International Journal of Analysis and Applications

Volume 17, Number 3 (2019), 440-447

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-440

ON THE BEHAVIORS OF ROUGH FRACTIONAL TYPE SUBLINEAR OPERATORS

ON VANISHING GENERALIZED WEIGHTED MORREY SPACES

FERİT GÜRBÜZ∗

Hakkari University, Faculty of Education, Department of Mathematics Education, Hakkari 30000, Turkey

∗Corresponding author: feritgurbuz@hakkari.edu.tr

Abstract. The aim of this paper is to get the boundedness of rough sublinear operators generated by

fractional integral operators on vanishing generalized weighted Morrey spaces under generic size conditions

which are satisfied by most of the operators in harmonic analysis. Also, rough fractional integral operator

and a related rough fractional maximal operator which satisfy the conditions of our main result can be

considered as some examples.

1. Introduction and useful informations

1.1. Background. The classical fractional integral (The classical fractional integral operator is also known

as Riesz potential.) was introduced by Riesz in 1949 [6], defined by

Iαf(x) = (−∆)
−α

2 f(x) 0 < α < n,

=
1

γ (α)

∫
Rn

f(y)

|x−y|n−α
dy

with

γ (α) =
π
n
2 2αΓ

(
α
2

)
Γ
(
n
2
− α

2

) ,
where Γ (·) is the standard gamma function and Iα plays an important role in partial diferential equation

as the inverse of power of Laplace operator. Especially, Its most significant feature is that Iα maps Lp(Rn)

Received 2018-12-19; accepted 2019-01-07; published 2019-05-01.

2010 Mathematics Subject Classification. 42B20, 42B25, 42B35.

Key words and phrases. fractional type sublinear operator; rough kernel; vanishing generalized weighted Morrey space;

A
(

p
s′
,

q
s′
)

weight.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

440

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-440


Int. J. Anal. Appl. 17 (3) (2019) 441

continuously into Lq(Rn), with 1q =
1
p
−α
n

and 1 < p < n
α

, through the well known Hardy-Littlewood-Sobolev

imbedding theorem (see pp. 119-121, Theorem 1 and its proof in [7]) for Iα.

Let Ω ∈ Ls(Sn−1), 1 < s ≤ ∞, Ω(µx) = Ω(x) for any µ > 0, x ∈ Rn \{0} and satisfy the cancellation

condition ∫
Sn−1

Ω(x′)dσ(x′) = 0,

where x′ = x|x| for any x 6= 0.

We first recall the definitions of rough fractional integral operator TΩ,α and a related rough fractional

maximal operator MΩ,α as follows:

Definition 1.1. Define

IΩ,αf(x) =

∫
Rn

Ω(x−y)
|x−y|n−α

f(y)dy 0 < α < n,

MΩ,αf (x) = sup
r>0

1

rn−α

∫
|x−y|<r

|Ω(x−y)| |f(y)|dy 0 < α < n.

Next, we give the definition of weighted Lebesgue spaces as follows:

Definition 1.2. (Weighted Lebesgue space) Let 1 ≤ p ≤∞ and given a weight w (x) ∈ Ap (Rn), we shall

define weighted Lebesgue spaces as

Lp(w) ≡ Lp(Rn,w) =


f : ‖f‖Lp,w =


∫
Rn

|f(x)|pw(x)dx




1
p

< ∞


 , 1 ≤ p < ∞.

L∞,w ≡ L∞(Rn,w) =
{
f : ‖f‖L∞,w = esssup

x∈Rn
|f(x)|w(x) < ∞

}
.

Here and later, Ap denotes the Muckenhoupt classes (see [2]).

