Multiplication Operators on Weighted Grand Lorentz Spaces with
Various Properties
İLKER ERYILMAZ
Department of Mathematics, Faculty of Sciences,
Ondokuz Mayıs University,
Kurupelit-Atakum, Samsun,
TURKEY
Abstract: - The concept of Lebesgue space has been generalized to the grand Lebesgue space with non-weight
and weight, and the classical Lorentz space concept has been generalized to grand Lorentz spaces with a similar
logic. In this study, instead of using rearrangement for a measurable function, weighted Grand Lorentz spaces
are defined by using the maximal function
1,pq
where the weight function is measurable, complex
valued, and locally bounded. In addition, multiplication operators on weighted grand Lorentz spaces are
defined and the fundamental properties of these operators such as boundedness, closed range, invertibility,
compactness, and closedness are characterized.
Key-Words: - Weighted Grand Lorentz space, Multiplication operator, Compact operator
Received: August 13, 2022. Revised: March 7, 2023. Accepted: April 13, 2023. Published: May 4, 2023.
1 Introduction
Iwaniec and Sbordone generalized the concept of
Lebesgue spaces and introduced a new space of
measurable, almost everywhere equal integrable
function classes, which they called grand Lebesgue
spaces. According to [12], grand Lebesgue spaces
are the space of equivalence classes of all
measurable functions defined on
0,1
and denoted
by
)p
L
with
1p
. For any
)p
fL
, the
function
1
1
)01
0
sup p
p
pp
f f x dx



defines a norm on
)p
L
and makes the space a
Banach function space. Also
)p p p
L L L

if
01p
. New results on grand Lebesgue
spaces can be observed in current studies, [8], [10],
[14], [15], [16], [21], [22], [27]. Presented in terms
of the Jacobian integrability problem, these works
have proven useful in various applications of partial
differential equations and variational problems,
where they are used in the study of maximum
functions, extrapolation theory, etc. The harmonic
analysis of these spaces, and the related small
Lebesgue spaces, has been intensively developed in
recent years and continues to attract the attention of
researchers due to various applications.
Let
and
w
be a weight function.
Weighted grand Lebesgue spaces denoted by
)p
w
L
are the space of measurable functions defined on
0,1
such that
1
1
), 01
0
sup p
p
pw p
f f x w x dx



is finite for any
)p
w
fL
. These spaces were defined
in the study, [9], and they also examined the
boundedness of the maximal operator on the space
)p
w
L
in there. The boundedness of the Riesz potential
on weighted grand Lebesgue spaces is characterized
in [16]. In addition to these, the classical weighted
Lorentz and grand Lorentz spaces were compared
and the boundedness of the maximal operator was
examined in [7], [25]. The basic properties of grand
Lorentz sequence spaces and the multiplication
operators on these spaces are examined in [24].
Let
,,X
and
,,Y
be two
-finite
measure spaces. A measurable transformation
:T Y X
is said to be non-singular if
10TA
, whenever
A
with
0A
. In
this case, we write
1
T

. Let
:uY
be a
measurable function. Then, the linear transformation
,: , , , ,
uT
W W L X L Y

is defined as
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,·
uT
W f x W f x u T x f T x
for all
xY
and for each
,,f L X

where
,,LX
and
,,LY
are the linear spaces of
all
measurable and
measurable functions on
X
and
Y
, respectively. Here, the non-singularity of
T
guarantees that the operator
W
is well defined as
a mapping of equivalence classes of functions. In
the case when
W
maps
L
into
L
, we call
,uT
WW
a weighted composition operator induced
by the pair
,uT
. If
1u
, then
T
WC
is called a
composition operator induced by
T
. If
T
is the
identity mapping, then
:·
u
W M f u f
is a
multiplication operator induced by
u
.
These operators are simple but have a wide
range of applications in ergodic theory, dynamical
systems, etc. The studies of (weighted) composition
and multiplication operators have a very long
history in Mathematics. From books on functional
analysis and papers related to these operators, one
can learn many properties of these operators on
various function spaces including Lebesgue and
Lorentz spaces. The study of these operators acting
on Lebesgue and Lorentz spaces has been made in
[5], [11], [13], [19], [20], [28], [29], [30], [1], [2],
[3], [17], [18], respectively.
2 Preliminaries
Throughout the paper
,,XX

