





































 

Suppose that α ∈R and 0 <α <n. The fractional integral operator or potential Riesz Iα is 

   

  for every x ∈ Rn. Size µ which satisfies the condition of growth, ie there are c> 0 and 0 

<n ≤ d so that µ (B (x, r)) ≤ crn (2)for each ball centered on x 

µ) is called a non-homogeneous space. In nonhomogeneous space, fractional integr

are defined 

as with

for 0 <α <n ≤ d and x ∈ 

sizes then they are obtained.

 
  

 

 
 

 

I. Introduction 

R and 0 <α <n. The fractional integral operator or potential Riesz Iα is 

) 

−α dy (1) n 

Rn. Size µ which satisfies the condition of growth, ie there are c> 0 and 0 

≤ d so that µ (B (x, r)) ≤ crn (2)for each ball centered on x ∈Rd and has radius r> 0 then (Rd, 

homogeneous space. In nonhomogeneous space, fractional integr

 

 Rd. It can be seen that if n = d and µ are Lebesgue 

sizes then they are obtained. 

 

 

 
 

BOUNDEDNESS OF GENERALLY FRACTIONAL INTEGRAL
OPERATOR ON GENERAL MORREY SPACE

Lina Nurhayati1, Hendra Gunawan2, Iwan Gunawan3,
Haryono Edi Hermawan4 ,

Universitas Sangga Buana1, Istitut Teknologi Bandung2, Universitas Langlang
Buana3

ABSTRACT. In this study  I  will discuss the limits  of fractional integral operators in the 
homogeneous  and  nonhomogeneous  Lebesgue  space,  the  Morrey  space  and  the  general 
Morrey  space.  In  particular,  in  this  study  it  will  be  proven  that  the  fractional  integral 
boundaries formulated in the Morrey space are generally not homogeneous. Evidence of 
integral  fractional  boundaries  formulated  in  the  Morrey  space  is  generally  not 
homogeneous  using  the  specified  maximum  operator  properties  in  space  and  using 
Hedberg's  inequality.  This  evidence  is  an  extension  of  Hardy-Littlewood-Sobolev's 
inequality  [11,  22].  My  research  related  to  BOUNDEDNESS  OF  GENERALLY 
FRACTIONAL  INTEGRAL  OPERATOR  ON  GENERAL  MORREY  SPACE  as  a 
scientific  work  that  must  be  published  in  an  international  journal,  as  for  the  results  I 
present in this journal, is the result of research

R and 0 <α <n. The fractional integral operator or potential Riesz Iα is  

Rn. Size µ which satisfies the condition of growth, ie there are c> 0 and 0 

Rd and has radius r> 0 then (Rd, 

homogeneous space. In nonhomogeneous space, fractional integral operators 

Rd. It can be seen that if n = d and µ are Lebesgue 

IJO- International Journal of Mathematics  (ISSN: 2805-413X)

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In [5], it is proven that, if 1 and, for 0 <α <n

 Then it is limited from Lebesgue non

in the wider space of the limited Lebesgue space from the non

space which is generally Lp, φ (µ) to Lq, ψ (µ). For any function f measured

borel size on Rd that satisfies the condition of growth (2), for 1 

(0, ∞) Morrey space is generally Lp, 

|| f || Lp, φ (µ) <∞} with 

for 0 <α <n ≤ d and x ∈ Rd. It can be seen that if n = d and µ are Lebesgue sizes then they are 

obtained.In [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from Lebesgue non

homogeneous space Lp (µ) to Lq (µ). Furthermore, in the wider space of the limited Lebesg

space from the non-homogeneous Morrey space which is generally Lp, φ (µ) to Lq, ψ (µ). For 

any function f measured-µ with µ borel size on Rd that satisfies the condition of growth (2), for 1 

≤ p <∞ and φ: (0, ∞) → (0, ∞) Morrey space is generally Lp, φ 

{f ∈ Lploc (µ): || f || Lp, φ (µ) <∞} with

With the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the 

following two conditions, 

1. The function φ (r) is almost down, namely there is a cons

applies φ (r) ≥ Cφ (s). 

2. The function rαφ (r) p almost rises, that is, there is a constant C> 0 such that for each r≤s 

applies rαφ (r) p ≤ Csαφ (s) p.

