Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras
Let $G$ be a locally compact Abelian group (noncompact and nondiscrete) with Haar measure. Suppose that $1 < p < ∞$ and $1 ≤ q ≤ ∞$. The purpose of the paper is to define temperate Lorentz spaces and study the spaces of multipliers on Lorentz spaces and characterize them as the spaces...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507891959595008 |
|---|---|
| author | Duyar, C. Eryılmaz, İ. Дуяр, С. Ерийлмаз, І. |
| author_facet | Duyar, C. Eryılmaz, İ. Дуяр, С. Ерийлмаз, І. |
| author_sort | Duyar, C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:48:08Z |
| description | Let $G$ be a locally compact Abelian group (noncompact and nondiscrete) with Haar measure. Suppose that $1 < p < ∞$ and $1 ≤ q ≤ ∞$. The purpose of the paper is to define temperate Lorentz spaces and study the spaces of multipliers on Lorentz spaces and characterize them as the spaces of multipliers of certain Banach algebras. |
| first_indexed | 2026-03-24T02:16:31Z |
| format | Article |
| fulltext |
UDC 517.9
İ. Eryılmaz, C. Duyar (Ondokuz Mayis Univ., Turkey)
A STUDY OF FUNCTIONAL PROPERTIES AND MULTIPLIERS SPACES
OF GROUP L (p, q) (G)-ALGEBRAS
ВИВЧЕННЯ ФУНКЦIОНАЛЬНИХ ВЛАСТИВОСТЕЙ ТА ПРОСТОРIВ
МНОЖНИКIВ ДЛЯ ГРУП L (p, q) (G)-АЛГЕБР
Let G be a locally compact Abelian group (noncompact and nondiscrete) with Haar measure and suppose that 1 < p <∞
and 1 ≤ q ≤ ∞. The purpose of the paper is to define temperate Lorentz spaces and study the spaces of multipliers on
Lorentz spaces and characterize them as the spaces of multipliers of certain Banach algebras.
Нехай G є локальною компактною абелiвською групою (некомпактною та недискретною) з мiрою Хаара. При-
пустимо, що 1 < p < ∞ та 1 ≤ q ≤ ∞. Наша метa — визначити помiрнi простори Лоренца, вивчити простори
множникiв на просторах Лоренца та охарактеризувати їх як простори множникiв на деяких банахових алгебрах.
1. Introduction and preliminaries. In this paper, we are interested in the relationship between the
spaces of multipliers on L (p, q) (G) and the spaces of multipliers Lt (p, q) (G) on group L (p, q) (G)-
algebra. The multipliers of type (p, p) and multipliers of the group Lp-algebras
(
Ltp (G)
)
were
studied and developed by Mckennon [11 – 13] and Griffin [6] where the multipliers are identified as
the operators commuting with the translation operators. The ideas in these papers are used frequently
in our paper for the generalization of the results of Mckennon and Griffin concerning multipliers of
type (p, p) to the Lorentz spaces L (p, q) (G) for 1 ≤ p <∞, 1 ≤ q ≤ ∞.
For the convenience of the reader, we now review briefly what we need from the theory of
L (p, q) (G) Lorentz spaces. Let (G,Σ, µ) be a measure space and let f be a measurable function on
G. For each y > 0, let
λf (y) = µ {x ∈ G : |f (x)| > y} .
The function λf is called the distribution function of f. The rearrangement of f is defined by
f∗ (t) = inf {y > 0 : λf (y) ≤ t } = sup {y > 0 : λf (y) > t } , t > 0,
where inf ∅ =∞. Also the average function of f is defined by
f∗∗(t) =
1
t
t∫
0
f∗ (s) ds, t > 0.
Note that λf (·) , f∗ (·) and f∗∗(·) are nonincreasing and right continuous on (0,∞) [2, 9]. For
p, q ∈ (0,∞) we define
‖f‖∗p,q =
q
p
∞∫
0
[f∗ (t)]q tq/p−1dt
1/q
, ‖f‖p,q =
q
p
∞∫
0
[f∗∗ (t)]q tq/p−1dt
1/q
.
Also, if 0 < p, q =∞ we define
‖f‖∗p,∞ = sup
t>0
t1/pf∗ (t) and ‖f‖p,∞ = sup
t>0
t1/pf∗∗ (t) .
c© İ. ERYILMAZ, C. DUYAR, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3 341
342 İ. ERYILMAZ, C. DUYAR
For 0 < p < ∞ and 0 < q ≤ ∞, the Lorentz spaces are denoted by L (p, q) (G,µ) (or shortly
L (p, q) (G)) is defined to be the vector space of all (equivalence classes of) measurable functions f on
G such that ‖f‖∗p,q <∞. We know that ‖f‖∗p,p = ‖f‖p and so Lp (G) = L (p, p) (G) , where Lp (G)
is the usual Lebesgue space. It is also known that if 1 < p <∞ and 1 ≤ q ≤ ∞, then
‖f‖∗p,q ≤ ‖f‖p,q ≤
p
p− 1
‖f‖∗p,q
for each f ∈ L (p, q) (G) and
(
L (p, q) (G) , ‖·‖p,q
)
is a Banach space [1, 2, 9].
2. Main results. Let G be a local compact Abelian group, λ be a Haar measure on G and
Cc (G) be the space of complex-valued, continuous functions with compact support. For 1 < p <∞,
1 ≤ q ≤ ∞, symbols Bp,q and Mp,q will stand for the following spaces:
Bp,q = {T | T : L (p, q) (G)→ L (p, q) (G) , T is bounded and linear } ,
Mp,q = {T ∈ Bp,q | T (Lx) = Lx (T ) for all x ∈ G} ,
where Lx (·) is the translation operator. It is so easy to show by usual techniques that the space
Bp,q is a Banach algebra with respect to composition under operator norm and the space Mp,q is a
complete subspace of Bp,q.
Let HomL1(G) (L (p, q) (G) , L (p, q) (G)) = HomL1(G) (L (p, q) (G)) denote the space of all
module homomorphisms of L1 (G)-module L (p, q) (G) , that is, the space of operators in Bp,q
satisfying T (f ∗ g) = f∗T (g) for all f ∈ L1 (G) and g ∈ L (p, q) (G) . The module homomorphisms
space is a Banach L1 (G)-module by (f ◦ T ) (g) = f ∗ T (g) = T (f ∗ g) for all g ∈ L (p, q) (G) .
