International Journal of Analysis and Applications

Volume 18, Number 3 (2020), 366-380

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-366

REPRODUCING FORMULAS FOR THE FOURIER-LIKE MULTIPLIERS

OPERATORS IN q-RUBIN SETTING

AHMED SAOUDI1,2,∗

1Northern Border University, College of Science, Arar, P.O. Box 1631, Saudi Arabia

2 Université de Tunis El Manar, Faculté des sciences de Tunis, Tunisie

∗Corresponding author: ahmed.saoudi@ipeim.rnu.tn

Abstract. The aim of this work is to study of the q2-Fourier multiplier operators on Rq and we give for

them Calderón’s reproducing formulas and best approximation on the q2-analogue Sobolev type space Hq

using the theory of q2-Fourier transform and reproducing kernels.

1. Introduction

The q2-analogue differential-difference operator ∂q, also called q-Rubin’s operator defined on Rq in [11, 12]

by

∂qf(z) =




f(q−1z) + f(−q−1z) −f(qz) + f(−qz) − 2f(−z)
2(1 −q)z

if z 6= 0

lim
z→0

∂qf(z) in Rq if z = 0.

This operator has correct eigenvalue relationships for analogue exponential Fourier analysis using the func-

tions and orthogonalities of [9].

The q2-analogue Fourier transform we employ to make our constructions and results in this paper is based

on analogue trigonometric functions and orthogonality results from [9] which have important applications to

Received January 13th, 2020; accepted January 30th, 2020; published May 1st, 2020.

2010 Mathematics Subject Classification. 46E35; 43A32.

Key words and phrases. q-Fourier analysis; q-Rubin’s operator; L2-multiplier operators; Calderón’s reproducing formulas;

extremal functions.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

366

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-366


Int. J. Anal. Appl. 18 (3) (2020) 367

q-deformed quantum mechanics. This transform generalizing the usual Fourier transform, is given by

Fq(f)(x) := K
∫ +∞
−∞

f(t)e(−itx; q2)dqt, x ∈ R̃q.

In this paper we study the Fourier multiplier operators Tm defined for f ∈ L2q by

Tmf(x) := F−1q (maFq(f)) (x), x ∈ Rq,

where the function ma is given by

ma(x) = m(ax).

These operators are a generalization of the multiplier operators Tm associated with a bounded function m

and given by Tm(ϕ) = F−1(mF(ϕ)), where F(ϕ) denotes the ordinary Fourier transform on Rn. These

operators made the interest of several Mathematicians and they were generalized in many settings, (see for

instance [1, 2, 14, 18]).

This paper is organized as follows. In Section 2, we recall some basic harmonic analysis results related

with the q-Rubin’s operator ∂q and we introduce preliminary facts that will be used later.

In section 3, we study the q2-Fourier L2-multiplier operators Tq and we give for them a Plancherel formula

and pointwise reproducing formulas. Afterward, we give Calderón’s reproducing formulas by using the theory

of q2-analogue Fourier transform.

The last section of this paper is devoted to giving best approximation for the operators Tq and good

estimates of the associated extremal function on the q2-analogue Sobolev type space Hq studied in [15–17].

2. Notations and preliminaries

Throughout this paper, we assume 0 < q < 1 and we refer the reader to [5, 7] for the definitions and

properties of hypergeometric functions. In this section we will fix some notations and recall some preliminary

results. We put Rq = {±qn : n ∈ Z} and R̃q = Rq ∪{0}. For a ∈ C, the q-shifted factorials are defined by

(a; q)0 = 1; (a; q)n =

n−1∏
k=0

(1 −aqk),n = 1, 2, ...; (a; q)∞ =
∞∏
k=0

(1 −aqk).

We denote also

[a]q =
1 −qa

1 −q
, a ∈ C and [n]q! =

(q; q)n
(1 −q)n

, n ∈ N.

