Adv Syst Sci Appl 2021; 02:133–141
Published online at https://ijassa.ipu.ru.

The Modular Inequalities for Hardy-type Operators on
Monotone Functions in Orlicz Space

Kh. Almohammad1*

1 Mathematical Institute named S.M. Nikolskii
Peoples’ Friendship University of Russia (RUDN University)

Moscow 117198, Miklukho-Maklaya str. 6, Russia

Abstract: This paper is devoted to the study of modular inequality for general Hardy–Copson
type operator restricted on the cone of monotone functions from weighted Orlicz space with
general weight.

Keywords: modular inequalities, norm inequalities, Orlicz space, cone of decreasing functions,
positively homogeneous operators

1. INTRODUCTION

In this paper, we consider modular inequalities for Hardy-type operators on the cone Ω of
positive decreasing functions from weighted Orlicz spaces. We use a general theorem (proved
in [10]) on the reduction of modular inequalities for positively homogeneous operators on the
cone Ω, which enables passing to modular inequalities for modified operators on the cone
of all positive functions from Orlicz space. It is based on the duality theorem describing the
associated norm for the cone Ω. We follow, mostly, the notation used in the book [2, Sec. 8,
Chap. 4] of Bennett and Sharpley. In the paper, we concretize modular inequalities for the
case in which the positive operator is a Hardy-type operator. It is shown that, in that case,
the modified operator is a generalized Hardy operator in the Jim Quile Sun notation [1]. This
allows us to use approaches developed in [8–10], as well as results obtained by Jim Quile
Sun [1] to establish the explicit criteria for the validity of modular inequalities.

2. AUXILIARY DEFINITIONS

Definition 2.1:
(i) A Banach function space, shortly BFS, E = E(Rn) is a Banach space of Lebesgue
measurable functions f : Rn → C with monotone norm, i.e. such that

|f | ≤ g, g ∈ E implies f ∈ E, ‖f‖E ≤ ‖g‖E. (2.1)

(ii) A BFS E is called a rearrangement-invariant space, shortly: RIS, if its norm is
monotone with respect to rearrangements,

f ∗ ≤ g∗, g ∈ E implies f ∈ E, ‖f‖E ≤ ‖g‖E. (2.2)

∗Corresponding author: khaleel.almahamad1985@gmail.com



134 KH. ALMOHAMMAD

Here f ∗ is the decreasing rearrangement of the function, i.e. a positive decreasing right
continuous function on R+ = (0,∞), which is equimeasurable with f :

µn
{
x ∈ Rn : |f(x)| > y

}
= µ1

{
t ∈ R+ : |f ∗(t)| > y

}
, y ∈ R+. (2.3)

where µn is n-dimensional Lebesgue measure.
Definition 2.2:
The potential space HG

E (Rn) on the n-dimensional Euclidean space Rn is defined by

HG
E (Rn) =

{
u = G ∗ f : f ∈ E(Rn)

}
, (2.4)

where E(Rn) — is a rearrangement-invariant space (shortly: RIS), and

‖u‖HG
E

= inf
{
‖f‖E : f ∈ E(Rn), G ∗ f = u

}
. (2.5)

Here G is an admissible kernel, that is G ∈ L1(Rn) + E ′(Rn), the convolution G ∗ f is
defined as the integral

(G ∗ f)(x) = (2π)−
n
2

∫
Rn

G(x− y)f(y) dy. (2.6)

Here E ′ = E ′(Rn) is the associated RIS, i.e. RIS with the norm:

‖g‖E′ = sup

{∫
Rn

|fg| dµ : f ∈ E, ‖f‖E ≤ 1

}
. (2.7)

Examples.

E = Lp, 1 ≤ p ≤ ∞⇒ E ′ = Lp′ ;
1

p
+

1

p′
= 1.

L′1 = L∞; L′∞ = L1.

For the RIS E(Rn), E ′(Rn), we consider the spaces Ẽ = Ẽ(R+), Ẽ ′ = Ẽ ′(R+) — their
Luxemburg representations [2], i.e. RIS for which the following equalities are satisfied

‖f‖E = ‖f ∗‖Ẽ, ‖g‖E′ = ‖g∗‖Ẽ′ .
We denote:

f ∗∗(t) =
1

t

t∫
0

f ∗(τ) dτ ; t ∈ R+. (2.8)

We introduce the class of monotone functions In(R), R > 0 as follows. The function
θ : (0, R)→ R+ belongs to the class In(R), if θ satisfies the following conditions:
decreasing and continuous at (0, R);
There is a constant c ∈ R+, such that

r∫
0

θ(ρ)ρn−1 dρ ≤ cθ(r)rn, r ∈ (0, R). (2.9)

Now, we introduce

ϕ(τ) = θ

(( τ
Vn

) 1
n

)
∈ I1(T ), T = VnR

n.

