function σβ is given by
σβ(λ) := σ(λβ). (1.5)These operators are a generalization of all classical multiplier operators introduced in [3, 10,26, 31, 32]. The remainder of this paper is arranged as follows, in section 2 we recall the mainresults concerning the harmonic analysis associated with the q-Bessel transform, in section 3,we introduce the q-Bessel L2α-multiplier operators Mq,σ,β and we give for them a Plancherel’s,point- wise reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Thelast section of this paper is devoted to give an application of the general theory of reproducingkernels to q-Bessel multiplier theory and to give best estimates and an integral representation ofthe extremal functions related to the q-Bessel L2α-multiplier operatorsMq,σ,β on weighted Sobolevspaces.

2. Harmonic Analysis Associated with the q-Bessel Transform
In this section we set some notations and we recall some results in harmonic analysis related tothe q-Bessel operator (1.3), all these results can be founded in [11,17,19–21,28].

2.1. Notations and preliminaries. In this subsection, we give some notations, definitions and prop-erties of the q-shifted factorial, the Jackson’s q-derivatives and the Jackson’s q-integrals introducedin [19].Let a ∈ C, the q-shifted factorial are defined by:
(a; q)0 = 1, (a; q)n =

n−1∏
k=0

(
1− aqk

)
, (a; q)∞ =

∞∏
k=0

(
1− aqk

)
.

The Jackson’s q-derivative of a function f is given by
Dqf (x) =

f (x)− f (qx)

(1− q)x
if x 6= 0.The q-Jackson’s integrals from 0 to a and from 0 to ∞ are defined by∫ a

0

f (x)dqx = (1− q)a

∞∑
0

f (aqn) qn,

∫ ∞
0

f (x)dqx = (1− q)

∞∑
n=−∞

f (qn) qn.

Provided the sums converge absolutely.
The normalized form of the q-Bessel kernel is defined in [14,17,28] by

jα(x ; q2) =

∞∑
n=0

(−1)n
q
n(n+1)
2

(qα+1; q)n (q; q)n
x2n. (2.1)

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 4It satisfies the following estimate [11]
∀x ∈ R+q ,

∣∣jα(x ; q2)
∣∣ ≤ 1. (2.2)

2.2. The q-Bessel transform. In this section, we define and give some basic properties of q-Besseltransform introduced in [11]. We first introduced the following spaces and norms• C0,q(R+q ) denotes the set of all functions defined on R+q continuous at zero and vanishing atinfinity, equiped with the induced topology of uniforme convergence.• Lpα(R+q ), 1 ≤ p ≤ ∞, denotes the space of measurable functions on R+q , satisfying
‖f ‖p,q,α =:

{ (∫∞
0 |f (x)|pdµq,α(x)

)1/p
<∞, 1 ≤ p <∞,

supx∈R+q
|f (x)| <∞, p =∞.where

dµq,α(x) =
1

1− q

(
q2α+2; q2

)
∞

(q2; q2)∞
x2α+1dq(x),

Definition 2.1. ( [11]) The q-Bessel transform Hq,α defined on L1α(R+q ) by

Hq,α(f )(λ) =

∫ ∞
0

jα(λx ; q2)f (x)dµq,α(x), for λ ∈ R+q

Some basic properties of this transform are as follows, for the proofs, we refer the reader to
[11,13,14,25].

Proposition 2.1.
(1) For every f ∈ L1α(R+q ) we have Hq,α(f ) ∈ C0,q(R+q ) and we have

‖Hq,α(f )‖∞,q,α ≤ Bq,α‖f ‖1,q,α. (2.3)
Where

Bq,α =
1

1− q

(
−q2α+2; q2

)
∞
(
−q2; q2

)
∞

(q2; q2)∞
(2.4)

(2)(q-Inversion formula) For f ∈
(
L1α ∩ L2α

)
(R+q ) such that Fα(f ) ∈ L1α(R+q ) we have

f (x) =

∫ ∞
0

jα(λx ; q2)Hq,α(f )(λ)dµq,α(λ), a.e x ∈ R+q . (2.5)
(3) (q-Parseval formula) For all f , g ∈ L2α(R+q ) we have

〈f , g〉q = 〈Hq,α(f ),Hq,α(g)〉q , (2.6)
In particular we have

‖f ‖2,q,α = ‖Hq,α(f )‖2,q,α . (2.7)
(4) (q-Plancherel theorem) The q-Bessel transform Hq,α can be extended to an isometric isomor-
phism from L2α(R+q ) into L2α(R+q ).

