The present paper is an offspring of [1]. We obtain some inequalities for generalized fractional integral operators on generalized Morrey spaces. We also show the boundedness property of the generalized fractional integral operators on the predual of the generalized Morrey spaces. They generalize what was shown in [1]. We will go through the same argument as [1].

For

the classical fractional integral operator

and the classical fractional maximal operator

are given by

In the present paper, we generalize the parameter

. Let

be a suitable function. We define the generalized fractional integral operator

and the generalized fractional maximal operator

by

Here, we use the notation
to denote the family of all cubes in
with sides parallel to the coordinate axes,
, to denote the sidelength of
and
to denote the volume of
. If
,
, then we have
and
.

A well-known fact in partial differential equations is that
is an inverse of
. The operator
admits an expression of the form
for some
. For more details of this operator we refer to [2]. As we will see, these operators will fall under the scope of our main results.

Among other function spaces, it seems that the Morrey spaces reflect the boundedness properties of the fractional integral operators. To describe the Morrey spaces we recall some definitions and notation. All cubes are assumed to have their sides parallel to the coordinate axes. For
we use
to denote the cube with the same center as
, but with sidelength of
.
denotes the Lebesgue measure of
.

Let

and

be a suitable function. For a function

locally in

we set

We will call the Morrey space

the subset of all functions

locally in

for which

is finite. Applying Hölder's inequality to (1.3), we see that

provided that

. This tells us that

when

. We remark that without the loss of generality we may assume

(See [1].) Hereafter, we always postulate (1.4) on
.

If

,

,

coincides with the usual Morrey space and we write this for

and the norm for

. Then we have the inclusion

when
.

In the present paper, we take up some relations between the generalized fractional integral operator
and the generalized fractional maximal operator
in the framework of the Morrey spaces
(Theorem 1.2). In the last section, we prove a dual version of Olsen's inequality on predual of Morrey spaces (Theorem 3.1). As a corollary (Corollary 3.2), we have the boundedness properties of the operator
on predual of Morrey spaces.

Let

be a function. By the Dini condition we mean that

fulfills

while the doubling condition on

(with a doubling constant

) is that

satisfies

We notice that (1.4) is stronger than the doubling condition. More quantitatively, if we assume (1.4), then

satisfies the doubling condition with the doubling constant

. A simple consequence that can be deduced from the doubling condition of

is that

The key observation made in [

1] is that it is frequently convenient to replace

satisfying (1.6) and (1.7) by

:

Before we formulate our main results, we recall a typical result obtained in [1].

Proposition 1.1 (see [1, Theorem 1.3]).

and

. Suppose that

is nonincreasing. Then

where the constant
is independent of
and
.

The aim of the present paper is to generalize the function spaces to which
and
belong. With theorem 1.2, which we will present just below, we can replace
with
and
with
. We now formulate our main theorems. In the sequel we always assume that
satisfies (1.6) and (1.7), and
is used to denote various positive constants.

Theorem 1.2.

Suppose that

and

are nondecreasing but that

and

are nonincreasing. Assume also that

where the constant
is independent of
and
.

Remark 1.3.

Let

and

. Then

and

satisfy the assumption (1.13). Indeed,

Hence, Theorem 1.2 generalizes Proposition 1.1.

Letting
and
in Theorem 1.2, we obtain the result of how
controls
.

Corollary 1.4.

Let

. Suppose that

Corollary 1.4 generalizes [

3, Theorem 4.2]. Letting

in Theorem 1.2, we also obtain the condition on

and

under which the mapping

is bounded.

Corollary 1.5.

In particular, if

, then

Here,

denotes the Hardy-Littlewood maximal operator defined by

We will establish that
is bounded on
when
(Lemma 2.2). Therefore, the second assertion is immediate from the first one.

Theorem 1.6.

Let

. Suppose that

and

are nondecreasing but that

and

are nonincreasing. Suppose also that

where the constant
is independent of
and
.

Theorem 1.6 extends [4, Theorem 2], [1, Theorem 1.1], and [5, Theorem 1]. As the special case
and
in Theorem 1.6 shows, this theorem covers [1, Remark 2.8].

Corollary 1.7 (see [1, Remark 2.8], see also [6–8]).

Let

. Suppose that

Nakai generalized Corollary 1.7 to the Orlicz-Morrey spaces ([9, Theorem 2.2] and [10, Theorem 7.1]).

We dare restate Theorem 1.6 in the special case when
is the fractional integral operator
. The result holds by letting
,
and
.

Proposition 1.8 (see [1, Proposition 1.7]).

Let

,

,

, and

. Suppose that

,

,

,

, and

then

where the constant
is independent of
and
.

Proposition 1.8 extends [4, Theorem 2] (see [1, Remark 1.9]).

Remark 1.9.

The special case
and
in Proposition 1.8 corresponds to the classical theorem due to Adams (see [11]).

The fractional integral operator
,
, is bounded from
to
if and only if the parameters
and
satisfy
and
.

Using naively the Adams theorem and Hölder's inequality, one can prove a minor part of

in Proposition 1.8. That is, the proof of Proposition 1.8 is fundamental provided

Indeed, by virtue of the Adams theorem we have, for any cube

,

The condition

,

reads

if

. In view of inclusion (1.5), the same can be said when

. Also observe that

Hence we have

. Thus, since the condition

, Proposition 1.8 is significant only when

The case

(the case of the Lebesgue spaces) corresponds (so-called) to the Fefferan-Phong inequality (see [

12]). An inequality of the form

is called the trace inequality and is useful in the analysis of the Schrödinger operators. For example, Kerman and Sawyer utilized an inequality of type (1.32) to obtain an eigenvalue estimates of the operators (see [13]). By letting
, we obtain a sharp estimate on the constant
in (1.32).

In [14], we characterized the range of
, which motivates us to consider Proposition 1.8.

Proposition 1.10 (see [14]).

Let

,

, and

. Assume that

(1)
is continuous but not surjective.

(2)Let

be an auxiliary function chosen so that

,

and that

,

,

. Then the norm equivalence

holds for
, where
denotes the Fourier transform.

In view of this proposition
is not a good space to describe the boundedness of
, although we have (1.29). As we have seen by using Hölder's inequality in Remark 1.9, if we use the space
, then we will obtain a result weaker than Proposition 1.8.

Finally it would be interesting to compare Theorem 1.2 with the following Theorem 1.11.

Theorem 1.11.

Let

. Suppose that

,

, and

are nondecreasing and that

and

are nonincreasing. Then

where the constant
is independent of
and
.

Theorem 1.11 generalizes [1, Theorem 1.7] and the proof remains unchanged except some minor modifications caused by our generalization of the function spaces to which
and
belong. So, we omit the proof in the present paper.