Now, let us consider the Muckenhoupt-Wheeden class A (p,q) in [5]. One says that w (x) ∈ A (p,q) for

1 < p < q < ∞ if and only if

[w]A(p,q) := sup
B


|B|−1∫

B

w(x)qdx




1
q

|B|−1∫

B

w(x)−p
′
dx




1
p′

< ∞, (1.1)

where the supremum is taken over all the balls B. Note that, by Hölder’s inequality, for all balls B we have

[w]A(p,q) ≥ [w]A(p,q)(B) = |B|
1
p
−1
q
−1‖w‖Lq(B)‖w

−1‖Lp′(B) ≥ 1. (1.2)

By (1.1), we have 
∫
B

w(x)qdx




1
q

∫
B

w(x)−p
′
dx




1
p′

. |B|
1
q

+ 1
p′ . (1.3)



Int. J. Anal. Appl. 17 (3) (2019) 442

On the other hand, let µ (x) = w (x)
s′

, p̃ = p
s′

and q̃ = q
s′

. If w (x)
s′ ∈ A

(
p
s′
, q
s′

)
, then we get µ (x) ∈ A (p̃, q̃).

By (1.2) and (1.3),

‖µ‖Lq̃(B)‖µ
−1‖Lp̃′(B) ≈ |B|

1+ 1
q̃
−1
p̃ (1.4)

is valid.

Now, we introduce some spaces which play important roles in PDE. Except the weighted Lebesgue space

Lp(w), the weighted Morrey space Lp,κ(w), which is a natural generalization of Lp(w) is another important

function space. Then, the definition of generalized weighted Morrey spaces Mp,ϕ (w) which could be viewed

as extension of Lp,κ(w) has been given as follows:

For 1 ≤ p < ∞, positive measurable function ϕ(x,r) on Rn×(0,∞) and nonnegative measurable function

w on Rn, f ∈ Mp,ϕ(w) ≡ Mp,ϕ(Rn,w) if f ∈ Llocp,w(Rn) and

‖f‖Mp,ϕ(w) = sup
x∈Rn,r>0

1

ϕ(x,r)
‖f‖Lp(B(x,r),w) < ∞.

is finite. Note that for ϕ(x,r) ≡ w(B(x,r))
κ
p , 0 < κ < 1 and ϕ(x,r) ≡ 1, we have Mp,ϕ(w) = Lp,κ(w) and

Mp,ϕ(w) = Lp(w), respectively.

Extending the definition of vanishing generalized Morrey spaces in [3] to the case of generalized weighted

Morrey spaces defined above, we introduce the following definition.

Definition 1.3. (Vanishing generalized weighted Morrey spaces) For 1 ≤ p < ∞, ϕ(x,r) is a positive

measurable function on Rn × (0,∞) and nonnegative measurable function w on Rn, f ∈ V Mp,ϕ (w) ≡

V Mp,ϕ(Rn,w) if f ∈ Llocp,w(Rn) and

lim
r→0

sup
x∈Rn

1

ϕ(x,r)
‖f‖Lp(B(x,r),w) = 0. (1.5)

Inherently, it is appropriate to impose on ϕ(x,t) with the following circumstances:

lim
t→0

sup
x∈Rn

(w(B(x,t)))
1
p

ϕ(x,t)
= 0, (1.6)

and

inf
t>1

sup
x∈Rn

(w(B(x,t)))
1
p

ϕ(x,t)
> 0. (1.7)

From (1.6) and (1.7), we easily know that the bounded functions with compact support belong to V Mp,ϕ (w).

On the other hand, the space V Mp,ϕ(w) is Banach space with respect to the following finite quasi-norm

‖f‖V Mp,ϕ(w) = sup
x∈Rn,r>0

1

ϕ(x,r)
‖f‖Lp(B(x,r),w),

such that

lim
r→0

sup
x∈Rn

1

ϕ(x,r)
‖f‖Lp(B(x,r),w) = 0,



Int. J. Anal. Appl. 17 (3) (2019) 443

we omit the details. Moreover, we have the following embeddings:

V Mp,ϕ (w) ⊂ Mp,ϕ (w) , ‖f‖Mp,ϕ(w) ≤‖f‖V Mp,ϕ(w).