will stand a
finite measure space,
L
will denote the
linear space of all equivalence classes of
-
measurable functions on
X
and
A
will be used for
the characteristic function of a set
A
. For any two
non-negative expressions (i.e. functions or
functionals),
A
and
B
, the symbol
AB
means
that
A cB
for some positive constant
c
independent
of the variables in the expressions
A
and
B
. If
AB
and
BA
, we write
AB
and say that
A
and
B
are equivalent.
Let
w
be a weight function, i.e. a
measurable, complex valued, and locally bounded
function on
X
, satisfying
1wx
for all
xX
.
Weighted Lorentz spaces or Lorentz spaces over
weighted measure spaces
,,L p q wd
are studied
and discussed in [6], [23], by taking the measure
wd
instead of the measure
. Then the
distribution function
f
which is considered
complex-valued, measurable, and defined on the
measure space
,X wd
,
:
:
, 0
fw
x X f x y
y w x X f x y
w x d x y


is found. The nonnegative rearrangement
f
is
given by
,
,
inf 0:
sup 0: , 0
w f w
fw
f t y y t
y y t t
where we assume that
inf
and
sup 0
.
Also, the average (maximal) function of
f
on
0,
is given by
0
1.
t
ww
f t f s ds
t

Note that
,·,
fw
·
w
f
and
·
w
f
are
non-increasing and right continuous functions. The
weighted Lorentz space
,,L p q wd
is the
collection of all the functions
f
such that
,pq
f
, where
1
1
0
,, 1
0
, 0 , .
sup , 0
qq
q
pw
p q w
pw
t
qt f t dt p q
p
f
t f t p q






(1)
In general, however,
,,
·p q w
is not a norm since the
Minkowski inequality may fail. But by replacing
w
f
with
w
f
in (1), we get that
,,L p q wd
is a
normed space, with the functional
,,
·p q w
defined
by
1
1
0
,, 1
0
, 0 , .
sup , 0
qq
q
pw
p q w
pw
t
qt f t dt p q
p
f
t f t p q








If
and
1q
, then
, , , , , ,
1
p q w p q w p q w
p
f f f
p


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where the first inequality is an immediate
consequence of the fact that
ww
ff

the second
follows from the Hardy inequality, [4]. For more on
weighted Lorentz spaces, one can refer to [6], [23],
and references therein.
Using the maximal function
·
w
f
, instead of the
nonnegative rearrangement
·
w
f
used in the
definition of grand Lorentz space defined in [25],
we defined the weighted grand Lorentz spaces as
follows.
Definition 1 The weighted grand Lorentz space
,)pq
w
L
is the collection of all the complex-valued,
measurable functions which are defined on the
measure space
0,1 ,wd
such that
,)
w
pq
f
where
1
1
1
0
01
,) 1
01
,1s ,up
sup , 1 ,
qq
q
p
w
q
w
pq
pw
t
fpq
qt t dt
p
f
t f t p q









for any
,)pq
w
fL
. In particular, if
any
1q
;
1pq
or
pq
, then the
normed space
,)pq
w
L
is a Banach space.
3 Multiplication Operators
Let
:uX
be a measurable function such that
u f L

whenever
fL
. This gives rise to
a linear transformation
:
u
M L L

defined
by
u
M f u f
, where the product is pointwise. In
case if
L
is a topological vector space and
u
M
is
a continuous (bounded) operator, then it is called a
multiplication operator induced by
u
.
Remark 1. In general, the multiplication operators
on measurable function’s spaces are not
injective. For example, let
G= X - supp u
where
= : 0supp u x X u x
. Then
0G
and
0
GGu x x u x

for all
xX
. This
implies that
0
uG
M
and
0
u
KerM
. Hence
u
M
is not injective.
On the contrary, if
u
M
is injective, then
X - supp u
must be 0. If
0X - supp u =
and
is a complete measure, then
0
u
Mf
implies that
0f x u x
for all
xX
and so
:0x X f x X - supp u
and
0f
(a.e.)
on
X
.
Proposition 1. The operator
u
M
is injective on
, ) , )
:.
p q p q
ww
usupp
K = L supp u f = f f L X

