C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat. Volum e 69, N umb er 1, Pages 73–87 (2020)

D O I: 10.31801/cfsuasm as.508702 ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr/index.php?series= A 1

GENERALIZED FRACTIONAL MAXIMAL OPERATOR ON GENERALIZED LOCAL MORREY SPACES

A. KUCUKASLAN, V.S. GULIYEV, AND A. SERBETCI

Abstract. In this paper, we study the boundedness of generalized fractional maximal operator M on generalized local Morrey spaces LMfx0g

p;' and

gen-eralized Morrey spaces Mp;', including weak estimates. Firstly, we prove the

Spanne type boundedness of M from the space LMfx0g

p;'1 to another LM fx0g q;'2, 1 < p < q < 1 and from LMfx0g

1;'1 to the weak space W LM fx0g

q;'2 for p = 1 and 1 < q < 1. Secondly, we prove the Adams type boundedness of M from the space M

p;' 1

p to another M_{q;'}1q for 1 < p < q < 1 and from M1;'to the weak
space W M

q;' 1

q for p = 1 and 1 < q < 1. In all cases the conditions for the boundedness of M are given in terms of supremal-type integral inequalities on ('1; '2; )and ('; ), which do not assume any assumption on monotonicity

of '1(x; r), '2(x; r)and '(x; r) in r.

1. Introduction

The classical Morrey spaces Mp; were …rst introduced by Morrey in [21] to study

the local behavior of solutions to second order elliptic partial di¤erential equations. The generalized Morrey spaces Mp;'are obtained by replacing r in the de…nition

of the Morrey space. During the last decades various classical operators, such as maximal, singular and potential operators were widely investigated in both in classical, generalized Morrey spaces and generalized local Morrey spaces. For the boundedness of the Hardy–Littlewood maximal operator, the fractional integral operator and the Calderón-Zygmund singular integral operators on these spaces, we refer the readers to [1, 9, 15, 16, 20, 22].

Received by the editors: January 07, 2019, Accepted: August 09, 2019. 2010 Mathematics Subject Classi…cation. 42B20, 42B25, 42B35.

Key words and phrases. Generalized fractional maximal operator, generalized local Morrey spaces, generalized Morrey spaces.

The research of A. Kucukaslan was totally supported by the grant of The Scienti…c and Tech-nological Research Council of Turkey (TUBITAK), [Grant-1059B191600675].

The research of V.S. Guliyev was partially supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008).

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For a measurable function _{: (0; 1) ! (0; 1) the generalized fractional maximal}
operator M and the generalized fractional integral operator I are de…ned by

M f (x) = sup t>0 (t) tn Z B(x;t)jf(y)jdy; I f (x) = Z Rn (jx yj) jx yjn f (y)dy

for any suitable function f on Rn_{. If (t)} _{t , then M} _{M}

t is the fractional

maximal operator and I It is the Riesz potential.

Spanne [24] and Adams [1] studied boundedness of the Riesz potential in Morrey spaces. Their results can be summarized as follows.

Theorem A.(Spanne, but published by Peetre [24]) Let 0 < < n, 1 < p < n, 0 < < n p. Moreover, let 1

p 1

q = n and p = q. Then for p > 1, the

operator I is bounded from Mp; to Mq; and for p = 1, I is bounded from M1;

to W Mq; .

Theorem B. (Adams [1]) Let 0 < < n, 1 < p < n, 0 < < n p and

1 p

1

q = n . Then for p > 1, the operator I is bounded from Mp; to Mq; and

for p = 1, I is bounded from M1; to W Mq; .

Nakai [22] proved the boundedness of the operators I and M from the general-ized Morrey spaces Mp;'1to the spaces Mq;'2for suitable functions '1and '2. The

boundedness of M and I from the generalized Morrey spaces Mp;'1 to the spaces

Mq;'2 is studied by Nakai [23], Eridani [10], Gunawan [18], Eridani, Gunawan and

Nakai [12], Sawano, Sugano, Tanaka [25], Eridani, Gunawan, Nakai, Sawano [11], Guliyev, Ismayilova, Kucukaslan, Serbetci [17], Kucukaslan, Hasanov, Aykol [19].

