J.
Indones.
Math.
Soc.
Vol.
No.
, pp.
1Ae7.
Pointwise Multipliers of Orlicz-Morrey Spaces Ifronika1O .
Denny Ivanal Hakim1 , and Wono Setya Budhi1
1,2,3
Department of Mathematics.
Institut Teknologi Bandung.
Indonesia ifronika@itb.
id, 2 dhakim@itb.
id, 3 wonosb@itb.
Abstract.
We investigate the space of pointwise multipliers of Orlicz-Morrey Using the HoOlder inequality in Orlicz-Morrey spaces, we prove that the space of pointwise multipliers of Orlicz-Morrey spaces contains an Orlicz-Morrey We also prove a partial reverse inclusion of this result.
In addition, we describe the space of pointwise multipliers of Orlicz-Morrey spaces by adding some growth conditions on Young functions.
Our results can be viewed as an extension of the results on pointwise multipliers of Morrey spaces.
Key words and phrases:
pointwise multipliers.
Morrey spaces.
Orlicz spaces.
Orlicz-Morrey spaces.
INTRODUCTION Suppose that we have two function spaces.
One way to compare these spaces is to define a simple transformation from one into another function space that is bounded with respect to the norms of the function spaces.
The transformation is the pointwise multiplier of a function from one function space to another, called a multiplier operator.
The multiplier function lies in the third Lebesgue space in the multiplier transformation in two Lebesgue spaces.
Explicitly, consider the multiplier operator T : Lp (Rn ) Ie Lq (Rn ) with p > q, and Tg .
) = gf , then g OO Lr (Rn ) with = .
In the case p = q, then r = O.
For the case p < q, we have g O 0, when the measure in Rn is a Lebesgue measure.
But this is not the case with the multiplier operator for two Morrey spaces.
This set of multiplier functions can only happen to certain Morrey spaces, either containing some Morrey spaces or the set is contained in a Morrey space.
See .
, 2, .
for the details.
The positive results also happen for multiplier operators in two different Orlicz spaces .
The set of multiplier functions is another Orlicz space with the O Corresponding author 2020 Mathematics Subject Classification: 46E30.
Received: 05-03-2025, accepted: 13-06-2025.
Young function that can be derived from two given Young functions with a similar relation to Lebesgue spaces.
We refer the reader to .
for the characterization of pointwise multiplier operators in weak Orlicz spaces and weak Morrey spaces, respectively.
In this paper, we generalize this multiplier transformation to some OrliczMorrey spaces .
ee Definition 2.
At this time, the results obtained are limited because the relation of the parameter in Lebesgue spaces and Morrey spaces cannot be directly extended in terms of Young functions.
Nevertheless, we get several results about spaces of pointwise multipliers of Orlicz-Morrey spaces.
Our first result is the fact that the space of pointwise multipliers from an Orlicz-Morrey space to another Orlicz-Morrey space contains the third Orlicz-Morrey spaces where their parameters are related by the usual assumption of the HoOlder inequality in these spaces .
and references therei.
The partial reverse inclusion of this inclusion is our second result.
Moreover, we prove a characterization of spaces of pointwise multipliers of Orlicz-Morrey spaces in terms of Orlicz-Morrey spaces by assuming additional growth conditions on Young functions .
ee Definition 2.
The remaining sections of this manuscript are organized as follows.
We recall the definition of the space of pointwise multipliers and several facts about OrliczMorrey spaces in Section 2.
Our main results and their proofs will be given in Section 3.
Throughout this paper, the notation A O B means the inequality A O CB holds for some C > 0, independent of A > 0 and B > 0.
Write A O B if the inequalities A O B and B O A hold.
PRELIMINARIES
This section recalls some definitions, notations, and basic facts.
Let X and Y be Banach spaces of measurable functions on Rn .
The space of pointwise multipliers from X to Y is defined by PWM(X.
Y ) := .
: f A g OO Y for all f OO X}.
For every g OO PWM(X.
Y ), we define OugOuPWM(X,Y ) := sup Ouf A gOuY : f OO X, f = 0 .