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THE MODULAR INEQUALITIES FOR HARDY-TYPE OPERATORS 135

where Vn is the volume of the unit ball in Rn.

fϕ(t; τ) = min
{
ϕ(t), ϕ(τ)

}
=

{
ϕ(t), t > τ,
ϕ(τ), τ > t.

(2.10)

Definition 2.3:
Let θ ∈ In(∞). The potentials u ∈ HG

E (Rn) are called generalized Riesz potentials, if

G(x) ∼= θ(|x|), x ∈ Rn, (∼= means two-sided estimate).

Definition 2.4:
Let θ ∈ In(R). The potentials u ∈ HG

E (Rn) are called generalized Bessel potentials, if

G(x) = G0
R(x) +G1

R(x);

BR =
{
x ∈ Rn : |x| < R

}
, R ∈ R+,

G1
R(x) = G(x)χBc

R
(x), G0

R(x) = G(x)χBR
(x),

G0
R(x) ∼= θ(|x|), x ∈ BR, G1

R(x) ∈ (L1 ∩ E ′)(Rn),

∫
Rn

Gdx 6= 0.

Definition 2.5:
Function Φ: [0,+∞)→ [0,+∞) is called N -function if

Φ(t) =

t∫
0

φ(τ) dτ ; where φ is continuous, 0 < φ ↑; φ(0) = 0, φ(∞) =∞.

Let φ−1 be the right continuous inverse function of φ, and define

Ψ(t) =

t∫
0

φ−1(τ) dτ.

Ψ is called the complementary function of Φ.
Definition 2.6:
a) AnN -function Φ is said to satisfy the ∆2-condition (we write Φ ∈ ∆2) if there is a constant
B > 0, such that

Φ(2t) ≤ BΦ(t), ∀t > 0. (2.11)
b) We write Φ1 ≺≺ Φ2 if there is a constant L0 > 0, such that inequality∑

i

Φ2 ◦ Φ−1
1 (ai) ≤ L0Φ2 ◦ Φ−1

1 (
∑
i

ai), (2.12)

holds for every sequence {ai} with ai ≥ 0.
c) Let v be a positive, measurable weight function and Φ be an N -function. The Orlicz

space LΦ,v consists of all measurable function f (modulo equivalence almost everywhere)
with

‖f‖Φ,v = inf

{
λ > 0,

∫ ∞
0

Φ
(
λ−1|f(x)|

)
v
(x) dx ≤ 1

}
<∞. (2.13)

We call ‖ · ‖Φ,v the Luxemburg norm.
The Orlicz norm of a function f is given by

‖f‖′Ψ,v = sup

{∫ ∞
0

|fg|v dx :

∫ ∞
0

Ψ(|g|)v dx ≤ 1

}
. (2.14)

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



136 KH. ALMOHAMMAD

Remark 2.1:
LΦ,v is a Banach space and the Luxemburg and Orlicz norms are equivalent . In fact,

‖f‖Φ,v ≤ ‖f‖′Ψ,v ≤ 2‖f‖Φ,v.

We assumeM(R+) is the set of Lebesgue-measurable almost everywhere finite functions,M+

is the cone of almost everywhere positive functions from M = M(R+);

M+ =
{
f ∈M(R+) : f > 0

}
.

Consider the cone of positive decreasing functions from the Orlicz space:

Ω =
{
f ∈ LΦ,v : 0 ≤ f ↓} (2.15)

For g ∈M+, we introduce the following associated norm on the cone Ω:

‖g‖′Ω = sup

{∫ ∞
0

fg dt : f ∈ Ω; ‖f‖Φ,v ≤ 1

}
. (2.16)

We formulate the result that generalize some previous results of papers [3], [5–7].

