Abstract and Applied Analysis

Volume 2013 (2013), Article ID 632384, 6 pages

http://dx.doi.org/10.1155/2013/632384

Research Article

Lipschitz Estimates for Fractional Multilinear Singular Integral on Variable Exponent Lebesgue Spaces

Hui-Ling Wu¹ and Jia-Cheng Lan²

¹College of Education, Lishui University, Lishui 323000, China

²College of Science, Lishui University, Lishui 323000, China

Received 6 February 2013; Accepted 11 August 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 Hui-Ling Wu and Jia-Cheng Lan. 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 obtain the Lipschitz boundedness for a class of fractional multilinear operators with rough kernels on variable exponent Lebesgue spaces. Our results generalize the related conclusions on Lebesgue spaces with constant exponent.

1. Introduction and Results

Let , is homogeneous of degree zero on , denotes the unit sphere in , the fractional multilinear singular integral operator with rough kernel is defined by

where denotes the th remainder of the Taylor series of a function defined on at about . More precisely,

and the corresponding fractional multilinear maximal operator is defined by

Multilinear operator was first introduced by Caldern in [1], and then Meyer [2] studied it in depth and extended such type of operators. Multilinear singular integral operator was later introduced by Professor Lu during 1999 [3]. Especially as , the fractional multilinear singular integral operator is obviously the commutator operator

the commutator is a typical non-convolution singular operator. Since the commutator has a close relation with partial differential equations and pseudo-differential operator, multilinear operator has been receiving more widely attention.

It is well known that the boundedness of and had been obtained on Lebesgue spaces in [4–7]. However, the corresponding results have not been obtained on . Nowadays, there is an evident increase of investigations related to both the theory of the spaces themselves and the operator theory in these spaces [8–11]. This is caused by possible applications to models with nonstandard local growth in elasticity theory, fluid mechanics, and differential equations [12–14]. The purpose of this paper is to study the behaviour of and on variable Lebesgue spaces.

To state the main results of this paper, we need to recall some notions.

Definition 1. Suppose a measurable function , for some , then, the variable exponent Lebesgue space is defined by

with norm

We denote

Using this notation we define a class of variable exponent as follows:

The exponent means the conjugate of , namely, holds.

Definition 2. For , the homogeneous Lipschitz space is the space of functions , such that

where , , .

Definition 3. For , the fractional integral operator with rough kernel is defined by

The corresponding fractional maximal operator with rough kernel is defined by

When , is much more closely related to the elliptic partial equations of second order with variable coefficients. In 1955, Calderón and Zygmund [15] proved the boundedness. In 1971, Muckenhoupt and Wheeden [16] proved the boundedness of with power weights.

In this paper, we state some properties of variable exponents belonging to class .

Proposition 4. If satisfies

Then, one has .

Recently, Mitsuo Izuki has proved the condition as below.

Theorem A (see [17]). Suppose that satisfies conditions (12) in Proposition 4. Let , and define the variable exponent by

Then, one has that for all ,

for all and .

Next, we will discuss the boundedness of and on variable Lebesgue spaces. We can get and are bounded from to . In fact, the results generalize Theorem A in [17] from classical Lebesgue spaces to variable exponent Lebesgue spaces. Now, let us formulate our results as follows.

Theorem 5. Suppose that satisfies conditions (12) in Proposition 4. Let , , , and , and define the variable exponent by

If , then, there is a , independent of and , such that

Theorem 6. Suppose that satisfies conditions (12) in Proposition 4. Let , , , and , and define the variable exponent by

If , then, there is a , independent of and , such that

Remark 7. We point out that will denote positive constants whose values may change at different places.

2. Lemmas and Proof of Theorems

Lemma 8 (see [15]). Let be a function on with th order derivatives in for some . Then,

where is the cube centered at and having diameter .

Lemma 9 (see [18]). For , , we have

Lemma 10 (see [18]). Let , , then,

We state the following important lemma.

Lemma 11. Suppose , , with , , . Then, there exists a constant only depends on , and , such that

Proof. For any , let the cube be centered at and having the diameter be , where , we have

Below, we give estimates of . Let

Note that . When , by Lemmas 8, 9, and 10, we have

Note that , we have , such that

Below, we give the estimates of . For , we get

For any ,

Thus, by Lemmas 8 and 9, we obtain

And for , we have . Hence,

From the proof above, we obtain

Lemma 12 (see [19]). If , for all , then, the norm has the following equivalence:

where .

Lemma 13 (see [19], the generalized Hlder inequality). If , then, for all and for all , we have

By a similar method of Ding and Lu [20], it is easy to verify the following result.

Lemma 14. For any with , we have

where depends only on , and .

Lemma 15 (see [19]). Given that , such that , then, if and only if . In particular, if either constant equals 1, one can make the other equals 1 as well.