Because both of these requirements must be fulfilled by the function

doubling condition, namely there is a constant C> 0 such that if

Proposition 1 and Lemma 2 below. Proposition 1. Suppose that ω is a non

f is neutralized locally at Rd, for

Z Z 

|Mµf(x)|pω(x)dµ(x) ≤ c |f(

Rd Rd

The above inequality is called the Fefferman

page 29. 

Lemma 2. If the function φ: (0, ∞) → (0, ∞) satisfies the

In [5], it is proven that, if 1 and, for 0 <α <n 

hen it is limited from Lebesgue non-homogeneous space Lp (µ) to Lq (µ). Furthermore, 

in the wider space of the limited Lebesgue space from the non-homogeneous Morrey 

space which is generally Lp, φ (µ) to Lq, ψ (µ). For any function f measured

l size on Rd that satisfies the condition of growth (2), for 1 ≤ p <∞ and 

∞) Morrey space is generally Lp, φ (µ ) = Lp, φ (Rd, µ) is Lp, φ (µ) = {f 

 

. 

Rd. It can be seen that if n = d and µ are Lebesgue sizes then they are 

obtained.In [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from Lebesgue non

homogeneous space Lp (µ) to Lq (µ). Furthermore, in the wider space of the limited Lebesg

homogeneous Morrey space which is generally Lp, φ (µ) to Lq, ψ (µ). For 

µ with µ borel size on Rd that satisfies the condition of growth (2), for 1 

φ: (0, ∞) → (0, ∞) Morrey space is generally Lp, φ (µ ) = Lp, φ (Rd, µ) is Lp, φ (µ) = 

Lploc (µ): || f || Lp, φ (µ) <∞} with 

With the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the 

 

1. The function φ (r) is almost down, namely there is a constant C> 0 such that for each r r s 

2. The function rαφ (r) p almost rises, that is, there is a constant C> 0 such that for each r≤s 

applies rαφ (r) p ≤ Csαφ (s) p. 

Because both of these requirements must be fulfilled by the function φ this function fulfills the 

doubling condition, namely there is a constant C> 0 such that if  2 then, for each r, s> 0. Note 

Proposition 1 and Lemma 2 below. Proposition 1. Suppose that ω is a non-negative function and 

f is neutralized locally at Rd, for 1 <p <∞, then there is c> 0 so that 

 

(x)|pMµω(x)dµ(x). (3) 

d 

The above inequality is called the Fefferman-Stein inequality and the proof can be seen in [22] 

Lemma 2. If the function φ: (0, ∞) → (0, ∞) satisfies the doubling condition then

 

homogeneous space Lp (µ) to Lq (µ). Furthermore, 

homogeneous Morrey 

space which is generally Lp, φ (µ) to Lq, ψ (µ). For any function f measured-µ with µ 

≤ p <∞ and φ: (0, ∞) → 

φ (µ ) = Lp, φ (Rd, µ) is Lp, φ (µ) = {f ∈ Lploc (µ): 

Rd. It can be seen that if n = d and µ are Lebesgue sizes then they are 

obtained.In [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from Lebesgue non-

homogeneous space Lp (µ) to Lq (µ). Furthermore, in the wider space of the limited Lebesgue 

homogeneous Morrey space which is generally Lp, φ (µ) to Lq, ψ (µ). For 

µ with µ borel size on Rd that satisfies the condition of growth (2), for 1 

(µ ) = Lp, φ (Rd, µ) is Lp, φ (µ) = 

With the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the 

tant C> 0 such that for each r r s 

2. The function rαφ (r) p almost rises, that is, there is a constant C> 0 such that for each r≤s 

φ this function fulfills the 

2 then, for each r, s> 0. Note 

negative function and 

Stein inequality and the proof can be seen in [22] 

doubling condition then 

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for every r> 0 and k positive integers.                                                                                      

Based on Proposition 1 and Lemma 2 it can be shown that the maximum operator Mµ is 

as 

for x ∈Rd and f ∈ L1loc (Rd), limited to Lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the 

following theorem. 

Theorem 3. Suppose f is integrally localized at Rd, φ: (0, ∞) → (0, ∞) satisfies doubling 

conditions and for a c1> 0

for every r> 0 and 1 ≤ p <∞, then

for a C> 0. 

