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On a Family of q-Weighted Bergman Spaces and Applications

Akram Nemri
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Kingdom of

Saudi Arabia

nakram@jazanu.edu.sa

Abstract. In this paper, we introduce a q-weighted Bergman spaces {Aα,n,q}n∈N. For n = 0 anuncertainty inequality of the Heisenberg-type for the space Aα,q is given by considering the operators
∇α,q := ∇α,q and Lα,q := Lα,q . Also, we study on this space the q-Toeplitz operators, the q-Hankeloperators. At the end, we study the theory of extremal function and reproducing kernel of Hilbertspace and we use it to establish the extremal function associated to an bounded linear operator
T : Aα,q → H, for any Hilbert space H.

1. Introduction
Many studies has happened in the last decade, characterizing the action of operators on Bergmanand weighted Bergman spaces. This line of inquiry has attracted interest owing to its intimate linkswith complex analysis, functional analysis, and operator theory [6]. Many techniques have beeninvestigate in different types of operators. For example, Hankel operators have been thoroughlyanalyzed using function-theoretic and operator-theoretic approaches ( [1], [12]); composition opera-tors have been studied through dynamical and analytic techniques ( [14]); and multiplier operatorshave been explored in the context of reproducing kernel Hilbert spaces and boundedness criteria.These developments have significantly enriched the theory and opened new directions for furtherinvestigation.The main results of this paper is to deal with operators acting on a general q-weighted Bergmanspaces {Aα,n,q}n∈N. We prove some properties concerning q-Toeplitz operators and q-Hankeloperators; we establish a more general Heisenberg-type uncertainty principle given in [16] for thespace Aα,q by considering the operators ∇α,q := ∇α,q and Lα,q := Lα,q ; we study the theoryof extremal function and reproducing kernel of Hilbert space, to establish the extremal functionassociated to a bounded linear operator T . Noting that, there exist many similar uncertainty
Received: 20 Apr 2025.
Key words and phrases. q-weighted Bergman spaces, uncertainty inequality, q-Toeplitz operators, q-Hankel opera-tors, extremal function. 1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.20
https://orcid.org/0000-0001-9195-5037


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 2principles, in physics [2], [4], [10], and mathematics [3], [19], that are based on position, momentum,energy, time, and so on.The weighted Bergman space is one of the complex analysis tools used in harmonic analysis [7].Let C be the complex plane, D =
{
z ∈ C : |z | < 1

} the open unit disk and H(D) the space of allanalytic functions on D. For any α > 0,
dνα(z) :=

1

π
α(1− |z |2)α−1dxdy

is the weighted Lebesgue measure on D. The weighted Bergman space Aα is the space
H(D)

⋂
L2(D, dνα). Noting that, it is an Hilbert when space equipped with the inner product

〈f , g〉Aα :=

∫
D

f (z)g(z)dνα(z),

and the norm ‖f ‖Aα = ‖f ‖L2α,q(D), see [8,16,20] for more details on the theory of Bergman spaces.The contents of the paper are as follows. Section 2 reviewers from [16] the q-analogue of the q-weighted Bergman spaceAα,q and we will introduce the q-analogue of q-weighted Bergman spaces
{Aα,n,q}n∈N. In Sect.3, we will study the q-derivative operator∇α,q and its adjoint operator Lα,q onthe q-weighted Bergman space Aα,q , we will prove some properties concerning q-Toeplitz operatorsand q-Hankel operators and we will establish at the end of this section a general uncertaintyinequality of Heisenberg type for the space Aα,q . In Sect.4, we will give an application of thetheory of extremal function and reproducing kernel of Hilbert space by establishing the extremalfunction associated to a bounded linear operator T .

2. Preliminaries
In all the sequel, assume that 0 < q < 1 and α > 0. The reader can refer to [9] and [13] formore details for the definitions and notations of the basic hypergeometric series, the Jackson’s

q-derivative and q-integrals, q-Gamma and q-Beta functions. The reference [16] is devoted to the
q-weighted Bergman space on the disk.

The standard Watson’s notation for the q-shifted factorials are defined for any complex number
a by

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

n∏
k=0

(1− aqk−1), n = 1, 2, ..., (a; q)∞ :=

∞∏
k=0

(1− aqk−1),

and [a]q is standing for the number associated to a,
[a]q :=

1− qa

1− q , [a]q! :=
(q; q)n

(1− q)n
, n ∈ N.

