Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 690258, 14 pages

http://dx.doi.org/10.1155/2013/690258

Research Article

Martingale Morrey-Hardy and Campanato-Hardy Spaces

Eiichi Nakai,¹ Gaku Sadasue,² and Yoshihiro Sawano³

¹Department of Mathematics, Ibaraki University, Mito, Ibaraki 310-8512, Japan

²Department of Mathematics, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan

³Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Osawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan

Received 10 June 2013; Accepted 17 August 2013

Academic Editor: Natasha Samko

Copyright © 2013 Eiichi Nakai 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 introduce generalized Morrey-Campanato spaces of martingales, which generalize both martingale Lipschitz spaces introduced by Weisz (1990) and martingale Morrey-Campanato spaces introduced in 2012. We also introduce generalized Morrey-Hardy and Campanato-Hardy spaces of martingales and study Burkholder-type equivalence. We give some results on the boundedness of fractional integrals of martingales on these spaces.

1. Introduction

Lebesgue spaces and Hardy spaces play an important role in martingale theory and in harmonic analysis as well. Morrey-Campanato spaces are very useful to know more precise properties of functions and martingales. It is known that Morrey-Campanato spaces contain , , and as special cases; see, for example, [1, 2].

In martingale theory, Weisz [3] introduced martingale Lipschitz spaces for general filtrations and proved the duality between martingale Hardy spaces and martingale Lipschitz spaces. This result was extended to generalized martingale Campanato spaces and martingale Orlicz-Hardy spaces in [4]. Recently, martingale Morrey-Campanato spaces were introduced in [5], where each sub--algebra is generated by countable atoms.

In this paper, we introduce martingale Morrey-Hardy and Campanato-Hardy spaces based on square functions and unify Hardy, Lipschitz, and Morrey-Campanato spaces in [3–5]. To do this, we first introduce generalized martingale Morrey-Campanato spaces by using subfamilies of the filtration with for each . We establish Burkholder-type equivalence and discuss equivalence between martingale Morrey spaces and martingale Campanato spaces in a suitable condition. We also establish a John-Nirenberg-type theorem for generalized martingale Campanato-Hardy spaces; see Theorem 15.

On these martingale spaces, we introduce generalized fractional integrals as martingale transforms and prove their boundedness. Our result extends several results in [5–7] to these spaces. The fractional integrals are very useful tools to analyse function spaces in harmonic analysis. Actually, on the Euclidean space, Hardy and Littlewood [8, 9] and Sobolev [10] investigated the fractional integrals to establish the theory of Lebesgue spaces and Lipschitz spaces. Stein and Weiss [11], Taibleson and Weiss [12], and Krantz [13] also investigated the fractional integrals to establish the theory of Hardy spaces; see also [14]. The - boundedness of the fractional integrals is well known as the Hardy-Littlewood-Sobolev theorem derived from [8–10]. This boundedness has been extended to Morrey-Campanato spaces by Peetre [1] and Adams [15]; see also Chiarenza and Frasca [16]. In martingale theory, based on the result on the Walsh multiplier by Watari [7, Theorem 1.1], Chao and Ombe [6] proved the boundedness of the fractional integrals for , , , and Lipschitz spaces of the dyadic martingale. The boundedness of the fractional integrals for martingale Morrey-Campanato spaces was established in [5]. For other types of operators for martingales, see the recent work by Tanaka and Terasawa [17].

At the end of this section, we make some conventions. Throughout this paper, we always use to denote a positive constant that is independent of the main parameters involved but whose value may differ from line to line. Constants with subscripts, such as , are dependent on the subscripts. If , we then write or ; and if , we then write .

2. Definitions and Notation

Let be a probability space and a nondecreasing sequence of sub--algebras of such that . For the sake of simplicity, let . The set is called atom, more precisely -atom, if any , , satisfying or . Denote by the set of all atoms in .

The expectation operator and the conditional expectation operators relative to are denoted by and , respectively. It is known from the Doob theorem that if , then any -bounded martingale converges in . Moreover, if , then, for any , its corresponding martingale with is an -bounded martingale and converges to in (see, e.g., [18]). For this reason a function and the corresponding martingale will be denoted by the same symbol .