A q-analogue of the classical exponential function is given by (see [11, 12])

e(z; q2) = cos(−iz; q2) + i sin(−iz; q2), (2.1)

where

cos(z; q2) =

+∞∑
n=0

qn(n+1)
(−1)nz2n

[2n]q!
, sin(z; q2) =

+∞∑
n=0

qn(n+1)
(−1)nz2n+1

[2n + 1]q!
, (2.2)



Int. J. Anal. Appl. 18 (3) (2020) 368

satisfying the following inequality for all x ∈ Rq

|cos(x; q2)| ≤
1

(q; q)∞
, sin(x; q2)| ≤

1

(q; q)∞
and |e(ix; q2)| ≤

2

(q; q)∞
. (2.3)

The q-differential-difference operators is defined as (see [11, 12])

∂qf(z) =




f(q−1z) + f(−q−1z) −f(qz) + f(−qz) − 2f(−z)
2(1 −q)z

if z 6= 0

lim
z→0

∂qf(z) in Rq if z = 0

and we denote a repeated application by

∂0qf = f, ∂
n+1
q f = ∂q(∂

n
q f).

The q-Jackson integrals are defined by (see [6])

∫ a
0

f(x)dqx = (1 −q)a
+∞∑
n=0

qnf(aqn),

∫ b
a

f(x)dqx = (1 −q)
+∞∑
n=0

qn(bf(bqn) −af(aqn))

and ∫ +∞
−∞

f(x)dqx = (1 −q)
+∞∑

n=−∞
qn{f(qn) + f(−qn)} ,

provided the sums converge absolutely.

In the following we denote by

• Cq,0 the space of bounded functions on Rq, continued at 0 and vanishing a ∞.

• Cpq the space of functions p-times q-differentiable on Rq such that for all 0 ≤ n ≤ p. ∂pqf is continuous

on Rq,

• Dq the space of functions infinitely q-differentiable on Rq with compact supports.

• Sq stands for the q-analogue Schwartz space of smooth functions over Rq whose q-derivatives of

all order decay at infinity. Sq is endowed with the topology generated by the following family of

semi-norms:

‖u‖M,Sq (f) := sup
x∈R;k≤M

(1 + |x|)M|∂kq u(x)| for all u ∈Sq and M ∈ N.

• S′q the space of tempered distributions on Rq, it is the topological dual of Sq.

• Lpq =
{
f : ‖f‖q,p =

(∫ +∞
−∞ |f(x)|

pdqx
)1

p

< ∞
}

.

• L∞q =
{
f : ‖f‖q,∞ = supx∈Rq |f(x)| < ∞

}
.



Int. J. Anal. Appl. 18 (3) (2020) 369

The q2-Fourier transform was defined by R. L. Rubin defined in [11], as follow

Fq(f)(x) = K
∫ +∞
−∞

f(t)e(−itx; q2)dqt, x ∈ R̃q

where

K =
(q; q2)∞

2(q2; q2)∞(1 −q)2
.

To get convergence of our analogue functions to their classical counterparts as q ↑ 1 as in [9,12], we impose

the condition that 1 − q = q2m for some integer m. Therefore, in the remainder of this paper, letting q ↑ 1

subject to the condition

log(1 −q)
log(q)

∈ 2Z.

It was shown in ( [4, 11]) that the q2-Fourier transform Fq verifies the following properties:

(a) If f, uf(u) ∈ L1q, then

∂q(Fq)(f)(x) = Fq(−iuf(u))(x).

(b) If f, ∂qf ∈ L1q, then

Fq(∂q(f))(x) = ixFq(f)(x). (2.4)

(c) If f ∈ L1q, then Fq(f) ∈Cq,0 and we have

‖Fq(f)‖q,∞ ≤
2K

(q; q)∞
‖f‖q,1. (2.5)

(d) If f ∈ L1q, then, we have the reciprocity formula

∀t ∈ Rq, f(t) = K
∫ +∞
−∞

Fq(f)(x)e(itx; q2)dqx. (2.6)

(e) The q2-Fourier transform Fq is an isomorphism from Sq onto itself and we have, for all f ∈Sq

F−1q (f)(x) = Fq(f)(−x) = Fq(f)(x). (2.7)

(f) Fq is an isomorphism from L2q onto itself, and we have

‖Fq(f)‖2,q = ‖f‖q,2, ∀f ∈ L2q (2.8)

and

∀t ∈ Rq, f(t) = K
∫ +∞
−∞

Fq(f)(x)e(itx; q2)dqx.