We briefly describe the content of this paper. In Subsection 2.1, we will construct pq-temperate
functions spaces for Lorentz spaces and give some properties. In Subsection 2.2, we will characterize
the multipliers space of L (p, q) (G) as a certain Banach algebra and generalize the results of Mcken-
non to L (p, q) (G) . Finally in Subsection 2.3, we will examine the multipliers spaces Lt (p, q) (G)
on group L (p, q) (G)-algebras.
2.1. The space Lt (p, q) (G) and its basic properties.
Definition 1. Let f ∈ L (p, q) (G) and (h ∗ f) (x) =
∫
G
h (t) f
(
t−1x
)
dλ (t) be defined for all
h ∈ L (p, q) (G) and almost all x ∈ G. If (h ∗ f) ∈ L (p, q) (G) for all h ∈ L (p, q) (G) and one of
the following conditions:
sup
{
‖h ∗ f‖p,q : h ∈ L (p, q) (G) , ‖h‖p,q ≤ 1
}
<∞, (1)
sup
{
‖h ∗ f‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
<∞ (2)
is satisfied, then the function f is called a pq-temperate and the space of these functions is showed
by Lt (p, q) (G) . The spaces Lt (p, q) (G) can be renamed as group L (p, q) (G)-algebras. The value
in (1) or (2) will be used for the norm of f ∈ Lt (p, q) (G) and showed by ‖f‖tp,q .
Proposition 1. Cc (G) is a subspace of
(
Lt (p, q) (G) , ‖·‖tp,q
)
. Also, Lt (p, q) (G) is a dense
subspace of L (p, q) (G) .
Remark 1. The space L1 (G) ∩ L (p, q) (G) is also contained by Lt (p, q) (G) . So we have
another evidence for nonempty property of the space Lt (p, q) (G) .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
A STUDY OF FUNCTIONAL PROPERTIES AND MULTIPLIERS SPACES OF GROUP L (p, q) (G)-ALGEBRAS 343
Proposition 2. For each f ∈ Lt (p, q) (G) , there is an operator Wf on L (p, q) (G) by Wf (·) =
= (·) ∗ f which also belongs to Bp,q.
Proof. Let’s take any f ∈ Lt (p, q) (G) . Since Wf (g) = g ∗ f for all g ∈ L (p, q) (G) , Wf is
well-defined by Definition 1. Also, the following equality:
‖Wf‖ = sup
‖g‖p,q≤1
‖Wf (g)‖p,q = sup
‖g‖p,q≤1
‖g ∗ f‖p,q = ‖f‖tp,q
says that Wf is bounded on L (p, q) (G) . The linearity property of Wf can be seen easily. Thus
Wf ∈ Bp,q.
Proposition 2 is proved.
We have also another good result here. That is, for all f, g ∈ Lt (p, q) (G)
Wf∗g = Wg ◦Wf = Wg∗f = Wf ◦Wg.
Proposition 3. The space
(
Lt (p, q) (G) , ‖·‖tp,q
)
is
(i) a normed algebra under convolution,
(ii) a Banach L1 (G)-module under convolution,
(iii) strongly invariant under translation.
Proposition 4. The space Lt (p, q) (G) is a Banach space under a new norm ‖| · |‖tp,q = ‖·‖p,q +
+ ‖·‖tp,q .
Proof. For any f ∈ Lt (p, q) (G) ⊂ L (p, q) (G) , we know that ‖f‖tp,q < ∞ and ‖f‖p,q < ∞.
Since the function ‖|·|‖tp,q = ‖·‖p,q + ‖·‖tp,q is a sum of two known norms, the function ‖|·|‖tp,q
is a norm on Lt (p, q) (G) . We will prove the proposition in a classical way. Let {fn}n∈N be a
Cauchy sequence in
(
Lt (p, q) (G) , |‖·‖|tp,q
)
. Then, for each ε > 0, there exists an N ∈ N such that
‖|fn − fm|‖tp,q < ε for all m,n ≥ N. The inequality ‖|·|‖tp,q ≥ ‖·‖p,q implies that ‖fn − fm‖p,q < ε
for all m,n ≥ N and so the sequence {fn}n∈N is also a Cauchy sequence in
(
L (p, q) (G) , ‖·‖p,q
)
.
The completeness of L (p, q) (G) shows that there exists a function f ∈ L (p, q) (G) such that fn → f
in L (p, q) (G) . If we consider the sequence {Wfn}n∈N corresponding to {fn}n∈N , then we see that
{Wfn}n∈N is a Cauchy sequence in Bp,q and converges to an operator W ∈ Bp,q such that
lim
n
‖fn − f‖p,q = 0 = lim
n
‖Wfn −W‖ .
Now, let’s take any g ∈ Cc (G) with ‖g‖p,q ≤ 1. Since L (p, q) (G) is a Banach L1 (G)-module, we
get fn ∗ g → f ∗ g as fn → f in L (p, q) (G) . Therefore, we have
‖g ∗ f‖p,q = lim
n
‖g ∗ fn‖p,q ≤ lim
n
‖g‖p,q ‖fn‖p,q ≤ lim
n
‖fn‖p,q = ‖f‖p,q
and
‖ f ‖tp,q = sup
‖g‖p,q≤1
‖g ∗ f‖p,q ≤ ‖f‖p,q <∞.
Also,
lim
n
‖|fn − f |‖tp,q = lim
n
(
‖fn − f‖p,q + ‖fn − f‖tp,q
)
=
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
344 İ. ERYILMAZ, C. DUYAR
= lim
n
(
‖fn − f‖p,q + sup
‖g‖p,q≤1
‖g ∗ (fn − f)‖p,q
)
≤
≤ lim
n
(
‖fn − f‖p,q + sup
‖g‖p,q≤1
‖g‖p,q ‖fn − f‖p,q
)
≤ lim
n
2 ‖fn − f‖p,q = 0
and for any h ∈ Cc (G)
W (h) = lim
n
Wfn (h) = lim
n
(h ∗ fn) = h ∗ f = Wf (h)
can be found. Density of Cc (G) in L (p, q) (G) implies that W = Wf and
(
Lt (p, q) (G) , ‖|·|‖tp,q
)
is a Banach space.