For any complex z , (a; q)z is defined by
(a; q)z :=

(a; q)∞
(aqz ; q)∞

, (1)

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 3and the q-binomial theorem [9] is given by
∞∑
n=0

(a; q)n
(q; q)n

zn =
(az ; q)∞
(z ; q)∞

. (2)
The q-analogue of the classical Euler Gamma and Beta functions defined by Jackson in [11] are

Γq(a) :=
(q; q)∞
(qa; q)∞

(1− q)1−a, <(a) > 0.

βq(a, b) :=

∫ 1
0

ta−1(qt; q)b−1dqt =
Γq(a)Γq(b)

Γq(a + b)
, <(a),<(b) > 0. (3)

The q-analogue exponential functions eq(z) and Eq(z) [9] are given by
eq(z) :=

∞∑
n=0

(1− q)zn

(q; q)n
=

1

(z ; q)∞
,

Eq(z) :=

∞∑
n=0

qn(n−1)/2(1− q)zn

(q; q)n
= (−z ; q)∞.

The q-derivative [9] on a subset of C is defined by
Dq,z f (z) :=

f (z)− f (qz)

(1− q)z
, z 6= 0. (4)

In all the sequel, we need the following spaces:
• H(D) the space of all analytic functions on the unit open disk D = {z ∈ C; |z | < 1}.
• L2α,q(D) := L2q(D, dνα,q) the space of measurable functions f on the unit disk D satisfying
‖ f ‖2L2α,q(D):=

[α]q
2π

∫ 1
0

(∫ 2π
0

| f (re iθ) |2 dθ
)

(qr2; q)α−1dq(r2) :=

∫
D
| f (z) |2 dνα,q(z)

is finite, where dνα,q [5] the measure defined on the unit disk D for α > 0 by
dνα,q(z) :=

[α]q
2π

(qr2; q)α−1dq(r2)dθ; z = re iθ,

and dθ is the usual Lebesgue measure on [0, 2π[ and the integral with respect to dq(r2) isrelated to the q-Jackson’s integral over [0, 1] defined by:∫ 1
0

f (t)dqt := (1− q)

∞∑
n=0

f (qn)qn.

• Aα,q := Aα,q(D) the q-weighted Bergman space of all functions in H(D)
⋂
L2α,q(D). It isa Hilbert space when equipped with the inner product

〈f , g〉Aα,q =

∫
D
f (z)g(z)dνα,q(z).

and the norm
‖f ‖Aα,q =

(∫
D
|f (z)|2dνα,q(z)

)1/2
.

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 4

• Aα,n,q := Aα,n,q(D), the Hilbert space of functions on H(D), such that
‖f ‖2Aα,n,q := |f (0)|2 +

∫
D
| Nnq f (z) |2 dνα,q, n = 1, 2, ...

‖f ‖2Aα,0,q := ‖f ‖2Aα,q ,

Nq is the q-multiplication operator on Aα,q given by Nq := zDq,z .

Moreover, if f (z) =
∑∞
k=0 akz

k then
‖f ‖2Aα,n,q = |a0|2 +

∞∑
k=1

[k ]2nq Ck(α; q)|ak |2,

where
Cn(α; q) :=

(q; q)n
(qα+1; q)n

.

3. Uncertainty inequality on the q-weighted Bergman space Aα,q
Consider the q-operator ∇α,q and Lα,q are the operators on Aα,q(D, dνα,q) defined by

∇α,q := q−α−1Dq,z , Nq := zDq,z Lα,q := z2Dq,z + [α+ 1]qq
−α−1z. (5)

So, we have the following q-commutation relation
Lemma 3.1. [∇α,q, Lα,q]q := ∇α,qLα,q − Lα,q∇α,q = q−α−1Λq

(
[α+ 1]qI + (1 + q−1)q−α−1Nq

)
,

where I is the identity operator and Λq is the q-shift operator given by Λqf (z) = f (qz).

We derive the following results
Proposition 3.1. Let f , g ∈ Aα,q(D, dνα,q) with f (z) =

∑∞
n=0 anz

n and g(z) =
∑∞
n=0 bnz

n, we
have(i) 〈f , g〉Aα,q(D,dνα,q) =

∞∑
n=0

anbn
(q; q)n

(qα+1; q)n
=

∞∑
n=0

anbn Cn(α; q).

(ii) ||f ||2Aα,q(D,dνα,q) =

∞∑
n=0

|an|2
(q; q)n

(qα+1; q)n
=

∞∑
n=0

|an|2 Cn(α; q).

(iii) The set
{
ξαn,q(z) :=

zn√
Cn(α; q)

}
n≥0

, forms a Hilbert’s basis for the space Aα,q(D, dνα,q).

