Jian Tan

Jiman Zhao

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1117–1122Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comHarmonic analysisRough fractional integrals and its commutators on variable Morrey spacesIntégrales fractionnaires rugueuses et ses commutateurs sur les espaces de Morrey avec exposant variableJian Tan, Jiman ZhaoSchool of Mathematical Sciences, Beijing Normal University, Key Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, PR Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 2 June 2015Accepted after revision 28 September 2015Available online 2 November 2015Presented by the Editorial BoardIn this paper, the authors obtain the boundedness of fractional integrals with rough kernel on variable Morrey spaces. The corresponding boundedness for commutators generalized by the fractional integral and BMO function is also considered.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cet article, les auteurs obtiennent la bornitude des intégrales fractionnaires avec un noyau singulier dans des espaces de Morrey (avec exposant variable). De plus, la bornitude des commutateurs généralisés entre ces opérateurs et la multiplication par une fonction BMO est aussi considérée.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionIn this paper, we are concerned with variable Morrey spaces Mp(·),u introduced in [8], which are equipped with a Morrey weight function u and a variable exponent function p(·). The main results of this paper consist of the boundedness of rough fractional integrals and its commutators with BMO functions together with the boundedness of the rough fractional maximal operators on variable Morrey spaces.Let Sn−1 denote the unit sphere in Euclidean space Rn and  ∈ Ls(Sn−1) (s ≥ 1) be homogeneous of degree zero on Rn . For 0 < α < n, the rough fractional integral is defined byT,α f (x) =∫Rn(y′)|y|n−α f (x − y)dyand a related fractional maximal operator M,α is defined byE-mail addresses: tanjian89@126.com (J. Tan), jzhao@bnu.edu.cn (J. Zhao).http://dx.doi.org/10.1016/j.crma.2015.09.0241631-073X/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1118 J. Tan, J. Zhao / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1117–1122M,α f (x) = supr>01rn−α∫|y|≤r|(y′) f (x − y)|dy,where y′ = y|y| for any y = 0.It is easy to see that T,α is just the Riesz potential Iα when  ≡ 1. The Hardy–Littlewood–Sobolev theorem (see [6,14]) states that Iα is a bounded operator from Lebesgue spaces Lp(Rn) to Lq(Rn) when 0 < α < n, 1 < p < q < ∞ and 1q = 1p − αn . The boundedness of T,α on Lebesgue spaces was first introduced by Muckenhoupt and Wheeden [11]. Recently, Wang studied the weighted norm inequalities for T,α on Morrey spaces in [17]. It extends Spanne’s result on the boundedness of the fractional integrals in [13]. On the other hand, moving in another direction, variable exponent function spaces theory has attracted much attention, due to its application to partial differential equations and the calculus of variations (see [3,4,12,18]). In many applications, a crucial step has been to show that the classical operators of harmonic analysis, such as maximal operators, singular integrals and fractional integrals, are bounded on variable exponent function