The q-translation operator τq;x,x ∈ Rq is defined on L1q by (see [11])

τq,y(f)(x) = K

∫ +∞
−∞

Fq(f)(t)e(itx; q2)e(ity; q2)dqt, y ∈ Rq,

τq,0(f)(x) = (f)(x).

It was shown in [11] that the q-translation operator can be also defined on L2q. Furthermore, it verifies the

following properties



Int. J. Anal. Appl. 18 (3) (2020) 370

(a) For f,g ∈ L1q, we have

τq,yf(x) = τq,xf(y), ∀x,y ∈ Rq

and ∫ +∞
−∞

τq,y(f)(−x)g(x)dqx =
∫ +∞
−∞

f(x)τq,y(g)(−x)dqx, ∀y ∈ R̃q.

(b) For all f ∈ L1q and all y ∈ Rq, we have(see [3])∫ +∞
−∞

τq,y(f)(x)dqx =

∫ +∞
−∞

f(x)dqx. (2.9)

(c) For all y ∈ Rq and for all f ∈ Lpq ,1 ≤ p ≤∞, we have τq,y(f) ∈ Lpq (see [3]) and

‖τq,yf‖q,p ≤ M‖f‖q,p, (2.10)

where

M =
4(−q,q)∞

(1 −q)2q(q,q)∞
+ 2C, with C = K2‖e(·,q2)‖∞,q‖e(·,q2)‖1,q. (2.11)

(d) τq;yf is an isomorphism for f ∈ L2q onto itself and we have

‖τq,yf‖q,2 ≤
2

(q,q)∞
‖f‖q,2, ∀y ∈ R̃q. (2.12)

(e) Let f ∈ L2q, then

Fq(τq,yf)(λ) = e(iλy; q2)Fq(f)(λ), ∀y ∈ R̃q. (2.13)

The q-convolution product is defined by using the q-translation operator, as follow For f ∈ L2q and g ∈ L1q,

the q-convolution product is given by

f ∗g(y) = K
∫ +∞
−∞

τq,yf(x)g(x)dqx.

The q-convolution product satisfying the following properties:

(a) f ∗g = g ∗f.

(b) ∀f,g ∈ L1q ∩L2q, Fq(f ∗q g) = Fq(f)Fq(g).

(c) ∀f,g ∈Sq, f ∗q g ∈Sq.

(d) f ∗g ∈ L2q if and only if Fq(f)Fq(g) ∈ L2q and we have

Fq(f ∗g) = Fq(f)Fq(g).

(e) Let f,g ∈ L2q. Then we have

‖f ∗g‖2q,2 = K‖Fq(f)Fq(g)‖
2
q,2, (2.14)

and

f ∗g = F−1q (Fq(f)Fq(g)) . (2.15)



Int. J. Anal. Appl. 18 (3) (2020) 371

(f) If f,g ∈ L1q then f ∗g ∈ L1q and

‖f ∗g‖q,1 = KM‖f‖q,1‖g‖q,1. (2.16)

3. L2-Multiplier operators for the q-Rubin-Fourier transform

In this section we study the q2-Fourier-multiplier operators and we establish theirs Calderón’s reproducing

formulas in L2-case.

Definition 3.1. Let a ∈ R+q , m ∈ L2q and f a smooth function on Rq. We define the q2-Fourier L2-multiplier

operators Tm for a regular function f on Rq as follow

Tmf(x) = F−1q (maFq(f)) (x), x ∈ Rq, (3.1)

where the function ma is given by

ma(x) = m(ax).

Remark 3.1. Let a ∈ R+q , m ∈ L2q and f, we can write the operator Tm as

Tmf(x) = F−1q (ma) ∗f(x), x ∈ Rq, (3.2)

where

F−1q (ma)(x) =
1

a
F−1q (m)(

x

a
).

Proposition 3.1. (i) If m ∈ L2q and f ∈ L1q, then Tmf ∈ L2q, and we have

‖Tmf‖q,2 ≤
2K

√
a(q,q)∞

‖m‖q,2‖f‖q,1.

(ii) If m ∈ L∞q and f ∈ L2q, then Tmf ∈ L2q, and we have

‖Tmf‖q,2 ≤‖m‖∞,q‖f‖q,2.