Henceforth, we denote by ϕ ∈B (w) if ϕ(x,r) is a positive measurable function on Rn × (0,∞) and positive

for all (x,r) ∈ Rn × (0,∞) and satisfies (1.6) and (1.7).

The purpose of this paper is to consider the mapping properties for the rough fractional type sublinear

operators TΩ,α satisfying the following condition

|TΩ,αf(x)|.
∫
Rn

|Ω(x−y)|
|x−y|n−α

|f(y)|dy, x /∈ supp f (1.8)

on vanishing generalized weighted Morrey spaces. Similar results still hold for the operators IΩ,α and MΩ,α,

respectively. On the other hand, these operators have not also been studied so far on vanishing generalized

weighted Morrey spaces and this paper seems to be the first in this direction.

At last, here and henceforth, F ≈ G means F & G & F ; while F & G means F ≥ CG for a constant

C > 0; and p′ and s′ always denote the conjugate index of any p > 1 and s > 1, that is, 1
p′

:= 1 − 1
p

and

1
s′

:= 1 − 1
s

and also C stands for a positive constant that can change its value in each statement without

explicit mention. Throughout the paper we assume that x ∈ Rn and r > 0 and also let B(x,r) denotes

x-centred Euclidean ball with radius r, BC(x,r) denotes its complement. For any set E, χ
E

denotes its

characteristic function, if E is also measurable and w is a weight, w(E) :=

∫
E

w(x)dx.

2. Main Results

Our result can be stated as follows.

Theorem 2.1. Suppose that 0 < α < n, 1 ≤ s′ < p < n
α

, 1
q

= 1
p
− α
n

, 1 < q < ∞, Ω ∈ Ls(Sn−1), 1 < s ≤∞,

Ω(µx) = Ω(x) for any µ > 0, x ∈ Rn\{0} such that TΩ,α is rough fractional type sublinear operator satisfying

(1.8). For p > 1, w (x)
s′ ∈ A

(
p
s′
, q
s′

)
and s′ < p, the following pointwise estimate

‖TΩ,αf‖Lq(B(x0,r),wq) . (w
q (B (x0,r)))

1
q

∞∫
2r

‖f‖Lp(B(x0,t),wp) (w
q (B (x0, t)))

−1
q
dt

t
(2.1)

holds for any ball B (x0,r) and for all f ∈ Llocp,w (Rn). If ϕ1 ∈ B (wp), ϕ2 ∈ B (wq) and the pair (ϕ1,ϕ2)

satisfies the following conditions

cδ :=

∞∫
δ

sup
x∈Rn

ϕ1 (x,t)

(wq (B (x,t)))
1
q

1

t
dt < ∞ (2.2)

for every δ > 0, and
∞∫
r

ϕ1 (x,t)

(wq (B (x,t)))
1
q

1

t
dt .

ϕ2(x,r)

(wq (B (x,t)))
1
q

, (2.3)



Int. J. Anal. Appl. 17 (3) (2019) 444

then for p > 1, w (x)
s′ ∈ A

(
p
s′
, q
s′

)
and s′ < p, the operator TΩ,α is bounded from V Mp,ϕ1 (w

p) to

V Mq,ϕ2 (w
q). Moreover,

‖TΩ,αf‖V Mq,ϕ2 (wq) . ‖f‖V Mp,ϕ1 (wp) · (2.4)

Proof. Since inequality (2.1) is the heart of the proof of (2.4), we first prove (2.1).

For any x0 ∈ Rn, we write as f = f1 + f2, where f1 (y) = f (y) χB(x0,2r) (y), f2 (y) = f (y) χ(B(x0,2r))C (y),

r > 0 and χB(x0,2r) denotes the characteristic function of B (x0, 2r). Then

‖TΩ,αf‖Lq(wq,B(x0,r)) ≤‖TΩ,αf1‖Lq(wq,B(x0,r)) + ‖TΩ,αf2‖Lq(wq,B(x0,r)) .

