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On a Generalization of (L1ω, L

p
ω)-Multipliers

Yaovi A. Tissinam1, Abudulaï Issa1, Yaogan Mensah1,2,∗
1Department of Mathematics, University of Lomé, Togo

asseketis@gmail.com, issaabudulai13@gmail.com, mensahyaogan2@gmail.com
2ICMPA, University of Abomey-Calavi, Benin

∗Correspondence: mensahyaogan2@gmail.com, ymensah@univ-lome.tg

Abstract. This paper deals with a generalized aspect of multipliers for the pair (L1ω, Lpω) of Beurlingspaces. Using the Fourier transform related to a Beurling weight, we give a characterization of theaforementioned multipliers. We also prove the identification of the space of the multipliers for thepair (L1ω, Lpω) with the Beurling space Lpω when 1 < p <∞.

1. Introduction
Multipliers are intensively studied by many researchers. They appear in several fields of math-ematics and in various contexts, namely : mobile communication, signal processing, stochasticprocess, partial differential equation etc. From a theoretical point of view, we refer to the source [9]for more details about multipliers for commutative Banach algebras.Like in [4], we are interested in the multipliers on a certain large class of Banach spaces related to alocally compact abelian group. Namely, multipliers of Beurling spaces are concerned. Some inter-esting publications about multipliers associated with locally compact groups are [1,6,11,12,14,18].In [4], we study the multipliers on the weighted group algebra L1ω(G) which is the Banach space

L1ω(G) endowed with a generalized convolution product ∗ω which depends on the weight ω. Thisgeneralized convolution product first appeared in [10]. The authors in [4] characterized the multi-pliers on this weighted group algebra.The present paper is the continuation of the study started in [4]. We consider a generalization ofthe multipliers for the pair (L1ω(G), Lpω(G)). That is, the linear maps T : L1ω(G) −→ Lpω(G)) thatcommute with a certain class of generalized translation operators denoted here by Γsω . If ω ≡ 1,then we recover the classical concept of multipliers. Via the weight Fourier transform, we obtain,among other results, a characterization of the multipliers for the pair (L1ω(G), Lpω(G)).
Received: 14 Jul 2023.
Key words and phrases. weight, convolution, multiplier, group algebra, Fourier transform, measure.1

https://adac.ee
https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 2The paper is organized as follows. In Section 2, the definition of Beurling spaces and someresults from [4, 7, 10] are recalled. In Section 3, we state our main results.
2. Preliminaries

2.1. The Beurling spaces. Let G be a group whose neutral element is denoted by e . A Beurling
weight on G is a continuous fonction ω : G → (0,∞) such that ∀x, y ∈ G,

ω(xy) 6 ω(x)ω(y),

ω(x) > 1,

ω(e) = 1.

For instance, for each α ≥ 0, the function ωα defined by
ωα(x) = (1 + ‖x‖)α,

where x = (x1, · · · , xn) ∈ Rn and ‖x‖ =

√
n∑
i=1

x2i , is a Beurling weight on (Rn,+).
Integration on G is taken with respect to a left Haar measure. Beurling spaces are defined tobe

Lpω(G) =

{
f : G → C :

∫
G

|f (x)|pω(x)dx <∞
}
, 1 6 p < +∞.

The case where p =∞ is defined in an obvious way by essential boundedness. The mapping
f 7−→ ‖f ‖p,ω =

(∫
G

|f (x)|pω(x)dx

) 1
p

is a norm on Lpω(G).It is well-known in the mathematical litterature that L1ω(G) is a Banach algebra under theconvolution product ∗ defined by
(f ∗ g)(x) =

∫
G

f (y)g(y−1x)dy.

The following sufficient condition for Lpω(G), 1 < p < ∞, to be a Banach algebra under theconvolution product ∗ can be found in [7] : the space Lpω(G), 1 < p <∞ is Banach algebra underthe convolution product ∗ if ω 1
1−p ∗ω

1
1−p 6 ω

1
1−p . For a general background and history on Beurlingspaces, we refer to [13,15].