(iii) If m ∈ L2q and f ∈ L2q, then Tmf ∈ L∞q , and we have

Tmf(x) = K
∫ ∞
−∞

m(aξ)Fq(f)(ξ)e(iξx; q2)dqξ, x ∈ Rq

and

‖Tmf‖q,∞ ≤
2K

√
a(q,q)∞

‖m‖q,2‖f‖q,2.



Int. J. Anal. Appl. 18 (3) (2020) 372

Proof. i) Let m ∈ L2q, and f ∈ L1. From the definition of the q2-Fourier L2-multiplier operators (3.1) and

relations (2.5) and (2.8) we get that the function Tmf belongs to L2q, and we have

‖Tmf‖q,2 = ‖maFq(f)‖q,2

≤
1
√
a
‖m‖q,2‖Fq(f)‖q,∞

≤
2K

√
a(q,q)∞

‖m‖q,2‖f‖q,1.

ii) The result follows from the Plancherel Theorem for the Rubin operator.

iii) Let m ∈ L2q, and f ∈ L2q, then from inversion formula we get Tmf ∈ L∞q , and by relation (2.5) we obtain

‖Tmf‖q,∞ ≤
2K

(q,q)∞
‖maFq(f)‖q,1

then, using Hölder’s inequality, we get

‖Tmf‖q,∞ ≤
2K

√
a(q,q)∞

‖m‖q,2‖f‖q,2.

�

In the following, we give Plancherel and pointwise reproducing inversion formulas for the q2-Fourier-

multiplier operators Tm.

Theorem 3.1. Let m be a function in L2q satisfying the admissibility condition:∫ ∞
0

|ma(x)|2
dqa

a
= 1, x ∈ Rq. (3.3)

i)Plancherel formula: For all f in L2q, we have∫ ∞
0

‖Tmf‖2q,2
dqa

a
= K

∫ ∞
−∞
|f(x)|2dq(x).

ii) First Calderón’s formula: Let f be a function in L1q such that Fqf in L1q then we have

f(x) =

∫ ∞
0

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
, x ∈ Rq.

Proof. i) According to identity (2.14) and relation (3.2) we have∫ ∞
0

‖Tmf‖2q,2
dqa

a
=

∫ ∞
0

‖F−1q (ma) ∗f‖
2
q,2

dqa

a

= K

∫ ∞
0

‖maFq(f)‖2q,2
dqa

a

= K

∫ ∞
−∞
|Fq(x)|

2

(∫ ∞
0

|ma|
2 dqa

a

)
dqx.



Int. J. Anal. Appl. 18 (3) (2020) 373

The result follows from Plancherel Theorem (2.8) and the assumption (3.3).

ii) Let f be a function in L1q, then∫ ∞
0

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
=

∫ ∞
0

(
K

∫ ∞
−∞
Tmf(y)τq,x

(
F−1q (ma)

)
(y)dqy

)
dqa

a
.

From Proposition 3.1 i), relation (2.12) and Plancherel Theorem, it is obvious that Tmf,τq,x
(
F−1q (ma)

)
∈ L2q.

After that, according to relation (2.13), identity (3.1) and Plancherel Theorem of the q2-Fourier transform,

we obtain∫ ∞
0

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
= K

∫ ∞
0

(∫ ∞
−∞

e(ixy; q2)Fq(f)(y)|ma(y)|2dqy
)
dqa

a
.

Since ∫ ∞
0

(∫ ∞
−∞
|e(ixy; q2)Fq(f)(y)||ma(y)|2dqy

)
dqa

a
≤‖Fq(f)‖q,1 ≤∞,

then, by Fubini’s theorem, we have∫ ∞
0

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
= K

∫ ∞
−∞

e(ixy; q2)Fq(y)
(∫ ∞

0

|ma(y)|2
dqa

a

)
dqy

= K

∫ ∞
−∞

e(ixy; q2)Fq(y)dqy = f(x).

�

We need the following technical lemma to establish the Calderón’s reproducing formulas for the q2-Fourier

L2-multiplier operators.