Proof. Given f (z) =
∑∞
k=0 akz

k and g(z) =
∑∞
k=0 bkz

k , the result follows by using dominateconvergence theorem and relation (4.6) in [5] we have
〈f , g〉Aα,q(D,dνα,q) =

∞∑
m,n=0

ambn

∫
D
zm zn dνα,q(z) =

∞∑
n=0

ambn
(q; q)n

(qα+1; q)n
. (6)

The last assertion follows directly from Proposition 4.1 in [5]. �

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 5

Theorem 3.1. The function Kα,q given for w, z ∈ D, by

Kα,q(z, w) = Kα,q(zw) =
1

(zw ; q)α+1
, (7)

is a reproducing kernel for the q-weighted Bergman space Aα,q(D, dνα,q).
That is(i) for all w ∈ D, z 7−→ Kα,q(z, w) belong to Aα,q(D, dνα,q).(ii) for all w, z ∈ D and f ∈ Aα,q(D, dνα,q), we have 〈f ,Kα,q(., w)〉Aα,q(D,dνα,q) = f (w).

(iii) For all f ∈ Aα,q(D, dνα,q) and z ∈ C, | f (z) |≤
[
eq(|z |2)Eq(qα+1|z |2)

]1/2
‖

f ‖Aα,q(D,dνα,q) .(iv) Let w ∈ D. The function u(z) = Kα,q(zw) is the unique analytic solution on D of the
initial problem

z∇α,qu(z) = wLα,qu(z), u(0) = 1.

Proof. To prove the first assertion (i), we use Proposition 3.1 (iii) the function ξαn,q(z) constitutean orthonormal basis of Aα,q(D, dνα,q). Therefore for any z, w ∈ D, Kα,q can be computed byevaluating the following sum
Kα,q(z, w) =

∞∑
n=0

ξαn,q(z)ξαn,q(w) =

∞∑
n=0

1

Cn(α; q)
znwn.

Hence by (1) combined with (2) we deduce easily
Kα,q(z, w) =

∞∑
n=0

(qα+1; q)n
(q; q)n

(zw)n =
(qα+1zw ; q)∞

(zw ; q)∞
=

1

(zw ; q)α+1
.

To prove (ii), we use the same as in Proposition 4.2 in [5]. The last assertion follows by using (4). �
The domain of the operator ∇α,q denoted by Dom(∇α,q) is defined by

Dom(∇α,q) :=
{
f ∈ Aα,q(D, dνα,q); ∇α,qf ∈ Aα,q(D, dνα,q)

}
,

and same for Domq(Nq) and Domq(Lα,q).
Lemma 3.2. The operators ∇α,q , Nq and Lα,q satisfies the following(i) Dom(∇α,q) = Dom(Lα,q) = Dom(Nq) = Aα,1,q.(ii) For any f , g in Aα,1,q we have: 〈∇α,qf , g〉Aα,q(D,dνα,q) = 〈f , Lα,qg〉Aα,q(D,dνα,q).(iii) For any f in Aα,1,q we have

‖ Lα,qf ‖2Aα,q(D,dνα,q)=‖ ∇α,qf ‖
2
Aα,q +q−α−1[α+ 1]q ‖ Λq1/2 f ‖2Aα,q +q−α−1(1 + q−1)〈NqΛq1/2 f ,Λq1/2 f 〉Aα,q .

Proof. Let f ∈ Aα,1,q , with f (z) =
∑∞
k=0 akz

k . Then using relation (4), we have respectively
∇α,qf (z) =

∞∑
k=1

q−α−1[k ]qakz
k−1 =

∞∑
k=0

q−α−1[k + 1]qak+1z
k (8)

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 6and
Lα,qf (z) =

∞∑
k=0

([k ]q + q−α−1[α+ 1]q)akz
k+1 =

∞∑
k=1

([k − 1]q + q−α−1[α+ 1]q)ak−1z
k . (9)

Thus from the previous relation, we get
‖ ∇α,qf ‖2Aα,q= 〈∇α,qf ,∇α,qf 〉Aα,q = 〈f , Lα,q∇α,qf 〉Aα,q =