Proposition 4 is proved.
Proposition 5. The space
(
Lt (p, q) (G) , ‖|·|‖tp,q
)
is a Banach algebra.
Proof. Let’s take any f, g ∈ Lt (p, q) (G) . Then
‖|f ∗ g|‖tp = ‖f ∗ g‖p,q + ‖f ∗ g‖tp,q ≤ ‖Wf (g)‖p,q + ‖f‖tp,q ‖g‖
t
p,q ≤
≤ ‖Wf‖ ‖g‖p,q + ‖f‖tp,q ‖g‖
t
p,q = ‖f‖tp,q ‖g‖p,q + ‖f‖tp,q ‖g‖
t
p,q =
= ‖f‖tp,q |‖g‖|
t
p,q ≤ |‖f‖|
t
p,q |‖g‖|
t
p,q .
Proposition 5 is proved.
Proposition 6. The set
Λ = span {Wf∗g | f ∈ Lt (p, q) (G) , g ∈ Cc (G)}
is a complete subalgebra of Bp,q and it has a minimal approximate identity.
Proof. By the definition of Λ, it is easy to see that Λ is a complete subalgebra of Bp,q. Let F be
the family of all neighbourhoods of the identity of G and say E2 ≺ E1 if E1 ⊂ E2 for E1, E2 ∈ F .
Then clearly (F ,≺) is a directed set. For every Ei ∈ F , there exists a positive continuous function
hEi onG such that
∫
G
hEi (x) dλ (x) = 1, the support of hEi is contained in Ei and ‖hEi‖1 = 1. This
net {hEi}i∈I ⊂ Cc (G) is a minimal approximate identity for L1 (G) [11, 16]. If {hγ} denotes the
product net of {hEi}i∈I with itself, i.e., hγ = hEi ∗ hEj , then {hγ} is again a bounded approximate
identity for L1 (G) . In other words, ‖hγ‖1 =
∥∥hEi ∗ hEj∥∥1
≤ ‖hEi‖1
∥∥hEj∥∥1
≤ 1,
lim
γ
‖f − f ∗ hγ‖1 = lim
i,j
∥∥f − f ∗ (fEi ∗ fEj)∥∥ ≤
≤ lim
i
‖f − f ∗ fEi‖1 + lim
j
∥∥(f ∗ fEi)− (f ∗ fEi) ∗ fEj
∥∥
1
≤
≤ lim
i
‖f − f ∗ fEi‖1 + lim
j
∥∥f − (f ∗ fEj)∥∥1
‖fEi‖1 = 0
and
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
A STUDY OF FUNCTIONAL PROPERTIES AND MULTIPLIERS SPACES OF GROUP L (p, q) (G)-ALGEBRAS 345
‖Wf‖ = sup
‖h‖p,q≤1
‖Wf (h)‖p,q = sup
‖h‖p,q≤1
‖h ∗ f‖p,q ≤ ‖f‖1 (3)
for all f ∈ L1 (G) . Hence, it is seen that the net
{
Whγ
}
is in Λ and limγ
∥∥Whγ
∥∥ ≤ 1. Since
Cc (G) ⊂ Lt (p, q) (G) , we have
(Wf −Wg) (h) = h ∗ f − h ∗ g = h ∗ (f − g) = Wf−g (h) (4)
for any f, g ∈ Cc (G) and h ∈ L (p, q) (G) . Therefore, by using (3), (4) and the equality Wf∗g =
= Wg ◦Wf , we get
lim
γ
∥∥Whγ ◦Wf∗g −Wf∗g
∥∥ = lim
γ
∥∥Wf∗g∗hγ −Wf∗g
∥∥ =
= lim
γ
∥∥Wg∗hγ ◦Wf −Wg ◦Wf
∥∥ =
= lim
γ
∥∥(Wg∗hγ −Wg
)
◦Wf
∥∥ ≤ lim
γ
∥∥Wg∗hγ −Wg
∥∥ ‖Wf‖ =
= lim
γ
∥∥Wg∗hγ−g
∥∥ ‖Wf‖ ≤ lim
γ
‖g ∗ hγ − g‖1 ‖Wf‖ = 0.
Therefore, the net
{
Whγ
}
is a minimal approximate identity for the space Λ.
Proposition 6 is proved.
Proposition 7. The space Λ is a complete subalgebra of HomL1(G) (L (p, q) (G)) .
Proof. Let W ∈ Λ. By the definition of the space Λ, there exists {Wfn}n∈N ⊂ span{Wf∗g | f ∈
∈ Lt (p, q) (G) , g ∈ Cc (G)} such thatWfn →W. Therefore for all g ∈ L1 (G) and h ∈ L (p, q) (G) ,
we can write that
W (g ∗ h) = lim
n
Wfn (g ∗ h) = lim
n
g ∗ h ∗ fn = lim
n
g ∗Wfn (h) = g ∗W (h) . (5)
From (5) we see thatW ∈ HomL1(G) (L (p, q) (G)) . Since the space HomL1(G) (L (p, q) (G)) is a Ba-
nach algebra under usual operator norm, the space Λ is a complete subspace of HomL1(G) (L (p, q) (G)) .
Now, let f1, f2 ∈ Lt (p, q) (G) with Wf1 ,Wf2 ∈ Λ. Since (f1 ∗ f2) ∈ Lt (p, q) (G) , we write that
(Wf1 ◦Wf2) (g ∗ h) = Wf1 (Wf2 (g ∗ h)) = Wf1 (g ∗ h ∗ f2) =
= g ∗ h ∗ f2 ∗ f1 = g ∗Wf1 (h ∗ f2) =
= g ∗Wf1 (Wf2 (h)) = g ∗ (Wf1 ◦Wf2) (h)
for all g ∈ L1 (G) and h ∈ L (p, q) (G) . This shows that Λ is a complete subalgebra of the space
HomL1(G) (L (p, q) (G)) .