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 52.3. The translation operator Associated with the q-Bessel transform.

Definition 2.2. ( [13]) Let x, y ∈ R+q and f is a measurable function on R+q the translation operator
is defined by

τxq,αf (y) =

∫ ∞
0

jα(λx ; q2)jα(λy ; q2)Hq,α(f )(λ)dµq,α(λ),

The following proposition summarizes some properties of the q-Bessel translation operator see [13].

Proposition 2.2. For all x, y ∈ R+q ,we have:
(1)

τxq,αf (y) = τyq,αf (x). (2.8)
(2) ∫ ∞

0

τxq,αf (y)dµq,α(y) =

∫ ∞
0

f (y)dµq,α(y). (2.9)
(3) for f ∈ Lpα(R+q ) with p ∈ [1; +∞] τxq,αf ∈ L

p
α(R+q ) and we have∥∥τxq,αf ∥∥p,q,α ≤ ‖f ‖p,q,α, (2.10)

(4) For f ∈ L1α(R+q ), τxq,αf ∈ L1α(R+q ) and we have

Hq,α
(
τxq,αf

)
(λ) = jα(λx ; q2)Hq,α(f )(λ), ∀λ ∈ R+q . (2.11)

The relation (2.11) shows that the translation operator τxq,α is a particular case of the q-Bessel
multiplier operator (1.4).

By using the q-Bessel translation operator, we define the generalized convolution product of
f , g by

(f ∗q g) (x) =

∫ ∞
0

τxq,α(f )(y)g(y)dµq,α(y).

This convolution is commutative, associative and its satisfies the following properties see [11,13].

Proposition 2.3.
(1)(q-Young’s inequality) for all p, q, r ∈ [1; +∞] such that: 1

p + 1
s = 1 + 1

r and for all f ∈
Lpα(R+q ), g ∈ Lsα(R+q ) the function f ∗α g belongs to the space Lrα(R+q ) and we have

‖f ∗α g‖r,q,α ≤ ‖f ‖p,q,α‖g‖s,q,α (2.12)
(2) For f , g ∈ L2α(R+q ) the function f ∗qg belongs to L2α(R+q ) if and only if the functionHq,α(f )Hq,α(g)

belongs to L2α(R+q ) and in this case we have

Hq,α (f ∗q g) = Hq,α(f )Hq,α(g). (2.13)
(3) For all f , g ∈ L2α(R+q ) then we have∫ ∞

0

|f ∗q g(x, t)|2 dµq,α(x) =

∫ ∞
0

|Hq,α(f )(λ)|2 |Hq,α(g)(λ)|2 dµq,α(λ), (2.14)
where both integrals are simultaneously finite or infinite.

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 63. The q-Bessel L2α-Multiplier Operators
The main purpose of this section is to introduce the q-Bessel L2α-multiplier operators on R+q and

to establish for them some uncertainty principles and Calderon’s reproducing formulas.

3.1. Calderon’s Reproducing Formulas for the q-Bessel L2α-multiplier operators.

Definition 3.1. Let σ ∈ L2α(R+q ) and β ∈ R+q , the q-Bessel L2α-multiplier operators are defined for
smooth function f on R+q as

Mq,σ,β(f )(x) := H−1q,α
(
σβHq,α(f )

)
(x), (3.1)

where the function σβ is given by the relation (1.5) and by a simple change of variable we find
that for all β ∈ R+q , σβ ∈ L2α(R+q ) and∥∥σβ∥∥2,q,α =

1

βα+1
‖σ‖2,q,α. (3.2)

Remark 3.1. According to the relation (2.13) we find that

Mq,σ,β(f )(x) =
(
H−1q,α

(
σβ
)
∗α f

)
(x), (3.3)

where
H−1q,α

(
σβ
)

(x) =
1

β2α+2
H−1q,α(σ)

(
x

β

)
. (3.4)

We give some properties of the q-Bessel L2α-multiplier operators.