In particular, the following statement containing both Theorem A and Theorem B was proved in [3, 4].

Theorem C.([3, 4]) Let 1 _{p < q < 1, 0 < ; < n and}
0 < =n
p
n
q <
n
p:

Then, for p > 1, the operator I is bounded from Mp; to Mq; , and, for p = 1, I

is bounded from M1; to W Mq; .

In [3, 4] it was also proved that, under the assumptions of Theorem C, the operator I , for p > 1, is bounded from the local Morrey space LMfx0g

p; to LMfx

0g

q; ,

and, for p = 1 from LMfx0g

1; to the weak local Morrey space W LMfx

0g

q; .

Since, for some c > 0, M f (x) _{c I (jfj) (x), x 2 R}n_{, it follows that in}

also the case p = q). For the operator M Theorem C was, in fact, earlier proved in [5, 6].

Guliyev [14] proved the Spanne and Adams type boundedness of I from the spaces Mp;'1(R

n_{) to M}
q;'2(R

n_{) without any assumption on monotonicity of '}
1,

'_{2}. Paper [7] should be mentioned where for = n 1
p

1

q necessary and su¢ cient

conditions of '_{1} and '_{2} are obtained. In [17], by using the method given in [13]
the Spanne and Adams type boundedness of the operator I from the generalized
local Morrey space LMfx0g

p;'1 to another one LM

fx0g

q;'2 were proved.

The main goal of this paper is to show that the boundedness of the generalized fractional maximal operator M in generalized local Morrey spaces LMfx0g

p;' and

generalized Morrey spaces Mp;' can be obtained under weaker assumptions on ,

namely in terms of the socalled supremal operators. More precisely, we …nd su¢
-cient conditions, in supremal terms, on the functions ('_{1}; '_{2}; ) which ensure the
boundedness of the operator M from one generalized local Morrey space LMfx0g

p;'1

to another LMfx0g

q;'2 for 1 < p < q < 1 and from LM

fx0g

1;'1 to the weak space

W LMfx0g

q;'2 for p = 1 and 1 < q < 1. We also …nd conditions on the pair ('; )

which ensure the Adams type boundedness of M from the spaces M

p;'p1 to another

M

q;'1q for 1 < p < q < 1 and from M1;'to the weak space W M_{q;'}1q for p = 1 and

1 < q < 1.

By A. B we mean that A CB with some positive constant C independent of appropriate quantities. If A. B and B . A, we write A B and say that A and B are equivalent.

2. Preliminaries

For x 2 Rn_{and r > 0, we denote by B(x; r) the open ball centered at x of radius}

r, and by {_{B(x; r) denote its complement. Let jB(x; r)j be the Lebesgue measure}
of the ball B(x; r). Therefore jB(x; r)j = wnrn, where wn denotes the volume of

the unit ball in Rn_{.}

De…nition 2.1. _{Let '(x; r) be a positive measurable function on R}n _{(0; 1) and}
1 _{p < 1. We denote by M}p;' Mp;'(Rn) the generalized Morrey space, the

space of all functions f 2 Lloc

p (Rn) with …nite norm

kfkMp;'= sup

x2Rn_{;r>0}

'(x; r) 1_{jB(x; r)j} p1_{kfk}

Lp(B(x;r)):

Also by W Mp;' W Mp;'(Rn) we denote the weak generalized Morrey space of all

functions f 2 W Llocp (Rn) for which

kfkW Mp;' = sup

x2Rn_{;r>0}

'(x; r) 1_{jB(x; r)j} 1p_{kfk}

According to this de…nition, we recover the Morrey space Mp; , the weak Morrey

space W Mp; respectively, under the choice '(x; r) = r

n
p _{:}
Mp; = Mp;'
'(x;r)=r
n
p ; W Mp; = W Mp;' _{'(x;r)=r}
n
p :

