Banach Function Space Property of A New Type of Grand Lorentz
Spaces
ILKER ERYILMAZ1, GOKHAN ISIK2
1Department of Mathematics, Faculty of Sciences,
Ondokuz Mayıs University,
Kurupelit-Atakum, Samsun,
TURKEY
2Samsun İl Milli Eğitim Müdürlüğü,
Samsun,
TURKEY
Abstract: - The concept of Lorentz space has been generalized to the grand Lorentz space. In this paper, a new
generalization of Lorentz spaces is defined as
,) ,,
pq
LX
with 0
by using the maximal function of a
measurable function. Besides an explicit proof of being Banach function space with a rearrangement invariant
norm seemed to be missing in the literature. Therefore, the authors provided such proof which can serve as a
reference for the next studies and literature and as a fundamental reference for subsequent results.
Key-Words: - Grand Lorentz space, Banach function space, Iwaniec-Sbordone space, Distribution function
Received: September 19, 2022. Revised: April 9, 2023. Accepted: May 2, 2023. Published: May 17, 2023.
1 Introduction and Preliminaries
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. Now, let X be a locally compact Hausdorff
space and suppose that

,,X
is a finite measure
space. According to [10], grand Lebesgue spaces are
the collection of equivalence classes of functions
obtained according to almost everywhere relation of
all
measurable functions defined on
,,X
and denoted by )p
L for1p. For any )p
vL,
the functional

1
1
)01
sup p
p
p
ppX
vvxd





defines a norm on )p
L and makes them Banach
function spaces with rearrangement invariant norm.
Also )pp p
LLL
 if 01p
. New results on
grand Lebesgue spaces can be observed in current
studies, [3], [6], [7], [9], [11], [12], [15]. 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.
There have been several generalizations of
the grand Lebesgue spaces in recent years. One such
generalization denoted by

),p
LX
is defined as
the set all functions belong to
01
p
p
L

in [8].
According to [1], [8],

),p
LX
is a rearrangement-
invariant space, i.e. Banach function space
generated with the rearrangement-invariant norm

1
), 01
01
sup
sup
p
p
p
ppX
p
p
p
vd
v
vx






for all
),p
vL X
where 1p and 0
.
),p
LX
reduces to classical Lebesgue spaces
p
LX
when 0
and reduces to grand Lebesgue
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spaces

)p
LX
when 1
, [3], [8]. Also, we have
),pp p
LX L X L X

 for 01p

and
  
12
), ),pp
p
LX L X L X

 for 12
0
, [1],
[8]. It is important to remember that the subspaces
of simple functions S and the subspace of test
functions

0
CX
is not dense in

),p
LX
. If we
call the closure of
0
CX
in

),p
LX
as
),p
E
X
,
then
 
), ),
0
:lim 0
pp
p
p
EX vLX v







and
 
12
), ),pp
LXE X

for 12
0
, [8]. The
Marcinkiewicz class, denoted by weak
p
LX or

,p
LX
, consists of all measurable functions
such that
0
sup pf
D

 where the
distribution function of
f
is

: , 0
f
DxXfx


.
Then
 
,),pp
LXLX
. It is commonly known
that in the sense of [2], the grand and small
Lebesgue spaces are Banach Function Spaces. An
explicit proof of these facts seemed to be missing in
the literature. In [1], the author provided such a
proof, which can serve as a reference for the next
studies and literature and as a fundamental reference
for subsequent results. The proofs in [1], show that
the grand Lebesgue spaces content all the axioms
which are necessary to be Banach Function Spaces,
including the Fatou property. These results are
important because they establish the grand Lebesgue
spaces as a well-behaved class of function spaces
that can be used to study various problems in
analysis and partial differential equations. Overall,
the results in [1], fill an important gap in the
literature and provide a solid foundation for further
research in this area.
Throughout this paper

,,XX

will
show a
finite measure space, and the collection
of all extended scalar-valued (real or complex)
measurable functions on X will be shown by M.
Also, 0
M will stand for the class of functions in M
which are finite-valued
a.e. The function U
will be employed for the characteristic function of
any subset U.
The distribution function of a complex-
valued, measurable function
f
defined on the
measure space X is

