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Abstract and Applied Analysis

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

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

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

^{1}College of Education, Lishui University, Lishui 323000, China^{2}College 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Meyer,
*Ondelettes et Opérateurs. I*, Hermann, Paris, France, 1990. View at MathSciNet - S. Lu, “Multilinear oscillatory integrals with Calderón-Zygmund kernel,”
*Science in China A*, vol. 42, no. 10, pp. 1039–1046, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. C. Lan, “Uniform boundedness of multilinear fractional integral operators,”
*Applied Mathematics*, vol. 21, no. 3, pp. 365–372, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Lu and P. Zhang, “Lipschitz estimates for generalized commutators of fractional integrals with rough kernel,”
*Mathematische Nachrichten*, vol. 252, pp. 70–85, 2003. View at Publisher · View at Google Scholar · View at Scopus - Y. Ding, “A note on multilinear fractional integrals with rough kernel,”
*Advances in Mathematics*, vol. 30, no. 3, pp. 238–246, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. X. Tao and Y. P. Wu, “BMO estimates for multilinear fractional integrals,”
*Analysis in Theory and Applications*, vol. 28, pp. 224–231, 2012. View at Google Scholar - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - L. Diening, “Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces,”
*Bulletin des Sciences Mathematiques*, vol. 129, no. 8, pp. 657–700, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable L
^{p}spaces,”*Annales Academiae Scientiarum Fennicae Mathematica*, vol. 31, no. 1, pp. 239–264, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - A. P. Calderón and A. Zygmund, “On a problem of Mihlin,”
*Transactions of the American Mathematical Society*, vol. 78, pp. 209–224, 1955. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Scopus - A. P. Calderón and A. Zygmund, “On singular integral with variable kernels,”
*Journal of Applied Analysis*, vol. 7, pp. 221–238, 1978. View at Google Scholar - O. Kováčik and J. Rákosník, “On spaces ${L}^{p(x)}$ and ${W}^{k,p(x)}$,”
*Czechoslovak Mathematical Journal*, vol. 41, no. 4, pp. 592–618, 1991. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Google Scholar