De…nition 2.2. _{Let '(x; r) be a positive measurable function on R}n _{(0; 1) and}

1 _{p < 1. We denote by LM}p;' LMp;'(Rn) the generalized local (central)

Morrey space, the space of all functions f 2 Lloc

p (Rn) with …nite norm

kfkLMp;' = sup

r>0

'(0; r) 1_{jB(0; r)j} 1p_{kfk}

Lp(B(0;r)):

Also by W LMp;' W LMp;'(Rn) we denote the weak generalized local (central)

Morrey space of all functions f 2 W Lloc

p (Rn) for which

kfkW LMp;'= sup

r>0

'(0; r) 1_{jB(0; r)j} 1p_{kfk}

W Lp(B(0;r))< 1:

De…nition 2.3. _{Let '(x; r) be a positive measurable function on R}n _{(0; 1) and}

1 _{p < 1. For any …xed x}0 2 Rn we denote by LMp;'fx0g LMp;'fx0g(Rn) the

generalized local Morrey space, the space of all functions f 2 Lloc

p (Rn) with …nite

norm

kfkLMp;'fx0g= kf(x0+ )kLMp;':

Also by W LMfx0g

p;' W LMp;'fx0g(Rn) we denote the weak generalized local Morrey

space of all functions f 2 W Lloc

p (Rn) for which

kfkW LMp;'fx0g= kf(x0+ )kW LMp;' < 1:

According to this de…nition, we recover the local Morrey space LMfx0g

p; and weak

local Morrey space W LMfx0g

p; under the choice '(x0; r) = r

n
p _{:}
LMfx0g
p; = LMp;'fx0g
'(x0;r)=r
n
p ; W LM
fx0g
p; = W LMp;'fx0g
'(x0;r)=r
n
p :

De…nition 2.4. Let M(0; 1) be the set of all Lebesgue-measurable functions on (0; 1) and M+(0; 1) its subset consisting of all non-negative functions on (0; 1). We de…ne a cone A by the set of the functions ' 2 M+(0; 1) which are non-decreasing on (0; 1) and such that limt!0+'(t) = 0, brie‡y

A = ' 2 M+(0; 1; ") : lim

t!0+'(t) = 0 :

De…nition 2.5. _{[8] Let u be a continuous and non-negative function on (0; 1).}
We de…ne the supremal operator Su on g 2 M(0; 1) by

Let v be a non-negative measurable function on (0; 1). We denote by L1;v(0; 1)

the space of all functions g(t), t > 0 with …nite norm kgkL1;v(0;1)= sup

t>0

v(t)g(t)

and L_{1}_{(0; 1)} L_{1;1}_{(0; 1). The following lemma is proved analogously to Lemma}
5.2 in [8].

Lemma 2.1. [8] Let v1and v2be weights and 0 < kv1kL1(t;1)< 1 for any t > 0

and let u be a continuous non-negative function on (0; 1): Then the operator Suis

bounded from L_{1;v}1(0; 1) to L1;v2(0; 1) on the cone A if and only if

v2Su kv1k_{L}_{1}1_{( ;}_{1)}

L1(0;1)

< 1: The following lemma was proved in [17].

Lemma 2.2. [17] Let v1, v2 be non-negative measurable functions satisfying 0 <

kv1kL1(t;1) < 1 for any t > 0. Then the identity operator I is bounded from

L_{1;v}1(0; 1) to L1;v2(0; 1) on the cone A if and only if

v2 kv1k_{L}1

1( ;1) _{L}

1(0;1)< 1:

3. Spanne type result for the operator M in the spaces LMfx0g

p;' We assume that sup 1 t<1 (t) tn < 1; (3.1)

so that the fractional maximal functions M f are well de…ned, at least for
charac-teristic functions 1=jxj2n _{of complementary balls:}

f (x) = RnnB(0;1)(x) jxj2n :

In addition, we shall also assume that satis…es the growth condition: there exist constants C1> 0 and 0 < 2k1< k2< 1 such that

sup r<s 2r (s) sn C1 sup k1r<t<k2r (t) tn ; r > 0: (3.2)

This condition is weaker than the usual doubling condition for the function _{t}(t)n

: there exists a constant C2> 0 such that

1 C2 (t) tn (r) rn C2 (t) tn ;

Remark 3.1. Typical examples of (t) that we envisage are, for 0 < < n
(t) n t log(e=t); 0 < t 1t
log(et); 1 t < 1
and, for c > 0
(t)
n _{t ;} _{0 < t} _{1}
ec_{e} ct2
; 1 _{t < 1:}

The second one is used to control the Bessel potential (see also [26]).