Proposition 3.1. (i) For every σ ∈ L2α(R+q ), and f ∈ L1α(R+q ), the function Mq,σ,β(f ) belongs to
L2α(R+q ), and we have ∥∥Mq,σ,β(f )

∥∥
2,q,α

≤
1

βα+1
‖σ‖2,q,α‖f ‖1,q,α.

(ii) For every σ ∈ L∞α (R+q ), and for every f ∈ L2α(R+q ), the function Mq,σ,β(f ) belongs to L2α(R+q ),
and we have ∥∥Mq,σ,β(f )

∥∥
2,q,α

≤ ‖σ‖∞,q,α‖f ‖2,q,α (3.5)
(iii) For every σ ∈ L2α(R+q ), and for every f ∈ L2α(R+q ), Mq,σ,β(f ) ∈ L∞α (R+q ), and we have

Mq,σ,β(f )(x) =

∫ ∞
0

σ(βλ)jα(λx ; q2)Hq,α(f )(λ)dµq,α(λ), a.e x ∈ R+q (3.6)
and ∥∥Mq,σ,β(f )

∥∥
∞,q,α ≤

1

βα+1
‖σ‖2,q,α‖f ‖2,q,α.

Proof. (i) By using the relations (2.12),(3.3) we find that∥∥Mq,σ,β(f )
∥∥2
2,q,α

=
∥∥H−1q,α (σβ) ∗q f ∥∥22,q,α ≤ ‖f ‖21,q,α ∥∥H−1q,α (σβ)∥∥22,q,αPlancherel’s formula (2.7) and the relation (3.2) gives the desired result.(ii) Is a consequence of Plancherel’s formula (2.7).

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 7(iii) Is a consequence of the relations (2.7),(2.12),(3.2) and (3.3), on the other hand the relation (3.6)follows from inversion formula (2.5). �

In the following result, we give Plancherel’s and pointwise reproducing inversion formula for the
q-Bessel L2α-multiplier operators.

Theorem 3.1. Let σ ∈ L2α(R+q ) satisfying the admissibility condition:∫ ∞
0

∣∣σβ(λ)
∣∣2 dq(β)

β
= 1, λ ∈ R. (3.7)

(i) (Plancherel formula) For all f in L2α(R+q ), we have∫ ∞
0

|f (x)|2dµq,α(x) =

∫ ∞
0

∥∥Mq,σ,β(f )
∥∥2
2,q,α

dq(β)

β
. (3.8)

(ii) (First calderón’s formula) Let f ∈ L1α(R+q ) such that Hq,α(f ) ∈ L1α(R+q ) then we have

f (x) =

∫ ∞
0

(
Mq,σ,β(f ) ∗α H−1q,α

(
σβ
))

(x)
dβ

β
, a.e. x ∈ R.

Proof. (i) By using the relations (2.14) and (3.3) we get∫ ∞
0

∥∥Mq,σ,β(f )
∥∥2
2,q,α

dq(β)

β
=

∫ ∞
0

[∫ ∞
0

∣∣Mq,σ,β(f )(x)
∣∣2 dµq,α(x)

]
dq(β)

β

=

∫ ∞
0

[∫ ∞
0

|Hq,α(f )(λ)|2 dµq,α(λ)

] ∣∣σβ(λ)
∣∣2 dq(β)

βthe admissibility condition (3.7) and Plancherel’s formula (2.7) gives the desired result.(ii) Let f ∈ L1α(R+q ) such that Hq,α(f ) ∈ L1α(R+q ), by using the relations (2.6),(2.11) we find that∫ ∞
0

(
Mq,σ,β(f ) ∗αH−1q,α

(
σβ
))

(x)
dβ

β

=

∫ ∞
0

[∫ ∞
0

∣∣σβ(λ)
∣∣2Hq,α(f )(λ)jα(λx ; q2)dµq,α(λ)

]
dq(β)

β

=

∫ ∞
0

[∫ ∞
0

Hq,α(f )(λ)jα(λx ; q2)dµq,α(λ)

] ∣∣σβ(λ)
∣∣2 dq(β)

βthe admissibility condition (3.7),inversion formula (2.5) gives the desired result. �

To establish the second Calderon’s reproducing formula for the q-Bessel L2α-multiplier operators,
we need the following technical result.