2.2. A Generalized convolution product. In [10], the author introduced a new convolution producton L1ω(G) which has the particularity to depend of the weight ω. That is,
f ∗ω g(x) =

∫
G

f (y)g(y−1x)
ω(y)ω(y−1x)

ω(x)
dy.

If ω ≡ 1, then one recovers the usual convolution
(f ∗ g)(x) =

∫
G

f (y)g(y−1x)dy.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 3Hence, the convolution product ∗ω is a generalization of the usual convolution product. It wasshown that L1ω(G) is a Banach algebra under this new convolution product [10]. We denote by
L1ω(G) this new Banach algebra ; in other words L1ω(G) = (L1ω(G), ‖ · ‖1,ω, ∗ω).For s ∈ G, define the operator Γsω by

Γsωf (x) =
τsMωf (x)

ω(x)
, f ∈ L1ω(G),

where Mω is the multiplication operator defined by
(Mωf )(x) = ω(x)f (x)

and τs is the translation operator defined by
(τs f )(x) = f (s−1x).

The operator Γsω appears first in [4] for the study of the multipliers for the algebra L1ω(G). Alinear map T : L1ω(G)→ L1ω(G) is called a multiplier if T commutes with the operators Γsω for all
s ∈ G. Since the operator Γsω is a generalization of the translation operator τs , the latter notion ofmultiplier covers the classical one related to commutation with translations.The natural next step is to investigate the multipliers for the pair (L1ω(G), Lpω(G)). This is themain purpose of the present article.We denote by M1ω(G) the Banach space of all complex bounded regular Borel measures µ on Gsuch that

‖µ‖ω =

∫
G

ω(x)d |µ|(x) <∞. (1)
We write M1(G) in the case where ω ≡ 1. For µ, ν ∈ M1ω(G), define µ ∗ω ν by

µ ∗ω ν(f ) =

∫
G

∫
G

f (xy)
ω(x)ω(y)

ω(xy)
dµ(x)dν(y), f ∈ Cc(G,ω−1)

where Cc(G,ω−1) is the set of complex functions f defined on G such that f ω−1 is of compactsupport. Also, define
µ ∗ω f (x) =

∫
G

f (y−1x)
ω(y)ω(y−1x)

ω(x)
dµ(y)

for f ∈ L1ω(G) and µ ∈ M1ω(G). Then, the Banach space M1ω(G) is a unital Banach algebra withrespect to the convolution product ∗ω and L1ω(G) is a closed ideal of M1ω(G) [10, Theorem 5.1].
2.3. Some useful facts. Let G be a locally compact abelian group with Pontryagin dual group Ĝ.We denote by M̂1(G) the collection of all the Fourier-Stieltjes transforms of elements of M1(G).That is,

M̂1(G) = {µ̂ : µ ∈ M1(G)}where µ̂ is defined by
µ̂(γ) =

∫
G

γ(x)dµ(x), γ ∈ Ĝ.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 4

For a function f ∈ L1ω(G), the Fourier transform of f , denoted F f or f̂ , is defined by
(F f )(γ) := f̂ (γ) =

∫
G

f (x)γ(x)dx

The following theorems will play an important role.
Theorem 2.1 ( [3] or [17]). Let G be a locally compact abelian group and let ϕ be a complex function
on Ĝ. Then, the following assertions are equivalent.(1) ϕ ∈ M̂1(G) and ‖ϕ‖∞ 6 C.(2) ϕ is continuous and there exists a constant C > 0 such that∣∣∣∣∣ n∑

i=1

ciϕ(γi)

∣∣∣∣∣ < C

∥∥∥∥∥ n∑
i=1

ciγi(·)

∥∥∥∥∥
∞

(2)
for all positive integer n and all choices of ci ∈ C and γi ∈ Ĝ, i = 1, 2, · · · , n.

Moreover, if ϕ = µ̂, then ‖µ‖ is the smallest constant C for which (2) holds.

Theorem 2.2. ( [9, page 252]) Let G be a locally compact abelian group. Then, for each compact
K ⊂ Ĝ and ε > 0, given an open set U containing K, there exists a function f ∈ L1(G) such that
0 6 f̂ (γ) 6 1 if γ ∈ Ĝ, f̂ (γ) = 1 if γ ∈ K, f̂ (γ) = 0 if γ /∈ U and ‖f ‖ 6 ε + 1. In particular,
given any open set U ⊂ Ĝ with compact closure, it is possible to find f ∈ L1(G) such that
f̂ (γ) = 1 if γ ∈ U.