Proposition 7 is proved.
Proposition 8. The space Λ is an essential Banach L1 (G)-module.
Proof. Let’s take any Wf ∈ Λ, g ∈ L1 (G) and define the mapping g ◦Wf : L (p, q) (G) →
→ L (p, q) (G) with (g ◦Wf ) (h) = Wf (g ∗ h) for all h ∈ L (p, q) (G) . Then
‖g ◦Wf‖ = sup
‖h‖p,q≤1
‖(g ◦Wf ) (h)‖p,q = sup
‖h‖p,q≤1
‖Wf (g ∗ h)‖p,q =
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
346 İ. ERYILMAZ, C. DUYAR
= ‖Wf‖ sup
‖h‖p,q≤1
‖g ∗ h‖p,q ≤ ‖Wf‖ ‖g‖1 = ‖f‖tp,q ‖g‖1
is found. Thus, Λ is a Banach L1 (G)-module. Since any bounded approximate identity {eα}α∈I of
L1 (G) is also an approximate identity for L (p, q) (G) , for any Wf ∈ Λ, we have
‖eα ◦Wf −Wf‖ = sup
‖h‖p,q≤1
‖(eα ◦Wf −Wf ) (h)‖p,q = sup
‖h‖p,q≤1
‖Wf (h ∗ eα)−Wf (h)‖p,q =
= sup
‖h‖p,q≤1
‖h ∗ eα ∗ f − h ∗ f‖p,q = sup
‖h‖p,q≤1
‖h ∗ (eα ∗ f − f)‖p,q =
= sup
‖h‖p,q≤1
‖h‖p,q ‖eα ∗ f − f‖p,q
for all h ∈ L (p, q) (G) . As a result, Λ is an essential Banach L1 (G)-module by [3] (Corollary 15.3).
Proposition 8 is proved.
Remark 2. If we consider any f ∈ L1 (G) and the net {Weα}α∈I ∈ Λ, then
lim
α
‖f − f ◦Weα‖ = lim
α
(
sup
‖h‖p,q≤1
‖(f − f ◦Weα) (h)‖p,q
)
=
= lim
α
(
sup
‖h‖p,q≤1
‖f ∗ h−Weα (f ∗ h)‖p,q
)
=
= lim
α
(
sup
‖h‖p,q≤1
‖f ∗ h− (eα ∗ f) ∗ h‖p,q
)
≤
≤ lim
α
(
sup
‖h‖p,q≤1
‖f − eα ∗ f‖1 ‖h‖p,q
)
≤
≤ lim
α
‖f − eα ∗ f‖1 sup
‖h‖p,q≤1
‖h‖p,q = 0
is found. Therefore, f ∈ L1 (G) ◦ Λ = Λ. In other words, L1 (G) ⊂ Λ.
Remark 3. If p = q = 1, then Lt (p, q) (G) = Lt (1, 1) (G) = Lt1 (G) = L1 (G) since L1 (G)
is a Banach algebra. If p = q, then Lt (p, q) (G) = Lt (p, p) (G) = Ltp (G) which is examined in
[6,11,12].
2.2. Identification of multipliers space of L1 (G)-module with the multipliers space of certain
normed algebra.
Proposition 9. Let f, g ∈ L (p, q) (G) and T ∈ HomL1(G) (L (p, q) (G)) .
(i) If f ∈ Lt (p, q) (G) , then T (f) ∈ Lt (p, q) (G) .
(ii) If g ∈ Lt (p, q) (G) , then T (f ∗ g) = f ∗ T (g) .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
A STUDY OF FUNCTIONAL PROPERTIES AND MULTIPLIERS SPACES OF GROUP L (p, q) (G)-ALGEBRAS 347
Proof. (i) Let’s take any f ∈ Lt (p, q) (G) . By the definition of T ∈ HomL1(G) (L (p, q) (G)) ,
we write
‖T (f)‖tp,q = sup
{
‖h ∗ T (f)‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
=
= sup
{
‖T (h ∗ f)‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
≤
≤ ‖T‖ sup
{
‖h ∗ f‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
≤ ‖T‖ ‖f‖tp,q <∞
and the result follows.
(ii) Let g ∈ Lt (p, q) (G) . Since Cc (G) is dense in L (p, q) (G) , we can find a sequence
{fn}n∈N ⊂ Cc (G) such that limn ‖fn − f‖p,q = 0 for all f ∈ L (p, q) (G) . Also, we can find
a bounded linear mapping Wg ∈ HomL1(G) (L (p, q) (G)) for all g ∈ Lt (p, q) (G) . Therefore,
lim
n
‖fn ∗ g − f ∗ g‖p,q = lim
n
‖Wg (fn)−Wg (f)‖p,q =
= lim
n
‖Wg (fn − f)‖p,q ≤ ‖Wg‖ lim
n
‖fn − f‖p,q = 0
is found. Since T (g) ∈ Lt (p, q) (G) by (i), we get
lim
n
‖fn ∗ T (g)− f ∗ T (g)‖p,q = 0.
Therefore, the continuity of T ∈ HomL1(G) (L (p, q) (G)) implies that
f ∗ T (g) = lim
n
fn ∗ T (g) = lim
n
T (fn ∗ g) = T (f ∗ g) .
Proposition 9 is proved.
Definition 2. For the space Λ, the space Λ∗ is defined as follows:
Λ∗ =
{
T ∈ HomL1(G) (L (p, q) (G)) : T ◦W ∈ Λ for all W ∈ Λ
}
.
Proposition 10. Λ∗ = HomL1(G) (L (p, q) (G)) .
Proof. Let T ∈ HomL1(G) (L (p, q) (G)) and Wf∗g ∈ Λ such that f ∈ Lt (p, q) (G) and
g ∈ Cc (G) . By using Proposition 9, since T (f) ∈ Lt (p, q) (G) , we have
(T ◦Wf∗g) (h) = T (Wf∗g (h)) = T (h ∗ f ∗ g) = h ∗ T (f ∗ g) = WT (f∗g) (h) =
= Wg∗T (f) (h)
for all h ∈ L (p, q) (G) . This means that T ∈ Λ∗. On the other hand, the inclusion Λ∗ ⊂
⊂ HomL1(G) (L (p, q) (G)) is obvious from the definition of Λ∗.