Let be the set of all martingales such that . For , let be the set of all such that . For any , its corresponding martingale with is an -bounded martingale in . For this reason, we regard as a subset of .

Let be subfamilies of with for each . We denote by this relation of and .

In this paper, we always postulate the following condition on

We first define generalized martingale Morrey-Campanato spaces with respect to as follows.

Definition 1. Let , , and . For , let

and define

Remark 2. By the condition (1), the functionals , , and are norms on .

Remark 3. Let . Then, if and only if its corresponding martingale is -bounded; that is, . The same conclusion holds for . Furthermore, if each sub--algebra is generated by countable atoms, and is almost decreasing, then the same conclusion holds for . More precisely, see Proposition 8.

Remark 4. In general, and hence . Actually, for any ,

Similarly, and .

Definition 5. For , denote and by and , respectively. For , , denote and by and , respectively.

If ,  , then we simply denote , , and by , , and , respectively.

A function is said to be almost increasing (resp., almost decreasing) if there exists a positive constant such that

For the case , the spaces and with , were introduced by Weisz [3].

Recall that is the set of all atoms in and let . Suppose that each sub--algebra is generated by countable atoms for the time being. Then, and ; see [5]. In general, if is almost increasing, then

with equivalent norms, respectively. However, if is not almost increasing, then these equalities fail in general; see [5].

In this paper, we do not always assume that each sub--algebra is generated by countable atoms. Let

and let

In this case, if is almost increasing, then we will show that

with equivalent norms, respectively (see Proposition 9). Moreover, if is nonatomic, then for all . If each sub--algebra is generated by countable atoms, then for all . Therefore, our definition generalizes those in [3–5].

Next we define Morrey-Hardy and Campanato-Hardy spaces, based on square functions, with respect to as follows. For , we denote by and the square function of :

where (, with convention and ). We further define

Definition 6. Let , , and . For , let

and define

By (1), the functionals , , and are quasinorms on .

Remark 7. If we take and , then the norm coincides with the norm in [19, Definition 2.45]. In this point, our notation is different from the one in [19].

In the end of this section, we present the definition of regularity on and the doubling condition on . The filtration is said to be regular, if there exists a constant such that

holds for all nonnegative martingales . We say the smallest constant satisfying (14) the regularity constant of . A function is said to satisfy the doubling condition if there exists a positive constant such that

The smallest constant satisfying (15) is called the doubling constant of .

3. Properties of Morrey-Hardy and Campanato-Hardy Spaces

In this section, we investigate the properties of Morrey-Hardy and Campanato-Hardy spaces. The proofs of the results in this section will be given in Section 6.

First we state basic properties of the norms.

Proposition 8. Let , and . Let and let be its corresponding martingale; . Then

Moreover, if each sub--algebra is generated by countable atoms, , and is almost decreasing; that is, there exists a positive constant , such that for , then

Proposition 9. Let . If is almost increasing; that is, there exists a positive constant , such that for , then

and the same conclusions hold for , , , , and . Consequently,

with equivalent norms, respectively.

For , let be the set of all such that . Let . Note that if and , then and .

The following is well known as Burkholder’s inequality.

Theorem 10 (Burkholder [20]). If , then there exist positive constants and , that depend only on , such that

for all .

For expressions of the constants and , see, for example, [21–23]. See also [24] for Burkholder’s inequality on Banach functions spaces.

Our first result is the following, which is an extension of Burkholder’s inequality to martingale Campanato spaces.

Theorem 11. Let , and . Then

for all , where and are the constants in Theorem 10.

Next we give the relations between and and between and . We consider the following condition on :

Theorem 12. Let and . Then

Conversely, if is regular, satisfies (23), and satisfies the doubling condition, then there exists a positive constant , dependent only on , the regularity constant of , and the doubling constant of , such that

We give a relation between martingale Morrey spaces and martingale Campanato spaces in the following form.

Theorem 13. Suppose that every -algebra is generated by countable atoms. Let , , and . Assume that satisfies the doubling condition and there exists a positive constant such that

Then, there exists a positive constant such that

Moreover, if is regular, then , , and are equivalent to each other, and , and are equivalent to each other.

Using Theorems 11–13, we have Burkholder-type equivalence for generalized martingale Morrey spaces.