∞∑
k=1

q−α−1[k]q

(
[k − 1]q + q−α−1[α+ 1]q

)
|ak |2Ck(α; q),

(10)
‖ Lα,qf ‖2Aα,q= 〈Lα,qf , Lα,qf 〉Aα,q = 〈f ,∇α,qLα,qf 〉Aα,q =

∞∑
k=1

q−α−1[k+1]q

(
[k]q +q−α−1[α+1]q

)
|ak |2Ck(α; q), (11)

and
‖ Nqf ‖2Aα,q(D,dνα,q)= 〈Nqf , Nqf 〉Aα,q(D,dνα,q) =

∞∑
k=1

[k ]2q|ak |2Ck(α; q). (12)
Therefore, from Proposition 3.1, (10), (11) and (12) we deduce easily
‖f ‖2Aα,q(D,dνα,q) − |f (0)|2 ≤ ‖∇α,qf ‖2Aα,q(D,dνα,q) ≤ (1 + q−α−1[α+ 1]q)‖f ‖2Aα,1,q(D,dνα,q)

‖f ‖2Aα,1,q(D,dνα,q) ≤ ‖Lα,qf ‖2Aα,q(D,dνα,q) ≤ [2]q(1 + q−α−1[α+ 1]q)‖f ‖2Aα,1,q(D,dνα,q)

‖f ‖2Aα,1,q(D,dνα,q) − |f (0)|2 ≤ ‖Nqf ‖2Aα,q(D,dνα,q) ≤ ‖f ‖
2
Aα,1,q(D,dνα,q).So, Dom(∇α,q) = Dom(Lα,q) = Dom(Nq) = Aα,1,q(D, dνα,q).

To prove (ii), let f , g in Aα,1,q(D, dνα,q) with f (z) =
∑∞
k=0 akz

k and g(z) =
∑∞
k=0 bkz

k . FromProposition 3.1, (8) and (9) we have
〈∇α,qf , g〉Aα,q(D,dνα,q) =

∞∑
k=0

q−α−1[k + 1]qak+1bkCk(α; q)

=

∞∑
k=0

ak+1bk
(q; q)k+1

(1− q)qα+1(qα+1; q)k

=

∞∑
k=1

akbk−1
(q; q)k

(1− q)qα+1(qα+1; q)k−1
,

on the other hand
〈f , Lα,qg〉Aα,q(D,dνα,q) =

∞∑
k=0

([k − 1]q + q−α−1[α+ 1]q)[k + 1]qakbk−1Ck(α; q)

=

∞∑
k=0

1− qα+k

1− q akbk−1Ck(α; q)

=

∞∑
k=1

[k + α]qakbk−1
(q; q)k

(qα+1; q)k

=

∞∑
k=1

akbk−1
(q; q)k

qα+1(1− q)(qα+1; q)k−1
= 〈∇α,qf , g〉Aα,q(D,dνα,q).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.20 7

Finally, to prove (iii), using [k + 1]q = [k ]q + qk we deduce easily that
[k + 1]q

(
[k ]q + q−α−1[α+ 1]q

)
=
(

[k ]q + qk
)(

[k − 1]q + qk−1 + q−α−1[α+ 1]q

)
= [k ]q

(
[k − 1]q + q−α−1[α+ 1]q

)
+ qk−α−1[α+ 1]q +

(
1 + q−1

)
qk [k ]q.

Which leads to the result using (10), (11), (12) and the fact that ΛqNq = NqΛq . �

Lemma 3.3. Dom(∇α,qLα,q) = Dom(∇α,qLα,q) = Aα,2,q(D, dνα,q).

Proof. Let f ∈ Aα,q(D, dνα,q), with f (z) =
∑∞
k=0 akz

k . Then using relation (8) and (9) weobtain
∇α,qLα,qf (z) =

∞∑
k=0

q−α−1[k + 1]q([k ]q + q−α−1[α+ 1]q)akz
k

and
Lα,q∇α,qf (z) =

∞∑
k=1

q−α−1[k ]q([k − 1]q + q−α−1[α+ 1]q)akz
k .

Therefore,
‖ ∇α,qLα,qf ‖Aα,q(D,dνα,q)=

∞∑
k=0

q−2(α−+)[k + 1]2q([k ]q + q−α−1[α+ 1]q)2|ak |2Ck(α; q)

and
‖ Lα,q∇α,qf ‖Aα,q(D,dνα,q)=

∞∑
k=1

q−2(α+1)[k ]2q([k − 1]q + q−α−1[α+ 1]q)2|ak |2Ck(α; q).