Evidence. Take any f ∈ Lp, φ (µ) and B (a, r) are open balls centered on a 

ω = χB (a, r) is a non-negative function. Then according to equality (3) is obtained,

|Mµf(x)|pdµ(x) 

B(a,r) 

Z 

≤ |Mµf(x)|pχB(a,r)dµ

Rd 

Z 

≤ C |f(x)|pMµχB(a,r)dµ

Rd 

" ∞ # 

Z Z 

≤ C |f(x)|pdµ(x) + X |f(

B(a,r) k=1 B(a,2k+1

 

 
for every r> 0 and k positive integers.                                                                                      

Based on Proposition 1 and Lemma 2 it can be shown that the maximum operator Mµ is 

 

L1loc (Rd), limited to Lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the 

Theorem 3. Suppose f is integrally localized at Rd, φ: (0, ∞) → (0, ∞) satisfies doubling 

conditions and for a c1> 0 

 

≤ p <∞, then||Mµf||
Lp,φ(µ) ≤ C ||f||

Lp,φ(µ) (4) 

Lp, φ (µ) and B (a, r) are open balls centered on a ∈Rd and radius r> 0 so 

negative function. Then according to equality (3) is obtained,

 

dµ(x) 

dµ(x) 

(x)|pMµχB(a,r)dµ(x) . 
+1r)−B(a,2kr)(5) 

 

for every r> 0 and k positive integers.                                                                                                          

Based on Proposition 1 and Lemma 2 it can be shown that the maximum operator Mµ is defined 

L1loc (Rd), limited to Lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the 

Theorem 3. Suppose f is integrally localized at Rd, φ: (0, ∞) → (0, ∞) satisfies doubling 

Rd and radius r> 0 so 

negative function. Then according to equality (3) is obtained,Z 

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Next, for

Based on (5) obtained, 

Z 

|Mµf(x)|pdµ(x)B(a,r

 

So, got it 

||f||
Lp,φ(µ)Maximum operator limitation Mµ above is needed in proving the 

fractional integral operators and fractional integral operators commonly from the Morrey 

space are generally Lp, φ to the Morrey space is generally for 1 <p <q <∞ with

5]. 

, we are estimatedMµχB(a,r)(x) as follows

 

a,r) 

There forMµf||
L

Maximum operator limitation Mµ above is needed in proving the boundedness

fractional integral operators and fractional integral operators commonly from the Morrey 

space are generally Lp, φ to the Morrey space is generally for 1 <p <q <∞ with

 

as follows. 

 

Lp,φ(µ) ≤ C 

boundedness of 

fractional integral operators and fractional integral operators commonly from the Morrey 

space are generally Lp, φ to the Morrey space is generally for 1 <p <q <∞ withp ψ = φq [3, 

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Fractional integral operators here are generally fraction

function ρ, which is a non

and satisfies doubling conditions. For 0 <n 

fractional integrals are generally Iρµ in nonhomogene

Lemma 4. Suppose φ: (0, ∞) → (0, ∞) with lim φ (R) = ∞ and lim φ (R) =
R → 0 + R → ∞ 
 
0 and fulfills doubling conditions so for every t 
 

 . (6)

Theorem 5. Suppose φ doubling and fulfilling

1. 

and inequality 

2. 

Where 1 < p < q <∞, then

Evidence. For each x ∈Rd and R> 0, we write
 

 

Note for I1 (x), obtained 

II. DISCUSSION 

Fractional integral operators here are generally fraction (integral) integrals using the 

function ρ, which is a non-negative function, namely ρ: (0, ∞) → (0, ∞) (also φ and ψ) 

and satisfies doubling conditions. For 0 <n ≤ d and the function ρ: (0, ∞) → (0, ∞) 

fractional integrals are generally Iρµ in nonhomogeneous space defined as

. 

Suppose φ: (0, ∞) → (0, ∞) with lim φ (R) = ∞ and lim φ (R) =

0 and fulfills doubling conditions so for every t ∈ R, t> 0 there is R> 0 so that 

(6) 

Theorem 5. Suppose φ doubling and fulfilling 

 

 

then 

||Iρ
µf|| q,φpq ≤ C||f||

Lp,φ(µ). 
L (µ) 

Rd and R> 0, we write 

 

(integral) integrals using the 

negative function, namely ρ: (0, ∞) → (0, ∞) (also φ and ψ) 

ρ: (0, ∞) → (0, ∞) 

ous space defined as 

Suppose φ: (0, ∞) → (0, ∞) with lim φ (R) = ∞ and lim φ (R) = 

 

 

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Next, for I2 (x) is obtained,

By adding I1 and I2, obtained

Next, assuming f 6 = 0, suppose 

As a result, 

Next, for I2 (x) is obtained,

By adding I1 and I2, obtained 

Next, assuming f 6 = 0, suppose 0. Based on (4), 

 . 

