#### Abstract

We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on generalized non-homogeneous Morrey spaces and extend to some singular integral operators. In addition, we also prove the boundedness of generalized fractional integral operators on generalized non-homogeneous Orlicz-Morrey spaces.

#### 1. Introduction

In this paper, we prove that the theory of generalized Morrey spaces can be staged on the nondoubling setting on , so that we assume that is a positive Borel measure on satisfying the growth condition; that is, there exist and such that for any ball centered at with radius (see [1–4]). For and a measurable function , we define the generalized fractional integral operator by for any suitable function on . This operator dates back to the book when for [5, Section 6.1]. Note that if is the Lebesgue measure, then is the fractional integral operator introduced in [6, 7]. See also [8, 9] for exhaustive and comprehensive explanation about the operator. Below, we will always assume the Dini condition, that is, and we also assume that satisfies the so-called growth condition; namely, there exist constants and such that for every . For convenience, write . Note that if satisfies the doubling condition, that is, there exists a constant such that whenever , then satisfies the growth condition. See [10–13] for discussion about , where satisfies the doubling condition.

Now, we say that a function belongs to the generalized nonhomogeneous Morrey space for a function and if Note that this definition is a special case of [14, Definition 1.1], where different types of operators are considered. In this paper, we will assume the following two conditions.(1.a)The function is almost decreasing; that is, there exists a constant such that for every .(1.b)The function is almost increasing; that is, there exists a constant such that for every .These two conditions imply that satisfies the doubling condition. Note that if , then is the nonhomogeneous Lebesgue space.

The study of the boundedness of the fractional integral operator on generalized Morrey spaces was initiated in [15, Theorem 3]. The following theorem presents the weak type inequalities for on generalized nonhomogeneous Morrey spaces.

Theorem 1 (see [16, Theorem 2.4]). *Let . Suppose that , , and there exist positive constants and such that
**
for every . Then, there exists a constant such that, for any function and any ball , one has
**
for every .*

*Remark 2. *Note that we can obtain the weak type inequalities for on nonhomogeneous Lebesgue spaces which are proved in [17, 18] by taking and in Theorem 1. By substituting for some to , we have for every , which is one of the hypotheses in the weak type inequalities for in [19].

The proof of Theorem 1 employs some inequalities involving the modified Hardy-Littlewood maximal operator (see [8]), which is defined for any locally integrable function by and the Chebyshev inequality which is presented in the following theorem.

Theorem 3 (see [20]). *Let be a measurable subset of . If is an integrable function on , then, for every , one has
*

One of the reasons why we are fascinated with the generalized fractional integral operators is that these operators appear naturally in the context of differential equations; see [21, Section 6.4] for a nice explanation in connection with the holomorphic calculus of operators and see [22, ] and [23, Lemma 2.5] for a detailed account that with satisfies the requirement of in the present paper. In addition, investigating generalized Morrey spaces is not a mere quest to the abstract theory; it arises naturally in the context of Sobolev embedding. In [24], the following proposition is proved.

Proposition 4 (see [24, Theorem 5.1]). *Let and . Then, there exists a positive constant such that
**
holds for all with and for all balls , where is the abbreviation of with .*

Later Proposition 4 is strengthened by [25, Example 5]. An example in [24] as well as the necessary and sufficient condition obtained in [25, Theorem 1.3] implicitly shows that the log factor above is absolutely necessary.

In this paper, we will prove the weak type inequalities for which is a generalization of Theorem 1. In Section 2, we will prove the weak type inequalities for by using the Chebyshev inequality and some inequalities involving operator . In Section 3, we will prove a Hedberg type inequality on generalized nonhomogeneous Morrey space by adapting the proof of a Hedberg type inequality on homogeneous setting in [25]. Through the weak type inequalities for , we then prove the weak type inequalities for on generalized nonhomogeneous Morrey spaces. In Section 4, we extend our results to the singular integral operators defined in [1]. Finally, in Section 5, we prove the boundedness of on generalized nonhomogeneous Orlicz-Morrey spaces. See [26–28] for related results.

Throughout the paper, denotes a positive constant which is independent of the function and the variable and may have different values from line to line. We also denote by () the fixed constants that satisfy certain conditions.

#### 2. Weak Type Inequalities for via the Chebyshev Inequality

Now, we give an inequality which is used in the proof of the weak type inequalities for in the following lemma.

Lemma 5. *Let . If and satisfy
**
for every , then, for any ball and every locally integrable function , one has
*

*Proof. *Let , for any ball . By the dyadic decomposition of the ball and the growth condition of , we have
Then, we use the overlapping property (see [29, 30]) to obtain
If we use (10) and the doubling condition of , then we have
Hence, .