Lemma 3.1. Let m be a function in L2q ∩L∞q satisfy the admissibility condition (3.3). Then the function

Φγ,δ(x) =

∫ δ
γ

|m(ax)|2
dqa

a

belongs to L2q for all 0 < γ < δ < ∞ and we have

Φγ,δ(x) ∈ L2q ∩L
∞
q .

Proof. Using Hölder’s inequality for the measure
dqa
a

, we get

|Φγ,δ(x)|2 ≤ ln (δ/γ)
∫ δ
γ

|m(ax)|4
dqa

a
, x ∈ Rq.

Therefore,

‖Φγ,δ‖2q,2 ≤ ln (δ/γ)
∫ δ
γ

(∫ ∞
−∞
|m(ax)|4dqx

)
dqa

a

≤ ln (δ/γ)
∫ δ
γ

(∫ ∞
−∞
|m(x)|4dqx

)
da

a2

≤
(

1

γ
−

1

δ

)
ln (δ/γ)‖m‖2q,2‖m‖

2
q,∞ < ∞.



Int. J. Anal. Appl. 18 (3) (2020) 374

On the other hand, from the admissibility condition (3.3), we get

‖Φγ,δ‖q,∞ ≤ 1,

which completes the proof. �

Theorem 3.2. (Second Calderón’s formula) Let f ∈ L2q, m ∈ L2q ∩ L∞q satisfy the admissibility condition

(3.3) and 0 < γ < δ < ∞. Then the function

fγ,δ(x) =

∫ δ
γ

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
, x ∈ Rq

belongs to L2q and satisfies

lim
(γ,δ)→(0,∞)

‖fγ,δ −f‖q,2 = 0. (3.4)

Proof. Let f be a function in L2q, and m ∈ L2q ∩L∞q , then∫ ∞
0

(
Tmf ∗F−1q (ma)

)
(x)

dqa

a
=

∫ ∞
0

(
K

∫ ∞
−∞
Tmf(y)τq,x

(
F−1q (ma)

)
(y)dqy

)
dqa

a
.

According to Proposition 3.1, relation (2.12) and Plancherel Theorem, it is obvious that

Tmf,τq,x
(
F−1q (ma)

)
∈ L2q. Then, from relation (2.13) and the identity (3.1), we obtain

fγ,δ(x) = K

∫ δ
γ

(∫ ∞
−∞

e(ixy,q2)Fq(f)(y)|ma(y)|2dqy
)
dqa

a
.

By Fubini-Tonnelli’s theorem, Hölder’s inequality and Lemma 3.1, we get∫ δ
γ

(∫ ∞
−∞
|e(ixy,q2)Fq(f)(y)||ma(y)|2dqy

)
dqa

a
≤

2

(q,q)∞

∫ ∞
−∞
|Fq(f)(y)|Φγ,δ(y)dqy

≤
2

(q,q)∞
‖f‖q,2‖Φγ,δ‖q,2 < ∞.

Then, according to Fubini’s theorem and the inversion formula, we have

fγ,δ(x) = K

∫ ∞
−∞

e(ixy,q2)Fq(f)(y)

(∫ δ
γ

|ma(y)|2
dqa

a

)
dqy

= K

∫ ∞
−∞

e(ixy,q2)Fq(f)(y)Φγ,δ(y)dqy

= F−1q [Fq(f)Φγ,δ] (x).

On the other hand, the function Φγ,δ belongs to L
∞
q which allows to see that fγ,δ belongs to L

2
q and using

the identity (2.15), we obtain

Fq(fγ,δ) = Fq(f)Φγ,δ.

By the Plancherel formula we get

‖fγ,δ −f‖2q,2 =
∫ ∞
−∞
|Fq(f)(y)|2(1 − Φγ,δ(y))2dqy.



Int. J. Anal. Appl. 18 (3) (2020) 375

The the admissibility condition (3.3) leads to

lim
(γ,δ)→(0,∞)

Φγ,δ(y) = 1, y ∈ Rq

and

|Fq(f)(y)|2(1 − Φγ,δ(y))2 ≤ |Fq(f)(y)|2.

Finally, the relation (3.4) follows from the dominated convergence theorem. �

4. The extremal function associated with q2-Fourier L2-multiplier operators

In this section, we study the extremal function associated to the q2-Fourier L2-multiplier operators.