spaces. In 2007, Capone et al. [1] established the boundedness of Iα on variable Lebesgue spaces. In 2013, the Hardy–Littlewood–Sobolev theorem was generalized in [8] by Ho to the case of variable Morrey spaces on unbounded domains. In 2015, one of the authors and Liu in [15] showed that the Hardy–Littlewood–Sobolev theorem still held for homogeneous fractional integrals on variable Lebesgue spaces as well as on variable Hardy spaces. These results leave open the question of the Morrey spaces estimates for T,α and related operators in the variable exponent setting. Before stating our results, we need some notations and definitions on variable exponent analysis.For a measurable subset E ⊂ Rn , we denote p−(E) = infx∈E p(x) and p+(E) = supx∈E p(x). Especially, we denote p− =p−(Rn) and p+ = p+(Rn). Let p(·): Rn → (0, ∞) be a measurable function with 0 < p− ≤ p+ < ∞ and P be the set of all measurable functions p(·) : Rn → [1, ∞) such that 1 < p− ≤ p+ < ∞. Let B be the set of p(·) ∈ P such that the Hardy–Littlewood maximal operator M is bounded on Lp(·) .An important subset of B is the class of globally log-Hölder continuous functions p ∈ LH, with 1 < p− ≤ p+ < ∞. Recall that p(·) ∈ LH(Rn), if p(·) satisfies|p(x) − p(y)| ≤ C− log(|x − y|) , |x − y| ≤ 1/2, |p(x) − p(y)| ≤Clog |x| + e , |y| ≥ |x|.Definition 1.1. (See [8].) Let q ∈ P . A Lebesgue measurable function u(x, r) : Rn × (0, ∞) → (0, ∞) is said to be a Morrey weight function Wq for Lp(·) if there exists a constant C > 0 such that for any x ∈ Rn and r ≥ 0, u fulfills∞∑j=0‖χB(x,r)‖Lq(·)‖χB(x,2 j+1r)‖Lq(·)u(x,2 j+1r) < C u(x, r).Definition 1.2. (See [2,5].) The variable Lebesgue space Lp(·) is defined as the set of all measurable functions f for which the quantity ∫Rn|ε f (x)|p(x)dx is finite for some ε > 0 and‖ f ‖L p(·) = inf⎧⎨⎩λ > 0 :∫Rn( | f (x)|λ)p(x)dx ≤ 1⎫⎬⎭.As a special case of the theory of Nakano and Luxemberg, we see that L p(·) is a quasi-normed space. Especially, when p− ≥ 1, Lp(·) is a Banach space.Definition 1.3. (See [7].) The variable Morrey space Mp(·),u is the collection of all Lebesgue measurable functions f satisfy-ing‖ f ‖Mp(·),u = supz∈Rn,R>01u(z, R)‖χB(z,R) f ‖L p(·) < ∞,for p(x) ∈P and u(x, r) :Rn × (0.∞) → (0, ∞).Throughout this paper, C or c denotes a positive constant that may vary at each occurrence, but is independent of the main parameter, and A ∼ B means that there are constants C1 > 0 and C2 > 0 independent of the main parameter such that C1 B ≤ A ≤ C2 B . Given a measurable set S ⊂ Rn , |S| denotes the Lebesgue measure and χS means the characteristic function.Now, let us formulate our main results.Theorem 1.1. Suppose that  ∈ Ls(Sn−1) with 1 < s ≤ ∞. Let α > 0, p(·), q(·) ∈ B, q(·)(n−α)n ∈ B and u ∈ Wq(·) . Suppose that 1 ≤ s′ < p− ≤ p+ < nα and 1p(x) − 1q(x) = αn , a.e. on Rn, then‖T,α f ‖M ≤ C‖‖Ls(Sn−1)‖ f ‖Mq(·),u p(·),uJ. Tan, J. Zhao / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1117–1122 1119and‖M,α f ‖Mq(·),u ≤ C‖‖Ls(Sn−1)‖ f ‖Mp(·),u .Furthermore, if b ∈ BMO and∞∑j=0( j + 1) ‖χB(x,r)‖Lq(·)‖χB(x,2 j+1r)‖Lq(·)u(x,2 j+1r) < Cu(x, r),then‖[b, T,α] f ‖Mq(·),u ≤ C‖b‖∗‖‖Ls(Sn−1)‖ f ‖Mp(·),u .2. Proof of the main resultsBefore we prove the main results, we need some more facts on variable exponent function