So, as in the previous lemma, from Proposition 3.1, we deduce easily
‖f ‖2Aα,2,q(D,dνα,q) − |f (0)|2 ≤ ‖Lα,q∇α,qf ‖2Aα,q(D,dνα,q) ≤ (1 + q−α−1[α+ 1]q)2‖f ‖2Aα,2,q(D,dνα,q),

and
‖f ‖2Aα,2,q(D,dνα,q) ≤ ‖∇α,qLα,qf ‖

2
Aα,q(D,dνα,q) ≤ [2]2q(1 + q−α−1[α+ 1]q)2‖f ‖2Aα,2,q(D,dνα,q).

Thus, Dom(∇α,qLα,q) = Dom(∇α,qLα,q) = Aα,2,q. �

We can now establish an uncertainty inequality of Heisenberg-type on the spaceAα,q(D, dνα,q),by the virtue of the following lemma:
Lemma 3.4. [6] Let X and Y be self-adjoint operators on Hilbert space H (i.e X∗ = X and Y ∗ = Y ).
Then

‖ (X − a)f ‖H‖ (Y − b)f ‖H≥
1

2
| 〈[X, Y ]f , f 〉H |,

for all f in Dom(XY ) ∩Dom(Y X) and a, b ∈ R

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Eur. J. Math. Anal. 10.28924/ada/ma.5.20 8

Theorem 3.2. Let f ∈ Aα,2,q(D, dνα,q). For all a, b ∈ R, we have

‖ (∇α,q + Lα,q − a)f ‖Aα,q‖ (∇α,q − Lα,q + ib)f ‖Aα,q

≥ q−α−1[α+ 1]q ‖ Λq1/2f ‖
2
Aα,q +q−α−1(1 + q−1)〈NqΛq1/2f ,Λq1/2f 〉Aα,q .

Proof. Consider X := ∇α,q +Lα,q and Y := i(∇α,q −Lα,q). By Lemma 3.2 and Lemma 3.3, theoperators X and Y verifies the following properties(a.) X∗ = X and Y ∗ = Y(b.) Dom(XY ) = Dom(Y X) = Aα,2,q(c.) [X, Y ]q = −2i [∇α,q, Lα,q]q .So, the result follows from Lemma 3.1 and Lemma 3.2. �

Proposition 3.2. Let a, b ∈ R.(i) For all f ∈ Aα,2,q(D, dνα,q), we have

‖ (∇α,q + Lα,q − a)f ‖Aα,q(D,dνα,q)‖ (∇α,q − Lα,q + ib)f ‖Aα,q(D,dνα,q)

≥‖ Lα,qf ‖2Aα,q(D,dνα,q) − ‖ ∇α,qf ‖
2
Aα,q(D,dνα,q) .

(ii) For all f ∈ Aα,1,q , we have

qα+1 ‖ (∇α,q + Lα,q − a)f ‖Aα,1,q‖ (∇α,q − Lα,q + ib)f ‖Aα,1,q

≥ [α+ 1]q ‖ Λq1/2f ‖
2
Aα,1,q +(1 + q−1)〈NqΛq1/2f ,Λq1/2f 〉Aα,1,q .

Proof. Let a, b ∈ R. The first inequality (i) hold from Lemma 3.2 (iii) and the second inequality(ii) hold by applying Lemma 3.2 (i). �

4. Operators on the q-weighted Bergman space Aα,q
4.1. q-Toepliz Operator on Aα,q . Consider the orthogonal projection operator Pα,q : L2α,q(D) →
Aα,q . Since L2α,q(D) = Aα,q ⊕A⊥α,q then for any f ∈ L2α,q(D), we have f = (f − f ⊥) + f ⊥ where
f − f ⊥ ∈ Aα,q and f ⊥ ∈ A⊥α,q . Furthermore, for z ∈ D,

Pα,qf (z) = (f − f ⊥)(z) = 〈(f − f ⊥)(z),Kα,q(z, .)〉L2α,q(D) = 〈f (z),Kα,q(z, .)〉L2α,q(D),

where Kα,q is the reproducing kernel given by (7). The following assertions then follow
Proposition 4.1. For all f , g ∈ L2α,q(D), we have:(i) Pα,q ◦ Pα,qf = Pα,qf .(ii) 〈Pα,qf , g〉L2α,q(D) = 〈f , Pα,qg〉L2α,q(D).(iii) The operator Pα,q is bounded with ‖ Pα,q ‖= 1 and ‖ I − Pα,q ‖≤ 1.

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 9Let φ ∈ L∞(D). The q-multiplication operators Mφ are the operators defined by
Mφ : L2α,q(D)→ L2α,q(D), Mφf (z) := φ(z)f (z), z ∈ D.

The q-Toepliz operators Tφ are the operators defined by
Tφ : Aα,q → Aα,q, Tφf (z) := Pα,qMφ(z)f (z), z ∈ D.