Corollary 14. Suppose that every -algebra is generated by countable atoms. Let , , and . Assume that satisfies the doubling condition and there exists a positive constant such that

If is regular, then there exist positive constants and such that, for all ,

For the martingale spaces based on square functions, the John-Nirenberg-type equivalence was established by Weisz [25] and [19, Theorem 2.50]. We extend this theorem to the spaces and .

Theorem 15. Let , and . Assume that is almost increasing and satisfies the doubling condition. Then, for all . If we further assume that is regular, then and are equivalent to .

4. Fractional Integrals

In this section, we state the results on the boundedness of fractional integrals as martingale transforms. The proofs of the results in this section will be given in Section 7.

Let be a sequence of nonnegative bounded functions adapted to ; that is, is -measurable for every . Let be the martingale transform associate to ; that is,

with convention . Note that if , then .

We now define a generalized fractional integral for martingales as a special case of under the assumption that every -algebra is generated by countable atoms. Our definition generalizes the fractional integral for dyadic martingales introduced in [6, 7]. The idea of comes from [26].

Suppose that every -algebra is generated by countable atoms. Let be an -measurable function such that

that is,

For a bounded function , we define a generalized fractional integral of by

The generalized fractional integral is obtained by taking in (32). If , then we simply denote by .

For quasinormed spaces and of martingales, we denote by the set of all bounded martingale transforms from to ; that is, means that there exists a positive constant such that

for all martingales .

We first study the boundedness on the spaces . On martingale Campanato-Hardy spaces, we consider the fractional integral as a martingale transform associated with monotone multipliers. We say a sequence of nonnegative measurable functions is almost decreasing if there exists a positive constant such that

For an almost decreasing sequence , we define by

In Theorem 16 below, we do not need any assumption on .

Theorem 16. Let , , and . Let be a sequence of nonnegative bounded almost decreasing adapted functions, and let be the martingale transform defined by (32). Assume that

Then

with

If every -algebra is generated by countable atoms, then we can apply Theorem 16 to the generalized fractional integral . The following corollary extends [5, Theorem 5.8] to the spaces .

Corollary 17. Assume that every -algebra is generated by countable atoms and . Let and . Suppose that is almost increasing and that

Then

If one further assumes that is regular and that is almost increasing and satisfies the doubling condition, then

We next study the boundedness on martingale Morrey-Hardy spaces and martingale Hardy spaces .

Recall that for all if is nonatomic.

Proposition 18. Let , , and . Let be a sequence of adapted functions. Suppose that is nonatomic and that . Assume in addition that is almost decreasing, that is almost increasing, and that . Then, .

According to Proposition 18, to consider the boundedness on and , we suppose that every -algebra is generated by countable atoms and that .

In this case, if and , then coincides with and . However, if , then does not coincide with in general. We do not always assume that .

Theorem 19. Suppose that every -algebra is generated by countable atoms, that , and that is regular. Let and , and let be a sequence of nonnegative bounded adapted functions. Assume that satisfies the doubling condition and that there exists a positive constant such that

for all , where is the measurable function defined by (33). Then

Furthermore, if , then

As a consequence of Theorem 19, we have the following corollary, which gives an extension of [5, Corollary 5.7] to the spaces and gives a martingale Morrey-Hardy version of Gunawan [27, Theorem ]:

Corollary 20. Suppose that every -algebra is generated by countable atoms, that , and that is regular. Let and . Assume that is bounded, that both and satisfy the doubling condition, and that there exists a positive constant such that

Then

The following extends the results for dyadic martingales in [6, 7] and the result for in [28].

Corollary 21. Suppose that every -algebra is generated by countable atoms, that , and that is regular. Let and . Then

5. Lemmas

We prepare some lemmas to prove the results in Sections 3 and 4.

Lemma 1. Let satisfy . Suppose that is almost increasing; that is, for all . Then, for all nonnegative functions ,

Proof. Let

For any , we can choose the sets , (finite or infinite) such that

In this case, , , since . Then

This shows the conclusion.

Lemma 2 (see [5, Lemma 3.3]). Suppose that every -algebra is generated by countable atoms and that is regular. Then, every sequence

has the following property: for each ,

where is the constant in (14).

Lemma 3. Suppose that every -algebra is generated by countable atoms and that is regular. For , let be

Let . Suppose that satisfies the doubling condition. Then, there exists a positive constant , that depends only on and the regularity constant , such that

where is the function defined by (33).