The boundedness of the operator I in the spaces Lp(Rn) can be found in [11].

Let _{t}(t)n be almost decreasing, that is, there exists a constant C such that

(t) tn

C _{s}(s)n for s < t. In this case we get

M f (x) = sup t>0 (t) tn Z B(x;t)jf(y)jdy . sup t>0 Z B(x;t) (jx yj) jx yjn jf(y)jdy = Z Rn (jx yj) jx yjn jf(y)jdy = I (jfj)(x):

For proving our main results, we need the following estimate.

Lemma 3.3. If B0:= B(x0; r0) B(x; r) and satis…es the doubling condition.

Then (r0). M B0(x) for every x 2 B0.

Proof. Let satisfy the doubling condition, then

M f(x) . M f(x); (3.3) where M (f)(x) = sup B3x (rB) jBj R

Bjf(y)jdy and rB is the center of the ball B.

Now let x 2 B0. By using (3.3), we get

M _{B}_{0}(x)& M _{B}_{0}(x) = sup
B3x
(rB)
jBj jB \ B0j
& (r0)
jB0j jB0\ B0j = (r0
):

The following lemma is valid.
Lemma 3.4. Let 1 _{p < q < 1.}

(1) The condition

(r) Crnp nq _{(3.4)}

for all r > 0, where C > 0 does not depend on r, is su¢ cient for the boundedness of M from Lp(Rn) to W Lq(Rn). Moreover, if p > 1, then the condition (3.4) is

(2) If satis…es the doubling condition, then the condition (3.4) is necessary for the boundedness of M from Lp(Rn) to W Lq(Rn) and from Lp(Rn) to Lq(Rn) for

p > 1.

(3) If satis…es the doubling condition and the supremal regularity condition sup

r<t<1

(t) t np _{C (r)r} np

holds for all r > 0, where C > 0 does not depend on r, then the condition (3.4) is necessary and su¢ cient for the boundedness of M from Lp(Rn) to W Lq(Rn).

Moreover, if p > 1, then the condition (3.4) is necessary and su¢ cient for the boundedness of M from Lp(Rn) to Lq(Rn).

Proof. (1) Suppose satis…es the condition (3.4). Then M f (x). Mn

p nqf (x): (3.5)

Since the operator Mn p

n

q is bounded from Lp(R

n_{) to W L}

q(Rn) and for p > 1 from

Lp(Rn) to Lq(Rn), then from (3.5) we get the statement (1).

(2) Now we shall prove the second part. Let B0 = B(x0; r0) and x 2 B0. By

Lemma 3.3, we have (r0). M B0(x). Therefore, we have

(r0). r n q 0 kM B0kW Lq(B0). r n q 0 kM B0kW Lq(Rn) . r nq 0 k B0kLp(Rn). r n p n q 0 and for p > 1 (r0). r n q 0 kM B0kLq(B0). r n q 0 kM B0kLq(Rn) . r nq 0 k B0kLp(Rn). r n p n q 0

holds for every r0> 0, hence the proof of statement (2) is completed.

(3) From the …rst and second statements the third statement of the lemma follows.

The following lemma is valid.

Lemma 3.5. Let 1 _{p < q < 1 and let (t) satisfy the conditions (3.1), (3.2)}
and (3.4). Then the inequality

kM fkW Lq(B(x0;r)). kfkLp(B(x0;2r))+ r
n
q _{sup}
t>rkfkLp(B(x0;t))
(t)
tnp

holds for any ball B(x0; r) and for all fploc(Rn).