Theorem 4.1. Let φ ∈ L∞(D).(i) The operators Tφ are bounded and ‖ Tφ ‖≤‖ φ ‖∞.(ii) For all f , g ∈ Aα,q , we have

〈Tφf , g〉Aα,q = 〈f , Tφg〉Aα,q .

Proof. Let φ ∈ L∞(D). To prove (i), let f ∈ Aα,q then from Proposition 4.1 (iii) we have
‖ Tφf ‖Aα,q=‖ Pα,qMφf ‖Aα,q=‖ Pα,q(φf ) ‖Aα,q≤‖ φf ‖L2α,q(D)≤‖ φf ‖L∞(D)‖ f ‖Aα,q .Thus, ‖ Tφ ‖≤‖ φ ‖.To prove the second assertion, we use the fact that for any f , g ∈ Aα,q , Pα,qf = f and Pα,qg = g.From Proposition 4.1 (ii), we obtain
〈Tφf , g〉Aα,q = 〈φf , Pα,qg〉L2α,q(D) = 〈f , φg〉L2α,q(D) = 〈Pα,qf , Tφg〉L2α,q(D) = 〈f , Tφg〉Aα,q .

�

Theorem 4.2. Let φ ∈ L∞(D) has compact support, then Tφ is a compact operator.

Proof. Let φ ∈ L∞(D) and n,m = 0, 1, 2, .... From Proposition 3.1, we have
Tφξ

α
n,q(z) =

∞∑
m=0

〈Tφξαn,q, ξαm,q〉L2α,q(D)
Cm(α; q)

zm.

So,
〈Tφξαn,q, ξαm,q〉Aα,q = 〈φξαn,q, ξαm,q〉L2α,q(D)Since φ ∈ L∞(D) with compact support, there exist a positive constant a and K such that | φ(z) |≤

a and φ(z) = 0, for any | z |> a. Then for all n,m ∈ N, we get from (3) and Proposition 3.1 (i),
〈φξαn,q, ξαm,q〉L2α,q(D) =

1√
Cn(α; q)Cm(α; q)

∫
|z |≤a

φ(z)znzmdνα,q(z)

Thus, we obtain∣∣∣∣〈φξαn,q, ξαm,q〉L2α,q(D)∣∣∣∣ ≤ K√
Cn(α; q)Cm(α; q)

∫
|z |≤a

|z |n+mdνα,q(z)

≤
2K√

Cn(α; q)Cm(α; q)

∫ a

0

rn+mdνα,q(z)

≤
2K[α]qa

n+m√
Cn(α; q)Cm(α; q)

∫ 1
0

(qr2; q)α−1dq(r2)

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Eur. J. Math. Anal. 10.28924/ada/ma.5.20 10

≤
Kan+m√

Cn(α; q)Cm(α; q)
.

Hence,
∞∑

n,m=0

∣∣〈Tφξαn,q, ξαm,q〉Aα,q ∣∣2
Cn(α; q)Cm(α; q)

≤ 4K2
( ∞∑
n=0

a2n

Cn(α; q)

)2
≤ 4K2eq(a2)(qα+1; q)2∞ <∞

Then Tφ is an Hilbert-Schmidt operator, and consequently it is compact. �

4.2. q-Hankel Operator on Aα,q . Let φ ∈ L∞(D). The q-Hankel operators Hφ are the operatorsdefined by
Hφ : Aα,q → Aα,q, Hφ := (I − Pα,q)Mφ.

Theorem 4.3. Let φ,ψ ∈ L∞(D).(i.) The operators Hφ are bounded and ‖ Hφ ‖≤‖ φ ‖∞.(ii) For all f ∈ Aα,q and g ∈ L2α,q(D), we have

〈Hφf , g〉L2α,q(D) = 〈f , H∗φg〉Aα,q , H∗φ = Pα,qMφ(I − Pα,q).

(iii) Tφψ − TφTψ = H∗
φ
Hφ.