Proposition 2.1:
([4]). Let Φ, Ψ be the complementary N -functions, the N -function Φ satisfies ∆2-condition,
let v ∈M+, and let

0 < V (t) :=

t∫
0

v dτ <∞, t ∈ R+, V (+∞) = +∞. (2.17)

The following two-sided estimate holds:

‖g‖′Ω ∼= ‖R0(g)‖Ψ,v = inf

{
λ > 0 :

∫ ∞
0

Ψ
(
λ−1|R0(g; t)|

)
v(t) dt ≤ 1

}
, (2.18)

where

R0(g; t) := V (t)−1

t∫
0

g(τ) dτ, t ∈ R+. (2.19)

Here and below we use the notation

A ∼= B ⇔ ∃ c ∈ [1,∞) : c−1 ≤ A/B ≤ c. (2.20)

In the following considerations, we will use the formula for the conjugate operator:

R∗0(f ; τ) =

∞∫
τ

f(t)

V (t)
dt, τ ∈ R+. (2.21)

Let us now state the main result of this section allowing us to reduce modular inequalities
for operators on the cone Ω to modular inequalities for modified operators on the cone M+.

Proposition 2.2:
( [10]). Let T and T ∗ be positively homogeneous operators that map M+ to M+ and are

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



THE MODULAR INEQUALITIES FOR HARDY-TYPE OPERATORS 137

adjoint, i.e., ∫
R+

gTf dτ =

∫
R+

fT ∗g dτ, f, g ∈M+. (2.22)

Let Φ1, Φ2 be N -functions, u, v, w ∈M+, and let condition (2.17) holds. Let the operator R0

be given by formula (2.19). Then the following inequalities are equivalent:

∃ c1 ∈ R+ : Φ−1
2

{∫
R+

Φ2(wTf)u dt

}
≤ Φ−1

1

{∫
R+

Φ1(c1f)v dt

}
, f ∈ Ω; (2.23)

∃ c3 ∈ R+ : Φ−1
2

{∫
R+

Φ2

(
wTR∗0(vf)

)
u dt

}
≤ Φ−1

1

{∫
R+

Φ1(c3f)v dt

}
, f ∈M+.

(2.24)

Definition 2.7:
The generalized Hardy Operators are operators of the form

Kf(x) =

x∫
0

k(x, t)f(t) dt, K∗g(t) =

+∞∫
t

k(x, t)g(x) dx, (2.25)

where

a) k :
{

(x, t) ∈ R2 : 0 < t < x < +∞
}
→ [0,+∞);

b) k(x, t) ≥ 0 is nondecreasing in x, nonincreasing in t;
c) k(x, y) ≤ D

(
k(x, t) + k(t, y)

)
, for some constant D, (2.26)

whenever 0 ≤ y ≤ t < x < +∞
Proposition 2.3:
( [1]). Let Φ1, Φ2 be N -function and Φ1 ≺≺ Φ2, and K be a generalized Hardy
operator (2.25). Let a, b, v and ω be positive weight functions. Then there exists a constant
A > 0 such that

Φ−1
2

( +∞∫
0

Φ2(aKf)ω dx

)
≤ Φ−1

1

( +∞∫
0

Φ1(Afb dx)v

)

for all positive, measurable functions f if and only there exists a constant C such that

Φ−1
2

( +∞∫
r

Φ2

(
a(x)

C

∥∥∥∥k(r; ·)χ(0,r)(·)
εvb

∥∥∥∥
Ψ1(εv)

)
ω(x) dx

)
≤ Φ−1

1

(1

ε

)
and

Φ−1
2

( +∞∫
r

Φ2

(
a(x)

C

∥∥∥∥χ(0,r)(·)
εvb

∥∥∥∥
Ψ1(εv)

k(x; r)

)
ω(x) dx

)
≤ Φ−1

1

(1

ε

)
holds for ε, r > 0.

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



138 KH. ALMOHAMMAD

Proposition 2.4:
( [1]). Let Φ1, Φ2 be N -function and Φ1 ≺≺ Φ2, and K∗ be a generalized Hardy
operator (2.25).
Let a, b, v and ω be positive weight functions. Then there exists a constant A > 0 such that

Φ−1
2

( +∞∫
0

Φ2(aK∗f)ω dt

)
≤ Φ−1

1

( +∞∫
0

Φ1(Abf)v dt

)
holds for all positive, measurable functions f if and only there exists a constant C such that

Φ−1
2

( r∫
0

Φ2

(
a(t)

C

∥∥∥∥k(·; r)χ(r,+∞)(·)
εvb

∥∥∥∥
Ψ1(εv)

)
ω(t) dt

)
≤ Φ−1

1

(1

ε

)
and

Φ−1
2

( r∫
0

Φ2

(
a(t)

C

∥∥∥∥χ(r,+∞)(·)
εvb

∥∥∥∥
Ψ1(εv)

k(r; t)

)
ω(t) dt

)
≤ Φ−1

1

(1

ε

)
holds for ε, r > 0.