 

 

 

 . (7) 

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for every x. Thus obtained,

Z Z 
|Iρµf

B(a,r) B(

So, 

As a result, 

Thus, it is evident that the 
Morrey space which is generally not homogeneous.
 
 

It can be seen that if the function ρ (t) = tα is chosen then for each x, y 
- y |) = | x - y | α, consequently

for every x. Thus obtained, 

 
Iρµf(x)|qdµ(x) ≤ ||f||Lq−p,φp (µ)Mµf(x)pdµ(x

(a,r) 

Z 
≤ C||f||qL−p,φp (µ) Mµf(x)pdµ(x

B(a,r) 

p ≤ C||f||
Lp,φ(µ). 

q 
(µ) 

Thus, it is evident that the generalized integral fractional operator is also bounded in the 
Morrey space which is generally not homogeneous. 

III. CONCLUSION 
 

It can be seen that if the function ρ (t) = tα is chosen then for each x, y ∈
y | α, consequently 

 

 

x) 

x). 

 

generalized integral fractional operator is also bounded in the 

∈Rd applies ρ (| x 

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Thus, the boundedness of the fractional integral operators that are generally formulated in the 

Morrey space are not homogeneous resulting in the boundedness of fractional integral operators 

in the Morrey space which are generally not homogeneous. In ad

in the limitation of the fractional integral operator Iα in the Morrey space. Next, with the 

selection of functions, for each f 

Thus, if then Lp, φ (µ) = Lp, λ (µ). Also, if selected φ (t) = 

whereas if dµ = dx, for Lp, φ (µ) = Lp, λ (Rn) and for Lp, φ ( µ) = Lp (Rn).

 
 

[1] Adams, D. R. dan L. I. Hedberg, (1975), ”A note on Riesz potentials”, Duke Math.
J., 42, 765-778. 

[2] Chiarenza, F dan M. Frasca, (1987), ”Morrey space and Hardy
 function”, Rend. Mat. 7, 273

[3] Eridani, (2002), ”On the boundedness of generalized fractional integral on generalized 
morrey spaces”, Tamkang J. Math. 33, 335

[4] Eridani, H. Gunawan dan E. Nakai, (2004), ”On generalized fractional integral operators”, 
Sci. Math. Jpn. 60, 539

[5] Eridani, H.Gunawan, (2006), ”Fractional integral and generalized olsen inequalities”, 
 ITB Research Grant. No. 0004/ K01.03.2/ PL 2.1.5/ I.

 

Thus, the boundedness of the fractional integral operators that are generally formulated in the 

Morrey space are not homogeneous resulting in the boundedness of fractional integral operators 

in the Morrey space which are generally not homogeneous. In addition, if dµ = dx then it results 

in the limitation of the fractional integral operator Iα in the Morrey space. Next, with the 

selection of functions, for each f ∈ Lp, φ (Rd) is obtained, 

Thus, if then Lp, φ (µ) = Lp, λ (µ). Also, if selected φ (t) = then Lp, φ (µ) = Lp (µ), 

whereas if dµ = dx, for Lp, φ (µ) = Lp, λ (Rn) and for Lp, φ ( µ) = Lp (Rn).

BIBLIOGRAPHY 

Adams, D. R. dan L. I. Hedberg, (1975), ”A note on Riesz potentials”, Duke Math.

Chiarenza, F dan M. Frasca, (1987), ”Morrey space and Hardy- Littlewood maximal 
function”, Rend. Mat. 7, 273-279. 

Eridani, (2002), ”On the boundedness of generalized fractional integral on generalized 
morrey spaces”, Tamkang J. Math. 33, 335-340. 

H. Gunawan dan E. Nakai, (2004), ”On generalized fractional integral operators”, 
Sci. Math. Jpn. 60, 539-550. 
Eridani, H.Gunawan, (2006), ”Fractional integral and generalized olsen inequalities”, 

ITB Research Grant. No. 0004/ K01.03.2/ PL 2.1.5/ I. 

