Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 809704, 12 pages

http://dx.doi.org/10.1155/2013/809704

Research Article

Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces

Hendra Gunawan,¹ Denny Ivanal Hakim,¹ Yoshihiro Sawano,² and Idha Sihwaningrum³

¹Department of Mathematics, Bandung Institute of Technology, Bandung 41032, Indonesia

²Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397, Japan

³Department of Mathematics, Jenderal Soedirman University, Purwokerto 53122, Indonesia

Received 9 September 2013; Accepted 20 November 2013

Academic Editor: Vagif Guliyev

Copyright © 2013 Hendra Gunawan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 in .

The proof of Theorem 18 is similar; we use the embedding

which follows from Lemma 22.

5.2. Proof of Theorem 19

We first concentrate on the maximal operator; we modify the argument to prove the boundedness of singular integral operators later.

The proof hinges upon the decomposition in Proposition 12, keeping the same notation as before. As for , we have a pointwise estimate, so that a small modification works. Also, we normalize .

Let us concentrate on . Let us establish

for any , where the constant is independent of and . Write . Then, we have

by virtue of the dominated convergence theorem. So, we have

So

Since

we have (by the convexity of )

In view of the doubling property, we are done with the maximal operator.

As for the singular integral operator, we combine the above proof and that of Theorem 15. We mimic the argument above for , while we use estimate (66) obtained in the proof of Theorem 15. We omit the further details.

5.3. Proof of Theorem 20

We start with the proof of a Hedberg type inequality. Let . Then, as in (37), we have

So we are led to

as we did in Proposition 11.

So we have to prove

As for the first inequality, we use the following observation:

In view of the definition of , we are done with the estimate.

As for the second inequality, we proceed as follows:

Here, for the last inequality, we used Theorem 19.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank Professor E. Nakai of Ibaraki University for his useful comments on the original paper. The first and second authors are supported by ITB Research and Innovation Program 2013. The third author is partially supported by Grant-in-Aid for Scientific Research (C), no. 24540174, Japan Society for the Promotion of Science. The fourth author is supported by Fundamental Research Program 2013, Directorate General of Higher Education, Ministry of Education and Culture, Indonesia.

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