spaces. First we state results concerning the boundedness of the rough fractional integrals on variable Lebesgue spaces given in [15].Proposition 2.1. Let p(·), q(·) ∈ P , 0 < α < n, 1 < p− ≤ p+ < nα and 1q(x) = 1p(x) − αn for any x ∈ Rn. If q(·)(n−α)n ∈ B,  ∈ Ls and 1 ≤ s′ < p− , then‖T α f ‖Lq(·)(Rn) ≤ C‖‖Ls(Sn−1)‖ f ‖L p(·)(Rn).Similarly, we can obtain the boundedness of the commutators of rough fractional integral on variable Lebesgue spaces. For the sake of brevity, here we state the result without proof.Proposition 2.2. Let b ∈ BMO, p(·), q(·) ∈ P , 0 < α < n, 1 < p− ≤ p+ < nα and 1q(x) = 1p(x) − αn for any x ∈ Rn. If q(·)(n−α)n ∈ B,  ∈ Ls(Sn−1) and 1 ≤ s′ < p− , then‖[b, T ]α f ‖Lq(·)(Rn) ≤ C‖b‖∗‖‖Ls(Sn−1)‖ f ‖L p(·)(Rn).The next basic property satisfied by Lp(·) is given in [8].Proposition 2.3. Let p(·) ∈B, then we have a constant C > 0 so that for any balls or cubes B,‖χB‖L p(·)‖χB‖L p′(·) ∼ |B|.Proof of Theorem 1.1. Let f ∈ Mp(·),u . For any z ∈ Rn , r > 0 and decompose f = f0 + ∑∞j=1 f j , where f0 = χB(z,2r) f and f j = χB(z,2 j+1r)\B(z,2 j r) f , j ∈N\{0}.Since T,α is a linear operator, then we can write1u(z, r)‖χB(z,r)T,α f ‖Lq(·) (1)≤ 1u(z, r)‖χB(z,r)T,α f0‖Lq(·) +1u(z, r)∞∑j=1‖χB(z,r)T,α f j‖Lq(·) = I + II. (2)Notice that p(·), q(·) ∈ P, 1p(x) − 1q(x) = αn . Proposition 2.1 shows that ‖T,α f ‖Lq(·) ≤ C‖‖Ls(Sn−1)‖ f ‖Lp(·) . Thus, we can obtain thatI ≤ 1u(z, r)‖T,α f0‖Lq(·) ≤C‖‖Ls(Sn−1)u(z,2r)‖ f χB(z,2r)‖L p(·) ≤ C‖‖Ls(Sn−1)‖ f ‖Mp(·),u . (3)We now turn to deal with the term II. By Hölder’s inequality,|T,α f j(x)| ≤⎛⎜⎝∫R j|(x − y)|sdy⎞⎟⎠1/s ⎛⎜⎝∫R j| f (y)|s′|x − y|(n−α)s′ dy⎞⎟⎠1/s′,where R j = B(z, 2 j+1r)\B(z, 2 jr). When x ∈ B(z, r) and y ∈ R j , it easy to see that |x − y| ∼ |y − z| ∼ 2 jr. Therefore, R j ⊂{y : |x − y| ≤ C2 jr}. Then we can get1120 J. Tan, J. Zhao / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1117–1122⎛⎜⎝∫R j|((x − y)′)|sdy⎞⎟⎠1/s≤⎛⎜⎝∫|x−y|≤C2 jr|((x − y)′)|sdy⎞⎟⎠1/s≤⎛⎜⎝∫Sn−1C2 j r∫0Rn−1|((x − y)′)|s dR dσ(y′)⎞⎟⎠1/s≤ C‖‖Ls(Sn−1)(2 jr)n/s.Moreover,⎛⎜⎝∫R j| f (y)|s′|x − y|(n−α)s′ dy⎞⎟⎠1/s′≤ 2− j(n−α)r−n+α‖ f χB(z,2 j+1r)‖Ls′ .The generalized Hölder inequality given in [10] ensures that‖ f χB(z,2 j+1r)‖Ls′ ≤ C(2 jr)−ns ‖χB(z,2 j+1r) f ‖L p(·)‖χB(z,2 j+1r)‖L p′(·) .In fact, we can choose p̃(·) > 0 such that 1s′ = 1p(x) + 1p̃(x) due to 1 ≤ s′ < p− , we have‖ f χB(z,2 j+1r)‖Ls′ ≤ C‖ f χB(z,2 j+1r)‖L p(·)‖χB(z,2 j+1r)‖L p̃(·) .Moreover, observing that 1p̃(·) = 1s′ − 1p(·) = 1p′(·) − 1s , by Lemma 2.4 in [16], we have‖χB(z,2 j+1r)‖L p̃(·) ∼ (2 jr)−ns ‖χB(z,2 j+1r)‖L p′(·) .Therefore, we get that|T,α f j(x)χB(z,r)(x)| ≤ C‖‖Ls(Sn−1)χB(z,r)(x)(2 jr)α−n‖χB(z,2 j+1r) f ‖L p(·)‖χB(z,2 j+1r)‖L p′(·)and the rest of the proof is the same as the proof of Theorem 2.1 in [8]. Thus, we can prove that‖T,α f ‖Mq(·),u ≤ C‖‖Ls(Sn−1)‖ f ‖Mp(·),u .It is not hard to check that M,α f (x) ≤ C T ||,α(| f |)(x), so the proof of rough fractional maximal operators follows from the boundedness of T,α f .Next, we prove that ‖[b, T,α] f ‖Mq(·),u ≤ C‖b‖∗‖‖Ls(Sn−1)‖ f ‖Mp(·),u .1u(z, r)‖χB(z,r)[b, T,α] f ‖Lq(·)≤ 1u(z, r)‖χB(z,r)[b, T,α] f0‖Lq(·) +1u(z, r)∞∑j=1‖χB(z,r)[b, T,α] f j‖Lq(·) = J1 + J2.Applying Proposition 2.2, we have J1 ≤ C‖b‖∗‖‖Ls(Sn−1)‖ f ‖Mq(·),u .Denote that f B(z,r) = 1|B(z,r)|∫B(z,r) f (y) dy. Now we consider J2.