 

Thus, the boundedness of the fractional integral operators that are generally formulated in the 

Morrey space are not homogeneous resulting in the boundedness of fractional integral operators 

dition, if dµ = dx then it results 

in the limitation of the fractional integral operator Iα in the Morrey space. Next, with the 

 

then Lp, φ (µ) = Lp (µ), 

whereas if dµ = dx, for Lp, φ (µ) = Lp, λ (Rn) and for Lp, φ ( µ) = Lp (Rn). 

Adams, D. R. dan L. I. Hedberg, (1975), ”A note on Riesz potentials”, Duke Math. 

Littlewood maximal 

Eridani, (2002), ”On the boundedness of generalized fractional integral on generalized 

H. Gunawan dan E. Nakai, (2004), ”On generalized fractional integral operators”, 

Eridani, H.Gunawan, (2006), ”Fractional integral and generalized olsen inequalities”, 

IJO- International Journal of Mathematics  (ISSN: 2805-413X)

Volume 02 |Issue 07 | July 2019                             www.ijojournals.com 8



 

[6] Garcia-Cuerva, J dan J. M . Martell, (2000), ”Two-weight norm inequalities for maximal 
 operators and fractional integrals on non-homogeneous space”, Departamento de 
 Matematicas, C-XV Universidad Autonoma de Madrid 28049 Madrid, Spain. 

[7] Gunawan, G., (2006), ”Boundness of fractional integral operator in lebesgue space and 
 morrey space”, Penelitian Program Magister, Institut teknologi Bandung. 

[8] Gunawan, H, (2000), ”Generalized fractional integral operators and their modified 
 versions”, Department of Mathematics, Bandung Institute of Technology, 
 Bandung. 

[9] Gunawan, H, (2003), ”A Note on the generalized fractional integral operators”, J. 
 Indonesia. Math. Soc. 9, 39-43. 

[10] Gunawan, H., Y. Sawano dan I. Sihwaningrum, (2009),” Fractional integral operators in 
 non homogeneous spaces”, Bull, austral. math. Soc. 80, 324-334. 

[11] Hardy, G. H dan J.E. Littlewood, (1927), ”Some properties of fractional integral I”,  Math. 
Zeith. 27, 565-606. 

[12] Lib, E. H dan M. Loss, (1997), ”Analysis”, American Mathematical Society. 
[13] Morrey, C. B., (1938), ”On the solutions of quasi-linear elliptic differential equations”, 

 Trans. Amer. Math, Soc. 43, 126-166. 
[14] Nakai, E., (1994), ”Hardy-Littelwood maximal operator, singular integral operators and the 

riesz potentials on generalized morrey space”, Math, nachr. 166, 95-103. 
[15] Nakai, E., (2001), ”On generalized fractional integrals”, Taiwanese J. Math. 5, 587- 

602. 
[16] Nakai, E., (2007), ”Recent topics of fractional integrals”, Sugaku Exposition, 20. 
[17] Nazarov, F, S. Treil dan A. Volberg, (1997), ”Cauchy integral and calderon-zygmund 

 operators on non homogeneous spaces”, Internat. Math. Notices (15),703-726. 
[18] Nazarov, F, S. Treil dan A. Volberg, (1998), ”Weak type estimetas and cotlar inequalities for 

calderon-zygmund operators on non homogeneous space”, Internat,  Math.Res.Notices, 463-
487. 

[19] Nazarov, F, S.Treil dan A. Volberg, (2003), ”The Tb-theorem on non homogeneous  space”, 
Acta Math. 190(2),151-239. 

[20] P. S, Herry, (2008), ”Keterbatasan operator integral fraksional di ruang lebesgue tak 
 homogen”, Universitas Sanata Dharma Yogyakarta. 

[21] Sawano, Y and H. Tanaka, (2006), ”Morrey space for non-doubling measure”, Acta  Math. 
Sinica,1, 153-172. 

[22] Sobolev, S.L., (1938), ”On a theorem in functional analysis ”, Math. sob. 46,471-497. 
[23] Stein, E. M., (1993), ”Harmonic analysis : real variable methods, orthogonality and 

 oscilatory integrals”, Princenton University University Press, Princenton, New jersey. 

IJO- International Journal of Mathematics  (ISSN: 2805-413X)

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