Theorem 2.3. ( [5, Theorem 3.2]) Let G be a locally compact group. Let f ∈ Lpω(G), 1 6 p < ∞.
Then, ∀s ∈ G,

[ω(s)]
1−p
p ‖f ‖p,ω 6 ‖Γsωf ‖p,ω 6

[
ω(s−1)

] p−1
p ‖f ‖p,ω. (3)

3. Multipliers for the pair (L1ω(G), Lpω(G))

In this section, we study a generalization of the concept of multipliers. Here, the multipliersare defined with respect to the generalized translation operators Γsω . Throughout this section, weassume that G is a locally compact abelian group. A look at Theorem 2.3 shows that f ∈ Lpω(G) ifand only if Γsωf ∈ L
p
ω(G). That is, the spaces Lpω(G) are stable under the action of the operators

Γsω . Therefore, we are able to define a concept of multiplier in the framework of this study.
Definition 3.1. A linear operator T : L1ω(G) −→ Lpω(G) is said to be a multiplier if T commutes
with all the operators Γsω, s ∈ G. That is,

∀s ∈ G, TΓsω = ΓsωT.

We denote by M1,p
ω (G) the set of such multipliers. We denote by ‖T‖ the operator norm of

T ∈M1,p
ω (G).

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 5We will use the fact that for 1 < p <∞, the following identification holds [7] :
(Lpω(G))′ = Lqw (G)

with 1

p
+

1

q
= 1 and w = ω−

q
p . From that, one may deduce that for 1 < p <∞, the space Lpω(G)is a reflexive space.For f ∈ L1ω(G), define the Fourier transform of f by
Fω(f )(γ) =

∫
G

f (x)γ(x)ω(x)dx, γ ∈ Ĝ.

In [4], the following convolution result was proved.
∀f , g ∈ L1ω(G), Fω(f ∗ω g) = Fω(f )Fω(g).

Set
Fω(L1ω(G)) =

{
Fω(f ) : f ∈ L1ω(G)

}
.

Let us remark that functions in Fω(L1ω(G)) are continuous and vanished at infinity by the Riemann-Lebesgue theorem. We fit out the space Fω(L1ω(G)) with the norm defined by
‖Fω(f )‖ = ‖f ‖1,ω, f ∈ L1ω(G).

Then, we have the following result.
Theorem 3.2. The space Fω(L1ω(G)) is a Banach algebra for the pointwise multiplication.

Proof. Let (Fω(fn)) be a Cauchy sequence in Fω(L1ω(G)). Let p, q ∈ N. The equality
‖Fω(fp)−Fω(fq)‖ = ‖fp − fq‖1,ω

and the fact that (L1ω(G), ‖ · ‖1,ω) is a Banach space show that there exists f ∈ L1ω(G) such that
(fn) converges to f in L1ω(G). Now, ‖Fω(fn)−Fω(f )‖ = ‖fn − f ‖1,ω . Thus, (Fω(fn)) converges to
(Fω(f )) in Fω(L1ω(G)). Thus, the space Fω(L1ω(G)) is a Banach space. Moreover,

‖Fω(f )Fω(g)‖ = ‖Fω(f ∗ω g)‖

= ‖f ∗ω g‖1,ω

6 ‖f ‖1,ω‖g‖1,ω = ‖Fω(f )‖‖Fω(g)‖.

Thus, the space (Fω(L1ω(G))), ·, ‖·‖1,ω
) is a Banach algebra. �

For f ∈ Lpω(G) and h ∈ Lqw (G) with 1

p
+

1

q
= 1, we set

〈f , h〉ω =

∫
G

f (x)h(x−1)ω(x)dx.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 6

Theorem 3.3. Let G be a locally compact abelian group. Let T : L1ω(G) −→ Lpω(G) be a bounded
linear transformation. Then, T ∈M1,p

ω (G) if and only if there exists a unique element ϕ such that
Tg = ϕ ∗ω g for all g ∈ L1ω(G), where ϕ ∈ M1ω(G) if p = 1 and ϕ ∈ Lpω(G) if 1 < p <∞.

