Boundedness of the Riesz Potential in Generalized Morrey Spaces
Hairur Rahman
1
, M. I. Utoyo
2
, and Eridani
2
1
Department of Mathematics, Universitas Islam Negeri Maulana Malik Ibrahim, Malang, 61515, Indonesia
2
Department of Mathematics, Analysis Mathematical Research Group, Airlangga University, Surabaya 60115,
Indonesia
Keywords: Riesz Potential, Morrey Spaces, Boundedness.
Abstract: The purpose of this paper is to prove the necessary and sufficient condition for the boundedness of Riesz
operators on homogeneous generalized Morrey spaces. Further, we will make use the Q-Ahlfors regularity
condition in the proof instead of usual doubling conditions.
1 INTRODUCTION
In this paper, we shall discuss about the
boundedness of a Riesz potential integral operator.
The boundedness of operator
on the several
homogeneous metric measure spaces has been
proved by some researchers (Eridani and Gunawan,
2009; Eridani, Kokilashvili and Meshky, 2009;
Nakai, 2000; Petree, 1969). Such boundedness
results have been obtained in the several kinds of
Morrey spaces thanks to the doubling condition
obeyed by the measure of homogeneous metric
measure spaces on the Euclid spaces (Adams, 1975;
Chiarenza et al. 1987; Petree, 1969). The Euclid
spaces combined with Lebesgue measure is the most
trivial example of the boundedness result of
on
homogeneous spaces. The generalized Morrey
spaces was introduced later by (Nakai, 2000) who
also proved the boundedness of
in those spaces.
Following from this progress, Eridani and Gunawan
obtained proof for the boundedness of the fractional
integral operator
on the generalized Morrey
spaces (Eridani & Gunawan, 2009). The further
results in the same line were obtained by Sobolev,
Spanne, Adams, Chiarensa dan Frasca, Nakai and
Gunawan and Eridani related to the boundedness of
on generalized Morrey spaces on Euclid spaces
equipped with Euclid norm
(Adams, 1975;
Chiarenza et al. 1987; Eridani and Gunawan, 2009;
Nakai, 2000). Furthermore, the result obtained by
Utoyo has described the generalized necessary and
sufficient condition for the boundedness of
on
classic and generalized Morrey spaces (Utoyo et al.
2012).
The boundedness of
in the results was
obtained using doubling condition obeyed by
measure on generalized Morrey spaces. This type of
spaces is called homogeneous spaces, the metric
measure spaces on which the measure obeys the
doubling condition. As the generalization of
homogenous properties of spaces, Ahlfors defined
the regularity condition
where
and
are some positive constants.
In this paper, we will prove the necessary and
sufficient conditions for the boundedness of
on
the generalizd Morrey spaces similar to the previous
results using Ahlfors regularity condition. All the
results in this article can be considered as the
alternative for the corresponding homogeneous
results.
2 LITERATURE REVIEW
Our result of the boundedness result of
on the
homogeneous generalized Morrey spaces
generalizes the following theorem about the
boundedness property of fractional integral operator
on the homogeneous classic Morrey space. The
theorem stated as the following.
Theorem 2.1. Let be a homogeneous metric
measure space, be a measure on ,
and
.
Then
is bounded from
to
if
and only if there is a constant such that for
every ball on
.
Rahman, H., Utoyo, M. and Eridani, .
Boundedness of the Riesz Potential in Generalized Morrey Spaces.
DOI: 10.5220/0008524405010505
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 501-505
ISBN: 978-989-758-407-7
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
501
The modification of the preceding theorem,
replacing the condition
, is
stated as the following.
Theorem 2.2. Let and
. The operator
is bounded from
to
if and only if
with
and
.
As the generalization of the above theorems, in this
article, we will prove the necessary and sufficient
conditions for the boundedness of
on the
homogeneous generalized Morrey space. The
generalized Morrey space is denoted by
defined as the set of functions
such
that
where
is a function satisfying
and . The
generalized Morrey space
is the strong
generalization of classic Morrey spaces
. By
choosing
where then the
corresponding generalized Morrey space reduces to
classic Morrey spaces
and hence is so for
Lebesgue spaces
.
Analysis of boundedness of
on the generalized
Morrey spaces requires two condition for function
, that is
(1) is said to satisfy doubling condition, denoted
by if there is a constant such
that for every and , if
then
,
(2) is said to satisfy integral condition
(integration condition) and denoted by
if there is a constant such that for
every ,
The boundedness results for fractional integral
operator
on generalized Morrey spaces has been
proven by (Nakai, E.) in the following theorem.
Theorem 2.3. (Kokilashvili and Meshky, 2005) If
with functions satisfy:
there is a constant such that for every
then
is bounded from
to
.
This result shows that
is bounded from
to
. Furthermore, the statement in the above
theorems is the implication statement, in sense that it
only says about the sufficient condition of the
boundedness of the operator. For that reason, in
determining of complete theory about the
boundedness of the fractional integral operator
on
the generalized Morrey spaces.In this article,we will
construct the necessary conditions for the
boundedness of the operator as acompanion to the
theorem above. Using the theorem from Adams-
Zhiarenza-Frasca, Gunawan and Eridani, which
states that (Eridani and Gunawan, 2009) shows that
is bounded from
to
. Their result is stated
by the following theorem.
Theorem 2.4. (Eridani and Gunawan, 2009) Let
and there is a
constant such that for every
where
. then
is bounded
from
to
where
.
