On the Boundedness of a Generalized Fractional Integral on Generalized Morrey Spaces

J. Math. and Its Appl.
ISSN: 1829-605X
Vol. 3, No. 1, May 2006, 11–17

On the Boundedness of a Generalized Fractional
Integral on Generalized Morrey Spaces
Eridani
Department of Mathematics
Airlangga University, Surabaya
dani@unair.ac.id

Abstract
In this paper we extend Nakai’s result on the boundedness of a generalized fractional integral operator from a generalized Morrey space to another
generalized Morrey or Campanato space.
keywords: Generalized fractional integrals, generalized Morrey spaces, generalized Campanato spaces

1. Introduction and Main results
For a given function ρ : (0, ∞) −→ (0, ∞), let Tρ be the generalized fractional
integral operator, given by
Tρ f (x) =

Z

Rn

f (y)ρ(|x − y|)
dy,
n
|x − y|

and put
Teρ f (x) =

Z

f (y)
Rn

µ

ρ(|x − y|) ρ(|y|)(1 − χBo (y))

n −
n
|x − y|
|y|
11

¶

dy,

12

On the Boundedness of a Generalized Fractional ...

the modified version of Tρ , where Bo is the unit ball about the origin, and χBo is
the characteristic function of Bo .
In [4], Nakai proved the boundedness of the operators Teρ and Tρ from a generalized Morrey space M1,φ to another generalized Morrey space M1,ψ or generalized
Campanato space L1,ψ . More precisely, he proved that
kTρ f kM1,ψ ≤ Ckf kM1,φ

and

kTeρ f kL1,ψ ≤ Ckf kM1,φ ,

where C > 0, with some appropriate conditions on ρ, φ and ψ. Using the techniques
developed by Kurata et.al. [1], we investigate the boundedness of these operators
from generalized Morrey spaces Mp,φ to generalized Morrey spaces Mp,ψ or generalized Campanato spaces Lp,ψ for 1 < p < ∞.
The generalized Morrey and Campanato spaces are defined as follows. For a
given function φ : (0, ∞) −→ (0, ∞), and 1 < p < ∞, let
µ
¶ p1
Z
1
1
p
kf kMp,φ = sup
|f (y)| dy ,
B φ(B) |B| B
and
kf kLp,φ

¶ p1
µ
Z
1
1
p
= sup
|f (y) − fB | dy ,
B φ(B) |B| B

where the supremum is taken over all open balls B = B(a, r) in Rn , |B| is the
R
1
f (y)dy. We define
Lebesgue measure of B in Rn , φ(B) = φ(r), and fB = |B|
B
the Morrey space Mp,φ by
Mp,φ = {f ∈ Lploc (Rn ) : kf kMp,φ < ∞},
and the Campanato space Lp,φ by

Lp,φ = {f ∈ Lploc (Rn ) : kf kLp,φ < ∞}.
Our results are the following:
Theorem 1.1 If ρ, φ, ψ : (0, ∞) −→ (0, ∞) satisfying the conditions below :
t
1
1
φ(t)
ρ(t)
1
≤ ≤2⇒
≤ A1 , and
≤ A2
≤
≤
2
r
A1
φ(r)
A2
ρ(r)

Z 1
Z ∞
p
ρ(t)
φ(t)
p
dt < ∞, and for allr > 0, we have
dt ≤ A3 φ(r) ,
t
t
0
r
Z ∞
Z r
ρ(t)φ(t)
ρ(t)
dt +
dt ≤ A4 ψ(r), for allr > 0,
φ(r)
t

t
r
0

(1)
(2)
(3)

where Ai > 0 are independent of t, r > 0, then for each 1 < p < ∞ there exists
Cp > 0 such that
kTρ f kMp,ψ ≤ Cp kf kMp,φ .

Eridani

13

Theorem 1.2 If ρ, φ, ψ : (0, ∞) −→ (0, ∞) satisfying the conditions below
1
t
1

φ(t)
1
ρ(t)
≤ ≤2⇒
≤
≤ A1 , and
≤
≤ A2
2
r
A1
φ(r)
A2
ρ(r)
Z ∞
Z 1
p
φ(t)
ρ(t)
p

dt < ∞, and for allr > 0, we have
dt ≤ A3 φ(r) ,
t
t
r
0
ρ(r)
1
t
ρ(r) ρ(t)
≤ ≤ 2,
| n − n | ≤ A4 |r − t| n+1 , f or
r Z t
2
r
Z ∞ r
r
ρ(t)φ(t)
ρ(t)
dt + r

dt ≤ A5 ψ(r), for allr > 0,
φ(r)
t
t2
r
0

(4)
(5)
(6)
(7)

where Ai > 0 are independent of t, r > 0, then for each 1 < p < ∞ there exists
Cp > 0 such that
kTeρ f kLp,ψ ≤ Cp kf kMp,φ .

2. Proof of the Theorems
To prove the theorems, we shall use the following result of Nakai [2] (in a slightly
modified version) about the boundedness of the standard maximal function M f on
a generalized Morrey space Mp,φ . The standard maximal function M f is defined

by
Z
1
M f (x) = sup
|f (y)|dy, x ∈ Rn ,
B∋x |B| B
where the supremum is taken over all open balls B containing x.
Theorem 2.1 (Nakai).If φ : (0, ∞) −→ (0, ∞) satisfying the conditions below :
(a).
(b).

1
2

≤

R∞
r

t
r

≤2⇒

φ(t)p
t dt

1
A1

≤

φ(t)
φ(r)

≤ A1 ,

p

≤ A2 φ(r) , for all r > 0,

where Ai > 0 are independent of t, r > 0, then for each 1 < p < ∞ there exists
Cp > 0 such that
kM f kMp,φ ≤ Cp kf kMp,φ .
From now on, C and Cp will denote positive constants, which may vary from
line to line. In general, these constants depend on n.

14

On the Boundedness of a Generalized Fractional ...

Proof of Theorem 1.1
For x ∈ Rn , and r > 0, write

Tρ f (x) =

Z

|x−y| 0 such that

ρ(2k r) ≤ C1

Z

2k+1 r

2k r

ρ(t)
dt
t

and

k

k

ρ(2 r)φ(2 r) ≤ C2

Z

2k+1 r
2k r

ρ(t)φ(t)
dt.
t

So, we have

|I1 (x)| ≤
≤

Z

|f (y)|ρ(|x − y|)
dy
n
|x − y|

|x−y|