Proof. (1) Suppose p = 1. Let T ∈ M1,1
ω (G). In [4, Proposition 5.4], it was shown that

T ∈ M1,1
ω (G) if and only if there exists a unique function B defined on Ĝ such that

Fω(T f ) = BFω(f ) for all f ∈ L1ω(G). Clearly, BFω(f ) ∈ Fω(L1ω(G)). Therefore, thefunction BFω(f ) is continuous for all f ∈ L1ω(G) (the Fourier transform of a function is acontinuous function). Moreover, for each open set in Ĝ with compact closure, there exists afunction f ∈ L1ω(G) such that Fω(f ) is constant on U [9, F.7e]. Thus, B is continuous on Ĝ.Let ε > 0 and let γ1, γ2, ...., γn ∈ Ĝ. Via Theorem 2.2, we can choose g ∈ L1(G) suchthat ‖F(g)‖ = ‖g‖1 < 1 + ε and F(g)(γi) = 1, i = 1, 2, 3, ....., n. Now, set f =
g

ω
. Then,

f ∈ L1ω(G), ‖Fω(f )‖ = ‖f ‖1,ω < 1 + ε and Fω(f )(γi) = 1, i = 1, 2, 3, ....., n.For zi ∈ C, i = 1, 2, ....., n, one has∣∣∣∣∣ n∑
i=1

ziB(γi)

∣∣∣∣∣ =

∣∣∣∣∣ n∑
i=1

ziFω(f )(γi)

∣∣∣∣∣ =

∣∣∣∣∣∫G
[

n∑
i=1

ziγi(x)

]
f (x)ω(x)dx

∣∣∣∣∣
6 ‖f ‖1,ω

∥∥∥∥∥ n∑
i=1

ziγi

∥∥∥∥∥
∞

< (1 + ε)

∥∥∥∥∥ n∑
i=1

ziγi

∥∥∥∥∥
∞

.

Since ε is chosen arbitrarily, it follows that∣∣∣∣∣ n∑
i=1

ziB(γi)

∣∣∣∣∣ <
∥∥∥∥∥ n∑
i=1

ziγi

∥∥∥∥∥
∞

.

We conclude via Theorem 2.1 (with C = 1) that there exists a unique bounded measure
µ such that B = F(µ). Now, if we set ϕ = ω−1µ, then ϕ ∈ M1ω(G) and B = Fω(ϕ).Therefore,

Fω(T f ) = Fω(ϕ)Fω(f ) = Fω(ϕ ∗ω f ).Since the Fourier transform is injective, it follows that T f = ϕ ∗ω f .(2) Assume that 1 < p < ∞. Let T ∈ M1,p
ω (G). The weighted group algebra (L1ω(G), ‖ ·

‖1,ω, ∗ω) has a bounded approximate identity [10, Theorem 2.2]. Let {υn} be a boundedapproximate identity for (L1ω(G), ‖ · ‖1,ω, ∗ω). Let g ∈ L1ω(G). Then,
‖Tg − Tυn ∗ω g‖p,ω = ‖Tg − T (υn ∗ω g)‖p,ω

6 ‖T‖‖g − υn ∗ω g‖1,ω.

Since ‖g − υn ∗ω g‖1,ω tends to 0 whenever n goes to ∞, then (Tυn ∗ω g)n converges to
Tg in Lpω(G).

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 7Moreover, ‖Tυn‖p,ω 6 ‖T‖‖υn‖1,ω = ‖T‖. Therefore, {Tυn} lies in a norm bounded subsetof Lpω(G) =
(
Lqw (G)

)′. So, by Alaoglu’s Theorem ( [16, page 299] or [9, Theorem D.4.3.]) andthe reflexivity of Lpω(G), we see that there exists a subnet {Tυm} of {Tυn} and ϕ ∈ Lpω(G)such that {Tυm} converges to ϕ in the weak∗-topology. That is, lim
m
〈Tυm, u〉ω = 〈ϕ, u〉ωfor all u ∈ Lqw (G). Then, for h, g ∈ Cc(G), we have

〈Th, g〉ω = lim
m
〈Tυm ∗ω h, g〉ω

= lim
m
〈Tυm, (h ∗ω

g

ω
)ω〉ω

= 〈ϕ, (h ∗ω
g

ω
)ω〉ω

= 〈ϕ ∗ω h, g〉ω.