Proof. To show that the operator
u
M
is injective, it
is enough to show that
0
u
KerM
. Indeed, if
=0
u
Mf
with
fK
, then
0
usupp
f x u x f x x u x
for all
xX
. From this, we get
0f x u x
for
all
x supp u
and so
0fx
. Therefore
0f
and
0
u
KerM
.
Theorem 1. The linear operator
u
M f u f
on
weighted grand Lorentz spaces
,)pq
w
L
is bounded for
1,pq
if and only if
u
is essentially bounded.
Moreover
q
p
u
uuM


.
Proof. Suppose that
,)pq
w
fL
and
u
is essentially
bounded, i.e.
uL
. Since
u x u
for all
xX
, it can be written that
u f x u f x

and so
,,
u
M f w f w y
yu





for all
0y
. Also, it is easy to see that
uw
w
M f t u f t
and
uw
w
t
M f t f u
 




.
Therefore
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11**
,) 01 0
11
01 0
11
01 0
,)
1
1
1
sup
sup
sup t
qq
q
wp
uu
pq q
qq
q
p
q
qq
qq
p
p
q
qw
ppq
qt f t dt
p
qt
t dt
pu
q
u t dt
p
uf
M f M
f
f




















can be obtained. For
q
, we have
**
,) 01
1
**
,)
01
1
1
sup
sup
wp
uu
w
pt
w
pp
wp
t
f t f t
t
tu
u
MM
ff







.
Thus,
u
M
is bounded for all
1,pq
.
Conversely, suppose that
u
M
is a bounded
operator on weighted grand Lorentz spaces for
1.q
If
u
is not an essentially bounded
function, then we can write a set
0,1 :
kx u x kG
that has a positive
measure for all
k
. Since the non-increasing
rearrangement of the characteristic function
k
G
is
*
0, ,
kw
GwG
k
tt

we can easily get that
,,
**
u
kk
kk
k w M w
uGG
ww
GG
y
t k t
y
M

and
kk
uww
GG
t
Mt k

  


.
Therefore
11**
,) 0
11**
0
,)
1
0 -1
1
0 -1
.
sup
sup
kk
k
k
qqq
wp
uu
p q w
qq
q
p
w
qw
p
pq
q
q
GG
G
G
qt
p
qkt
p
k
M t M dt
t dt















Besides these,
q
we have
**
,)
1
**
,)
1
1
01
1
01
1.
sup
sup
kk
kk
wp
uu
pq w
t
w
pp
pq
w
t
q
pp
k
GG
GG
t
k t k
k q w G
M t M
t






This contradicts the boundedness of
u
M
. Hence
u
must be essentially bounded. Now for any
0
, let
:F = x X u x u
.
Then
: - :
FF
x X u x x X u x
and so
,
,-F
Fw
wuu
yy

. Therefore, we
get
**
FF
w
w
u
M t u t

and
u
Mu

. As a result,
u
Mu
.
Remark 2. If
X
and
Y
are Banach spaces and
,T X YB
, then
T
is bounded below if and only
if
T
is 1-1 and has a closed range. According to this
knowledge, we can give the following corollaries.
Corollary 1.
u
M
has a closed range on
,)pq
w
Lsupp u
if and only if
u
M
is bounded below
on
,)pq
w
Lsupp u
.
Corollary 2. If
is a complete measure and
0u
a.e., then
, ) , )
: , , , ,
p q p q
u w w
M L X L X

has
closed range if and only if
u
M
is bounded below on
,) ,,
pq
w
LX
.
Theorem 2. The set of all multiplication operators
on weighted grand Lorentz spaces
,)pq
w
L
for
1,pq
is a maximal Abelian sub algebra of
, ) , )
,
p q p q
ww
LLB
, Banach algebra of all bounded linear
operators on
,)pq
w
L
.
Proof. Let
:
u
H M u L