Proof. Let . Then, by Lemma 2, we have

In Theorem 13, we do not assume that is regular. Hence, we need the following lemma.

Lemma 4. Let . Suppose that every -algebra is generated by countable atoms. For , let be

For the sequence above, one defines a decreasing sequence of integers inductively by

where one uses the convention . One further defines

Suppose that satisfies the doubling condition. Then, there exists a positive constant , that depends only on , such that

where is the function defined by (33).

Note that this lemma is the counterpart to the technique in [29, page 1104, line 5].

Proof. By the definition of , if , then

where we use the convention .

Using the doubling condition on , we have

because the intervals are disjointed by (64).

In the proof of Theorem 19, we need the following estimates for the square function of .

Lemma 5. Suppose that every -algebra is generated by countable atoms and that is regular. Let with . Let be a sequence of nonnegative bounded adapted functions. Suppose that satisfies the doubling condition. Assume that there exists a positive constant such that

for all , where is the measurable function defined by (33). Then, for with ,

for all , where is a positive constant independent of .

Proof. Let such that . We first show that

where is a positive constant that depends only on and the regularity constant . Let . Then, on the set , keeping in mind that

we have

We have obtained (68). We now show (67). Using (68) and the assumption (66), we have

Remark 22. In the course of the proof, the embedding is used. If one does not use the embedding, then

6. Proofs of the Results in Section 3

In this section, we prove the results in Section 3.

Proposition 8 can be proved in the same way as [5, Proposition 2.2], so we omit the proof. Proposition 9 is a direct consequence of Lemma 1. Then, we will prove Theorems 11, 12, and 13.

Recall that is defined by (11).

6.1. Proof of Theorem 11

We first show Theorem 11, Burkholder’s inequality on generalized martingale Campanato spaces.

Proof of Theorem 11. Let and . Then, and

Therefore, we have . Hence, using Theorem 10, we have

We have obtained (21).

We next show (22). Using (74), we have

Therefore,

For the converse part, using the inequality

which we have mentioned in Remark 4, we obtain

That is,

6.2. Proof of Theorem 12

We next show Theorem 12, a relation between and , , and .

Proof of Theorem 12. Inequality (24) was mentioned in Remark 4. Inequality (25) is deduced from the inequality .

We now show (26). Let and . Since , we have

that is, .

Suppose that is regular. To show (26), we first prove

where is the regularity constant. By the definition of , we have . We will show the converse. By the regularity, we have . This implies , or equivalently,

Operating , we have

We have obtained (81).

From (81), we deduce that

Hence, using the assumption (23) and the doubling condition on with (84), we have

We have obtained (26).

By the same way as above, we have (27). The proof is completed.

6.3. Proof of Theorem 13

We now prove Theorem 13, a relation between martingale Morrey spaces and martingale Campanato spaces.

Proof of Theorem 13. The part was shown in Remark 4, and the part is obvious. We now show the part .

Let . We take such that . Let be the decreasing sequence of integers defined in Lemma 4, with convention . Since , the function is constant on . Therefore, on the set , we have

where is the same as in  (62).

Let be the same as in Lemma 4 and let . Using Lemma 4 and the assumption (28), we have

For , we may assume that . By the definition of , we have . Therefore,

Hence, on the set , the constant has the following bound:

Combining (87) and (89), we have

Using (24) in Theorem 12, we have

that is,

We can show by the same way. Indeed, in (86), we can replace and by and , respectively. The rest is similar and we can obtain .

If , then satisfies (23). Therefore, if is regular, we can apply Theorem 12 to obtain the equivalence of , , and and the equivalence of , , and .

6.4. Proof of Theorem 15

We will now prove Theorem 15, the John-Nirenberg-type theorem for martingale Campanato-Hardy spaces. Following Weisz [19, Definition 2.45], we define

for and .

Proof of Theorem 15. We may assume that by Proposition 9.

By Hölder’s inequality and Theorem 12, we only need to show that for .