If p > 1, then the inequality

kM fkLq(B(x0;r)). kfkLp(B(x0;2r))+ r
n
q _{sup}
t>rkfkLp(B(x0;t))
(t)
tnp
(3.6)
holds for any ball B(x0; r) and for all fploc(Rn).

Proof. Let 1 _{p < q < 1 and let (t) satisfy the conditions (3.1), (3.2) and (3.4).}
For arbitrary x02 Rn, set B = B(x0; r) for the ball centered at x0 and of radius r.

Write f = f1+ f2 with f1= f 2B and f2= f {

(2B). Hence

kM fkW Lq(B) kM f1kW Lq(B)+ kM f2kW Lq(B):

Since f1 2 Lp(Rn), M f1 2 W Lq(Rn) and by Lemma 3.4 M is bounded from

Lp(Rn) to W Lq(Rn). Thus it follows that

kM f1kW Lq(B) kM f1kW Lq(Rn) Ckf1kLp(Rn)= CkfkLp(2B);

where constant C > 0 is independent of f .

Let x be an arbitrary point from B: If B(x; t) \ {(2B) 6= ;; then t > r: Indeed, if y 2 B(x; t) \ {(2B); then t > jx yj jx0 yj jx0 xj > 2r r = r.

On the other hand, B(x; t) \ {(2B) B(x0; 2t): Indeed, y 2 B(x; t) \

{ (2B), then we get jx0 yj jx yj + jx0 xj < t + r < 2t. Hence M f2(x) = sup t>0 (t) tn Z B(x;t)\{(2B)jf(y)jdy . sup t>r (2t) (2t)n Z B(x0;2t) jf(y)jdy = sup t>2r (t) tn Z B(x0;t) jf(y)jdy: Therefore, for all x 2 B we have

M f2(x). sup
t>2r
(t)
tn
Z
B(x0;t)
jf(y)jdy: (3.7)
Thus
kM fkW Lq(B) kM fkLq(B). kfkLp(2B)+ jBj
1
q sup
t>2rkfkL1(B(x0;t))
(t)
tn
. kfkLp(2B)+ r
n
q _{sup}
t>2rkfkLp(B(x0;t))
(t)
tnp
: (3.8)
Let p > 1. From the (p; q) boundedness of M and (3.4) it follows that:

kM f1kLq(B) kM f1kLq(Rn). kf1kLp(Rn)= kfkLp(2B): (3.9)

Then by (3.8) and (3.9) we get the inequality (3.6).

The following theorem is one of the main results of the paper in which we get the Spanne type boundedness of the generalized fractional maximal operator M in the generalized local Morrey spaces LMfx0g

Theorem 3.1. Let x0 2 Rn, 1 p < q < 1, and let the function satisfy the

conditions (3.1), (3.2) and (3.4). Let also ('_{1}; '_{2}) satisfy the conditions
ess inf
t<s<1'1(x0; s)s
n
p _{C '}
2 x0;
t
2 t
n
q_{;} _{(3.10)}
sup
t>r
ess inf
t<s<1'1(x0; s)s
n
p (t)
tnp C '2(x0; r);

where C does not depend on x0 and r. Then the operator M is bounded from

LMfx0g

p;'1 to W LM

fx0g

q;'2 and for p > 1 from LM

fx0g
p;'1 to LM
fx0g
q;'2. Moreover,
kM fkW LM_{q;'2}fx0g. kfkLM_{p;'1}fx0g;
and for p > 1
kM fkLM_{q;'2}fx0g . kfkLM_{p;'1}fx0g:

Proof. Let the function satisfy the conditions (3.1), (3.2), (3.4), and also ('_{1}; '_{2})
satisfy the conditions (3.10) and (3.11). By Lemmas 2.1, 2.2 and 3.5 we have

kM fkW LM_{q;'2}fx0g. sup
r>0
'_{2}(x0; r) 1r
n
q_{kfk}
Lp(B(x0;2r))
+ sup
r>0
'_{2}(x0; r) 1sup
t>rkfkLp(B(x0;t))
(t)
tnp
t sup
r>0
'_{1}(x0; r) 1r
n
p_{kfk}
Lp(B(x0;r))= kfk_{LM}fx0g
p;'1
and for p > 1
kM fk_{LM}fx0g
q;'2 . sup_{r>0}'2(x0; r)
1_{r} n
q_{kfk}
Lp(B(x0;2r))
+ sup
r>0
'_{2}(x0; r) 1sup
t>rkfkLp(B(x0;t))
(t)
tnp
t sup
r>0
'_{1}(x0; r) 1r
n
p_{kfk}
Lp(B(x0;r))= kfk_{LM}_{p;'1}fx0g:

In the following corollary we get the boundedness of the generalized fractional maximal operator M on generalized Morrey spaces Mp;'.

Corollary 3.1. Let 1 _{p < q < 1, the function} satisfy the conditions (3.1),
(3.2) and (3.4). Let also ('_{1}; '_{2}) satisfy the following conditions

ess inf r<t<1'1(x; t)t n p C ' 2 x; r 2 r n q; sup t>r ess inf t<s<1'1(x; s)s n p (t) tnp C '2(x; r);

where C does not depend on x and r. Then the operator M is bounded from Mp;'1

In the case (t) = t from Theorem 3.1 we get new Spanne type result for fractional maximal operator M on generalized local Morrey spaces.

Corollary 3.2. Let x0 2 Rn, 0 < < n, 1 p < q < 1 and 1=p 1=q = =n.

Let also ('1; '2) satisfy the condition

sup t>r ess inf t<s<1'1(x0; s)s n p t n q C ' 2(x0; r); (3.11)

where C does not depend on r. Then the operator M is bounded from LMfx0g

p;'1 to

LMfx0g

q;'2 for p > 1 and from LM

fx0g

1;'1 to W LM

fx0g

q;'2 for p = 1.

Also in the case (t) = t and '(x; t) = t pn_{, 0 <} _{< n from Theorem 3.1 we}

get local Morrey space variant of Theorem A.

Corollary 3.3. Let x02 Rn, 0 < < n, 1 < p < n, 0 < < n p. Moreover,

let = n_{p} n_{q} and _{p} = _{q}. Then for p > 1, the operator M is bounded from
LMfx0g

p; to LM

fx0g

q; and for p = 1, M is bounded from LM_{1;}fx0g to W LMq;fx0g.

Remark 3.2. For this case = n_{p} n_{q} necessary and su¢ cient conditions for the
boundedness of I from Mp;'1 to Mq;'2 are obtained in [4].

4. Adams type result for the operator M in the spaces Mp;'

The following theorem was proved in [2].

Theorem D.Let 1 _{p < 1 and ('}_{1}; '_{2}) satisfy the condition
sup
r<t<1
t np _{ess inf}
t<s<1'1(x; s) s
n
p _{C '}
2(x; r);

where C does not depend on x and r. Then the operator M is bounded from Mp;'1

to W Mp;'2 and for p > 1, the operator M is bounded from Mp;'1 to Mp;'2.

The following theorem is another main result of the paper, in which we get the Adams type boundedness of the generalized fractional maximal operator M in the generalized Morrey spaces Mp;'.

Theorem 4.2. Let 1 _{p < q < 1,} _{t}(t)n be almost decreasing, and let (t) satisfy

the condition (3.2) and the inequality Z k2r

0

(s)

s ds C (r);

where k2 is given by the condition (3.2) and C does not depend on r > 0. Let also

'(x; t) satisfy the conditions sup

r<t<1

t n ess inf

t<s<1'(x; s) s

and

(r)'(x; r) + sup

t>r

(t)'(x; t) C'(x; r)pq_{;} _{(4.2)}

where C does not depend on x 2 Rn _{and r > 0.}

Then the operator M is bounded from M

p;'1p to W M_{q;'}1q and for p > 1 from

Mp;'to M q;'1q.