: , 0
fyxXfxyy


.
The nonnegative rearrangement function
*
f
of
f
is given by



*inf 0 :
sup 0: , 0
f
f
ft y y t
yytt


where we assume that inf  and
sup 0
.
Likewise, the (average) maximal function
**
f
of
f
which is defined on
0, is given by
 
** *
0
1.
t
f
tfsds
t
Note that
ꞏ,
f
*ꞏf and
** ꞏf are right
continuous and non-increasing functions.
The generalization of ordinary Lebesgue
spaces are Lorentz spaces

,Lpq which are the
collection of all classes of the functions
f
such that
,pq
f
, where


1
1*
0
,1
*
0
,0 ,
sup , 0 , .
qq
q
p
pq
p
t
qtftdt pq
p
f
tf t p q

 





(1)
In general, however, ,
ꞏ
p
q
is not a norm if not
1qp
 or
p
q
 since the Minkowski
inequality may fail. But by replacing
*
f
with
**
f
in (1), we get that

,Lpq is a normed
space, with the functional ,
ꞏpq defined by


1
1**
0
,1
**
0
,0 ,
sup , 0 , .
qq
q
p
pq
p
t
qtftdt pq
p
f
tf t p q

 





(2)
This functional defined in (2) is sub-additive and
equivalent to (1) if 1p
 and1 q, that is
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,, ,
1
pq pq pq
p
f
ff
p


.
Here the (left) first inequality is coming from the
fact that
 
***
ff and the second (right) is an
immediate consequence of Hardy inequality. For
detailed knowledge of Lorentz spaces, we can refer
to [3], [4], [5], [13], [14], and references therein.
2 Main Results
In [14], the grand Lorentz space ,)
p
q
L
is defined as
the collection of all the complex-valued, measurable
functions which are defined on
0,1 such that the
quasi-norm *
,)
pq
f where


1
1
0
1
01
*
,) 1
*
01
*
*
sup ,,1
sup , 1 , (3)
qq
q
p
q
pq
p
t
qtft d pqt
p
f
tf t p q










for any ,)
p
q
f
L
.
Using the maximal function
** ꞏf, instead of
the nonnegative rearrangement

*
f used in the
definition of grand Lorentz space defined in [14],
we generalized the grand Lorentz spaces as follows.
Definition 1 The grand Lorentz space ,)
p
q
L
is
classes of all the complex-valued, measurable
functions which are defined on the measure space
0,
X
such that ,)pq
f
 where




1
1
0
01
**
,) 1
**
0
sup ,1 ,
sup , 1 , (4)
qq
q
p
q
X
pq
p
tX
qttq
ppft d
f
tf t p q









for any ,)
p
q
f
L
. In particular, if 1p
and
1q
; 1pq
or pq
, then the normed
space
,)
,)
,
pq
pq
L
is a Banach space.
Proposition 1 For any ,)
p
q
f
L
, the inequality
,* ,*
,) ,) ,)pq pq pq
p
ff f
pq


exists, i.e. the quasi-norm ,*
,)
p
q
f
and ,)
p
q
f
are
equivalent.
Proof. Since
***
ff

, for any ,)
p
q
f
L
, we
have




1
1*
,) 0
01
1
1**
0
01
,)
,*
.
sup
sup
qq
q
p
pq q
qq
X
Xq
p
q
pq
q
ftftdt
p
qtft dt
p
f













On the other side, if one uses, [3], then






1
1**
,) 0
01
1
1
0
01
1
1
0
01
*
0
0
*
1
0
1
sup
sup
sup
sup
qq
q
p
pq q
q
qq
p
q
Xq
q
p
X
t
X
q
q
q
t
q
q
ft
fsd
tft d
p
qtdt
p
qtdt
p
qpqp
ppqqp
s
t
fsds





