Proposition 3.2. Let σ ∈ L2α(R+q ) ∩ L∞α (R+q ) satisfy the admissibility condition (3.7) then the
function defined by

Φγ,δ(λ) =

∫ δ

γ

∣∣σβ(λ)
∣∣2 dq(β)

β

belongs to L2α(R+q ) ∩ L∞α (R+q ) for all 0 < γ < δ <∞.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.8 8

Proof. Using Hölder’s inequality for the measure dq(β)
β and the relation (3.2) we find that∥∥Φγ,δ

∥∥2
2,q,α

≤ log(δ/γ)‖σ‖22,q,α‖σ‖2∞,q,α
∫ δ

γ

dq(β)

βα+2
<∞

So Φγ,δ ∈ L2α(R+q ), furthermore by using the relation (3.7) we get ∥∥Φγ,δ

∥∥
∞,q,α < ∞ therefore

Φγ,δ belongs to L2α(R+q ) ∩ L∞α (R+q ). �

Theorem 3.2. (Second Calderón’s formula). Let f ∈ L2α(R+q ) and σ ∈ L2α(R+q ) ∩ L∞α (R+q ) satisfy
the admissibility condition (3.7) and 0 < γ < δ <∞. Then the function

fγ,δ(x) =

∫ δ

γ

(
Mq,σ,β(f ) ∗α H−1q,α

(
σβ
))

(x)
dq(β)

β
, x ∈ R+q

belongs to L2α(R+q ) and satisfies

lim
(γ,δ)→(0,∞)

∥∥fγ,δ − f ∥∥2,q,α = 0 (3.9)
Proof. By a simple computation we find that

fγ,δ(x) =

∫ ∞
0

Φγ,δ(λ)jα(λx ; q2)Hq,α(f )(λ)dµq,α(λ) = H−1q,α
(

Φγ,δHq,α(f )
)

(x),

by using proposition 3.2 we find that Φγ,δ ∈ L∞α (R+q ) then we have fγ,δ ∈ L2α(R+q ) and
Hq,α

(
fγ,δ
)

(λ) = Φγ,δ(λ,m)Hq,α(f )(λ)

on the other hand by using Plancherel’s formula (2.7) we find that
lim

(γ,δ)→(0,∞)

∥∥fγ,δ − f ∥∥22,q,α = lim
(γ,δ)→(0,∞)

∫ ∞
0

|Hq,α(f )(λ)|2
(

1−Φγ,δ(λ)
)2
dµq,α(λ)

by using the admissibility condition (3.7), the relation (3.9) follows from the dominated convergencetheorem. �

3.2. Uncerainty principles for the q-Bessel L2α-multiplier operators. The main purpose of this
subsection is to establish Heisenberg’s and Donoho-Stark’s uncertainty principles for the q-Bessel
L2α-multiplier operators Mq,σ,β .

3.2.1. Heisenberg’s uncertainty principle forMq,σ,β . Heisenberg’s uncertainty principle for the q-
Bessel Fourier transform Hq,α has been established in [9, 11] as follows, for all f ∈ L2α(R+q ) we
have

‖|x |f ‖2,q,α ‖|λ|Hq,α(f )‖2,q,α ≥ kq,v‖f ‖
2
2,q,α, (3.10)

where kq,α =
[1+
√
q×qα+1]

1−q2(α+1) .
The inequality (3.10) says that if f is highly localized, then Hq,α(f ) cannot be concentrated

near a single point. We will generalize this inequality for Mq,σ,β , we have the following result

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Eur. J. Math. Anal. 10.28924/ada/ma.5.8 9

Theorem 3.3. For all f ∈ L2α(R+q ) we have

‖f ‖22,q,α ≤

∥∥|λ|2Hq,α(f )
∥∥
2,q,α

kq,α

[∫ ∞
0

∥∥|x |Mq,σ,β(f )
∥∥2
2,q,α

dq(β)