However, Cc(G) is norm dense in Lqw (G). Therefore, Th = ϕ ∗ω h for each h ∈ Cc(G).Moreover, Cc(G) is norm dense in L1ω(G). Thus, Th = ϕ ∗ω h for all h ∈ L1ω(G).(3) Conversely, let 1 ≤ p < ∞. Assume that there exists a measure µ ∈ M1ω(G) or a function
ϕ ∈ Lpω(G) such that Th = ϕ ∗ω h for all h ∈ L1ω(G). Then,

(TΓsω)h = T (Γsωh)

= ϕ ∗ω Γsωh

= Γsω(ϕ ∗ω h)

= Γsω(Th) = (ΓsωT )h.

Thus, T ∈M1,p
ω (G).(4) Concerning the uniqueness statement, let us consider ϕ and ψ be such that Th = ϕ∗ω h =

ψ ∗ω h for all h ∈ L1ω(G). Then, using the Fourier transform, we obtain Fω(ϕ)Fω(h) =

Fω(ψ)Fω(h). So, Fω(ϕ) = Fω(ψ). Finally, ϕ = ψ by the injectivity of the Fouriertransform.
�

Theorem 3.4. Let G be a locally compact abelian group. Then,M1,1
ω (G) is isometrically isomorphic

to M1ω(G).

Proof. We have seen in Theorem 3.3 that T ∈M1,1
ω (G) if and only if there exists a unique measure

µ ∈ M1ω(G) such that T f = µ∗ω f for all f ∈ L1ω(G). Then, the mapping T 7−→ µ defines a bijectionfrom M1,1
ω (G)) onto M1ω(G). Moreover,

‖T f ‖1,ω = ‖µ ∗ω f ‖1,ω

=

∫
G

∣∣∣∣∫
G

f (y−1x)
ω(y−1x)ω(y)

ω(x)
dµ(y)

∣∣∣∣ω(x)dx

6
∫
G

∫
G

∣∣f (y−1x)
∣∣ ω(y−1x)ω(y)

ω(x)
ω(x)d |µ|(y)dx

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 8

6

(∫
G

|f (x)|ω(x)dx

)(∫
G

ω(y)d |µ|(y)

)
(invariance of the Haar measure)

6 ‖f ‖1,ω‖µ‖ω.

Then, ‖T‖1,ω 6 ‖µ‖ω .In the converse, for γ1, · · · , γn ∈ Ĝ, z1, · · · , zn ∈ C, and ε > 0, let us choose f ∈ L1ω(G) suchthat ‖Fω(f )‖ = ‖f ‖1,ω < 1 + ε and Fω(f )(γi) = 1, i = 1, 2, 3, ....., n. Then,∣∣∣∣∣ n∑
i=1

ziFω(µ)(γi)

∣∣∣∣∣ =

∣∣∣∣∣ n∑
i=1

ziFω(µ)(γi)Fω(f )(γi)

∣∣∣∣∣
=

∣∣∣∣∣ n∑
i=1

ziFω(µ ∗ω f )(γi)

∣∣∣∣∣
=

∣∣∣∣∣ n∑
i=1

ziFω(T f )(γi)

∣∣∣∣∣
6 ‖T‖(1 + ε)

∥∥∥∥∥ n∑
i=1

ziγi

∥∥∥∥∥
∞

.

Since ε is arbitrary, then ‖T‖ > ‖µ‖ω by the use of Theorem 2.1 applied with Fω instead of F . �
Theorem 3.5. Let G be a locally compact abelian group. Let 1 6 p < ∞. If f ∈ Lpω(G), then the
mapping s 7−→ Γsωf is continuous from G into Lpω(G).