3. APPLICATIONS FOR HARDY-TYPE OPERATORS

Let us now state the main result of this section allowing us to reduce modular inequalities for
operators on the cone Ω to modular inequalities for modified operators on the cone M+.

I(f ; t) =

+∞∫
0

fϕ(t; τ)f(τ) dτ, τ ∈ R+, (3.27)

where fϕ(t; τ) = min
{
ϕ(t), ϕ(τ)

}
.

Theorem 3.1:
Let Φ1, Φ2 be N -function and Φ1 ≺≺ Φ2, w, u, v be positive weight functions, I be Hardy-
type operators (3.27). Let the condition be satisfied

Aϕ = sup
t∈R+

1

tϕ(t)

(∫ t

0

ϕ dτ

)
<∞. (3.28)

Then there exists a constant C > 0 such that inequality

Φ−1
2

{∫
R+

Φ2

(
w(t)If

)
u(t) dt

}
≤ Φ−1

1

{∫
R+

Φ1(Cf)v dt

}
, f ∈ Ω, (3.29)

holds for all positive, nonincreasing functions f if and only if there is a constant B such that
the following inequalities hold for all ε, r > 0:

Φ−1
2

{ ∞∫
0

Φ2

(
w(t)

B
· fϕ(t, r)

ϕ(r)

∥∥∥∥fϕ(·, r)(·)
εV

∥∥∥∥
Ψ1(εv)

)
u(t) dt

}
≤ Φ−1

1

(
1

ε

)
. (3.30)

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



THE MODULAR INEQUALITIES FOR HARDY-TYPE OPERATORS 139

Proof
The purpose of the first step is to reduce estimate(2.24) to the estimate for the Hardy-type

operator given in [11]. For the Hardy-type operator (3.27), by using (2.21), we obtain

I
(
R∗0(vf ; t)

)
=

∞∫
0

fϕ(t; τ)R∗0(vf ; τ) dτ =

∞∫
0

fϕ(t; τ)

( ∞∫
τ

f(ξ)v(ξ)

V (ξ)
dξ

)
dτ, τ ∈ R+ .

By changing the order of integration, we have

I
(
R∗0(vf ; t)

)
=

∞∫
0

f(ξ)v(ξ)

V (ξ)

( ξ∫
0

fϕ(t; τ) dτ

)
dξ. (3.31)

Let us show that
ξ∫

0

fϕ(t; τ) dτ ∼= fϕ(t; ξ)ξ; t, ξ ∈ R+ = (0,+∞). (3.32)

From the decrease of ϕ and from the condition Aϕ <∞ it follows that

ϕ(t)t ≤
t∫

0

ϕ dτ ≤ Aϕϕ(t)t, t ∈ R+. (3.33)

1) For ξ ≤ 1 we have fϕ(t; τ) = ϕ(t), τ ∈ (0, ξ), so that

ξ∫
0

fϕ(t; τ) dτ = ϕ(t)

ξ∫
0

dτ = ϕ(t)ξ = fϕ(t; ξ)ξ. (3.34)

2) For ξ > t we have

ξ∫
0

fϕ(t; τ) dτ =

t∫
0

ϕ(t) dτ +

ξ∫
t

ϕ(τ) dτ =

= ϕ(t)t+

ξ∫
t

ϕ(τ) dτ ∼=
(3.34)

t∫
0

ϕ(τ) dτ +

ξ∫
t

ϕ(τ) dτ =

ξ∫
0

ϕ(τ) dτ ∼=
(3.34)

ϕ(ξ)ξ,

that is, for ξ > t
ξ∫

0

fϕ(t; τ) dτ ∼= ϕ(ξ)ξ = fϕ(t, ξ)ξ. (3.35)

From (3.34), (3.35) it follows (3.32).
Substitute (3.32) in (3.31):

I
(
R∗0(vf ; t)