Proof. Let φ,ψ ∈ L∞(D). To prove (i), from Proposition 4.1 (iii) for any f ∈ Aα,q
‖ Hφ ‖L2α,q(D)=‖ (I − Pα,q) ‖L2α,q(D)≤‖ φf ‖L2α,q(D)≤‖ φ ‖L∞α,q(D)‖ φf ‖L2α,q(D) .So, ‖ Hφ ‖≤‖ φ ‖L∞(D).(ii) Let f ∈ Aα,q and g ∈ L2α,q(D). From Proposition 4.1 (ii) and the fact that Pα,qf = f we obtain

〈Hφf , g〉L2α,q(D) = 〈φf , g〉L2α,q(D) − 〈φf , Pα,qg〉L2α,q(D)

= 〈f , φ(I − Pα,q)g〉L2α,q(D)

= 〈Pα,qf , φ(I − Pα,q)g〉L2α,q(D)

= 〈f , Pα,qMφ(I − Pα,q)g〉Aα,q .(iii) Let φ,ψ ∈ L∞(D). Then
H∗
φ
Hψ = Pα,qMφ(I − Pα,q)2Mψ = Pα,qMφ(I − Pα,q)Mψ = Pα,qMφψ − Pα,qMφPα,qMψ = Tφψ − TφTψ. �

5. Extremal function on the q-weighted Bergman space Aα,q
Let η > 0 and T : Aα,q → H be a bounded operator from Aα,q into a Hilbert space H. Wedenote by 〈., .〉T,η,q the inner product defined on the q-weighted Bergman space Aα,q by

〈f , g〉T,η,q := η〈f , g〉Aα,q + 〈T f , Tg〉H,and ‖ f ‖T,η,q:=
√
〈f , f 〉T,η,q .

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 11By the virtue of the theory of reproducing kernels of Hilbert space, we study the extremal functionassociated to the operator T on the q-weighted Bergman space Aα,q .
Theorem 5.1. Let η > 0. The space (Aα,q, 〈., .〉T,η,q) possesses a reproducing kernel
KT,η,q(z, w); z, w ∈ D which satisfies the equation (ηI + T ∗T )KT,η,q(z, .) = Kα,q(z, .), where
Kα,q is the kernel given by (7). Moreover, the kernel KT,η,q satisfies the following properties

(i) ‖ KT,η,q(z, .) ‖Aα,q≤
1

η

√
eq(|z |2)Eq(qα+1|z |2).

(ii) ‖ TKT,η,q(z, .) ‖H≤

√
eq(|z |2)Eq(qα+1|z |2)

2η
.

(iii) ‖ T ∗TKT,η,q(z, .) ‖Aα,q≤
√
eq(|z |2)Eq(qα+1|z |2),

Proof. Let f ∈ Aα,q . Using Theorem 3.1 (iii), the map f 7→ f (z) is a continuous linear functionalon (Aα,q, 〈., .〉T,η,q). Thus, (Aα,q, 〈., .〈T,η,q) has a reproducing kernel denoted KT,η,q . Now usingthe fact that
f (z) = η〈f ,KT,η,q(z, .)〉Aα,q + 〈T f , TKT,η,q(z, .)〉H = 〈f , (ηI + T ∗T )KT,η,q(z, .)〉Aα,q ,

we deduce easily that (ηI + T ∗T )KT,η,q(z, .) = Kα,q(z, .). So the previous relation implies that
η2 ‖ KT,η,q(z, .) ‖2Aα,q +2η ‖ TKT,η,q(z, .) ‖2H + ‖ T ∗TKT,η,q(z, .) ‖2Aα,q=‖ Kα,q(z, .) ‖2Aα,q .

So we obtain the properties (i), (ii) and (iii) by using relation (7). �

Since relations (10), (11) and (12), we get
Example 5.1. For any w, z ∈ D, let H = Aα,q .(a) If T = ∇α,q , then

KT,η,q(z, w) =
1

ηC0(α; q)
+

∞∑
n=1

(zw)n(
η + q−α−1[n]q([n − 1]q + q−α−1[α+ 1]q)

)
Cn(α; q)

.

(b) If T = Lα,q , then

KT,η,q(z, w) =
1

ηC0(α; q)
+

∞∑
n=1

(zw)n(
η + q−α−1[n + 1]q([n]q + q−α−1[α+ 1]q)

)
Cn(α; q)

.

(c) If T = Nq , then

KT,η,q(z, w) =
1

ηC0(α; q)
+

∞∑
n=1

(zw)n(
η + [n]2q

)
Cn(α; q)

.

We can state now the main result of this section.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.20 12

Theorem 5.2. For any h ∈ H and η > 0, there exists a unique function f ∗η,h, where the infimum

inf
f ∈Aα,q

{
η ‖ f ‖2Aα,q + ‖ h − T f ‖2H

} (13)
is attained. Moreover, the extremal function f ∗η,h is given by

f ∗η,h(z) = 〈h, TKT,η,q(z, .)〉H, (14)
and satisfies the following | f ∗η,h(z) |≤

√
eq(|z |2)Eq(qα+1|z |2)

2η
‖ h ‖H .