β

] 1
2

Proof. Let us suppose that ∥∥|λ|2Hq,α(f )
∥∥
2,q,α

+
[∫∞
0

∥∥|x |2Mq,σ,β(f )
∥∥2
2,q,α

dq(β)
β

]
< ∞, by usingthe relation (3.10) we find that

kq,α

∫ ∞
0

|Mq,σ,β(f )(x)|2dµq,α(x) ≤
∥∥|x |Mq,σ,β(f )

∥∥
2,q,α

∥∥|λ|σβHq,α(f )
∥∥
2,q,α

,

integrating over ]0,+∞[ with respect to measure dq(β)
β and using Plancherel’s formula (3.8) andSchwartz’s inequality we get

kq,α‖f ‖22,q,α

≤
[∫ ∞
0

‖|x |Mq,σ,β(f )‖22,q,α
dq(β)

β

] 1
2
[∫ ∞
0

[∫ ∞
0

||λσβ(λ)|2 |Hq,α(f )(λ)|2|(λ)|dµq,α(λ)

]
dq(β)

β

] 1
2

the admissibility condition (3.7) gives the desired result. �

3.2.2. Donoho-Stark’s uncertainty principle forMq,σ,β . Building on the ideas of Donoho and Stark
In [3], the main purpose of this subsection is to give an uncertainty inequality of concentration type
in L2θ(R+q ) where L2θ(R+q ) the space of measurables functions on R+q × R+q such that

‖f ‖2,θα =

[∫ ∞
0

‖f (β, .)‖22,q,α
dq(β)

β

] 1
2

.

We denote by θα the measure defined on R+q × R+q by

dθα(β, x) = dµq,α(x)⊗
dq(β)

β
,

Definition 3.2. [12]
(i) Let E be a measurable subset of R+q , we say that the function f ∈ L2α(R+q ) is ε-concentrated

on E if
‖f − 1Ef ‖2,q,α ≤ ε‖f ‖2,q,α, (3.11)

where 1E is the indicator function of the set E.
(ii) Let F be a measurable subset of R+q × R+q , we say that the function Tσ,β(f ) is ρ-concentrated
on F if

‖Mq,σ,β(f )− 1FMq,σ,β(f )‖2,θα ≤ ρ‖Mq,σ,β(f )‖2,θα . (3.12)
We have the following result

Theorem 3.4. Let f ∈ L2α(R+q ) and σ ∈ σ ∈ L2α(R+q )) ∩ L∞α (R+q ) satisfying the admissibility
condition (3.7), if f is ε-concentrated on E and Mq,σ,β(f ) is ρ-concentrated on F then we have

‖σ‖2,q,α(µα(E))
1
2

[∫
F

dθα(β, x)

β4α+2

] 1
2

≥ 1− (ε+ ρ).

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 10

Proof. Let f ∈ L2α(R+q ) and σ ∈ L2α(R) ∩ L∞α (R+q ) satisfying (3.7) and assume that µα(E) < ∞and [∫F dθα(β,x)
β4α+2

] 1
2
<∞. According to the relations (3.11),(3.12) we have

‖Mq,σ,β(f )−1FMq,σ,β(1Ef )‖2,θα ≤ ‖Mq,σ,β(f )−1FMq,σ,β(f )‖2,θα + ‖1FMq,σ,β(f −1Ef )‖2,θα

≤ ρ‖Mq,σ,β(f )‖2,θα + ‖Mq,σ,β(f − 1Ef )‖2,θα ,by using Plancherel’s relation (3.8) we get
‖Mq,σ,β(f )‖2,θα ≤ ‖Mq,σ,β(f )− 1FMq,σ,β(1Ef )‖2,θα + ‖1FMq,σ,β(1Ef )‖2,θα

≤ (ε+ ρ)‖f ‖2,q,α + ‖1FMq,σ,β(1Ef )‖2,θα , (3.13)on the other hand by using the relation (3.6) and Hölder’s inequality we find that
‖1FMq,σ,β(1Ef )‖2,θα ≤ ‖f ‖2,q,α‖σ‖1,q,α(µ(E))

1
2

[∫
F

dθα(β, x)

β4α+2

] 1
2

, (3.14)
by the relations (3.13),(3.14) we deduce that

‖Mq,σ,β(f )‖2,θα ≤ ‖f ‖2,q,α

[
(ε+ ρ) + ‖σ‖1,q,α(µα(E))

1
2

[∫
F

dθα(β, x

β4α+2

] 1
2

]
Plancherel’s formula (3.8) for Mσ,β gives the desired result. �

4. Extremal Functions Associated with the q-Bessel L2α-Multiplier Operators
In the following, we study the extremal functions associated with the the q-Bessel L2α-multiplier

operators.

