Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 735795, 11 pages

http://dx.doi.org/10.1155/2013/735795

Research Article

Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

Hua Wang¹ and Wentan Yi²

¹College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

²Department of Applied Mathematics, Zhengzhou Information Science and Technology Institute, Zhengzhou 450002, China

Received 2 May 2013; Accepted 12 September 2013

Academic Editor: John R. Akeroyd

Copyright © 2013 Hua Wang and Wentan Yi. 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 will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.

1. Introduction and Main Results

Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. Let be the -dimensional Euclidean space, and let be the -fold product space (). We denote by the space of all Schwartz functions on and by its dual space, the set of all tempered distributions on . Let and be an -linear operator initially defined on the -fold product of Schwartz spaces, and taking values into the space of tempered distributions,

Following [2], for given , we say that is an -linear Calderón-Zygmund operator if, for some and with , it extends to a bounded multilinear operator from into and if there exists a kernel function in the class -, defined away from the diagonal in such that

whenever and . We say that is a kernel in the class - if it satisfies the size condition

for some and all with for some . Moreover, for some , it satisfies the regularity condition that

whenever and also that, for each fixed with ,

whenever . In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calderón-Zygmund singular integral operators were established in [9] by Lerner et al. New more refined multilinear maximal function was defined and used in [9] to characterize the class of multiple weights.

Theorem A (see [9]). Let and be an -linear Calderón-Zygmund operator. If and with and satisfies the condition, then there exists a constant independent of such that

where .

Theorem B (see [9]). Let , and let be an -linear Calderón-Zygmund operator. If , and with , and satisfies the condition, then there exists a constant independent of such that

where .

Let , and let . For given , the -linear fractional integral operator is defined by

For the boundedness properties of multilinear fractional integrals on various function spaces, we refer the reader to [10–16]. In 2009, Moen [17] considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple weights (see also [18]).

Theorem C (see [17, 18]). Let , , and let be an -linear fractional integral operator. If , and , and satisfies the condition, then there exists a constant independent of such that

where .

Theorem D (see [17, 18]). Let , , and let be an -linear fractional integral operator. If , , and , and satisfies the condition, then there exists a constant independent of such that

where .

On the other hand, the classical Morrey spaces were originally introduced by Morrey in [19] to study the local behavior of solutions to second-order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [20–22]. For the properties and applications of classical Morrey spaces, one can see [23–25] and the references therein.

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces which could be viewed as an extension of weighted Lebesgue spaces and studied the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [27–34], we have established the continuity properties of some other operators and their commutators on the weighted Morrey spaces .

The main purpose of this paper is to establish the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. We now formulate our main results as follows.

Theorem 1. Let , and let be an -linear Calderón-Zygmund operator. If and with and with , then for any , there exists a constant independent of such that

where .

Theorem 2. Let and be an -linear Calderón-Zygmund operator. If , and with , and with , then for any , there exists a constant independent of such that

where .

Theorem 3. Let , let , and let be an -linear fractional integral operator. If , , and , and with , then for any , there exists a constant independent of such that

where .

Theorem 4. Let , let , and let be an -linear fractional integral operator. If , , , and , and with , then, for any , there exists a constant independent of such that

where .

2. Notations and Definitions

The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [35]. A weight is a nonnegative, locally integrable function on ; denotes the ball with the center and radius . For , a weight function is said to belong to if there is a constant such that, for every ball ,

where denotes the Lebesgue measure of . For the case , , if there is a constant such that for every ball ,

A weight function if it satisfies the condition for some . We also need another weight class introduced by Muckenhoupt and Wheeden in [36]. A weight function belongs to for if there is a constant such that, for every ball ,

When , is in the class with if there is a constant such that, for every ball ,

Now let us recall the definitions of multiple weights. For exponents , we will write for the vector . Let , and let with . Given , set . We say that satisfies the condition if it satisfies

When , is understood as .

Let , let , and let . Given , set . We say that satisfies the condition if it satisfies

When , is understood as .

Given a ball and , denotes the ball with the same center as whose radius is times that of . For a given weight function and a measurable set , we also denote the Lebesgue measure of by and the weighted measure of by , where .