Proposition 10 is proved.
Proposition 11. M (Λ,Λ) , the space of multipliers on Banach algebra Λ, is isometrically iso-
metric to the space Λ∗.
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Proof. Define a mapping F : Λ∗ → M (Λ,Λ) by letting F (T ) = ρT for each T ∈ Λ∗, where
ρT (S) = T ◦ S for all S ∈ Λ. Note that, if ρT (S ◦K) = T ◦ S ◦K = ρT (S) ◦K for all S,K ∈
∈ Λ, then we see that F (T ) = ρT ∈M (Λ,Λ) and so F is well-defined. Also, it is obvious that the
mapping F is linear.
For any T1, T2 ∈ Λ∗, if F (T1) = F (T2) , then ρT1 = ρT2 and ρT1 (S) = ρT2 (S) for all S ∈ Λ.
This means that T1 ◦ S = T2 ◦ S and T1 = T2. Hence, the mapping F is one-to-one.
Beside these, for any T ∈ Λ∗ = HomL1(G) (Λ,Λ) and for all S ∈ Λ, we get
‖T ◦ S‖ = sup
‖g‖p,q≤1
‖(T ◦ S) (g)‖p,q = sup
‖g‖p,q≤1
‖T (S (g))‖p,q ≤
≤ ‖T‖ sup
‖g‖p,q≤1
‖S (g)‖p,q = ‖T‖ ‖S‖.
Therefore, we have
‖ρT ‖ = sup
S∈Λ
‖ρT (S)‖
‖S‖
= sup
S∈Λ
‖T ◦ S‖
‖S‖
≤ sup
S∈Λ
‖T‖ ‖S‖
‖S‖
= ‖T‖ . (6)
On the other hand, since the net
{
Whγ
}
is a minimal approximate identity for Λ, we can write the
following inequality:
‖ρT ‖ = sup
S∈Λ
‖T ◦ S‖
‖S‖
≥ sup
γ
∥∥T ◦Whγ
∥∥∥∥Whγ
∥∥ ≥ sup
γ
∥∥T ◦Whγ
∥∥ ≥
≥ lim
γ
∥∥T ◦Whγ
∥∥ = ‖T‖ . (7)
By (6) and (7), we see that ‖ρT ‖ = ‖T‖ .
Lastly, we need to show that the mapping F is onto. Let’s take any ρ ∈M (Λ,Λ) and the minimal
approximate identity {eα}α∈I of L1 (G) . Since, it is known that Λ ⊂ HomL1(G) (L (p, q) (G)) and
ρeα ∈ Λ, we get
ρeα (f ∗ g) = (f ◦ (ρeα)) (g) , (8)
where f ∈ L1 (G) and g ∈ L (p, q) (G) . If we use the property M (Λ,Λ) ⊂ HomL1(G) (Λ,Λ) , then
we can write the equalities
ρ (f ◦ eα) (g) = ρ (f ∗ eα) (g) = (f ◦ ρeα) (g) . (9)
From (8) and (9), we have
ρeα (f ∗ g) = (f ◦ (ρeα)) (g) = ρ (f ∗ eα) (g) .
Therefore, for all f ∈ L1 (G) and g ∈ L (p, q) (G)
lim
α
‖ρ (f ∗ eα) (g)− ρf (g)‖p,q = lim
α
‖(ρ (f ∗ eα)− ρf) (g)‖p,q = lim
α
‖ρ (f ∗ eα − f) (g)‖p,q ≤
≤ lim
α
‖ρ (f ∗ eα − f)‖ ‖g‖p,q ≤ lim
α
‖ρ‖ ‖(f ∗ eα − f)‖ ‖g‖p,q ≤
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≤ lim
α
‖ρ‖ ‖(f ∗ eα − f)‖1 ‖g‖p,q = 0
is found and we can write that
lim
α
(ρeα) (f ∗ g) = lim
α
(f ◦ (ρeα)) (g) = lim
α
ρ (f ∗ eα) (g) = ρf (g) .
Since L (p, q) (G) is an essential Banach L1 (G)-module, the following expression:
(ρeα) (f ∗ g) = (f ◦ ρeα) (g) = f ∗ (ρeα) (g)
converges to such a f ∗ T (g) ∈ L (p, q) (G) where T ∈ HomL1(G) (L (p, q) (G)) . Since
limα (ρeα) (f ∗ g) = limα (f ◦ (ρeα)) (g) = ρf (g) , we write f ◦ T = ρf for all f ∈ L1 (G) .
Thus, for any W ∈ Λ, we have
eα ◦ T ◦W = (ρeα) ◦W = ρ (eα ◦W ) .
Since Λ is an essential Banach L1 (G)-module, we have T ◦W = ρ (W ) and ρ (W ) = ρT (W ) for
all W ∈ Λ. This means that ρ = ρT .
Proposition 11 is proved.
Conclusion 1. The space M (Λ,Λ) is isometrically isomorphic to HomL1(G) (L (p, q) (G)) .
2.3. Multipliers on group L (p, q) (G)-algebras. If an operator T ∈ Bp,q satisfies the condition
T (f ∗ g) = f ∗ T (g) for all f, g ∈ Lt (p, q) (G) , then we will call the operator T as a multiplier on
the space Lt (p, q) (G) . The space of all multipliers on Lt (p, q) (G) will be showed by the symbol
mp,q. Since, one can show by using usual techniques that the space mp,q is a Banach algebra under
operator norm with composition, we will omit its proof.
Lemma 1. For any f ∈ L (p, q) (G) with f 6= 0, there exists a g ∈ Cc (G) such that f ∗ g 6= 0.
Proof. Let’s take a function f ∈ L (p, q) (G) with f 6= 0 and assume that f ∗ g = 0 for all
g ∈ Cc (G) . We know by [5] that
‖f‖p,q ≤ p
′ ‖f‖∗p,q ≤ p
′C sup
∣∣∣∣∣∣
∫
G
f (x) g (−x) dµ (x)
∣∣∣∣∣∣ : g ∈ Cc (G) , ‖g‖∗p′,q′ ≤ 1
=
= p′C sup
{
|(f ∗ g) (0)| : g ∈ Cc (G) , ‖g‖∗p′,q′ ≤ 1
}
.