As the preceding results, the boundedness theorem
of
on the generalized Morrey spaces stated above
is the implication statement. Then, also will be
developed in this article to be, the boundedness from
to
and
to
with biimplication form on the metric measure
space.
3 RESULTS
The first result in our paper is the boundedness
property of fractional integral operator similar to that
of Theorem 2.1, and 2.2. The difference is that the
measures used in the spaces are made to be different
cause maximal operator to be unusable to prove the
boundedness properties of the operator. Also, the
condition of the boundedness of the fractional
integral operator
in our result uses Ahlfors
regularity condition instead of the traditional
doubling condition. The following is the de_nition
of generalized Morrey spaces equipped with
ICMIs 2018 - International Conference on Mathematics and Islam
502
measures and which is alowed to be different in
the later boundedness results.
Definition 3.1. Let be a measure on , ,
and function
. The generalized
Morrey space
is de_ned as the set of
functions
, such that the following
equation holds
,
with the supremum is evaluated over every ball
on .
Remark 3.2. If , then
.
In the above equation, and later on this article,
is always assumed to satisfy the following both
conditions:
1. is almost decreasing function, that is, there
is a constant such that for every
2.
is almost increasing function, that is,
there is a constant such that for every
The above conditions ensure that the functions
and , appearing in the boundedness property, does
not too rapidly blow up to infinity nor rapidly decay
to zero respectively. The following theorem states
about the condition that must be satisfied by the
functions and , and also measure appearing in
the spaces, in order to ensure the boundedness
property of
form the spaces
to
.
Theorem 3.3. Let be a homogeneous
metric space, and
. If
and
,, that is satisfies the Q-Ahlfors
regularity condition, and
then,
is bounded from
to
.
Proof. Necessity. Suppose that
is bounded from
to
such that
Then,
.
were and thus,
Since
and
,
,
,
Sufficiency. Let ball B be an arbitrary ball on that
is
. Assume that
. and
. then we write
,
Boundedness of the Riesz Potential in Generalized Morrey Spaces
503
.
If
then, according to Hardy-
Littlewood-Sobolev inequality, we obtain
.
Next, we estimate
. According to definition
of
, we have
Then, using Holder’s inequality, we obtain
.
Hence, according to the hypothesis of the theorem,
we obtain
and
Thus, we obtain the following inequality
.
The above result can be written as
.
Following the above results, the next corollary is
the simple implication of Theorem 3.3.
Theorem 3.4. Let be a homogeneous metric
spaces,
, and satisfies Q-
Ahlfors regularity condition. if
ICMIs 2018 - International Conference on Mathematics and Islam
504
, and
, for some constant
, that is satisfies the Q-Ahlfors regularity
condition, and
.
Then,
is bounded from
to
if and only if
When , the above theorem implies the
following corollary.
Corollary 3.5. Let
be a homogeneous
metric space, and
. If
, and
, that is satisfies the - Ahlfors regularity
condition, and
.
Then,
is bounded from
to
if
and only if
.
Corollary 3.6. Let
be a homogeneous
metric space, and
. If
, and, that is satisfies the -
Ahlfors regularity condition, and
.
Then,
is bounded from
to
if
and only if
.
ACKNOWLEDGEMENTS
This research article was developed with a certain
purpose related to doctorate program.
REFERENCES
Adams, R. A., 1975. Sobolev Spaces, Pure and Applied
Mathematics. Academic Press, London.
Chiarenza, F. and Frasca, 1987. Morrey Spaces and
Hardy-Littlewood Maximal Function. Rend. Mat.,
7(1), 273-279.
Edmunds D, Kokilashvili V. and Meshky, A., 2002.
Bounded and Compac integral Operator (Mathematics
and It's Applications). Springer, p. 543.
Eridani and Gunawan, H., 2009. Fractional Integrals and
Generalized Olsen Inequalities. Kyungpook
Mathematical Journal, 49(1), 31-39.
Eridani and Gunawan, H., 2002. On Generalized
Fractional Integrals, J. Indonesian Math. Soc.
(MIHMI), 8(1), 25-28.
Eridani, V. Kokilashvili, and Meshky, A., 2009. Morrey
Spaces and Fractional Integral Operators, Expo. Math,
27(3), 227-239.
Kokilashvili, V and Meshky, A., 2005. On Some
Weighted Inequlities for Fractional integrals on Non-
homogeneous Spaces, Journal for Analysis and It's
Applications, 24(4), 871- 885.
Nakai, E., 1994. Hardy-Littlewood Maximal Operator,
Singular integral Operators, and the Riesz Potential on
the Generalized Morrey Spaces, Math. Nachr., 166,
95-103.
Nakai, E., 2000. On Generalized Fractional Integrals in the
Orlicz Spaces, Proc. of the Second ISAAC Congress
2000, 75-81.
Petree, J., 1969. On the Theory of L Spaces, J. Funct. Anal
4, 71-87.
Stein, E. M., 1970. Singular Integrals and Differentiability
Properties of Functions. Princeton University Press,
New Jersey.
Stein, E. M., 1993. Harmonic Analysis; Real Variable
methods, Orthogonality, and Oscillatory Integrals.
Princeton University Press, New Jersey.
Utoyo, M. I., Nusantara, T., Widodo, B. and
Suhariningsih, 2012. Fractional integral operator and
Olsen inequality in the non-homogeneous classic
Morrey space, Int. J. Math. Anal. 6 (31), 1501-1511.
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