Proof. The set of complex continuous functions on G with compact support Cc(G) is dense in Lpω(G)under the norm ‖ · ‖p,ω . Let ε > 0. Consider g ∈ Cc(G) and set C1 = supp(g). Let us choose acompact neighborhood C2 of the neutral element e . Set C = C1∪C2∪ (C1C2). We have for s ∈ C2,
‖Γsωg − g‖pp,ω =

∫
C

|Γsωg(x)− g(x)|pω(x)dx

6
∫
C

|g(s−1x)ω(s−1x)− g(x)ω(x)|pdx.

The mapping x 7−→ (gω)(x) is uniformly continuous on G. Thus, there exists a neighborhood U of
e which we may assume to be contained in C2, such that

∀s ∈ U, |(gω)(s−1x)− (gω)(x)|p <
εp

|C|where |C| is the measure of the compact set C. Then, for s ∈ U , we have
‖Γsωg − g‖pp,ω 6

∫
C

|(gω)(s−1x)− (gω)(x)|pdx <
ε|C|
|C| = ε.

We will show the claim for f ∈ Lpω(G). Let K be a compact neighborhood of e . Since Cc(G) isdense in Lpω(G), then there exists g ∈ Cc(G) such that
‖f − g‖p,ω <

ε

3
.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 9There exists a compact neighborhood V of e which we may assume to be contained in K, such that
‖Γsωg − g‖p,ω <

ε

3
for all s ∈ V .Then, for s ∈ V , we have

‖Γsωf − f ‖p,ω 6 ‖Γsωf − Γsωg‖p,ω + ‖Γsωg − g‖p,ω + ‖f − g‖p,ω

<
1

ω(s)

∫
G

|(f − g)(t)|pω(st)dt +
ε

3
+
ε

3

6
1

ω(s)

∫
G

|(f − g)(t)|pω(s)ω(t)dt +
ε

3
+
ε

3

<

∫
G

|(f − g)(t)|pω(t)dt +
ε

3
+
ε

3
‖f − g‖p,ω +

ε

3
+
ε

3
=
ε

3
+
ε

3
+
ε

3
= ε.

�

Theorem 3.6. Let G be a locally compact abelian group. Let f ∈ Lpω(G), 1 6 p < ∞. Let ε > 0.
Then, there exists a positive function g ∈ Cc(G) such that ‖g‖1,ω = 1 and ‖f ∗ω g − f ‖p,ω 6 ε.

Proof. Let f ∈ Lpω(G) and ε > 0. According to Theorem 3.5, the mapping s 7−→ Γsωf is continuousat the neutral element e of G. Then, there exists a compact neighborhood K of e such that
‖Γsωf − f ‖p,ω 6 ε, ∀s ∈ K.

Consider a positive function g such that supp(g) ⊂ K and ∫
G

g(y)ω(y)dy = 1 (that is ‖g‖1,ω = 1).
Then, |(f ∗ω g)(x) − f (x)| 6

∫
G

|Γsωf (x) − f (x)|g(s)ω(s)ds. Using the Hölder’s inequality withrespect to the measure g(s)ω(s)ds , one has
|(f ∗ω g)(x)− f (x)| 6

(∫
G

|Γsωf (x)− f (x)|pg(s)ω(s)ds

) 1
p
(∫

G

g(s)ω(s)ds

) 1
q

6

(∫
G

|Γsωf (x)− f (x)|pg(s)ω(s)ds

) 1
p

,

where q is such that 1

p
+

1

q
= 1. Then,

‖f ∗ω g − f ‖pp,ω =

∫
G

|(f ∗ω g)(x)− f (x)|pω(x)dx

6
∫∫

G×G
|Γsωf (x)− f (x)|pg(s)ω(s)dsω(x)dx

6
∫
G

‖Γsωf − f ‖pp,ωg(s)ω(s)ds

= ‖Γsωf − f ‖pp,ω
∫
G

g(s)ω(s)ds = ‖Γsωf − f ‖pp,ω 6 εp.

Thus, ‖f ∗ω g − f ‖p,ω 6 ε. �

Theorem 3.7. Let G be a locally compact abelian group. If T ∈M1,p
ω (G), then ‖T f ‖p 6 ‖T‖‖f ‖1.