Therefore ‖f‖p,q = 0 and so f = 0 (a.e.). This contradiction proves the lemma.
Lemma 2. For any T ∈ mp,q and V ∈ Λ, we have
sup
{
‖T ◦ V (h)‖p,q : h ∈ Lt (p, q) (G) , ‖h‖p,q ≤ 1
}
≤ ‖T‖ ‖V ‖ .
Proof. Let’s define a set D =
{
Wf : f ∈ Lt (p, q) (G) , Wf ∈ Λ
}
and take any W ∈ Λ.
By the definiton of Λ, for all ε > 0, we can find f ∈ Lt (p, q) (G) and g ∈ Cc (G) such that
‖W −Wf∗g‖ < ε. Since f ∗ g ∈ Lt (p, q) (G) , we can conclude that Wf∗g ∈ D and so D̄ = Λ.
Now, define a map ϕ′ : D → Bp,q such that ϕ′ (Wf ) = WT (f) for any T ∈ mp,q. Since it is easy to
show that ϕ′ is bounded, it has a continuous extension ϕ : Λ→ Bp,q with ‖ϕ′‖ = ‖ϕ‖ .
Now, we will show the operators ϕ (V ) and (T ◦ V ) coinciding on Lt (p, q) (G) . Let’s take any
h ∈ Lt (p, q) (G) , ‖h‖p,q ≤ 1 and V ∈ Λ. Since D̄ = Λ, we can write
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lim
n
‖Wfn − V ‖ = 0, (10)
where {Wfn}n∈N ∈ D and {fn}n∈N ∈ Lt (p, q) (G) for all n ∈ N. Also, since
Wg (Lxf) = g ∗ Lxf = Lx (f ∗ g) = LxWg (f)
for all f ∈ L (p, q) (G) and Wg ∈ Λ, we say that Λ ⊂Mp,q and so V ∈Mp,q. By Proposition 9, we
have
(V ◦Wh) (g) = V (Wh (g)) = V (g ∗ h) = g ∗ V (h) = WV (h) (g)
for all g ∈ L (p, q) (G) and so V ◦Wh = WV (h). Again, the equality WWfn (h) = Wfn ◦Wh implies
that
‖Wfn (h)− V (h)‖tp,q =
∥∥∥WWfn (h) −WV (h)
∥∥∥ = ‖Wfn ◦Wh − V ◦Wh‖ =
= ‖(Wfn − V ) ◦Wh‖
for all n ∈ N. Therefore
lim
n
‖Wfn (h)− V (h)‖tp,q = lim
n
‖(Wfn − V ) ◦Wh‖ ≤ lim
n
‖Wfn − V ‖ ‖Wh‖ = 0
is found. As a result, for T ∈ mp,q,
lim
n
‖T (Wfn (h))− T (V (h))‖tp,q = 0 (11)
can be written. For any g ∈ Lt (p, q) (G) and n ∈ N, we have
WT (fn) (g) = g ∗ T (fn) = T (g ∗ fn) = T (Wfn (g)) = (T ◦Wfn) (g)
or simply WT (fn) = T ◦Wfn . This says that ϕ (Wfn) = ϕ′ (Wfn) = WT (fn) = T ◦Wfn . Therefore,
we get
lim
n
‖T ◦Wfn − ϕ (V )‖ = lim
n
‖ϕ (Wfn)− ϕ (V )‖ ≤
≤ ‖ϕ‖ lim
n
‖Wfn − V ‖ = 0
by (10). Since limn ‖(T ◦Wfn) (h)− ϕ (V ) (h)‖p,q = 0, we can easily obtain that limn ‖g ∗ ((T ◦
◦Wfn)(h))− g ∗ (ϕ (V ) (h)) ‖p,q = 0 and g ∗ϕ (V ) (h) = g ∗ T (V (h)) by (11) for all g ∈ Cc (G) .
By Lemma 1, we get ϕ (V ) (h) = (T ◦ V ) (h) . Since Lt (p, q) (G) is a normed algebra, it is easy to
see that
‖(T ◦ V ) (h)‖p,q = ‖ϕ (V ) (h)‖p,q = lim
n
‖ϕ (Wfn) (h)‖p,q = lim
n
∥∥WT (fn) (h)
∥∥
p,q
≤
≤ lim
n
∥∥WT (fn)
∥∥ ‖h‖p,q = ‖h‖p,q lim
n
‖T (fn)‖tp,q .
Therefore, by using equalities ‖Wf‖ = ‖f‖tp,q , ‖h‖p,q ≤ 1 and (10), we see that
‖(T ◦ V ) (h)‖p,q ≤ ‖T‖ lim
n
‖fn‖tp,q = ‖T‖ lim
n
‖Wfn‖ ≤ ‖T‖ ‖V ‖ .
Lemma 2 is proved.
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Proposition 12. For any T ∈ mp,q, V ∈ Λ and f ∈ Lt (p, q) (G) , we get
‖(T ◦ V ) (f)‖p,q ≤ ‖T‖ ‖V (f)‖p,q .
Proof. Let ε > 0. Since Λ is a Banach algebra with an approximate identity, for any V ∈ Λ, we
can find P, S ∈ Λ such that ‖P‖ = 1, ‖S − V ‖ < ε and V = P ◦ S by Cohen factorization theorem
in [8]. Therefore, we get
‖S (f)‖p,q = ‖(S − V + V ) (f)‖p,q ≤ ‖(S − V ) (f)‖p,q + ‖V (f)‖p,q ≤
≤ ε ‖f‖p,q + ‖V (f)‖p,q
and
‖(T ◦ V ) (f)‖p,q = ‖(T ◦ (P ◦ S)) (f)‖p,q = ‖T (P (S (f)))‖p,q ≤
≤ ‖T‖ ‖P‖ ‖S (f)‖p,q ≤ ‖T‖
(
ε ‖f‖p,q + ‖V (f)‖p,q
)
by Lemma 2. Since ε is arbitrary, we have ‖(T ◦ V ) (f)‖p,q ≤ ‖T‖ ‖V (f)‖p,q .