Let s ∈ R and 1 ≤ p < ∞, the q2-analogue Sobolev type spaces is defined in [15] by

Ws,pq =
{
u ∈S′q : (1 + |ξ|

2)
s
2Fq(u) ∈ Lpq

}
.

In the particular case p = 2, we denote Ws,pq by Hsq which provided with the inner product

〈u,v〉Hsq =
∫ +∞
−∞

(1 + ξ2)sFq(u)(ξ)Fq(v)(ξ)dqξ

and the norm

‖u‖Hsq :=
√
〈u,u〉Hsq.

Hsq is a Hilbert space satisfying the following properties

(a) H0q = L2q.

(b) For all s > 0 the space Hsq is continuously contained in L2q and we have

‖f‖q,2 ≤‖f‖Hsq. (4.1)

Proposition 4.1. Let m be a function in L∞q . Then the q
2-Fourier L2-multiplier operators Tm are bounded

and linear from Hsq into L2q and we have for all f ∈Hsq

‖Tmf‖q,2 ≤‖m‖q,∞‖f‖Hsq.

Proof. Let f ∈Hsq. According to Proposition 3.1 (ii), the operator Tm belongs to L2q and we have

‖Tmf‖q,2 ≤‖m‖q,∞‖f‖q,2.

On the other hand, by the inequality (4.1) we have ‖f‖q,2 ≤‖f‖Hsq , which gives the result. �



Int. J. Anal. Appl. 18 (3) (2020) 376

Definition 4.1. Let η > 0 and let m be a function in L∞q . We denote by 〈u,v〉Hsq,η the inner product defined

on the space Hsq by

〈f,g〉Hsq,η = η〈f,g〉Hsq + 〈Tmf,Tmg〉q,2 (4.2)

and the norm

‖f‖Hsq,η =
√
〈f,f〉Hsq,η.

It is easy to show the following results.

Proposition 4.2. Let m be a function in L∞q and f in Hsq
(i) The norm ‖ ·‖Hsq,η satisfies:

‖f‖2Hsq,η = η‖f‖
2
Hsq

+ ‖Tmf‖2q,2.

(ii) The norms ‖ ·‖Hsq,η and ‖ ·‖Hsq are equivalent and we have

√
η ‖f‖Hsq ≤‖f‖Hsq,η ≤

√
η + ‖m‖2q,∞ ‖f‖Hsq.

Theorem 4.1. Let s > 1
2

and m be a function in L∞q . Then the Hilbert space (Hsq,〈·, ·〉Hsq,η) has the following

reproducing Kernel

Ψs,η(x,y) =

∫ ∞
−∞

e(ixξ,q2)e(−iyξ,q2)
η(1 + |ξ|2)s + |ma(ξ)|2

dq(ξ), (4.3)

such that

(i) For all y ∈ Rq, the function x 7→ Ψs,η(x,y) belongs to Hsq.

(ii) For all f ∈Hsq and y ∈ Rq, we have the reproducing property

〈f, Ψs,η(·,y)〉Hsq,η = f(y).

(iii) The Hilbert space (Hsq,〈·, ·〉Hsq ) has the following reproducing Kernel

Ψs(x,y) =

∫ ∞
−∞

e(ixξ,q2)e(−iyξ,q2)
(1 + |ξ|2)s

dq(ξ). (4.4)

Proof. (i) Let y ∈ Rq and s > 12 . From the relation (2.3), we show that the function

ϕy : ξ −→
e(−iyξ,q2)

η(1 + |ξ|2)s + |ma(ξ)|2

belongs to L1q ∩L2q. Hence the function Ψs,η is well defined and by the inversion formula, we obtain

Ψs,η(x,y) = F−1q (ϕy)(x), x ∈ Rq.

On the other hand, using Plancherel theorem, we get that Ψs,η(·,y) belongs to L2q and we have

Fq (Ψs,η(·,y)) (ξ) =
e(−iyξ,q2)

η(1 + |ξ|2)s + |ma(ξ)|2
, ξ ∈ Rq. (4.5)

Therefore, by the identity (2) we obtain



Int. J. Anal. Appl. 18 (3) (2020) 377

|Fq (Ψs,η(·,y)) (ξ)| ≤
(q,q)−1∞

2η(1 + |ξ|2)s
,

and

‖Ψs,η(·,y)‖2Hsq ≤ (2η(q,q)∞)
−2‖(1 + | · |2)−s‖q,1 < ∞.