Definition 4.1. Let ψ be a positive function on R+q satisfying the following conditions
1

ψ
∈ L1α(R+q ) (4.1)

and
ψ(λ) ≥ 1, λ ∈ R+q . (4.2)

We define the Sobolev-type space Sψ(R+q ) by

Sψ(R+q ) =
{
f ∈ L2α(R+q ) :

√
ψHq,α(f ) ∈ L2α(R+q )

}
provided with inner product

〈f , g〉ψ =

∫ ∞
0

ψ(λ,m)Hq,α(f )(λ)Hq,α(g)(λ)dµq,α(λ),

and the norm
‖f ‖ψ =

√
〈f , f 〉ψ.

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 11

Proposition 4.1. Let σ be a function in L∞α (R+q ). Then the q-Bessel L2α-multiplier operatorsMq,σ,β

are bounded and linear from Sψ(R+q ) into L2α(R+q ) and we have for all f ∈ Sψ(R+q )∥∥Mq,σ,β(f )
∥∥
2,q,α

≤ ‖σ‖∞,q,α‖f ‖ψ. (4.3)
Proof. By using the relations (2.8),(3.5),(4.2) we get the result �

Definition 4.2. Let η > 0 and let σ be a function in L∞α (R+q ). We denote by 〈f , g〉ψ,η the inner
product defined on the space Sψ(R+q ) by

〈f , g〉ψ,η =

∫ ∞
0

(
ηψ(λ) +

∣∣σβ(λ)
∣∣2)Hq,α(f )(λ)Hq,α(g)(λ)dµq,α(λ),

and the norm
‖f ‖ψ,η =

√
〈f , f 〉ψ,η

Theorem 4.1. Let σ ∈ L∞α (R+q ) the Sobolev-type space
(
Sψ(R+q

)
, 〈·, ·〉ψ,η) is a reproducing kernel

Hilbert space with kernel

Kq,ψ,η(x, y) =

∫ ∞
0

jα(λx ; q2)jα(λy ; q2)

ηψ(λ) +
∣∣σβ(λ)

∣∣2 dµq,α(λ),

that is
(i) For all y ∈ R+q , the function x 7→ Kq,ψ,η (x, y) belongs to Sψ(R+q ).
(ii) For all f ∈ Sψ(R+q ) and y ∈ R+q , we have the reproducing property

f (y) =
〈
f ,Kq,ψ,η(·, (y))

〉
ψ,η

.

Furthermore the kernel Kq,ψ,η is a positive definite function.

Proof. (i) Let y ∈ R+q , from the relations (2.2),(4.1) we have the function
gy : λ −→

jα(λy ; q2)

ηψ(λ) +
∣∣σβ(λ)

∣∣2
belongs to L1α(R+q ) ∩ L2α(R+q ). Hence the function Kq,ψ,η is well defined and by the inversionformula (2.5), we get

Kq,ψ,η(x, y) = H−1q,α(gy )(x)by using Plancherel’s theorem for Hq,α we find that Kq,ψ,η(·, y) belongs to L2α(R+q ) and we have
Hq,α(Kq,ψ,η(·, y))(λ) =

jα(λy ; q2)

ηψ(λ) +
∣∣σβ(λ)

∣∣2 (4.4)
by using the relations (2.2),(4.1) and (4.4) we find that

‖
√
ψHq,α(Kq,ψ,η(·, y))‖2,q,α ≤

1

η2

∥∥∥∥ 1

ψ

∥∥∥∥
1,q,α

<∞,

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 12this prove that for every y ∈ R+q the function x 7→ Kq,ψ,η (x, y) belongs to Sψ(R+q ).(ii) By using the relation (4.4) we find that for all f ∈ Hψ(R) ,
〈f ,Kq,ψ,η (·, y)〉ψ,η =

∫ ∞
0

(
ηψ(λ) +

∣∣σβ(λ)
∣∣2)Hq,α(f )(λ)Hq,α(Kq,ψ,η) (·, y))(λ)dµq,α(λ)