Remark 16. We denote .

Lemma 17 (see [21]). Suppose that satisfies conditions (12) in Proposition 4. Let , and define the variable exponent by

Then, one has that for all ,

Lemma 18. Let , , then, for ,

where

Proof. Since

then,

Proof of Theorem 5. Since

by Lemma 12, then, we have

Using the generalized Hlder inequality, then,

Next, we will prove . Fix , without loss of generality we may assume that . Since , by Lemma 15 it will suffice to prove that .

Fix , , such that

define by

Then, by (44), we have . Moreover, by elementary algebra, for all ,

So that by Lemma 14, we have

By Lemma 13, then,

Without loss of generality, we may assume that each is greater than 1, since, otherwise, there is nothing to prove. In this case, in the definition of each norm we may assume that the infimum is taken over by values of which are greater than 1. But then, since for all and , , we have

Therefore, by (46) and Lemma 17, we have

In the same way, we have

Therefore, by (47) and Lemma 17, then,

Hence,

So, we have

This completes the proof of Theorem 5.

By Lemmas 15 and 18 and Theorem 5, the proof of Theorem 6 is directly deduced.

Acknowledgments

This paper is supported by the Natural Science Foundation of Zhejiang Province (M6090681) and supported by the Education Deptartment of Zhejiang Province (Y201120509).

References

A. P. Calderón, “Algebras of singular integral operators,” in Proceedings of Symposia in Pure Mathematics, vol. 10, pp. 18–55, American Mathematical Society, Providence, RI, USA, 1967.

Y. Meyer, Ondelettes et Opérateurs. I, Hermann, Paris, France, 1990.

S. Lu, “Multilinear oscillatory integrals with Calderón-Zygmund kernel,” Science in China A, vol. 42, no. 10, pp. 1039–1046, 1999.

J. C. Lan, “Uniform boundedness of multilinear fractional integral operators,” Applied Mathematics, vol. 21, no. 3, pp. 365–372, 2006.

S. Lu and P. Zhang, “Lipschitz estimates for generalized commutators of fractional integrals with rough kernel,” Mathematische Nachrichten, vol. 252, pp. 70–85, 2003.

Y. Ding, “A note on multilinear fractional integrals with rough kernel,” Advances in Mathematics, vol. 30, no. 3, pp. 238–246, 2001.

X. X. Tao and Y. P. Wu, “BMO estimates for multilinear fractional integrals,” Analysis in Theory and Applications, vol. 28, pp. 224–231, 2012.

V. Kokilashvili and S. Samko, “On sobolev theorem for Riesz-Type potentials in Lebesgue spaces with variable exponent,” Journal for Analysis and its Applications, vol. 22, no. 4, pp. 899–910, 2003.

L. Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,” Bulletin des Sciences Mathematiques, vol. 129, no. 8, pp. 657–700, 2005.

D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable Lp spaces,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 31, no. 1, pp. 239–264, 2006.

W. Wang and J. Xu, “Commutators of multilinear singular integrals with Lipschitz functions on products of variable exponent Lebesgue spaces,” Advances in Mathematics, vol. 38, no. 6, pp. 669–677, 2009.

E. Acerbi and G. Mingione, “Regularity results for stationary electro-rheological fluids,” Archive for Rational Mechanics and Analysis, vol. 164, no. 3, pp. 213–259, 2002.

M. Ružička, Electroreological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, 2000.

E. Acerbi and G. Mingione, “Regularity results for a class of functionals with non-standard growth,” Archive for Rational Mechanics and Analysis, vol. 156, no. 2, pp. 121–140, 2001.

A. P. Calderón and A. Zygmund, “On a problem of Mihlin,” Transactions of the American Mathematical Society, vol. 78, pp. 209–224, 1955.

B. Muckenhoupt and R. L. Wheeden, “Weighted norm inequalities for singular and fractional integrals,” Transactions of the American Mathematical Society, vol. 161, pp. 249–258, 1971.

M. Izuki, “Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent,” Rendiconti del Circolo Matematico di Palermo, vol. 59, no. 3, pp. 461–472, 2010.

A. P. Calderón and A. Zygmund, “On singular integral with variable kernels,” Journal of Applied Analysis, vol. 7, pp. 221–238, 1978.

O. Kováčik and J. Rákosník, “On spaces and ,” Czechoslovak Mathematical Journal, vol. 41, no. 4, pp. 592–618, 1991.

Y. Ding and S. Lu, “Higher order commutators for a class of rough operators,” Arkiv for Matematik, vol. 37, no. 1, pp. 33–44, 1999.

H. L. Wu and J. C. Lan, “The boundedness of rough fractional integral operators on variable exponent Lebesgue spaces,” Anyalsis in Theory and Applications, vol. 28, pp. 286–293, 2012.