This proves that for every y ∈ Rq, the function Ψs,η(·,y) belongs to Hsq.

(ii) Let f ∈ Hsq and y ∈ Rq. According to the definition of inner product (4.2) and identity (4.5), we

obtain

〈f, Ψs,η(·,y)〉Hsq,η =
∫ ∞
−∞

e(ixξ,q2)Fq(ξ)dq(ξ).

On the other hand, the function ξ 7−→ (1 + |ξ|2)−s/2 belongs to L2q for all s > 1/2. Therefore, the function

Fq(f) belongs to L1q and we have

〈f, Ψs,η(·,y)〉Hsq,η = f(y).

(iii) The result is obtained by taking m a null function and η = 1. �

The main result of this section can be stated as follows.

Theorem 4.2. Let s > 1
2

and m be a function in L∞q and a > 0. For any h ∈ L2q and for any η > 0, there

exists a unique function f∗η,h,a, where the infimum

inf
f∈Hsq

{
η‖f‖2Hsq + ‖h−Tmf‖

2
q,2

}
(4.6)

is attained. Moreover the extremal function f∗η,h,a is given by

f∗η,h,a(y) =

∫ ∞
∞

h(x)Θs,η(x,y)dqx, (4.7)

where

Θs,η(x,y) =

∫ ∞
−∞

ma(ξ)e(ixξ,q
2)

η(1 + |ξ|2)s + |ma(ξ)|2
e(−iyξ,q2)dqξ.

Proof. The existence and unicity of the extremal function f∗η,h,a satisfying (4.6) is given by [8, 10, 13]. On

the other hand from Theorem 4.1 we have

f∗η,h,a(y) = 〈h,Tm(Ψs,η(·,y))〉q,2.

From Proposition 3.1 and identity (4.5) we obtain

Θs,η(x,y) = Tm(Ψs,η(·,y))(x)

=

∫ ∞
−∞

ma(ξ)e(ixξ,q
2)

η(1 + |ξ|2)s + |ma(ξ)|2
e(−iyξ,q2)dqξ.

�



Int. J. Anal. Appl. 18 (3) (2020) 378

Theorem 4.3. Let s > 1
2

and m be a function in L∞q and h ∈ L2q. Then the extremal function f∗η,h,a satisfies

the following properties:

Fq(f∗η,h,a)(ξ) =
ma(ξ)

η(1 + |ξ|2)s + |ma(ξ)|2
Fq(h)(ξ), ξ ∈ Rq

and

‖f∗η,h,a‖
2
Hsq
≤

1

4η
‖h‖2q,2.

Proof. Let y ∈ Rq, then the function

gy : ξ 7−→
ma(ξ)e(−iyξ,q2)

η(1 + |ξ|2)s + |ma(ξ)|2

belongs to L1q ∩L2q and by the inversion formula we obtain

Θs,η(x,y) = F−1q (gy)(x), x ∈ Rq.

Hence, by Plancherel formula, we have Θs,η(·,y) belongs to L2q and

f∗η,h,a(y) =

∫ ∞
−∞
Fq(h)(ξ)gy(ξ)dqξ

=

∫ ∞
−∞

ma(ξ)Fq(h)(ξ)
η(1 + |ξ|2)s + |ma(ξ)|2

e(iyξ,q2)dqξ.

On the other hand, the function

F : ξ 7−→
ma(ξ)Fq(h)(ξ)

η(1 + |ξ|2)s + |ma(ξ)|2

belongs to L1q ∩L2q and by the inversion formula we obtain

f∗η,h,a(y) = F
−1
q (F)(y).

Afterwards, by Plancherel formula, it follows that f∗η,h,a belongs to L
2
q and we have

Fq(f∗η,h,a)(ξ) =
ma(ξ)Fq(h)(ξ)

η(1 + |ξ|2)s + |ma(ξ)|2
, ξ ∈ Rq.

