Journal of Function Spaces

Volume 2016, Article ID 2438157, 7 pages

http://dx.doi.org/10.1155/2016/2438157

## Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

^{1}Department of Mathematics, Dalian Maritime University, Dalian, Liaoning 116026, China^{2}Department of Mathematics, Hainan Normal University, Haikou 571158, China

Received 1 April 2016; Accepted 8 May 2016

Academic Editor: Yoshihiro Sawano

Copyright © 2016 Canqin Tang 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

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

#### 1. Introduction

It is well known that the boundedness of fractional integral operators is the focus of study in the classical Lebesgue spaces. Sobolev [1] showed that the fractional integral operator is bounded from the classical Lebesgue space to . In 1999, Kenig and Stein [2] have obtained the boundedness of multilinear fractional integral operator from to . During the recent three decades, variable exponent function spaces have been studied extensively; see, for example, [3–15]. The characterization and the boundedness of the classical operators on variable exponent function spaces were systemically studied; see [5, 6, 16–21]. For example, Capone et al. [22] proved that Riesz fractional integral satisfies the weak-type inequality on variable exponent Lebesgue spaces in 2007.

Motivated by the aforementioned results, we will consider the variable fractional integral operator in this paper. Before stating our results, we need to recall some notions firstly.

Given a bounded open set and a measurable function The variable exponent Lebesgue space is equipped with the norm

Denote by the set of measurable functions on with values in satisfyingwhere we use the standard notation

*Definition 1. *(i) If satisfies then we say satisfies the Log-Hölder condition.

(ii) If satisfiesthen we say satisfies the Log-Hölder decay condition.

Given , , the variable fractional operator can be defined by Now the first result is the following theorem.

Theorem 2. *Given an open set and , let be a function from to such that and . Suppose further that satisfies condition (5) and satisfies conditions (3), (5), and (6). Define by Then, the fractional integral operator satisfies the weak-type inequality *

*Remark 3. *If , Theorem 2 is consistent with Theorem in [22].

Given with , the multilinear variable fractional integral operators can be defined byThe following is our second result.

Theorem 4. *Let and satisfy the Log-Hölder condition, If satisfies the Log-Hölder condition and , for , and then if each ,*

The following sections are the proofs of Theorems 2 and 4. Here we point out that denotes a positive constant, but it may vary from line to line.

#### 2. Proof of Theorem 2

All cubes are assumed to have their sides parallel to the coordinate axes; denotes that a cube is centered at with side length . is the Lebesgue measure of . denotes that a ball is centered at with radius

Define the variable fractional maximal operator by If , becomes which is the usual Hardy-Littlewood maximal function.

Given a function , denote

Lemma 5 (see [4]). *Given such that , then if and only if . In particular, if either constant equals 1, we can take the other equal to 1 as well.*

Lemma 6 (see [4]). *Given a set with finite measure and exponent functions such that , *

The next lemma is the generalized Hölder inequality on variable exponent Lebesgue spaces.

Lemma 7 (see [12]). *If , there is a constant such that, for all and all , where .*

Lemma 8 (see [23]). *Given an open set and a function which satisfies the Log-Hölder condition, then, for any ball such that , *

Lemma 9 (see [22]). *Given an open set and a function which satisfies the Log-Hölder condition, then, for any ball such that and , *

Here and below, for , let

Lemma 10 (see [22]). *Given a set and two nonnegative functions and , suppose that, for each , Then, there exists a constant such that, for every function obeying , *

Lemma 11. *Given , fix . Then, there exists a constant such that, for all and all ,*

*Proof. *Let be given. Then, Let ; we have where . Hence, Similarly, By choosing , one finds

Lemma 12. *Given an open set and , let be such that , , and such that satisfies the Log-Hölder condition. Define by Then, for all such that and such that or ,*

*Proof. *Fix , and fix a ball containing . Then, by the definition of , To complete the proof we will show that We consider it in two cases depending on the size of Suppose first that . Let . By Chebyschev’s inequality and Lemma 5, Therefore, since , by Lemma 6, Now suppose . If , then and satisfies the Log-Hölder condition, so by Lemmas 7 and 9The argument is the same when except that instead of applying Lemma 9 we note that, by Lemma 6, Then, by Lemma 8, This completes the proof.