Recall the notation . Let , , and . By (73), we have . Hence, for , , we have

Therefore, for the -martingale , we have

By [19, Theorem 2.50], there exists a positive constant that depends only on , , and such that

Combining (96) and the fact that , we have

Therefore, for , we have

that is,

for . For general , applying (99) to the martingale , we have

Taking in (100), we have that is an -bounded martingale. Therefore, we have (99) for all . The proof is completed.

7. Proofs of the Results in Section 4

In this section, we prove the results in Section 4.

7.1. Proofs of Theorem 16 and Corollary 17

Recall that .

Proof of Theorem 16. Using the assumption that is almost decreasing, we have

Then, for , using the assumption (39), we have

Therefore, we have

and . The proof is completed.

Proof of Corollary 17. Let . Then, we have that is almost decreasing and that for . Hence,

Therefore, we can apply Theorem 16 to obtain . If we further assume that is regular and that is almost increasing and satisfies the doubling condition, then, by the John-Nirenberg-type equivalence (Theorem 15), we have for all .

Remark 23. Assume that (39) holds. Suppose further that there exists a positive number such that a.s. for all . Then, in the light of Remark 22, we see that

7.2. Proofs of Theorem 19 and Corollaries 20 and 21

Proof of Theorem 19. We first show the part . Assume (45). Let such that . We need only to show that there exists independent of such that

To obtain (106), we first show that

Let . We define measurable subsets , , and by

Let . Then, we can take infinitely many integers such that . For such , we have

by Lemma 5. Letting along that satisfies , we have (107) on .

On , again by Lemma 5, we have

Let . Then, we can take an integer such that

Hence, by Lemma 5, we have

We have obtained (107).

We now show (106). Let . Using (107), we have

We have obtained (106).

We now show the part . Let . We simply denote by . Let such that . Observe that . By the assumption that (45) holds for , we can apply (107) to , and we have

Hence, we obtain

The proof is completed.

Proof of Corollary 20. Let . We only have to verify (45). Using Lemma 3 and the assumption (48), we have

By Theorem 19, we have the conclusion.

Proof of Corollary 21. If and , then

Observing (116) and applying Theorem 19 to , we have the conclusion.

Remark 24. In the light of Remark 22, we see that

which is similar to (105).

In words of harmonic analysis, this corresponds to the embedding for , , and ; see [30, Section 2.3] and [30, page 129] for the definition of the space and the above embedding, respectively. It may be interesting to observe that this embedding is translated into the fact that makes functions have bounded variation.

7.3. Proof of Proposition 18

In this subsection, we prove Proposition 18.

Proof of Proposition 18. To prove , we only need to show the following for any :

We now show (119) for . We may assume that . For , define

If , then the function is -measurable. Therefore, we have

To complete the proof of (119), we only have to show the following:

Assume that . We fix and such that . Let and . Note that , , and .

To prove (122), we define two decreasing sequences of measurable sets and that satisfy

for every , inductively as follows.

Suppose that we can choose and that satisfy

By the assumption that is nonatomic, is also nonatomic. Hence, there exists such that , and . Let . Then, we have and with and that satisfy (123).

For the set defined above, let . Since is -measurable, we have

By (125) and the assumption that is almost increasing, we have, for any ,

We now show that . Suppose that ; that is, there exists a positive number such that

for all .

Since , we have , and for .

Therefore, we have

For , we take such that

By (126), we may assume that . As a consequence, we have , and by (123). Then, using (127) with (128), we have

Therefore, we have

However, this contradicts and . We have (122) and hence have (119).

Conflict of Interests

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

Acknowledgments

The authors are thankful to Professor Masaaki Fukasawa in Osaka University for his kind hint about Remarks 22, 23, and 24. The first author was supported by Grant-in-Aid for Scientific Research (C) (no. 24540159), Japan Society for the Promotion of Science. The second author was supported by Grant-in-Aid for Scientific Research (C) (no. 24540171), Japan Society for the Promotion of Science. The third author was supported by Grant-in-Aid for Young Scientists (B) (no. 24740085), Japan Society for the Promotion of Science.

References

J. Peetre, “On the theory of spaces,” Journal of Functional Analysis, vol. 4, no. 1, pp. 71–87, 1969.

W. Yuan, W. Sickel, and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2010.

F. Weisz, “Martingale Hardy spaces for ,” Probability Theory and Related Fields, vol. 84, no. 3, pp. 361–376, 1990.

T. Miyamoto, E. Nakai, and G. Sadasue, “Martingale Orlicz-Hardy spaces,” Mathematische Nachrichten, vol. 285, no. 5-6, pp. 670–686, 2012.