Proposition 12 is proved.
Lemma 3. The set ℘ =
{
V (f) : f ∈ Lt (p, q) (G) , V ∈ Λ
}
is dense in L (p, q) (G) .
Proof. Let’s take any g ∈ L (p, q) (G) and ε > 0. Since Cc (G) = L (p, q) (G) , there exists a
function f ∈ Cc (G) such that ‖g − f‖p,q <
ε
2
. If we take the approximate identity
{
Whγ
}
of Λ into
consideration, then Whγ (f) ∈ Lt (p, q) (G) for each γ by Proposition 9. Since the net {hγ} is an
approximate identity for L (p, q) (G) , we have limγ
∥∥Whγ (f)− f
∥∥
p,q
= limγ ‖hγ ∗ f − f‖p,q = 0
and so
{
Whγ (f)
}
⊂ ℘. As a result, we get
∥∥Whγ (f)− g
∥∥
p,q
≤
∥∥Whγ (f)− f
∥∥
p,q
+‖f − g‖p,q < ε
and ℘ = L (p, q) (G) .
Lemma 3 is proved.
Lemma 4. Let = be a dense subspace of L (p, q) (G) and V ∈ Bp,q. If V (h ∗ f) = h ∗ V (f)
for all h ∈ Cc (G) and f ∈ =, then V ∈Mp,q.
Proof. Let’s take any x ∈ G. Then by [7], we can find a net {uα}α∈I ⊂ Cc (G) such
that limα ‖Lxh− uα ∗ h‖p,q = 0 for all h ∈ =. Since V ∈ Bp,q is bounded, limα ‖V (Lxh) −
− V (uα ∗ h) ‖p,q = 0 can be written. Also by the hypothesis, we get limα ‖LxV (h) − uα ∗
∗ V (h) ‖p,q = 0 as V (h) ∈ L (p, q) (G) . Therefore, we have
‖V (Lxh)− LxV (h)‖p,q = ‖V (Lxh)− V (uα ∗ h) + V (uα ∗ h)− LxV (h)‖p,q ≤
≤ ‖V (Lxh)− V (uα ∗ h)‖p,q + ‖V (uα ∗ h)− LxV (h)‖p,q ≤
≤ ‖V (Lxh)− V (uα ∗ h)‖p,q + ‖uα ∗ V (h)− LxV (h)‖p,q
for all h ∈ =. It means that V (Lxh) = LxV (h) and V ∈Mp,q as = = L (p, q) (G) .
Lemma 4 is proved.
Lemma 5. Let f, g ∈ L (p, q) (G) and T ∈Mp,q.
(i) If f ∈ L1 (G) , then T (f ∗ g) = f ∗ T (g) .
(ii) If f ∈ Lt (p, q) (G) , then T (f) ∈ Lt (p, q) (G) .
(iii) If g ∈ Lt (p, q) (G) , then T (f ∗ g) = f ∗ T (g) .
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Proof. (i) is proved in [4] (Lemma 2.1).
(ii) Let f ∈ Lt (p, q) (G) and T ∈Mp,q. Then by (i), we get
‖T (f)‖tp,q = sup
{
‖h ∗ T (f)‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
=
= sup
{
‖T (h ∗ f)‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
≤
≤ ‖T‖ sup
{
‖h ∗ f‖p,q : h ∈ Cc (G) , ‖h‖p,q ≤ 1
}
≤ ‖T‖ ‖f‖tp,q <∞.
As a result, T (f) ∈ Lt (p, q) (G) .
(iii) Let g ∈ Lt (p, q) (G) . Since Cc (G) is dense in L (p, q) (G) , we can find a sequence
{fn}n∈N ⊂ Cc (G) such that limn ‖fn − f‖p,q = 0. If we remember Proposition 7, then it says that
Wg ∈ HomL1(G) (L (p, q) (G)) and so
lim
n
‖fn ∗ g − f ∗ g‖p,q = lim
n
‖Wg (fn)−Wg (f)‖p,q =
= lim
n
‖Wg (fn − f)‖p,q ≤ ‖Wg‖ lim
n
‖fn − f‖p,q = 0.
Since T (g) ∈ Lt (p, q) (G) by (ii), we get
lim
n
‖fn ∗ T (g)− f ∗ T (g)‖p,q = 0.
Therefore,
f ∗ T (g) = lim
n
fn ∗ T (g) = lim
n
T (fn ∗ g) = T (f ∗ g)
by the continuity of T and (i).
Lemma 5 is proved.
Proposition 13. Let ω be a map such that ω : Mp,q → mp,q, ω (T ) = ωT and ω (T ) (f) =
= ωT (f) = T (f) for all T ∈Mp,q and f ∈ Lt (p, q) (G) . Then ω is an isometric isomorphism.
Also, for T ∈ mp,q, there exists at least one S ∈ Mp,q such that ωS (V (f)) = (T ◦ V ) (f) for
all V ∈ Λ and f ∈ Lt (p, q) (G) .
Proof. We showed that T (f) ∈ Lt (p, q) (G) for T ∈ Mp,q and f ∈ Lt (p, q) (G) in Lemma 5.
If we use Λ ⊂Mp,q, then
ω (T ) (f ∗ g) = ωT (f ∗ g) = T (f ∗ g) = T (Wf (g)) = Wf (T (g)) =
= f ∗ T (g) = f ∗ ωT (g)
for all f, g ∈ Lt (p, q) (G) and T ∈ Mp,q by Lemma 5. This means that ω (T ) ∈ mp,q and ω is
well-defined.
On the other hand, we see the linearity of ω by the following equality:
ω (αT1 + βT2) (f) = ωαT1+βT2 (f) = (αT1 + βT2) (f) = αT1 (f) + βT2 (f) =
= αωT1 (f) + βωT2 (f) = αω (T1) (f) + βω (T2) (f) ,
where α, β numbers, f ∈ Lt (p, q) (G) and T1, T2 ∈Mp,q.