=

∫ ∞
0

jα(λy ; q2)Hq,α(f )(λ)dµq,α(λ),

inversion formula (2.5) gives the desired result. On the other hand since 1ψ is positive function thenfor all z1, . . . ., zn complex numbers and x1, . . . . . . , xn in R+q , we obtain
n∑
r=1

n∑
l=1

zrzlKq,ψ,η(xr , xl) =

∫ +∞
0

[
n∑
r=1

n∑
l=1

zrzl jα
(
xrλ; q2

)
jα
(
xlλ; q2

)] 1

ψ
(λ)dµq,α(λ)

=

∫ +∞
0

∣∣∣∣∣ n∑
r=1

zr j
(
xrλ; q2

)∣∣∣∣∣
2

1

ψ
(λ)dµq,α(λ) ≥ 0

Which proves that the kernel Kq,ψ,η is positive definite. �

The main result of this section can be stated as follows

Theorem 4.2. Let σ ∈ L∞α (R+q ) and β ∈ R+q , for any h ∈ L2α
(
R+q
)

and for any η > 0, there exist a
unique function f ∗q,η,β,h where the infimum

inf
f ∈Sψ(R+q )

{
η‖f ‖2ψ +

∥∥h −Mq,σ,β(f )
∥∥2
2,q,α

} (4.5)
is attained. Moreover the extremal function f ∗q,η,β,h is given by

f ∗q,η,β,h(y) =

∫ ∞
0

h(x)Θq,η,β(x, y)dµq,α(x),

where Θq,η,β is given by

Θq,η,β(x, y) =

∫ ∞
0

σβ(λ)jα(λx ; q2)jα(λy ; q2)

ηψ(λ) + |σβ(λ)|2 dµq,α(λ)

Proof. The existence and the unicity of the extremal function f ∗q,η,β,h satisfying (4.5) is given in[23,30], furthermore f ∗q,η,β,h is given by
f ∗q,η,β,h(y) = 〈h,Mq,σ,β(Kq,ψ,η (·, y))〉q

, by using inversion formula (2.5) and the relation (4.4) we get
Mq,σ,β(Kq,ψ,η (·, y) (x) =

∫ ∞
0

σβ(λ)jα(λx ; q2)jα(λy ; q2)

ηψ(λ) + |σβ(λ)|2 dµq,α(λ)

= Θq,η,β(x, y)and the proof is complete. �

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 13

Theorem 4.3. σ ∈ L∞α (R+q ) and h ∈ L2α
(
R+q
)

then the function f ∗q,η,β,h satisfies the following
properties

Hq,α(f ∗q,η,β,h)(λ) =
σβ(λ)

ηψ(λ) + |σβ(λ)|2Hq,α(h)(λ) (4.6)
and

‖f ∗q,η,β,h‖ψ ≤
1√
2η
‖h‖2,q,α.

Proof. Let y ∈ R+q then the function
ky : λ −→

σβ(λ)jα(λy ; q2)

ηψ(λ) +
∣∣σβ(λ)

∣∣2
belongs to L2α(R+q ) ∩ L1α(R+q ) and by using inversion formula (2.5) we get

Θq,η,β(x, y) = H−1q,α(ky )(x)

using Plancherel’s theorem and Parseval’s relation (2.6) we find that Θq,η,β(·, y) ∈ L2α(R+q ) and
f ∗q,η,β,h(y) =

∫ ∞
0

Hq,α(λ)ky (λ)dµq,α(λ) =

∫ ∞
0

σβ(λ)

ηψ(λ) + |σβ(λ)|2Hq,α(h)(λ)dµq,α(λ)

on the other hand the function
F : λ −→

σβ(λ)Hq,α(h)(λ)

ηψ(λ) +
∣∣σβ(λ)

∣∣2
belongs to L1α(R+q ) ∩ L2α(R+q ), by using inversion formula (2.5), Plancherel’s theorem we find that
f ∗q,η,β,h belongs to L2α(R+q ) and

Hq,α(f ∗q,η,β,h)(λ) = F (λ)on the other hand we have
|Hq,α(f ∗q,η,β,h)(λ)|2 =

∣∣σβ(λ)
∣∣2(

ηψ(λ) +
∣∣σβ(λ)

∣∣2)2 |Hq,α(h)(λ)|2 ≤
1

2ηψ(λ)
|Hq,α(h)(λ)|2

by Plancherel’s formula (2.7) we find that
‖f ∗q,η,β,h‖ψ ≤

1√
2η
‖h‖2,q,α.