Proof. Let x0 2 Rn, 1 p < q < 1 and f 2 M

p;'1p. Write f = f1+ f2, where

B = B(x; r), f1= f 2B and f2= f {_{(2B)}. Then we have

M f (x) M f1(x) + M f2(x):

For M f1(y), y 2 B(x; r), following Hedberg’s trick (see for instance [27], p. 354),

we obtain
M f1(y) = sup
t>0
(t)
tn
Z
B(y;t)\B(x;2r)jf(z)jdz
. sup
t>0
Z
B(y;t)\B(x;2r)
(jy zj)
jy zjn jf(z)jdz
t sup
t>0
0
X
k= 1
Z
B(y;t)\ B(x;2k+1_{r)}_{nB(x;2}k_{r)}
(jy zj)
jy zjn jf(z)jdz
. sup
t>0
0
X
k= 1
Z 2kk2r
2k_{k}
1r
(s)
sn+1ds
Z
B(y;t)\B(x;2k+1_{r)}jf(z)jdz
t Mf(x) sup
t>0
0
X
k= 1
Z 2kk2r
2k_{k}_{1}_{r}
(s)
s ds
= M f (x)
Z k2r
0
(s)
s ds. Mf(x) (r): (4.3)

For M f2(y), y 2 B(x; r) from (3.7) we have

M f2(y). sup t>2r (t) tn Z B(x;t)jf(z)jdz . sup t>2rkfkLp(B(x;t)) (t) tnp : (4.4)

Then from condition (4.2) and inequalities (4.3), (4.4) for all y 2 B(x; r) we get M f (y). (r) Mf(x) + sup t>rkfkLp(B(x;t)) (t) tnp (r) M f (x) + kfkM p;' 1 p sup t>r '(x; t) (t): (4.5)

Thus, by (4.2) and (4.5) we obtain
M f (y). min '(x; t)pq 1M f (x); '(x; t)
p
qkfk
M
p;'
1
p
. sup
s>0
min spq 1_{M f (x); s}
p
q_{kfk}
M
p;'
1
p
= (M f (x))pq _{kfk}1
p
q
M
p;'
1
p
;

where we have used that the supremum is achieved when the minimum parts are balanced. Hence for all y 2 B(x; r) , we have

M f (y). (Mf(x))pq_{kfk}1
p
q
M
p;'
1
p
:

Consequently the statement of the theorem follows in view of the boundedness of the maximal operator M in M

p;'1p provided by Theorem D in virtue of condition

(4.1).
kM fkW M
q;'
1
q
= sup
x2Rn_{;t>0}
'(x; t) 1q_{t}
n
q_{kM fk}
W Lq(B(x;t))
. kfk1 pq
M
p;'
1
p
sup
x2Rn_{;t>0}
'(x; t) 1q_{t} nq_{kMfk}
p
q
W Lp(B(x;t))
= kfk1
p
q
M
p;'
1
p
sup
x2Rn_{;t>0}
'(x; t) 1p_{t} np_{kMfk}
W Lp(B(x;t))
p
q
= kfk1
p
q
M
p;'
1
p kMfk
p
q
W M
p;'
1
p
. kfkM
p;'
1
p
;
and
kM fkM
q;'
1
q
= sup
x2Rn_{;t>0}
'(x; t) 1q_{t} nq_{kM fk}
Lq(B(x;t))
. kfk1 pq
M
p;'
1
p
sup
x2Rn_{;t>0}
'(x; t) 1q_{t} nq_{kMfk}
p
q
Lp(B(x;t))
= kfk1
p
q
M
p;'
1
p
sup
x2Rn_{;t>0}
'(x; t) 1pt
n
pkMfk
Lp(B(x;t))
p
q
= kfk1
p
q
M
p;'
1
p
kMfk
p
q
M
p;'
1
p
. kfkM
p;'
1
p
;
if 1 < p < q < 1 .

In the case (t) = t from Theorem 4.2 we get the Adams type result on gener-alized Morrey spaces (see [16, Theorem 5.7, p. 182]).