Lemma 5 (see [37]). Let with . Then, for any ball , there exists an absolute constant such that

Lemma 6 (see [38]). Let . Then for all balls , the following reverse Jensen inequality holds:

Lemma 7 (see [37]). Let . Then, for all balls and all measurable subsets of , there exists such that

Lemma 8 (see [9]). Let , and let . Then if and only if

where and the condition in the case is understood as .

Lemma 9 (see [17, 18]). Let , and , let , and let . If , then

where .

Given a weight function on , for , the weighted Lebesgue space    is defined as the set of all functions such that

We also denote by the weighted weak space consisting of all measurable functions such that

In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces for . In order to deal with the multilinear case , we will define for all .

Definition 10. Let , let , and let be a weight function on . Then the weighted Morrey space is defined by

where

and the supremum is taken over all balls in .

Definition 11. Let , let , and let be a weight function on . Then the weighted weak Morrey space is defined by

where

Furthermore, in order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weights.

Definition 12. Let and . Then for two weights and , the weighted Morrey space is defined by

where

Throughout this paper, we will use to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. Moreover, we will denote the conjugate exponent of by .

3. Proofs of Theorems 1 and 2

Before proving the main theorems of this section, we need to establish the following lemma.

Lemma 13. Let , let , and let with . Assume that and ; then, for any ball , there exists a constant such that

Proof. Since , then, by using Lemma 6, we have

Note that and . Then, by Jensen inequality, we obtain

We are done.

Proof of Theorem 1. For any ball and letting  , where , , and denotes the characteristic function of , then we write

where each term of contains at least one . Since is an -linear operator, then we have

In view of Lemma 8, we have that . Applying Theorem A and Lemmas 13 and 5, we get

For the other terms, let us first consider the case when . By the size condition, for any , we obtain

where we have used the notation . Furthermore, by using Hölder’s inequality, the multiple condition, and Lemma 13, we deduce that

Since , then it follows directly from Lemma 7 that

Hence,

where the last inequality holds since and . We now consider the case where exactly of the are for some . We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the size condition again, we deduce that, for any ,

and we arrive at the expression considered in the previous case. So for any , we also have

Therefore, by the inequality (42) and the above pointwise inequality, we have

Combining the above estimates and then taking the supremum over all balls , we complete the proof of Theorem 1.

Proof of Theorem 2. For any ball and decomposing , where , , then, for any given , we can write

where each term of contains at least one . By Lemma 8 again, we know that with . Applying Theorem B and Lemmas 13 and 5, we have

In the proof of Theorem 1, we have already showed the following pointwise estimate (see (40) and (44)). Consider

Without loss of generality, we may assume that and . Using Hölder’s inequality, the multiple condition, and Lemma 13, we obtain

Observe that with . Thus, it follows from the inequality (42) that, for any ,

If , then the inequality

holds trivially. Now, if instead we suppose that , then, by the pointwise inequality (51), we have

which is equivalent to

Therefore,

Summing up all the above estimates and then taking the supremum over all balls and all , we complete the proof of Theorem 2.

By using Hölder’s inequality, it is easy to check that if each is in , then

and this inclusion is strict (see [9]). Thus, as direct consequences of Theorems 1 and 2, we immediately obtain the following.

Corollary 14. Let , and let be an -linear Calderón-Zygmund operator. If and with and , then, for any , there exists a constant independent of such that

where .

Corollary 15. Let and let be an -linear Calderón-Zygmund operator. If , and with , and , then, for any , there exists a constant independent of such that

where .

4. Proofs of Theorems 3 and 4

Following along the same lines as those of Lemma 13, we can also show the following result, which plays an important role in our proofs of Theorems 3 and 4.