�

Theorem 4.4. (Third Calderón’s formula) Let σ ∈ L∞α (R+q ) and f ∈ Sψ(R+q ) then the extremal
function given by

f ∗q,η,β(y) =

∫ ∞
0

Mq,σ,β(f )(x)Θq,η,β(x, y)dµq,α(x),

satisfies
lim
η→0+

∥∥f ∗q,η,β − f ∥∥2,q,α = 0 (4.7)
moreover we have f ∗q,η,β −→ f uniformly when η −→ 0+.

https://doi.org/10.28924/ada/ma.5.8


Eur. J. Math. Anal. 10.28924/ada/ma.5.8 14

Proof. f ∈ Sψ(R+q ), we put h =Mq,σ,β(f ) and f ∗q,η,β,h = f ∗q,η,β in the relation (4.6) we find that
Hq,α(f ∗q,η,β − f )(λ) =

−ηψ(λ)Hq,α(f )(λ)

ηψ(λ) +
∣∣σβ(λ)

∣∣2 (4.8)
therefore ∥∥f ∗q,η,β − f ∥∥2ψ =

∫ ∞
0

η2 (ψ(λ))3

ηψ(λ) + |σβ(λ)|2 |Hq,α(f )(λ)|2 dµq,α(λ)On the other hand we have
η2 (ψ(λ))3

ηψ(λ) + |σβ(λ)|2 |Hq,α(f )(λ)|2 ≤ ψ(λ) |Hq,α(f )(λ)|2 (4.9)
the result (4.7) follows from (4.9) and the dominated convergence theorem. Now, for all f ∈ Sψ(R+q )we have Hq,α(f ) ∈ L2α(R+q ) ∩ L1α(R+q ) and by using the relations (2.5), (4.8) we find that

f ∗q,η,β(y)− f (y) =

∫ ∞
0

−ηψ(λ)Hq,α(f )(λ)

ηψ(λ) +
∣∣σβ(λ)

∣∣2 jα(λy ; q2)dµq,α(λ)

and ∣∣∣∣∣−ηψ(λ)Hq,α(f )(λ)

ηψ(λ) +
∣∣σβ(λ)

∣∣2 jα(λy ; q2)

∣∣∣∣∣ ≤ |Hq,α(f )(λ)| (4.10)
By using the relation (4.10) and the dominated convergence theorem we deduce that

lim
η→0+

∣∣f ∗q,η,β(y)− f (y)
∣∣ = 0

which complete the proof of the theorem. �

Acknowledgments: The authors are deeply indebted to the referees for providing constructivecomments and helps in improving the contents of this article.
Authors’ Contributions: Both authors contributed equally to this work.

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[32] A. Saoudi, Reproducing Formulas for the Fourier-Like Multipliers Operators in q-Rubin Setting, Int. J. Anal. Appl.18 (2020), 366-380.[33] F. Soltani, I. Maktouf, Dunkl–Weinstein Multiplier Operators and Applications to Reproducing Kernel Theory,Mediterranean J. Math. 21 (2024), 80. https://doi.org/10.1007/s00009-024-02623-2.

https://doi.org/10.28924/ada/ma.5.8
https://doi.org/10.1007/s00009-024-02623-2

	1. Introduction
	2. Harmonic Analysis Associated with the q-Bessel Transform
	2.1. Notations and preliminaries
	2.2. The q-Bessel transform
	2.3. The translation operator Associated with the q-Bessel transform

	3.  The q-Bessel L2-Multiplier Operators
	3.1. Calderon's Reproducing Formulas for the q-Bessel L2-multiplier operators
	3.2.  Uncerainty principles for the q-Bessel L2-multiplier operators

	4. Extremal Functions Associated with the q-Bessel L2-Multiplier Operators
	Acknowledgments:
	Authors' Contributions:

	References