Hence

(1 + |ξ|2)s
∣∣Fq(f∗η,h,a)(ξ)∣∣2 = (1 + |ξ|2)s

∣∣∣∣∣ ma(ξ)Fq(h)(ξ)η(1 + |ξ|2)s + |ma(ξ)|2
∣∣∣∣∣
2

≤ (1 + |ξ|2)s

∣∣∣ma(ξ)Fq(h)(ξ)∣∣∣2
4η(1 + |ξ|2)s|ma(ξ)|2

≤
1

4η
|Fq(h)(ξ)|

2
.

Finally, using Plancherel theorem, we obtain

‖f∗η,h,a‖
2
Hsq
≤

1

4η
‖h‖2q,2.



Int. J. Anal. Appl. 18 (3) (2020) 379

�

Theorem 4.4. (Third Calderón’s formula). Let s > 1
2

, m be a function in L∞q and f ∈ Hsq. The extremal

function given by

f∗η,a(y) =

∫ ∞
−∞
Tmf(x)Θs,η(x,y)dqx (4.8)

satisfies

lim
η→0+

‖f∗η,a −f‖Hsq = 0.

Moreover, {f∗η,a}η>0 converges uniformly to f when η converge to 0+.

Proof. Let f ∈ Hsq, h = Tmf and f∗η,a = f∗η,h,a. According to Proposition 4.1 the function h belongs to L
2
q.

From the definition of the q2-Fourier-multiplier operators Tm and Theorem 4.3, we obtain

Fq(f∗η,a)(ξ) =
|ma(ξ)|2

η(1 + |ξ|2)s + |ma(ξ)|2
Fq(f)(ξ), ξ ∈ Rq.

Hence, it follows that

Fq(f∗η,a −f)(ξ) =
−η(1 + |ξ|2)s

η(1 + |ξ|2)s + |ma(ξ)|2
Fq(f)(ξ), ξ ∈ Rq. (4.9)

Therefore,

‖f∗η,a −f‖
2
Hsq

=

∫ ∞
−∞

η2(1 + |ξ|2)3s(ξ)|Fq(f)(ξ)|2

(η(1 + |ξ|2)s + |ma(ξ)|2)
2
dqx.

Then, from the dominated convergence theorem and the following inequality

η2(1 + |ξ|2)3s|Fq(f)(ξ)|2

(η(1 + |ξ|2)s + |ma(ξ)|2)
2
≤ (1 + |ξ|2)s|Fq(f)(ξ)|2,

we deduce that

lim
η→0+

‖f∗η,a −f‖Hsq = 0.

On the other hand, the function ξ 7−→ (1 + |ξ|2)−s/2 belongs to L2q for all s > 1/2. Therefore, the function

Fq(f) belongs to L1q∩L2q for all f ∈Hsq. Then, according to (4.9) and the inversion formula for the q2-Fourier

transform, we get

f∗η,a(y) −f(y) = K
∫ ∞
−∞

−η(1 + |ξ|2)sFq(f)(ξ)
η(1 + |ξ|2)s + |ma(ξ)|2

e(iyξ,q2)dqx.

By using the dominated convergence theorem and the fact

η(1 + |ξ|2)s|Fq(f)(ξ)|2

η(1 + |ξ|2)s + |ma(ξ)|2
≤ |Fq(f)(ξ)|,

we deduce that

lim
η→0+

sup
y∈Rq

‖f∗η,a(y) −f(y)‖ = 0.

which completes the proof of the Theorem. �



Int. J. Anal. Appl. 18 (3) (2020) 380

Acknowledgement 4.1. The author gratefully acknowledge the approval and the support of this research

study by the grant no. 7909-SCI-2018-3-9-F from the Deanship of Scientific Research at Northern Border

University, Arar, K.S.A.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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https://doi.org/10.1142/S1793557121500030
https://doi.org/10.1142/S1793557121500030
https://doi.org/10.1080/00036811.2018.1555321
https://doi.org/10.1080/00036811.2018.1555321

	1. Introduction
	2. Notations and preliminaries
	3.  L2-Multiplier operators for the q-Rubin-Fourier transform
	4. The extremal function associated with q2-Fourier L2-multiplier operators
	References