In the case (t) = t , '(x; t) = t n_{, 0 <} _{< n from Theorem 4.2 we get the}

Corollary 4.4. Let 0 < < n, 1 < p < n_{, 0 <} _{< n} _{p and} 1
p

1

q = n .

Then for p > 1, the operator M is bounded from Mp; to Mq; and for p = 1, M

is bounded from M1; to W Mq; .

Remark 4.3. Note that, the condition (3.1) is weaker than the following condition which was given in [17] for I :

Z _{1}
1
(t)
tn
dt
t < 1: (4.6)

For example, the function

(t) = t

n

log(e + t); t > 0

satis…es (3.1), but not (4.6). This example shows that the function satis…es Theorems 3.1 and 4.2, but does not satisfy the assumptions of Theorems 16 and 22 in [17]. In other words, the condition (3.1) which satis…es our main theorems, is better (more general and comprehensive) than the condition (4.8) which satis…es the main theorems were given in [17].

References

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[2] Akbulut, A., Guliyev, V.S. and Mustafayev, R., On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces, Math. Bohem. 137 (1) (2012), 27-43.

[3] Burenkov, V., Guliyev, H.V. and Guliyev, V.S., Necessary and su¢ cient conditions for bound-edness of the Riesz operator in the local Morrey-type spaces, Doklady Ross. Akad. Nauk. Matematika 412 (5) (2007), 585-589 (Russian). English trans. in Acad. Sci. Dokl. Math. 76 (2007).

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bound-edness of the fractional maximal operator in the local Morrey-type spaces, Doklady Ross. Akad. Nauk. Matematika 409 (4) (2006), 443-447 (Russian). English trans. in Acad. Sci. Dokl. Math. 74 (2006).

[6] Burenkov, V., Guliyev, H.V. and Guliyev, V.S., Necessary and su¢ cient conditions for bound-edness of the fractional maximal operator in the local Morrey-type spaces, J. Comput. Appl. Math. 208 (2007), no. 1, 280-301.

[7] Burenkov, V., Gogatishvili, A., Guliyev, V. and Mustafayev, R.Ch., Boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 35 (2011), no. 1, 67-87.

[8] Burenkov, V., Gogatishvili, A., Guliyev, V.S. and Mustafayev, R., Boundedness of the frac-tional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ. 55 (8-10) (2010), 739-758.

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[10] Eridani, A., On the boundedness of a generalized fractional integral on generalized Morrey spaces, Tamkang J. Math. 33 (2002), no. 4, 335-340.

[11] Eridani, A., Gunawan, H., Nakai, E. and Sawano, Y., Characterizations for the generalized fractional integral operators on Morrey spaces, Math. Inequal. Appl. 17 (2014), no. 2, 761-777. [12] Eridani, A., Gunawan, H. and Nakai, E., On generalized fractional integral operators, Sci.

Math. Jpn. 60 (2004), no. 3, 539-550.

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Russian)

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[17] Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A. and Serbetci, A., Generalized fractional integral operators on generalized local Morrey spaces, Journal of Function Spaces, Volume 2015, Article ID 594323, 8 pages.

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Current address : A. Kucukaslan(Coressponding author): Institute of Mathematics, Czech Academy of Sciences, 11567, Prague, Czech Republic, School of Applied Sciences, Pamukkale University, 20680, Denizli, Turkey

E-mail address : kucukaslan@pau.edu.tr

ORCID Address: http://orcid.org/0000-0002-9207-8977

Current address : V.S. Guliyev: Nikolskii Institute of Mathematics at RUDN University, 117198, Moscow, Russia, Department of Mathematics, Dumlupinar University, 43100, Kutahya, Turkey

E-mail address : vagif@guliyev.com

ORCID Address: http://orcid.org/0000-0001-7486-0298

Current address : A. Serbetci: Ankara University, Faculty of Sciences, Dept. of Mathematics, Ankara, TURKEY, Department of Mathematics, Cankiri Karatekin University, Cankiri, Turkey

E-mail address : serbetci@ankara.edu.tr