Let us estimate ‖TΩ,αf1‖Lq(wq,B(x0,r)) and ‖TΩ,αf2‖Lq(wq,B(x0,r)), respectively.

Since f1 ∈ Lp (wp,Rn), by the boundedness of TΩ,α from Lp (wp,Rn) to Lq (wq,Rn) (see Theorem 3.4.2

in [4]), (1.4) and since 1 ≤ s′ < p < q we get

‖TΩ,αf1‖Lq(wq,B(x0,r)) ≤‖TΩ,αf1‖Lq(wq,Rn)

. ‖f1‖Lp(wp,Rn)

= ‖f‖Lp(wp,B(x0,2r))

. rn−αs
′
‖f‖Lp(wp,B(x0,2r))

∞∫
2r

dt

tn−αs
′+1

≈‖ws
′
‖L q

s′
(B(x0,r))‖w

−s′‖L
( ps′ )

′(B(x0,r))

∞∫
2r

‖f‖Lp(wp,B(x0,t))
dt

tn−αs
′+1

. (wq (B(x0,r)))
1
q

∞∫
2r

‖f‖Lp(wp,B(x0,t)) ‖w
−s′‖L

( ps′ )
′(B(x0,t))

dt

tn−αs
′+1

. (wq (B(x0,r)))
1
q

∞∫
2r

‖f‖Lp(wp,B(x0,t))

[
‖ws

′
‖L

( qs′ )
(B(x0,t))

]−1
1

t
dt

. (wq (B(x0,r)))
1
q

×
∞∫

2r

‖f‖Lp(wp,B(x0,t)) (w
q (B(x0, t)))

−1
q

1

t
dt.

Now, let’s estimate the second part (= ‖TΩ,αf2‖Lq(wq,B(x0,r))). For the estimate used in ‖TΩ,αf2‖Lq(wq,B(x0,r)),

we first have to prove the below inequality:

|TΩ,αf2 (x)|.
∞∫

2r

‖f‖Lp(wp,B(x0,t)) (w
q (B(x0, t)))

−1
q

1

t
dt. (2.5)

By [1] (see pp. 7 in the proof of Lemma2:), we get

|TΩ,αf2 (x)|.
∞∫

2r

‖Ω (x−·)‖Ls(B(x0,t)) ‖f‖Ls′(B(x0,t))
dt

tn+1−α
. (2.6)



Int. J. Anal. Appl. 17 (3) (2019) 445

On the other hand, by Hölder’s inequality we have

‖f‖Ls′(B(x0,t)) =


 ∫
B(x0,t)

|f (y)|s
′
dy




1
s′

≤


 ∫
B(x0,t)

|f (y)|p |µ (y)|p̃ dy




1
p

 ∫
B(x0,t)

|µ (y)|−p̃
′
dy




1
p̃′s′

≤


 ∫
B(x0,t)

|f (y)|p |µ (y)|p̃ dy




1
p

(wq (B(x0, t)))
−1
q |B(x0, t)|

1
s′ +

1
q
−1
p

= ‖f‖Lp(wp,B(x0,t)) (w
q (B(x0, t)))

−1
q |B(x0, t)|

1
s′ +

1
q
−1
p , (2.7)

where in the second inequality we have used the following fact:

By (1.4), we get the following:


 ∫
B(x0,t)

|µ (y)|−p̃
′
dy




1
p̃′s′

≈
[
‖µ‖Lq̃(B(x0,t))

]− 1
s′
[
|B (x0, t) |1+

1
q̃
−1
p̃

] 1
s′

=

[(
‖ws

′
‖Lq̃(B(x0,t))

)−1
|B (x0, t) |1+

1
q̃
−1
p̃

] 1
s′

=




 ∫
B(x0,t)

|w (y)|q dy



−s
′
q

|B (x0, t) |1+
s′
q
−s
′
p




1
s′

= (wq (B(x0, t)))
−1
q |B(x0, t)|

1
s′ +

1
q
−1
p . (2.8)

At last, substituting (3.10) in [1] and (2.7) into (2.6), the proof of (2.5) is completed. Thus, by (2.5) we get

‖TΩ,αf2‖Lq(wq,B(x0,r)) . (w
q (B(x0,r)))

1
q

×
∞∫

2r

‖f‖Lp(wp,B(x0,t)) (w
q (B(x0, t)))

−1
q

1

t
dt.