E. Nakai and G. Sadasue, “Martingale Morrey-Campanato spaces and fractional integrals,” Journal of Function Spaces and Applications, vol. 2012, Article ID 673929, 29 pages, 2012.

J.-A. Chao and H. Ombe, “Commutators on dyadic martingales,” Proceedings of the Japan Academy, vol. 61, no. 2, pp. 35–38, 1985.

C. Watari, “Multipliers for Walsh-Fourier series,” The Tohoku Mathematical Journal, vol. 16, pp. 239–251, 1964.

G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals. I,” Mathematische Zeitschrift, vol. 27, no. 1, pp. 565–606, 1928.

G. H. Hardy and J. E. Littlewood, “Some properties of fractional integrals. II,” Mathematische Zeitschrift, vol. 34, no. 1, pp. 403–439, 1932.

S. L. Sobolev, “On a theorem in functional analysis,” Matematicheskii Sbornik, vol. 4, pp. 471–497, 1938.

E. M. Stein and G. Weiss, “On the theory of harmonic functions of several variables. I. The theory of -spaces,” Acta Mathematica, vol. 103, pp. 25–62, 1960.

M. H. Taibleson and G. Weiss, “The molecular characterization of certain Hardy spaces,” in Representation theorems for Hardy spaces, vol. 77 of Astérisque, pp. 67–149, Soc. Math. France, Paris, 1980.

S. G. Krantz, “Fractional integration on Hardy spaces,” Studia Mathematica, vol. 73, no. 2, pp. 87–94, 1982.

E. Nakai, “Recent topics of fractional integrals,” Sugaku Expositions, vol. 20, no. 2, pp. 215–235, 2007.

D. R. Adams, “A note on Riesz potentials,” Duke Mathematical Journal, vol. 42, no. 4, pp. 765–778, 1975.

F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni, vol. 7, no. 3-4, pp. 273–279, 1987.

H. Tanaka and Y. Terasawa, “Positive operators and maximal operators in a filtered measure space,” Journal of Functional Analysis, vol. 264, no. 4, pp. 920–946, 2013.

J. Neveu, Discrete-Parameter Martingales, North-Holland, Amsterdam, The Netherlands, 1975.

F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier Analysis, vol. 1568 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1994.

D. L. Burkholder, “Martingale transforms,” Annals of Mathematical Statistics, vol. 37, pp. 1494–1504, 1966.

D. L. Burkholder, “Sharp inequalities for martingales and stochastic integrals, Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987),” Astérisque, no. 157-158, pp. 75–94, 1988.

D. L. Burkholder, “Explorations in martingale theory and its applications,” in École d’Été de Probabilités de Saint-Flour XIX 1989, vol. 1464 of Lecture Notes in Mathematics, pp. 1–66, Springer, Berlin, Germany, 1991.

R. L. Long, Martingale Spaces and Inequalities, Peking University Press, Beijing, China, 1993.

M. Kikuchi, “Characterization of Banach function spaces that preserve the Burkholder square-function inequality,” Illinois Journal of Mathematics, vol. 47, no. 3, pp. 867–882, 2003.

F. Weisz, “Interpolation between martingale Hardy and BMO spaces, the real method,” Bulletin des Sciences Mathématiques, vol. 116, no. 2, pp. 145–158, 1992.

E. Nakai, “On generalized fractional integrals,” Taiwanese Journal of Mathematics, vol. 5, no. 3, pp. 587–602, 2001.

H. Gunawan, “A note on the generalized fractional integral operators,” Journal of the Indonesian Mathematical Society, vol. 9, no. 1, pp. 39–43, 2003.

G. Sadasue, “Fractional integrals on martingale Hardy spaces for ,” Memoirs of Osaka Kyoiku University, vol. 60, no. 1, pp. 1–7, 2011.

Y. Sawano and H. Tanaka, “Sharp maximal inequalities and commutators on Morrey spaces with non-doubling measures,” Taiwanese Journal of Mathematics, vol. 11, no. 4, pp. 1091–1112, 2007.

H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, Switzerland, 1983.