In other words, T : L1ω(G) −→ Lpω(G) is a bounded operator.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 10

Proof. Let ε > 0. Via Theorem 3.6, there exists a positive function g in Cc(G) such that∫
G

g(t)ω(t)dt = 1 and ‖g ∗ω T f − T f ‖p 6 ε because ‖·‖p 6 ‖·‖p,ω . We have,
‖g ∗ω T f − T f ‖p > ‖T f ‖p − ‖g ∗ω T f ‖p.

Therefore,
‖T f ‖p 6 ‖g ∗ω T f ‖p + ε

= ‖Tg ∗ω f ‖p + ε

6 ‖Tg‖p‖f ‖1 + ε

6 ‖Tg‖p,ω‖f ‖1 + ε

6 ‖T‖‖‖1,ω‖f ‖1 + ε

= ‖T‖‖f ‖1 + ε.

Since the latter inequality is true for arbitrary ε > 0, then we obtain ‖T f ‖p 6 ‖T‖‖f ‖1. �

For a function f in Lpω(G), we define the convolution operator Tf by
Tf g = f ∗ω g.

Theorem 3.8. Let G be a locally compact abelian group. Let 1 < p < ∞. Let f be a function in
Lpω(G). Then, ‖Tf ‖ = ‖f ‖p,ω.

Proof. Let f ∈ Lpω(G) and let ε > 0. From Theorem 3.6, there exits a positive function g such that∫
G

g(t)ω(t)dt = 1 and ‖f ∗ω g − f ‖p,ω 6 ε. Then,
‖f ‖p,ω 6 ε+ ‖f ∗ω g‖p,ω = ε+ ‖Tf g‖p,ω

6 ε+ ‖Tf ‖‖g‖1,ω = ε+ ‖Tf ‖.

Thus ‖f ‖p,ω 6 ‖Tf ‖.Let us prove the inverse inequality. Let g ∈ L1ω(G). Applying the Hölder’s inequality withrespect to the measure g(y)ω(y)dy , one has
‖f ∗ g‖pp,ω =

∫
G

|g ∗ω f |pω(x)dx

=

∫
G

∣∣∣∣∫
G

g(y)Γyωf (x)ω(y)

∣∣∣∣p ω(x)dx

6
∫
G

[∫
G

|Γyωf (x)|p|g(y)|ω(y)dy

] [∫
G

|g(y)|ω(y)dy

] p
q

ω(x)dx

6
∫
G

(|f |p ∗ω |g|)ω(x)dx

[∫
G

|g(y)|ω(y)dy

] p
q

6 ‖|f |p ∗ω |g|‖1,ω‖g‖
p
q

1,ω 6 ‖f ‖
p
p,ω‖g‖1,ω‖g‖

p
q

1,ω = ‖f ‖pp,ω‖g‖
p
1,ω.

https://doi.org/10.28924/ada/ma.3.27


Eur. J. Math. Anal. 10.28924/ada/ma.3.27 11Then, ‖Tf g‖pp,ω 6 ‖f ‖pp,ω‖g‖p1,ω. Thus, ‖Tf ‖ 6 ‖f ‖p,ω . �

As a consequence of Theorem 3.3 and Theorem 3.8, we have the following result.
Corollary 3.9. Let G be a locally compact abelian group. Let 1 < p < ∞. Then, the multipliers
space M1,p

ω (G) and the Beurling space Lpω(G) are isometricaly identified by the mapping T :

f 7−→ Tf .

Conclusion
In this paper, we obtain a characterization of multipliers for the pair (L1ω, L

p
ω) using the Fouriertransform related to a Beurling weight. We also obtain the identification of the space of suchmultipliers with the Beurling space Lpω when 1 < p < ∞. It would be interesting in the future toconsider the case of the pair (Lpω, L

q
ω) in this framework of the weight dependent convolution.

Competing Interests
The authors declare that no competing interests exist.

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https://doi.org/10.1016/j.jfa.2014.11.019
https://doi.org/10.1016/j.acha.2005.06.003

	1. Introduction
	2. Preliminaries
	2.1. The Beurling spaces
	2.2. A Generalized convolution product
	2.3. Some useful facts

	3. Multipliers for the pair (L1(G),Lp(G))
	Conclusion
	Competing Interests
	References