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Now, let’s take any T1, T2 ∈ Mp,q and assume that ω (T1) = ω (T2) . Then, by definition of ω,
we get T1 (f) = T2 (f) for all f ∈ Lt (p, q) (G) . Since Lt (p, q) (G) = L (p, q) (G) , we get T1 = T2
on L (p, q) (G) and ω is injective.
For any T ∈ Mp,q, we now know that ωT ∈ mp,q. From this point, since Lt (p, q) (G) =
= L (p, q) (G) , we have
‖ωT ‖ = sup
‖ωT (f)‖tp,q
‖f‖tp,q
= sup
‖T (f)‖tp,q
‖f‖tp,q
= ‖T‖tp,q =
∥∥∥T̃∥∥∥
p,q
,
where T̃ is extension of T. Therefore, ‖ω‖ = 1 and ω is continuous. Let’s take any T ∈ mp,q. By
Proposition 12 and Lemma 3, there exists a S ∈ Bp,q such that S (V (f)) = T (V (f)) for all V ∈ Λ
and f ∈ Lt (p, q) (G) . Therefore, we get
S (h ∗ V (f)) = S (V (h ∗ f)) = T (V (h ∗ f)) = T (h ∗ V (f)) =
= h ∗ T (V (f)) = h ∗ S (V (f))
for all h ∈ Cc (G) , V ∈ Λ and f ∈ Lt (p, q) (G) by Proposition 9 and Lemma 5. Besides these,
S ∈Mp,q by Lemmas 3 and 4. As a result, we obtain
ω (S) (V (h)) = ωS (V (h)) = S (V (h)) = T (V (h)) (12)
for all h ∈ Lt (p, q) (G) and V ∈ Λ as S ∈Mp,q.
Now, let’s assume that there is a function h ∈ Lt (p, q) (G) such that ωS (h) 6= T (h) . Then
ωS (h)−T (h) 6= 0 and g ∗ (ωS (h)− T (h)) 6= 0 for all g ∈ Cc (G) by Lemma 1. Since we can find
a sequence {hn}n∈N ⊂ Cc (G) with limn ‖hn − h‖p,q = 0 for h ∈ Lt (p, q) (G) ⊂ L (p, q) (G) , we
write the following:
‖g ∗ (ωS (h)− T (h))‖p,q = ‖g ∗ (ωS − T ) (h)‖p,q = ‖(ωS − T ) (g ∗ h)‖p,q =
= ‖h ∗ (ωS − T ) (g)‖p,q = lim
n
‖hn ∗ (ωS − T ) (g)‖p,q =
= lim
n
‖(ωS − T ) (hn ∗ g)‖p,q = lim
n
‖(ωS − T ) (Whn (g))‖p,q .
Therefore, Whn (g) ∈ ℘ and ℘ = L (p, q) (G) imply that ‖g ∗ (ωS (h)− T (h))‖p,q = 0 by (12). This
contradiction shows that ω (S) = ωS = T, i.e., ω is surjective.
For any T ∈ Mp,q, f ∈ Lt (p, q) (G) and ε > 0, there exists a g ∈ L (p, q) (G) with ‖g‖p,q ≤ 1
such that ‖ωT (f)‖tp,q < ‖g ∗ ωT (f)‖p,q + ε by the property of ‖·‖tp,q norm. Since Lemma 5 says
that
g ∗ ωT (f) = g ∗ T (f) = T (g ∗ f) = T (Wf (g)) = (T ◦Wf ) (g) ,
we get
‖ωT (f)‖tp,q ≤ ‖g ∗ ωT (f)‖p,q + ε ≤
≤ ‖T‖ ‖Wf‖ ‖g‖p,q + ε ≤ ‖T‖ ‖f‖tp,q + ε.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
354 İ. ERYILMAZ, C. DUYAR
Therefore, ‖ω (T )‖ = ‖ωT ‖ ≤ ‖T‖ . Conversely, with Proposition 12 and Lemma 3, we have
‖T‖ = sup
{
‖T (V (h))‖p,q : V (h) ∈ ℘, ‖V (h)‖p,q ≤ 1
}
=
= sup
{
‖ωT (V (h))‖p,q : V (h) ∈ ℘, ‖V (h)‖p,q ≤ 1
}
≤ ‖ωT ‖
for T ∈Mp,q. Lastly, we have ‖ω (T )‖ = ‖T‖ and Mp,q
∼= mp,q.
Proposition 13 is proved.
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Received 12.02.12
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
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| id | umjimathkievua-article-1987 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:31Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/0a/732722d1b11b224b7522bdd91550850a.pdf |
| spelling | umjimathkievua-article-19872019-12-05T09:48:08Z Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras Вивчення функціональних властивостей та просторів множників для груп $L(p, q)(G)$-алгебр Duyar, C. Eryılmaz, İ. Дуяр, С. Ерийлмаз, І. Let $G$ be a locally compact Abelian group (noncompact and nondiscrete) with Haar measure. Suppose that $1 < p < ∞$ and $1 ≤ q ≤ ∞$. The purpose of the paper is to define temperate Lorentz spaces and study the spaces of multipliers on Lorentz spaces and characterize them as the spaces of multipliers of certain Banach algebras. Нехай $G$ є локальною компактною абелiвською групою (некомпактною та недискретною) з мірою Хаара. Припустимо, що $1 < p < ∞$ та $1 ≤ q ≤ ∞$. Наша метa — визначити помірні простори Лоренца, вивчити простори множників на просторах Лоренца та охарактеризувати їх як простори множників на деяких банахових алгебрах. Institute of Mathematics, NAS of Ukraine 2015-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1987 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 3 (2015); 341–354 Український математичний журнал; Том 67 № 3 (2015); 341–354 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1987/993 https://umj.imath.kiev.ua/index.php/umj/article/view/1987/994 Copyright (c) 2015 Duyar C.; Eryılmaz İ. |
| spellingShingle | Duyar, C. Eryılmaz, İ. Дуяр, С. Ерийлмаз, І. Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title | Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title_alt | Вивчення функціональних властивостей та просторів множників для груп $L(p, q)(G)$-алгебр |
| title_full | Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title_fullStr | Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title_full_unstemmed | Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title_short | Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras |
| title_sort | investigation of the functional properties and spaces of multipliers for group $l(p, q)(g)$-algebras |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1987 |
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