1
1
0
*
q
qq
q
qp
Xsf s dss




01
1
1
0
,)
*
,*
sup
q
qq
qp
X
pq
pq
pq q p
pfs
qds
f
s
p
pq









can be found.
Definition 2 Let X be a measure space and
M
be the cone of M. A mapping
:0,
M is
named as Banach function norm if, for all
,, , 1,2,3,...,
n
uvu n in
M, for all constants
0
, and for all
measurable subsets U of X,
the following properties hold:
(P1)
0u
(P2)
00uu
a.e. in X
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(P3)

uu

(P4)
uv u v


(P5) if vu a.e. in X, then

vu

(P6) if 0n
uu
in X, then

n
uu

(P7) if
U
, then

U


(P8) if
U
, then
U
U
vd C v

for some constant U
C depending on Uand
but
independent of v, [2].
Lemma 1 If ,0xy, 1r and

0,1
are real
numbers, then

1
11
rr
rr r
x
yx y

 .
The equality holds if and only if 1
x
y
.
Proof. If 1r, then

r
x
x
is strictly convex.
Therefore ((1)) (1)
rr r
aba b

  . Setting
x
a
and

1yb
 , we get the result
immediately.
Theorem 1 If 1p and 1q; 1pq
or
pq
, then the normed space
,)
,)
,
pq
pq
L
is a
rearrangement-invariant Banach function space.
Proof. The first three (P1-P3) properties of being
Banach function norm come from the identical
properties true for Lorentz spaces.
Proof of P4.
For any ,)
,
p
q
f
L
g
, we have



 
1
1**
0
01
1
1** **
0
01
,) sup
sup
Xq
qq
p
q
qq
p
Xq
q
pq
qtfgtdt
p
qtftgt dt
p
fg















by the property of maximal function, [3]. If we use
Lemma 1, then we get

 





,)
1
1** **
0
01
11**
0
01
1
1**
sup
sup
(1 )
pq
qq
q
p
q
qq
q
p
q
qq
X
X
q
fg
qtftgt dt
p
qtft
gdt
p
t











because 1q
. Finally with 11
q
,
,)
p
q
fg














11**
0
01
1
1**
11**
0
01
1
11**
0
11**
0
01
sup
(1 )
sup
(1 )
sup
qq
q
p
q
qq
q
qq
q
p
q
qq
q
q
p
q
q
X
X
X
q
Xq
p
qtft
p
gt
qtft
p
qtgt
p
q
p
dt
ttft
dt
dt
d



























1
1
11**
0
1
11**
0
01
1
11**
0
(1 )
sup
(1 )
q
qq
q
q
p
qq q
q
qp
p
X
X
qq q
q
qX
q
d
qtgt
p
qtft
t
t
t
p
q
d
dtgt
p





















can be written. If we take the supremum of
11
,(1 )
qq
qq



over 01q
, then we get
that
,) ,) ,)
p
qpqpq
fg f g

 .
Proof of P5.
Let
g
f
a.e. inX. By [3], it is known that
**
g
tft if
g
f
. Then
  
** * * **
00
11
tt
g
t g sds f sds f t
tt


and so the result comes from the definition of the
norm in (4).
Proof of P6.
Let 0n
f
in
X. Then

**
n
f
tft for
1, 2, 3,n
by [3]. At the same time, if
liminf
nn
uuxx
for all
x
X
a.e., then
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liminf n
uu
n
DD

for any 0
, where
nn
u is a sequence of measurable functions in
M and so
 
**
n
f
tft for 0t. Since
 
** **
n
f
tft, we get
 
** **
n
f
tft for all
0t by Monotone Convergence Theorem.
Therefore








,) ,)
1
1**
0
01
1
1**
0
01
1
1**
0
01
,)
sup
sup
s
sup
sup
sup
.
up
nn
pq pq
qq
q
pn
q
qq
q
pn
q
qq
q
p
n
X
n
q
X
X
p
n
q
ff
qtftdt
p
qtft dt
p
qtftdt
p
f

















Proof of P7.
Let E be a measurable subset of X with
.E
 Then

*
0,
EE
tt

,

** min 1,
E
E
tt



and so







1
1**
,) 0
01
1**
0
01
1
1**
sup
sup
qq
q
pE
pq q
q
X
E
E
X
q
pE
q
qq
q
pE
E
qttdt
p
qttdt
p
qttdt
p























 



1
11
0
01
1
1
01
1
1
01
0
sup
sup
sup
su1p
1
q
qqq
pp
q
q
qq
qp
p
q
qq
qq
q
EX
E
E
p
p
q
X
E
q
X
p
E
qq
tdt t dt
ppt
E
qp q tdt
pq p t
qEtt
E
dqE
q
p
E