. Then
H
is a vector
space under addition and scaler multiplication. Also,
it is a sub algebra of
, ) , )
,
p q p q
ww
LLB
according to
multiplication. Let
T
be any operator on
,)pq
w
L
such
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that
uu
T M M T
for every
uL
and
:eX
be the unit function defined by
1ex
for all
xX
with
v Te
. Then
EE
E E E
T T M e M T e M
and so
TM
for all measurable set
E
. If possible,
the set
:
kx X v x kG
has a positive
measure for each
k
, then
,)
, ) , ) .
k k k
q
ww
w
p
vpq
p q p q
G G G
kTM

Therefore
T
is an unbounded operator that is a
contradiction to the fact that
T
is bounded.
Therefore
vL
and
v
M
is bounded by
Theorem 1. Now, let
,)pq
w
fL
and
nn
s
be a
nondecreasing sequence of measurable simple
functions such that
lim n
sf
. Then
n n n
n
lim lim lim
lim
n n v n
v n v
T f T s T s s
sf
M
MM




.
Therefore, we can conclude that
TH
.
Corollary 3. The multiplication operator
u
M
on
,)pq
w
L
for
1,pq
is invertible if and only if
u
is
invertible in
L
.
Proof. Let
u
M
be invertible. Then there exists a
, ) , )
,
p q p q
ww
T L LB
such that
uu
T M M T I
.
Let
MH
. Then
u v v u
M M M M
and
uu
T M T M I
T M M T T M M T
I M T M T





.
Therefore, we can conclude that
T
commute with
M
and so
TH
by Theorem 2. Then there exists
a
wL
such that
w
T = M
and
u w w u
M M M M I
.
This implies that
1uw wu
a.e, which means that
u
is invertible on
L
. On the other hand, assume
that
u
is invertible on
L
, that is
1
uL
. Then
11
uu
uu
M M M M I


which means that
u
M
is
invertible on
, ) , )
,
p q p q
ww
LLB
.
4 Compact Multiplication Operators
In this section, compact multiplication operators are
characterized.
Definition 2. Let
T
be an operator on a normed
space
X
. A subspace
K
of
X
is said to be
invariant under
T
(or simply
T
-invariant) if
T K K
.
Lemma 1. Let
:T X X
be an operator. If
T
is
compact and
N
is a closed
T
-invariant subspace of
X
, then
N
T
is also compact.
Proof. Let
kk
g
be a sequence in
NX
. Then
compactness property of
T
implies that there exists
a subsequence
n
kn
g
of
kk
g
such that
n
kn
Tg
converges in
X
. Since
n
k
gN
and
n
kn
T g T N
, then
n
kn
Tg
converges
on
N
. Hence
N
T
is compact.
Theorem 4. Let
u
M
be a compact operator,
:u x X u xG
and
, ) , )
:
p q p q
ww
Gu
L G u f f L

for any
0
.
Then
,)pq
w
LuG
is a closed invariant subspace of
,)pq
w
L
under
u
M
and
u
M
is a compact operator on
,)pq
w
LuG
.
Proof. We first show that
,)pq
w
LuG
is a
subspace of
,)pq
w
L
. Let
,)
,pq
w
f g L uG
and
,ab
. Since
Gu
f = f
and
Gu
=gg
for
any
,)
,pq
w
f g L
, we have
G u G u G u
+b +baf g=af g af bg