By letting or , we have the following.

Corollary 6. *Let and be any ball in . If the functions and satisfy inequality (10), then
**
and, for any ball , one has
*

*Remark 7. *These two inequalities will be used later to prove one of our main theorems. The next lemma presents an inequality involving the modified Hardy-Littlewood maximal operator . This inequality is an important part of the proof of the weak type inequalities for in [16, 19]. See [8] for similar results.

Lemma 8 (see [16]). *Let . If satisfies for every , then, for any function and any ball , one has
*

With Theorem 3, Corollary 6, and Lemma 8, we are now ready to prove the weak type inequalities for on generalized nonhomogeneous Morrey spaces.

Theorem 9 (see [31]). *Let and assume that . If and satisfy
**
for every , then, for any function and any ball , one has
**
for every .*

*Proof. *Let be any ball in . For every and , let
Let , for any . Since , we have
By the dyadic decomposition of and the growth condition of , we have
We use Hölder’s inequality, the growth condition of , and the definition of to obtain
By using the doubling condition of and the overlapping property, we have
Now, we invoke the integral assumption on :
Hence,
Let . Remark that implies that . Otherwise,
which is impossible. Now, from , we can find such that, for and , we have
Since , there exists such that . Hence,
Taking , we obtain
Consequently,
We combine Hölder’s inequality and the inequality (15) to obtain
Finally, by using the last inequality and the Chebyshev inequality, we get
By virtue of the inequalities (16), (17), and (29) as well as the definition of , we get
as desired.

*Remark 10. *Note that , where satisfies the condition of Theorem 9 and, for this , we obtain the weak type inequalities for in Theorem 1.

#### 3. Weak Type Inequalities for via a Hedberg Type Inequality and Weak Type Inequalities for

In this section, we will prove weak type inequalities for using a different technique, namely, via a Hedberg type inequality and weak type inequalities for . It turns out that some hypotheses can be removed. The Hedberg type inequality is presented in the following proposition.

Proposition 11 (see [25, 31]). *Let . If and satisfy
**
for every , then, for every and , one has
*

*Proof. *We adapt the proof of a Hedberg type inequality on generalized Morrey space in [25]. For every and , write , where and are defined in the proof of Theorem 9. By using the inequalities (11) and (26), we get
Next, we separate the proof into the following two cases.*First Case *. In this case, we have
for every . Hence, .*Second Case. *. We use Hölder’s inequality, the growth condition of , and the definition of to obtain
for every . Hence,
Since , we have
Thus, there exists such that
for and . Since , there exists such that
By choosing in the inequality (37) and using the inequality (43), we have
From these two cases, we obtain the inequality (36).

Sihwaningrum et al. [19] proved the weak type inequalities for on generalized nonhomogeneous Morrey space by assuming that satisfies the integral condition; that is, for every . In [19], the weak type inequalities for are also proved by using the weak type inequalities for . In this paper, we remove the integral condition of in the hypothesis of our proposition below. See [32, Theorem 2.3] and [33, Theorem 2.3] for such attempts.

Proposition 12 (see [31]). *Let ; then, there exists a constant such that, for any function and any ball , one has
**
for every .*

When , we have the strong boundedness; see [15, Theorem 1] and [34, Lemma 2.4] for the Lebesgue case and see [35, Theorem 4.3] for the strong to result and the weak to result with equal to the Lebesgue measure.

*Proof. *The proof is similar to that of strong boundedness of maximal operator on generalized nonhomogeneous Morrey spaces which is discussed in [34]. The difference is that in the final step we use the Chebyshev inequality, as we will see below. Consider the ball . Let and let be any positive real number. For , define and . Note that
Since , for every , we have
for every . Hence,
Observe that, for every , we have
Thus,
Since , we have
For the first term, we use the weak type inequalities for on the nonhomogeneous Lebesgue space (see [18]) to obtain
Meanwhile, for the second term, by using the Chebyshev inequality and the inequality (50), we have
Finally, by combining these two estimates, we obtain inequality (45).

With Propositions 11 and 12, we are now ready to prove the weak type inequalities for on generalized nonhomogeneous Morrey spaces.

Theorem 13 (see [31]). *Let . If and satisfy inequality (35), then, for any function and any ball , one has
**for every .*

*Proof. *This proof is adapted from [25]. We replace by . Consider the ball . By applying Proposition 11, we have
Observe that the second term in the most right-hand side of the above inequality vanishes, when
So, to estimate the term, we can suppose that
With this in mind, we calculate
Meanwhile, by using Proposition 12, we have
By summing the two previous estimates, we get the desired inequality.