 

  
1
1
11
1
1
11
q
q
qq
q
qq
pp
q
p
qq
pp
p
XE
q
q
E
qE
p
XE
E
pqqpp
p




 









can be found if q
. Otherwise


 


111
1
,) 00
**
1
sup sup sup
2
ppp
ptX tE EtX
p
EE
tt t tE
E


  

when q
. As a result, ,)
E
p
q
is finite.
Proof of P8.
Let E
be a measurable subset of X with
.E
By using Hardy-Littlewood inequality,
one can write that
    


 



**
0
**
00
**
0
0,
.
EE
X
E
E
E
E
f
dfttdtfttdt
f t dt f t dt
ftt
t
d





Now fix 01q
. If one uses Hölder’s
inequality in the preceding inequality, then
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.41
Ilker Eryilmaz, Gokhan Isik
E-ISSN: 2224-2880
352
Volume 22, 2023

 














** **
00
1
**
0
1
1
0
1
)
*
1
,
*
0
1
0
1
qp pq
pq pq
E
qp q
q
p
q
pq q q
pq q
Xqp q
q
p
q
p
EE
q
E
q
q
E
E
p
q
p
f d f t dt t f t t dt
tftdt
tdt
pq tftdt
qp
tdt
pq f
q
pq































 
0
,)
1
,
1
)
1
1
1
sup
1
q
pq p q q
p
p
qp p
q
p
qp q
p
pq
qp
q
p
pq
q
pp
pq p q
pq f
q
pq p p
pq p q
pf
E
q
q
E
E



















is found.
References:
[1] G. Anatriello, Iterated grand and small
Lebesgue spaces, Collect. Math. 65, 2014,
273-284.
[2] C. Bennett, R.C. Sharpley. Interpolation of
operators. Academic Press, 1988.
[3] R.E. Castillo, H. Rafeiro, An introductory
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Mathematics/Ouvrages de Mathematiques de
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[4] C. Duyar and A.T. Gürkanlı, Multipliers and
Relative completion in weighted Lorentz
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[5] İ. Eryılmaz, G. Işık, Some Basic Properties of
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[6] A. Fiorenza, Duality and reflexivity in grand
Lebesgue spaces, Collect. Math., 51(2), 2000,
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[7] A. Fiorenza, G.E. Karadzhov, Grand and
small Lebesgue spaces and their analogs, Z.
Anal. Anwendungen, 23(4), 2008, 657--681.
[8] L. Greco, T. Iwaniec and C. Sbordone,
Inverting the p-harmonic operator,
Manuscripta Mathematica, 92, 1997, no. 1,
249-258.
[9] A.T. Gürkanlı, Multipliers of grand and small
Lebesgue
spaces,https://arxiv.org/abs/1903.06743v1.
[10] T. Iwaniec, C. Sbordone, On the integrability
of the Jacobian under minimal hypotheses,
Arch. Rational Mech. Anal., 119(2), 1992,
129-143.
[11] A. Meskhi, Weighted criteria for the Hardy
transform under the p
B
condition in grand
Lebesgue spaces and some applications, J.
Math. Sci., 178(6), 2011, 622-636.
[12] A. Meskhi, Criteria for the boundedness of
potential operators in grand Lebesgue spaces,
Proc. A. Razmadze Math. Inst. 169, 2015,
119-132.
[13] O. Oğur, Grand Lorentz sequence space and
its multiplication operator, Communications
Faculty of Sciences University of Ankara
Series A1 Mathematics and Statistics, 69 (1),
2020, 771-781.
[14] J. Pankaj, S. Kumari, On grand Lorentz
spaces and the maximal operator, Georgian
Math. J., 19(2), 2012, 235-246.
[15] S.G. Samko, S.M. Umarkhadzhiev, On
Iwaniec-Sbordone spaces on sets which may
have infinite measure, Azerb. J. Math., 1(1),
2011, 67-84.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.41
Ilker Eryilmaz, Gokhan Isik
E-ISSN: 2224-2880
353
Volume 22, 2023