Lemma 16. Let , and with . Assume that and ; then, for any ball , there exists a constant such that

Proof of Theorem 3. Arguing as in the proof of Theorem 1, fix a ball and decompose , where , . Since is an -linear operator, then we have

where each term of contains at least one . In view of Lemma 9, we can see that . Using Theorem C and Lemmas 16 and 5, we get

For the other terms, let us first deal with the case when . By the definition of , for any , we obtain

Moreover, by using Hölder’s inequality, the multiple condition, and Lemma 16, we deduce that

Since , then it follows immediately from Lemma 7 that

Hence,

where in the last inequality we have used the fact that and . We now consider the case where exactly of the are for some . We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the definition of again, we can see that, for any ,

and we arrive at the expression considered in the previous case. Thus, for any , we also have

Therefore, by the inequality (64) and the above pointwise inequality, we obtain

Summarizing the estimates derived above and then taking the supremum over all balls , we finish the proof of Theorem 3.

Proof of Theorem 4. As before, fix a ball and split into , where , . Then for each fixed , we can write

where each term of contains at least one . By Lemma 9 again, we know that with . Using Theorem D and Lemmas 16 and 5, we have

In the proof of Theorem 3, we have already proved the following pointwise estimate (see (62) and (66)). Consider

Without loss of generality, we may assume that and . By using Hölder’s inequality, the multiple condition, and Lemma 16, we obtain

Note that with . Hence, it follows from the inequality (64) that, for any ,

If , then the inequality

holds trivially. Now, if instead we assume that , then, by the pointwise inequality (73), we get

which in turn gives that

Therefore,

Collecting all the above estimates and then taking the supremum over all balls and all , we conclude the proof of Theorem 4.

By using Hölder’s inequality, it is easy to verify that if , , and each is in , then we have

and this inclusion is strict (see [17]). Also, recall that if and only if (see [36]). Thus, as straightforward consequences of Theorems 3 and 4, we finally obtain the following.

Corollary 17. Let , let , and let be an -linear fractional integral operator. If , , and , and , then, for any , there exists a constant independent of such that

where .

Corollary 18. Let , let , and let be an -linear fractional integral operator. If , , , and , and , then, for any , there exists a constant independent of such that

where .

References

R. R. Coifman and Y. Meyer, “On commutators of singular integrals and bilinear singular integrals,” Transactions of the American Mathematical Society, vol. 212, pp. 315–331, 1975.

L. Grafakos and R. H. Torres, “Multilinear Calderón-Zygmund theory,” Advances in Mathematics, vol. 165, no. 1, pp. 124–164, 2002.

L. Grafakos and R. H. Torres, “Maximal operator and weighted norm inequalities for multilinear singular integrals,” Indiana University Mathematics Journal, vol. 51, no. 5, pp. 1261–1276, 2002.

L. Grafakos and R. H. Torres, “On multilinear singular integrals of Calderón-Zygmund type,” Publicacions Matemàtiques, pp. 57–91, 2002.

L. Grafakos and N. Kalton, “Multilinear Calderón-Zygmund operators on Hardy spaces,” Collectanea Mathematica, vol. 52, no. 2, pp. 169–179, 2001.

G. E. Hu and Y. Meng, “Multilinear Calderón-Zygmund operator on products of Hardy spaces,” Acta Mathematica Sinica (English Series), vol. 28, no. 2, pp. 281–294, 2012.

W. J. Li, Q. Y. Xue, and K. Yabuta, “Maximal operator for multilinear Calderón-Zygmund singular integral operators on weighted Hardy spaces,” Journal of Mathematical Analysis and Applications, vol. 373, no. 2, pp. 384–392, 2011.

W. J. Li, Q. Y. Xue, and K. Yabuta, “Multilinear Calderón-Zygmund operators on weighted Hardy spaces,” Studia Mathematica, vol. 199, no. 1, pp. 1–16, 2010.

A. K. Lerner, S. Ombrosi, C. Pérez, R. H. Torres, and R. Trujillo-González, “New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory,” Advances in Mathematics, vol. 220, no. 4, pp. 1222–1264, 2009.

L. Grafakos, “On multilinear fractional integrals,” Studia Mathematica, vol. 102, no. 1, pp. 49–56, 1992.

T. Iida, E. Sato, Y. Sawano, and H. Tanaka, “Weighted norm inequalities for multilinear fractional operators on Morrey spaces,” Studia Mathematica, vol. 205, no. 2, pp. 139–170, 2011.