.
By the definition of
, ) , )
:p q p q
u w w
M L u L XG
,
we have
uGu
M f uf uf

. Therefore
,)pq
w
LuG
is an invariant subspace of
,) ,,
pq
w
LX
under
u
M
. Now, let us show that
, ) , )p q p q
ww
L G u L G u

. Let
,)pq
w
g L G u
.
Then there exists a sequence
k
g
in
,)pq
w
L G u
such that
k
gg
where
kk
Gu
gg
for each
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k
. Since
k
g
is a Cauchy sequence in
,)pq
w
L G u
, it can be written that for all
0
,
there exists a
0
k
such that
,)
w
kr
pq
gg

for
all
0
,k r k
. Hence for all
0
,k r k
, we can find a
0
such that
k r k r Gu
g g g g

and
**
0,
k r k r
ww
w G u
t t tg g g g

.
Then,
, ) , )
1
q
q
ww
p
p
k r k r
p q p q
w G u qg g g g
can be written. Therefore
kk
g
is also a Cauchy
sequence in
,)pq
w
L
. Since
,)pq
w
L
is a Banach space, we
can write that
k
gg
for an element
,)pq
w
Lg
.
Thus, we have
,)
,)
ww
kk
G u G u pq
pq
g g g g

and
k
gg
. Consequently
,)pq
w
LugG
and
,)pq
w
L G u
u
M
is a compact operator by Lemma 1.
Theorem 5. A multiplication operator
u
M
on
,)pq
w
L
is compact if and only if
,)pq
w
LuG
is finite
dimensional for each
0
, where
uG
and
,)pq
w
LuG
as in Theorem 4.
Proof. If
u
M
is a compact operator, then
,)pq
w
LuG
is a closed invariant subspace of
,)pq
w
L
under
u
M
and
,)pq
w
uL G u
M
is a compact operator
by Theorem 4. Let’s take any
xX
. If
uxG
then for each
,)pq
w
Lf
, we can obtain
,) 0
pq
w
uGu
L G u ttM f u f

and so
,) 0.
pq
w
uL G u
M
If
uxG
, then we have
ux
and
G u G u
u f x f x


,
,,
Gu Gu
fw u f w
yy



. Therefore
,
,
0: 0: Gu
Gu fw
u f w
y
y y t y t







for all
0.t
By using this, we have
**
** **
G u G u
ww
G u G u
ww
f t u f t
f t u f t





and
11**
,) 10
11**
10
,)
1
0
1
0
sup
sup
.
qq
q
wp
uu
G u G u
p q w
q
qq
q
p
Gu w
q
w
Gu pq
qtt
p
qtt
p
M f M f dt
f dt
f















Thus, in either case
,)pq
w
L G u
u
M
has a closed range
in
,)pq
w
LuG
and invertible. Being compact
implies that
,)pq
w
LuG
is finite dimensional.
Conversely, suppose that
,)pq
w
LuG
is
finite dimensional for each
0
. In particular,
,) 1
pq
wn
LuG
is finite dimensional for each
.n
Define a sequence
:
n
uX
as
,1
0, 1
n
u x u x n
ux u x n
for all
n
. Since
uL
, it’s easy to see that
n
uL
for each
n
. Moreover for any
,)pq
w
Lf
,
nn
u u f,w y w u u yx X f x
 :
and
*
,
inf :0n
nu u f w
w
u u t y tfy

.
If
1n
xuG
then
*
n0
w
u u tf
and
0
n
uuf
. If
1n
xuG
, then we get
* **
* **
11
n w n w
ww
u u t t u u t t
nn
f f f f
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DOI: 10.37394/23206.2023.22.32
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and
,)
,)
1
n
ww
uu pq
pq n
M f f
. This implies that
n
u
M
converges to
u
M
uniformly. Since
,) 1
pq
w
n
LuG



is finite-dimensional,
n
u
M
is a finite
rank operator. Therefore,
n
u
M
is a compact
operator’s sequence and so
u
M
is.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
İlker ERYILMAZ contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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DOI: 10.37394/23206.2023.22.32
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Volume 22, 2023