|[b, T,α] f j(x)χB(z,r)(x)|≤ χB(z,r)(x)2( j+1)(α−n)rα−n∫B(z,2 j+1r)|b(x) − bB(z,r) + bB(z,r) − b(y)||(x − y)|| f (y)| dy≤ χB(z,r)(x)2( j+1)(α−n)rα−n|b(x) − bB(z,r)|∫B(z,2 j+1r)|(x − y)|| f (y)| dy+ χB(z,r)(x)2( j+1)(α−n)rα−n∫B(z,2 j+1r)|bB(z,r) − b(y)||(x − y)|| f (y)| dy = J3 + J4.It is easy to check thatJ3 ≤ C‖‖Ls(Sn−1)χB(z,r)(x)(2 jr)α−n|b(x) − bB(z,r)|‖χB(z,2 j+1r) f ‖L p(·)‖χB(z,2 j+1r)‖ p′(·) .LJ. Tan, J. Zhao / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 1117–1122 1121In the other hand, we estimate J4.J4 ≤ (2 jr)α−nχB(z,r)(x)∫B(z,2 j+1r)|(x − y)||b(y) − bB(z,2 j+1r)|| f (y)|dy+ (2 jr)α−nχB(z,r)(x)|bB(z,2 j+1r) − bB(z,r)|∫B(z,2 j+1r)|(x − y)| f (y)|dy = K1 + K2.As an application of Hölder’s inequality yieldsK1 ≤ C(2 jr)α−n+n/s‖‖Ls(Sn−1)χB(z,r)(x)‖(b − bB(z,2 j+1r)) f χB(z,2 j+1r)‖Ls′≤ C(2 jr)α−n‖‖Ls(Sn−1)χB(z,r)(x)‖ f χB(z,2 j+1r)‖L p(·)‖(b − bB(z,2 j+1r))χB(z,2 j+1r)‖L p′(·)≤ C(2 jr)α−n‖‖Ls(Sn−1)‖b‖∗χB(z,r)(x)‖ f χB(z,2 j+1r)‖L p(·)‖χB(z,2 j+1r)‖L p′(·) ,where the last inequality follows from the fact that ‖b‖∗ ∼ supQ 1‖χQ ‖L p(·) ‖(b − bQ )χQ ‖Lp(·) holds for all b ∈ BMO (see Lemma 3 in [9]).On the other hand, in view of |bB(z,2 j+1r) − bB(z,r)| ≤ C j‖b‖∗ , we can get thatK2 ≤ C(2 jr)α−nχB(z,r)(x) j‖b‖∗‖‖Ls(Sn−1)‖ f χB(z,2 j+1r)‖L p(·)‖χB(z,2 j+1r)‖L p′(·) .Therefore, by Proposition 2.3, we can obtain that‖[b, T,α] f j(x)χB(z,r)(x)‖Lq(·)≤ C( j + 1)(2 jr)α−n‖b‖∗‖‖Ls(Sn−1)‖χB(z,r)‖Lq(·)‖ f χB(z,2 j+1r)‖L p(·)‖χB(z,2 j+1r)‖L p′(·)≤ C‖b‖∗‖‖Ls(Sn−1)( j + 1)‖χB(z,r)‖Lq(·)‖χB(z,2 j+1r)‖Lq(·)u(z,2 j+1r) supy∈Rn,R>01u(y, R)‖ f χB(y,R)‖L p(·)≤ C‖b‖∗‖‖Ls(Sn−1)( j + 1)‖χB(z,r)‖Lq(·)‖χB(z,2 j+1r)‖Lq(·)u(z,2 j+1r)‖ f ‖Mp(·),u .Notice that ∑∞j=1( j + 1) ‖χB(z,r)‖Lq(·)‖χB(z,2 j+1r)‖Lq(·)u(z,2 j+1r)u(z,r) ≤ C .Thus,J2 ≤ C‖b‖∗‖‖Ls(Sn−1)∞∑j=1( j + 1) ‖χB(z,r)‖Lq(·)‖χB(z,2 j+1r)‖Lq(·)u(z,2 j+1r)u(z, r)‖ f ‖Mp(·),u ≤ C‖b‖∗‖‖Ls(Sn−1)‖ f ‖Mp(·),u .So we have completed the proof of Theorem 1.1. AcknowledgementsThe authors are very grateful to the referees for their valuable suggestions and comments. The authors also would like to express their thanks to Prof. Frédéric Bernicot for his warm help.The project is supported by National Natural Science Foundation of China (Grant No. 11471040), the Fundamental Re-search Funds for the Central Universities (No. 2014KJJCA10).References[1] C. Capone, D. Cruz-Uribe, A. Fiorenza, The fractional maximal operators and fractional integrals on variable Lp spaces, Rev. Mat. Iberoam. 23 (2007) 743–770.[2] D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser, Basel, Switzerland, 2013.[3] D. Cruz-Uribe, A. Fiorenza, J. Martell, C. Pérez, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn., Math. 31 (2006) 239–264.[4] D. Cruz-Uribe, L. Wang, Variable Hardy spaces, Indiana Univ. Math. J. 63 (2) (2014) 447–493.[5] L. Diening, P. Harjulehto, P. Hästö, M. 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Rough fractional integrals and its commutators on variable Morrey spaces

Tan, Jian

Zhao, Jiman

Numérisation de Documents Anciens Mathématiques

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Rough fractional integrals and its commutators on variable Morrey spaces

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