Combining all the estimates for ‖TΩ,αf1‖Lq(wq,B(x0,r)) and ‖TΩ,αf2‖Lq(wq,B(x0,r)), we get (2.1).



Int. J. Anal. Appl. 17 (3) (2019) 446

Now, let’s estimate the second part (2.4) of Theorem 2.1. Indeed, by the definition of vanishing generalized

weighted Morrey spaces, (2.1) and (2.3), we have

‖TΩ,αf‖V Mq,ϕ2 (wq) = sup
x∈Rn,r>0

1

ϕ2(x,r)
‖TΩ,αf‖Lq(wq,B(x0,r))

. sup
x∈Rn,r>0

1

ϕ2(x,r)
(wq (B (x0,r)))

1
q

×
∞∫
r

‖f‖Lp(B(x0,t),wp) (w
q (B (x0, t)))

−1
q
dt

t

. sup
x∈Rn,r>0

1

ϕ2(x,r)
(wq (B (x0,r)))

1
q

×
∞∫
r

(wq (B (x0, t)))
−1
q ϕ1 (x,t)

[
ϕ1 (x,t)

−1 ‖f‖Lp(B(x0,t),wp)
] dt
t

. ‖f‖V Mp,ϕ1 (wp) supx∈Rn,r>0
1

ϕ2(x,r)
(wq (B (x0,r)))

1
q

×
∞∫
r

(wq (B (x0, t)))
−1
q ϕ1 (x,t)

dt

t

. ‖f‖V Mp,ϕ1 (wp) .

At last, we need to prove that

lim
r→0

sup
x∈Rn

1

ϕ2(x,r)
‖TΩ,αf‖Lq(wq,B(x0,r)) . lim

r→0
sup
x∈Rn

1

ϕ1(x,r)
‖f‖Lp(wp,B(x0,r)) = 0.

But, because the proof of above inequality is similar to Theorem 2 in [3], we omit the details, which completes

the proof. �

Corollary 2.1. Under the conditions of Theorem 2.1, the operators MΩ,α and IΩ,α are bounded from

V Mp,ϕ1 (w
p) to V Mq,ϕ2 (w

q).

Corollary 2.2. For w ≡ 1, under the conditions of Theorem 2.1, we get the Theorem 2 in [3].

References

[1] A. S. Balakishiyev, E. A. Gadjieva, F. Gürbüz and A. Serbetci, Boundedness of some sublinear operators and their

commutators on generalized local Morrey spaces, Complex Var. Elliptic Equ. 63(11) (2018), 1620-1641.

[2] F. Gürbüz, On the behaviors of sublinear operators with rough kernel generated by Calderón-Zygmund operators both on

weighted Morrey and generalized weighted Morrey spaces, Int. J. Appl. Math. Stat. 57(2) (2018), 33-42.

[3] F. Gürbüz, A class of sublinear operators and their commutators by with rough kernels on vanishing generalized Morrey

spaces, J. Sci. Eng. Res. 5(5) (2018), 86-101.

[4] S. Z. Lu, Y. Ding and D. Yan, Singular integrals and related topics, World Scientific Publishing, Singapore, 2006.

[5] B. Muckenhoupt and R.L. Wheeden, Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math.

Soc. 161 (1971), 249-258.



Int. J. Anal. Appl. 17 (3) (2019) 447

[6] M. Riesz, L´intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1-222.

[7] E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton N J, Princeton Univ Press, 1970.


	1. Introduction and useful informations
	1.1. Background

	2. Main Results
	References