Lemma 13. *Given an open set and , let be such that , , and satisfy the Log-Hölder decay condition. Define by Then, for all such that and , ,where .*

*Proof. *Fix and let be any ball containing . It will suffice to show that where is independent of .

It follows at once from the definition of that . Therefore, by Hölder’s inequality, Since , , so . Therefore, to complete the proof we only need to show that Forit suffices to prove Define the setsThen, The integral in the right-hand side is easy to estimate. If , then . Thus, since , by Lemma 5, To estimate the first integral, we will apply Lemma 10. If , then, by the Log-Hölder decay condition, Therefore, for any and , is integrable, andThis completes the proof.

Lemma 14 (see [22]). *Given an open set , let satisfy the Log-Hölder decay condition. Suppose such that . Then, for every ball and every , where and *

*Proof of Theorem 2. *Fix ; without loss of generality we may assume that . Since , by Lemma 5 it will suffice to prove that Fix , such that Define by Since and , by (23) and Young’s inequality, for each , Therefore,Denote , where and .

Therefore, by (29) and (38), Therefore, We estimate each integral in turn.

For each , define , where . Then, for each the set is bounded, and by the monotone convergence theorem, the first integral is immediate: So, it will suffice to prove that where is independent of Fix ; for each there exists a ball containing such that Therefore, by the Besicovitch covering lemma (see [24]), there exists a sequence of such that and such that the balls have finite overlap uniformly bounded by a constant depending only on the dimension. Hence, Forby Lemma 14, we have where, in the third inequality, since , we used the boundedness of maximal operator on , and in the last inequality, we used the fact that the balls have uniformly bounded finite overlap.

For , ; then, it will suffice to prove We consider the two cases and to estimate it.

If , since , therefore If , since , by Hölder’s inequality To estimate the second term, we need to show that We first need to control the quantity Fix ; then, there exists , such that Then, by the Log-Hölder decay condition, we have Since is arbitrary and , it follows that For , is integrable, and by Lemma 10The proof now proceeds as it does for the first term.

We estimate as we did before; here we omit the detail. Then, we finish the proof of Theorem 2.

#### 3. Proof of Theorem 4

To prove Theorem 4, we need the following lemma.

Lemma 15 (see [16]). *Let be bounded. Suppose that satisfies conditions (3) and (5) and also satisfies the Log-Hölder condition: If , then there exists a positive constant such that for all *

*Proof of Theorem 4. *Since , some . If , , since , we have . Then, integration in reduces matters to the case when all are finite (and ). Thus, we can assume that all .

Now, observe that ; we can find such that . Let . Since and it follows that where is the variable exponent fractional integral operator.

Because , by Hölder’s inequality and (72), we obtain This finishes the proof.

#### Competing Interests

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

#### Acknowledgments

The first author is supported by the National Natural Science Foundation of China (Grant no. 11021043) and the Fundamental Research Funds for the Central Universities (no. 3132014324). The third author is supported by the National Natural Science Foundation of China (Grant no. 11361020) and the Natural Science Foundation of Hainan Province (Grant no. 20151011).