*Remarks 1. *(i) Note that the hypotheses in Theorem 9 are not included in Theorem 13, since we can prove the weak type inequalities for without this condition.

(ii) The conditions on , namely, and , are not included in the hypotheses in Theorem 13. However, we have to use the weak type inequalities for on generalized nonhomogeneous Morrey spaces and a Hedberg type inequality for in the proof of Theorem 13.

#### 4. Boundedness of Singular Integral Operators

Proposition 12 carries over to the singular integral operator whose definition is given in [1]. Recall that the singular integral operator is a bounded linear operator on for which there exists a function that satisfies three properties listed below.(4.a)There exists such that for all .(4.b)There exist and such that if with .(4.c)If is a bounded -measurable function with a compact support, then we have

As for this singular integral operator , the following result is due to Nazarov, Treil, and Volberg.

Proposition 14 (see [1, 2]). *The singular operator is bounded on for . Moreover, there exists a constant such that
**
for every and every .*

Theorem 15. *Let be a singular integral operator. Let . In addition to the doubling condition, assume that
**
for every . Then, there exists a constant such that, for any function and any ball , one has
**
for every .*

*Proof. *The proof is a modification of that of Proposition 12. We decompose as before. The treatment of is the same as that in Proposition 12 but by using the weak type inequality for in Proposition 14. We need to take care of . By the condition , Hölder’s inequality, and the growth condition of , we have
If we use our integrability assumption, then we have a pointwise estimate:
So, we are done.

*Remark 16. *If we define the generalized weak Morrey space of nonhomogeneous type to be the set of all -measurable functions such that
then the inequality (64) amounts to the boundedness of from to . Similarly, our previous results can be translated into this language. In the following section, we will use these notations for convenience.

#### 5. Generalized Nonhomogeneous Orlicz-Morrey Spaces

Our results above can be carried over to generalized nonhomogeneous Orlicz-Morrey spaces. We first formulate our main results and then prove them later in Sections 5.1–5.3.

Recall that is a Young function, if is bijective and convex. We define the -average of over a ball as follows: The generalized nonhomogeneous Orlicz-Morrey space is the set of all for which the norm is finite. Note that if , then . About the structure of this function space, we have the following.

Theorem 17. *Let be a Young function and let satisfy the two conditions (1.a) and (1.b) as usual. Then, is a Banach space.*

We define the generalized weak Orlicz-Morrey spaces of nonhomogeneous type as follows. For a Young function , the generalized weak Orlicz-Morrey space of nonhomogeneous type is the set of all -measurable functions for which the norm is finite. Write , when . It is not so hard to prove for all -measurable functions from the inequality By taking and and is the Lebesgue measure on , we see that , showing that is a proper superset of .

Theorem 18. *Let be a Young function and let satisfy the two conditions (1.a) and (1.b) as usual. Then, is a quasi-Banach space. More precisely,*(1)* if and only if ;*(2)* for all and ;*(3)*if is a sequence in such that
Then, there exists such that
*

We prove the following boundedness result on generalized nonhomogeneous Orlicz-Morrey spaces.

Theorem 19. *Let be a Young function and let satisfy the two conditions (1.a) and (1.b) as usual. Then, the maximal operator is bounded from to . If we assume that
**
and that satisfies the doubling condition, then the singular integral operator is bounded from to .*

Theorem 20. *Let and, for some , , satisfy
**
for every . Suppose that is a Young function with the doubling condition. Set
**
Then,
*

##### 5.1. Proof of Theorems 17 and 18

We start with a lemma.

Lemma 21. *Let be a Young function with the doubling property:
**
For -measurable functions and and a ball , one has
*

*Proof. *If -a.e. or -a.e., then we have the equality trivially; so let us assume that -a.e. and -a.e. Then, by virtue of the convexity, we have
From the definition of the quantity , we obtain the inequality.

Lemma 22. *If is a Young function, then
**
for any ball and -measurable function .*

*Proof. *A normalization allows us to assume that ; our target will be to prove
In view of the growth condition, we may suppose that assumes its value in . Since is a Young function, we have
Therefore,
So, we are done.

Now, we are ready for the proof of Theorem 17.

*Proof of Theorem 17. *In view of Lemma 21, is a normed space. So, we need to prove the completeness. To this end, we choose a sequence of -measurable function such that
Denote by the origin. Then, we have
from Lemma 22. This implies that is finite -a.e. on for all . Hence, is finite -a.e. on . With this in mind, let us set whenever the series is absolutely convergent; otherwise set .

We fix a ball . Then, we have
As a result, and
So converges to