Ouf OuX Let us recall the definition of Orlicz spaces.
The function : .
O) Ie .
O) is called a Young function if is continuous, convex, and increasing with .
= 0.
Assume that is not identically zero.
The Orlicz space L is defined to be the set of all measurable functions f for which .
| dx < O, for > 0.
The norm in this space is defined by .
| Ouf OuL := inf > 0 :
dx O 1 .
If .
:= tp with 1 O p < O, then L = Lp .
The characterization of the spaces of pointwise multipliers of Orlicz spaces can be found in .
The result of the characterization of spaces of pointwise multipliers on Musielak-Orlicz spaces can be seen in .
Related results in the setting of Musielak-Orlicz-Morrey spaces can be found in .
As mentioned in the first section, we will investigate the pointwise multipliers in Orlicz-Morrey spaces.
These spaces are introduced in .
and can act as a generalization of Morrey spaces.
We recall the definition of these spaces below.
Definition 2.
Let B.
, .
be any ball centered at x OO Rn and r > 0.
Define .
| Ouf Ou.
:= inf > 0 :
dy O 1 .
|B.
, .
| B.
Let I : .
O) Ie .
O) be an increasing function and suppose that the function t 7Ie tOen I.
is decreasing.
The Orlicz-Morrey space MI is the space of all measurable functions f such that Ouf OuMI := xOORn ,r>0 I.
Ouf Ou.
< O.
If .
:= tq and I.
:= tOe p where 1 O q O p < O, then MI is equal to the Morrey space ! q1 Mpq = f : sup |B.
, .
| p Oe q .
dy <O .
xOORn ,r>0 B.
Remark that there are two other variants of Orlicz-Morrey spaces .
, 12, 13, .
In this paper, we only consider Definition 2.
We also assume the following growth conditions in describing spaces of pointwise multipliers of Orlicz-Morrey spaces in Theorem 3.
Definition 2.
The Young function is said to satisfy the OIA -condition if .
O .
for every s, t Ou 0.
If there exists a constant C > 0 such that inequality .
O C.
holds for every s, t Ou 0, then we call satisfies the ONA -condition.
MAIN RESULTS
Our first result is that the space of a pointwise multiplier from an OrliczMorrey space to another Orlicz-Morrey space contains certain Orlicz-Morrey spaces.
Theorem 3.
Let 0 and 1 be Young functions.
Let 2 be a Young function satisfying the inequality 1 .
O 0 .
for every s, t Ou 0.
Suppose that I0 and I1 are positive and increasing functions on .
O) such that tOen I0 .
and tOen I1 .
are decreasing functions.
If I2 is a function that satisfies I1 .
O I0 .
I2 .
for every r > 0, then I0 2 OI PWM(M0 .
M1 ).
Proof.
Let f OO MI 0 and g OO M2 .
We will show that Tg : f 7Ie g A f is a bounded operator from M0 to MI 1 .
We show that the inequality Ouf A gOu1 .
O 2Ouf Ou0 .
OugOu2 .
holds for every x OO Rn and r > 0.
Note that, for every A > 0 we have .
| 0 dy O 1 |B.
, .
| B.
Ouf Ou0 ,B.
A .
| dy O 1.
|B.
, .
| B.
OugOu2 ,B.
A By the relations of 0 , 1 , and 2 , we get .
| dy O 1.
|B.
, .
| B.
2(Ouf Ou0 ,B.
A)(OugOu2 ,B.
A) Therefore, we obtain the inequality .
Finally, we have that the multiplier operator is bounded from MI00 into MI g OO PWM(MI 0 .
M1 ).
n We now prove a partial reverse inclusion of Theorem 3.
1, namely, the space of pointwise multipliers of Orlicz-Morrey spaces can be realized as a subset of OrliczMorrey spaces.
Theorem 3.
Let 0 and 1 be Young functions.
Assume that I0 and I1 are positive, increasing on .
O), and also the functions tOen I0 .
and tOen I1 .
are If I2 satisfies I1 .
O I0 .
I2 .
for every r > 0, then I1
PWM(MI
M1 ) OI M1 .