T. Iida, E. Sato, Y. Sawano, and H. Tanaka, “Multilinear fractional integrals on Morrey spaces,” Acta Mathematica Sinica (English Series), vol. 28, no. 7, pp. 1375–1384, 2012.

T. Iida, E. Sato, Y. Sawano, and H. Tanaka, “Sharp bounds for multilinear fractional integral operators on Morrey type spaces,” Positivity, vol. 16, no. 2, pp. 339–358, 2012.

C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,” Mathematical Research Letters, vol. 6, no. 1, pp. 1–15, 1999.

G. Pradolini, “Weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators,” Journal of Mathematical Analysis and Applications, vol. 367, no. 2, pp. 640–656, 2010.

L. Tang, “Endpoint estimates for multilinear fractional integrals,” Journal of the Australian Mathematical Society, vol. 84, no. 3, pp. 419–429, 2008.

K. Moen, “Weighted inequalities for multilinear fractional integral operators,” Collectanea Mathematica, vol. 60, no. 2, pp. 213–238, 2009.

X. Chen and Q. Y. Xue, “Weighted estimates for a class of multilinear fractional type operators,” Journal of Mathematical Analysis and Applications, vol. 362, no. 2, pp. 355–373, 2010.

C. B. Morrey, “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126–166, 1938.

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.

J. Peetre, “On the theory of spaces,” Journal of Functional Analysis, vol. 4, pp. 71–87, 1969.

D. S. Fan, S. Z. Lu, and D. C. Yang, “Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coefficients,” Georgian Mathematical Journal, vol. 5, no. 5, pp. 425–440, 1998.

G. Di Fazio and M. A. Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients,” Journal of Functional Analysis, vol. 112, no. 2, pp. 241–256, 1993.

G. Di Fazio, D. K. Palagachev, and M. A. Ragusa, “Global Morrey regularity of strong solutions to the Dirichlet problem for elliptic equations with discontinuous coefficients,” Journal of Functional Analysis, vol. 166, no. 2, pp. 179–196, 1999.

Y. Komori and S. Shirai, “Weighted Morrey spaces and a singular integral operator,” Mathematische Nachrichten, vol. 282, no. 2, pp. 219–231, 2009.

H. Wang, “Some estimates for commutators of Calderón-Zygmund operators on the weighted Morrey spaces,” Scientia Sinica Mathematica, vol. 42, no. 1, pp. 31–45, 2012.

H. Wang, “The boundedness of some operators with rough kernel on the weighted Morrey spaces,” Acta Mathematica Sinica (Chinese Series), vol. 55, no. 4, pp. 589–600, 2012.

H. Wang, “Intrinsic square functions on the weighted Morrey spaces,” Journal of Mathematical Analysis and Applications, vol. 396, no. 1, pp. 302–314, 2012.

H. Wang, “Boundedness of fractional integral operators with rough kernels on weighted Morrey spaces,” Acta Mathematica Sinica (Chinese Series), vol. 56, pp. 175–186, 2013.

H. Wang, “Some estimates for commutators of fractional integral operators on weighted Morrey spaces,” Acta Mathematica Sinica (Chinese Series), vol. 56, pp. 889–906, 2013.

H. Wang, “Some estimates for commutators of fractional integrals associated to operators with Gaussian kernel bounds on weighted Morrey spaces,” Analysis in Theory and Applications, vol. 29, pp. 72–85, 2013.

H. Wang, “Weak type estimates for intrinsic square functions on weighted Morrey spaces,” Analysis in Theory and Applications, vol. 29, pp. 104–119, 2013.

H. Wang and H. P. Liu, “Some estimates for Bochner-Riesz operators on the weighted Morrey spaces,” Acta Mathematica Sinica (Chinese Series), vol. 55, no. 3, pp. 551–560, 2012.

B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Transactions of the American Mathematical Society, vol. 165, pp. 207–226, 1972.

B. Muckenhoupt and R. Wheeden, “Weighted norm inequalities for fractional integrals,” Transactions of the American Mathematical Society, vol. 192, pp. 261–274, 1974.

J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam, The Netherlands, 1985.

J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2000.