)
=

∞∫
0

f(ξ)v(ξ)

V (ξ)
fϕ(t; ξ)ξ dξ, (3.36)

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



140 KH. ALMOHAMMAD

so

I
(
R∗0(vf ; t)

)
=

∞∫
0

fϕ(t; ξ)g(ξ) dξ, (3.37)

where

g(ξ) =
f(ξ)v(ξ)ξ

V (ξ)
. (3.38)

We obtain the equivalence of (2.24) and (3.39), where (3.39) is of the form

∃ c3 ∈ R+ :

Φ−1
2

{∫
R+

Φ2

(
w(t)

∫ ∞
0

fϕ(t; ξ)g(ξ) dξ

)
u(t) dt

}
≤ Φ−1

1

{∫
R+

Φ1(c3σg)v dt

}
,

g ∈M+ (3.39)

and σ(t) = V (t)v−1(t)t−1. As a result, introducing the operator

I0(g; t) =

∞∫
0

fϕ(t; ξ)g(ξ) dξ, t ∈ R+ (3.40)

where the kernel

fϕ(t; τ) = min
{
ϕ(t), ϕ(τ)

}
=

{
ϕ(t), t > τ,
ϕ(τ), τ > t

(operator I0 in the notations of [11]) we obtain the equivalence of the modular
inequalities (2.24) and (3.39).

2. We now pass to the proof of the equivalence of inequality (3.39) and the set of
conditions (3.29). To this end, we use a known result due to Jim Quile Sun which was given
in our paper [11] combined with the generalizations given in [4]. Denote

Φ−1
2

{∫ r

0

Φ2

(
w(t)

B
· tfϕ(t, r)

ϕ(r)

∥∥∥∥fϕ(·, r)
εV

∥∥∥∥
Ψ1(εv)

)
u(t) dt

}
≤ Φ−1

1

(1

ε

)
. (3.41)

Thus, we have shown that (2.24)⇔ (3.39)⇔ (3.29). Theorem 3.1 is proved.

ACKNOWLEDGEMENTS
The author thanks Professor M.L. Goldman for helpful discussions.

REFERENCES

1. Jim Quile Sun (1995) Hardy type inequalities on weighted Orlicz spaces. Ph.D. Thesis,
The Univ. of Western Ontario, London, Canada, 1995.

2. Bennett, C., & Sharpley, R. (1988) Interpolation of operators. Boston: Acad. Press,
1988. (Pure and Appl. Math.; V. 129).

3. Bloom, S., & Kerman, R. (1994) Weighted Orlicz space integral inequalities for the
Hardy–Littlewood maximal operator. Studia Math. 1994, 110(2), 149–167.

Copyright © 2021 ASSA. Adv Syst Sci Appl (2021)



THE MODULAR INEQUALITIES FOR HARDY-TYPE OPERATORS 141

4. Goldman, M., & Kerman, R. (1998) On the principle of duality in Orlicz-Lorentz spaces,
Function spaces. Differential Operators. Problems of mathematical Education. Proc.
Intern. Conf. dedicated to 75-th birthday of prof. Kudrjavtsev 1998, 1, 179–183.

5. Drabek, P., Heinig, H.P., & Kufner, A. (1997) Weighted modular inequalities for
monotone functions. J. Inequal. and Applications 1997, 1, 183–197.

6. Sawyer, E. (1990) Boundedness of classical operators on classical Lorentz spaces.
Studia Math., 1990, 96, 145–158.

7. Heinig, H.P., & Kufner, A. (1996) Hardy operators on monotone functions and
sequences in Orlicz spaces. J. London Math. Soc. 1996, 53(2), 256–270.

8. Goldman, M. (2016) Estimates for Restrictions of Monotone Operators on the Cone of
Decreasing Functions in Orlicz Spaces. Mathematical Notes 2016, 100(1), 24–37.

9. Goldman, M. (2016) Estimates for the norms of monotone operators on weighted
Orlicz-Lorentz classes. Doklady Mathematics 2016, 94(3), 627–631.

10. Bakhtigareeva, E., & Goldman, M. (2017) Inequalities for Hardy-Type Operators on the
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11. Almohammad, Kh. On Modular Inequalities for Generalized Hardy Operators on
Weighted Orlicz Spaces. Eurasian Mathem. Journal, (to appear).

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	Introduction
	Auxiliary definitions
	Applications for Hardy-type operators