#### References

- S. L. Sobolev, “On a theorem of functional analysis,”
*Matematicheskii Sbornik*, vol. 4, pp. 417–497, 1938. View at Google Scholar - C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,”
*Mathematical Research Letters*, vol. 6, no. 1, pp. 1–15, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. Almeida and P. Hästö, “Besov spaces with variable smoothness and integrability,”
*Journal of Functional Analysis*, vol. 258, no. 5, pp. 1628–1655, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - D. Cruz-Uribe and A. Fiorenza,
*Variable Lebesgue Spaces*, Springer, Basel, Switzerland, 2013. - L. Diening, P. Harjulehto, P. Hästö, and M. Růžička,
*Lebesgue and Sobolev Spaces with Variable Exponents*, Springer, Berlin, Germany, 2011. - L. Diening, P. Hästö, and A. Nekvinda, “Open problems in variable exponent Lebesgue and Sobolev spaces,” in
*Function Spaces, Differential Operators and Nonlinear Analysis*, pp. 38–52, Academy of Sciences of the Czech Republic, Prague, Czech Republic, 2005. View at Google Scholar - L. Diening, P. Hästö, and S. Roudenko, “Function spaces of variable smoothness and integrability,”
*Journal of Functional Analysis*, vol. 256, no. 6, pp. 1731–1768, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. Dong and J. Xu, “New Herz type Besov and Triebel-Lizorkin spaces with variable exponents,”
*Journal of Function Spaces and Applications*, vol. 2012, Article ID 384593, 27 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - B. Dong and J. Xu, “Local characterizations of Besov and Triebel-Lizorkin spaces with variable exponent,”
*Journal of Function Spaces*, vol. 2014, Article ID 417341, 8 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus - H. Kempka, “2-Microlocal besov and triebel-lizorkin spaces of variable integrability,”
*Revista Matematica Complutense*, vol. 22, no. 1, pp. 227–251, 2009. View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Kempka, “Atomic, molecular and wavelet decomposition of generalized 2-microlocal Besov spaces,”
*Journal of Function Spaces and Applications*, vol. 8, no. 2, pp. 129–165, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 - E. Nakai and Y. Sawano, “Hardy spaces with variable exponents and generalized Campanato spaces,”
*Journal of Functional Analysis*, vol. 262, no. 9, pp. 3665–3748, 2012. View at Publisher · View at Google Scholar · View at Scopus - T. Noi, “Duality of variable exponent Triebel-Lizorkin and Besov spaces,”
*Journal of Function Spaces and Applications*, vol. 2012, Article ID 361807, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus - Y. Sawano, “Atomic decompositions of hardy spaces with variable exponents and its application to bounded linear operators,”
*Integral Equations and Operator Theory*, vol. 77, no. 1, pp. 123–148, 2013. View at Publisher · View at Google Scholar · View at Scopus - A. Almeida, J. Hasanov, and S. Samko, “Maximal and potential operetors in variable exponent Morrey spaces,”
*Georgian Mathematical Journal*, vol. 15, no. 2, pp. 195–208, 2008. View at Google Scholar - D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Pérez, “The boundedness of classical operators on variable
*L*^{p}spaces,”*Annales Academiæ Scientiarum Fennicæ Mathematica*, vol. 31, no. 1, pp. 239–264, 2006. View at Google Scholar · View at Scopus - 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 - S. G. Samko, “Convolution and potential type operators in ${L}^{p\left(x\right)}({R}^{n})$,”
*Integral Transforms and Special Functions*, vol. 7, no. 3-4, pp. 261–284, 1998. View at Publisher · View at Google Scholar - V. Kokilashvili, “On a progress in the theory of integral operators in weighted Banach function spaces,” in
*Function Spaces, Differential Operators and Nonlinear Analysis,*, pp. 152–175, Academy of Sciences of the Czech Republic, Praha, Czech Republic, 2005. View at Google Scholar - W. Wang and J. S. Xu, “Commutators of multilinear singular integrals with Lipschitz functions on products of variable exponent Lebesgue spaces,”
*Advances in Mathematics*, vol. 38, pp. 669–677, 2009 (Chinese). View at Google Scholar - C. Capone, D. Cruz-Uribe, and A. Fiorenza, “The fractional maximal operator and fractional integrals on variable ${L}^{p}$spaces,”
*Revista Matemática Iberoamericana*, vol. 23, no. 3, pp. 743–770, 2007. View at Google Scholar · View at Scopus - L. Diening, “Maximal function on generalized ${L}^{p(\bullet )}$ spaces,”
*Mathematical Inequalites and Applications*, vol. 7, pp. 245–253, 2004. View at Google Scholar - M. De Guzman,
*Differentiation of Integrals in R*, vol. 481 of^{n}*Lecture Notes in Mathematics*, Springer, New York, NY, USA, 1975.