Proof. The existence and unicity of the extremal function f ∗η,h satisfying (13) is obtained in [15,17].In particular, f ∗η,h is given by the reproducing kernel of Aα,q with ‖ . ‖T,η,q norm as f ∗η,h(z) =

〈h, TKT,η,q(z, .)〉H . This yields the result, by using relation (14), Theorem 5.1 (ii) and the fact that
| f ∗η,h(z) |≤‖ h ‖H‖ TKT,η,q(z, .) ‖H≤

√
eq(|z |2)Eq(qα+1|z |2)

2η
‖ h ‖H,

which completes the proof of the theorem. �

5.1. Applications. Let H be the prehilbertian space of analytic functions on the disk D equippedwith the inner product
〈f , g〉H :=

∫
D
f (z)g(z)|z |2dνα,q(z).

For any f , g ∈ H with f (z) =
∑
n≥0 anz

n and g(z) =
∑
n≥0 bnz

n we have from Proposition 3.1and relation (6)
〈f , g〉H =

∑
n≥0

anbnCn+1(α; q), ‖ f ‖H=
∑
n≥0
| an |2 Cn+1(α; q).

The space H is a Hilbert space with Hilbert’s basis { zn√
Cn+1(α; q)

}
n≥0

and reproducing kernel
Sα,q(z, w) =

∞∑
n=0

(zw)n

Cn+1(α; q)
=
Kα,q(zw)− 1

zw
. (15)

5.1.1. Application 1. Let T be the q-difference operator defined on Aα,q by
T f (z) :=

1

z
(f (z)− f (0)).

The operator T maps continuously from Aα,q into H and ‖ T f ‖H≤‖ f ‖Aα,q . So, if f , g ∈ Aα,qwith f (z) =
∑
n≥0 anz

n and g(z) =
∑
n≥0 bnz

n we can deduce easily that
〈f , g〉T,η = ηa0b0 + (η + 1)

∞∑
n=1

anbnCn(α; q).

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 13Thus, for z, w ∈ D we have
KT,η,q(z, w) =

1

η
+

1

η + 1
(Kα,q(zw)− 1),

TKT,η,q(z, .)(w) =
1

η + 1

Kα,q(zw)− 1

w
,

hence for all h ∈ H we deduce that
f ∗η,h(z) =

1

η + 1
zh(z).

Figure 1. The following is the color function of f ∗η,h(z) associated to the q-difference operator T f (z) := 1
z (f (z) − f (0)) for λ = 10, z = x + iy , (x, y) ∈

[−5, 5] × [−5, 5] and respectively h(z) = 1, z, z2, z3, z4, z5. The argument of acomplex value is encoded by the hue of a color (red = positive real, and thencounterclockwise through yellow, green, cyan, blue and purple; cyan stands fornegative real). Strong colors denote points close to the origin, black = 0, weakcolors denote points with large absolute value, white = ∞.

https://doi.org/10.28924/ada/ma.5.20


Eur. J. Math. Anal. 10.28924/ada/ma.5.20 14

Acknowledgments. The author appreciates anonymous referees and the handling editor fortheir careful corrections to and valuable comments on the original version of this paper.
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https://doi.org/10.28924/ada/ma.5.20
https://doi.org/10.2307/2374685
https://doi.org/10.1103/PhysRevA.89.012129
https://doi.org/10.1137/0515012
https://doi.org/10.1137/0149053
https://doi.org/10.1016/j.jmaa.2016.06.013
https://doi.org/10.1016/j.jmaa.2016.06.013
https://www.jstor.org/stable/24715892
https://doi.org/10.1119/1.18410
https://doi.org/10.1119/1.18410
https://www.jstor.org/stable/116654
https://www.jstor.org/stable/116654
https://doi.org/10.1007/BF02386120
https://doi.org/10.1007/BF02386120
https://doi.org/10.4153/CJM-1986-043-4
https://doi.org/10.1080/27684830.2022.2066812
https://doi.org/10.4134/CKMS.c240150
https://doi.org/10.1177/1081286516657686
https://www.jstor.org/stable/24103129
https://www.jstor.org/stable/24103129
https://doi.org/10.1007/s13370-021-00924-3
https://doi.org/10.1090/surv/138
https://doi.org/10.1090/surv/138

	1. Introduction
	2. Preliminaries
	3. Uncertainty inequality on the q-weighted Bergman space A,q
	4. Operators on the q-weighted Bergman space A,q
	4.1. q-Toepliz Operator on A,q
	4.2. q-Hankel Operator on A,q

	5. Extremal function on the q-weighted Bergman space A,q
	5.1. Applications.

	References