Proof.
Let g OO PWM(MI 0 .
M1 ), x OO R , and r > 0.
Since NB.
OO M0 , we OugNB.
OuMI1 I1 .
I2 .
OugOu1 .
O OugOu1 .
I0 .
I0 .
I1 Since g OO PWM(MI 0 .
M1 ) and OuNB.
OuM0 O I0 .
, we see that OugNB.
OuMI1 O OugOuPWM(MI0 ,MI1 ) I0 .
Combining the inequalities .
, we obtain I2 .
OugOu1 .
O OugOuPWM(MI0 ,MI1 ) .
Therefore, g OO MI 2 with OugOuMI2 O OugOuPWM(MI0 ,MI1 ) .
This completes the proof.
n Under some additional growth assumptions on Young functions, we give the following characterization of the space of pointwise multipliers of Orlicz-Morrey Theorem 3.
Let 0 and 1 be Young functions.
Assume that 0 and 1 satisfy OIA and ONA conditions.
Define I2 and 2 by I2 .
:= I1 .
I0 .
Oe1 2 .
:= Oe1 1 .
Oe1 0 .
0 Oe1 2 (I0 .
) is increasing.
If I0 .
I1 .
I2 , 0 , 1 , and 2 also satisI2 .
fy the condition in Theorem 3.
1, then Assume that I1
PWM(MI
M1 ) = M2 .
Proof of Theorem 3.
The inclusion MI 2 OI PWM(M0 .
M1 ) is a consequence I1 of Theorem 3.
To prove the reverse inclusion, let g OO PWM(MI 0 .
M1 ), we will I2 show that g OO M2 .
For every L > 0, define gL := gNB.
,L)O{.
OL} .
Since gL OO LO c (R ) and Lc (R ) OI M2 (R ), we see that gL OO M2 (R ).
Observe Oe1 1 (Oe1 2 (Oug A .
0 (.
L |)NB.
)OuM1 )) OugL Ou2 .
O 1 (Oe1 (I .
)) I0 The assumption g OO PWM(M .
MI 1 ) implies Oe1 Oug A .
Oe1 0 (.
L |)NB.
)OuM1 O OugOuPWM(M0 ,M1 ) Ou2 0 (.
L |)NB.
OuM0 .
Using the OIA -condition of 0 , we get Oe1 Ou2 Oe1 0 (.
L |)NB.
OuM0 O .
0 )(OugL NB.
OuMO ), 2 where O := Oe1 2 0 I0 .
We calculate the right-hand side of the last inequality to obtain OugL NB.
OuMO O OugL OuMI2 I2 .
Combining the previous estimates, we have Oe1 I2 .
OugL Ou2 .
O .
Oe1 2 )(OugOuPWM(M0 ,M1 ) ).
0 )(OugL OuM2 ).
Taking supremum over all x OO Rn and r > 0 and then using the ONA -condition of 1 , we get Oe1 OugL OuMI2 O 1 Oe1 2 (OugOuPWM(M0 ,M1 ) )0 (OugOuM2 ) .
Combining the last inequality with the definition of 2 , we obtain OugL OuMI2 O OugOuPWM(MI0 ,MI1 ) .
Thus, g OO MI 2 follows from the last inequality and FatouAos lemma.
that PWM(M0 .
MI 1 ) OI M2 .
This shows n CONCLUDING REMARKS In this paper, we obtain three results about the space of pointwise multipliers between two Orlicz-Morrey spaces.
Our first result is these spaces contain an Orlicz-Morrey space.
The second result is a partial reverse inclusion of the first result, namely the space of pointwise multipliers of Orlicz-Morrey spaces, which can be recognized as a subset of an Orlicz-Morrey space.
The last result is a characterization of pointwise multipliers of two Orlicz-Morrey spaces as a third Orlicz-Morrey space under some additional growth conditions of Young functions.
The complete description of pointwise multipliers between Orlicz-Morrey spaces with weaker assumptions can be further investigated in future research.
Acknowledgement.
This research is supported by PPMI-ITB 2021 Program.
REFERENCES