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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 961502, 20 pages doi:10.1155/2010/961502 Research Article Commutators of Littlewood-Paley Operators on the Generalized Morrey Space Yanping Chen, 1 Yong Ding, 2 and Xinxia Wang 3 1 Department of Mathematics and Mechanics, Applied Science School, University of Science and Technology Beijing, Beijing 100083, China 2 Laboratory of Mathematics and Complex Systems (BNU), School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, China 3 The College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang 830046, China Correspondence should be addressed to Yanping Chen, yanpingch@126.com Received 6 May 2010; Accepted 11 July 2010 Academic Editor: Shusen Ding Copyright q 2010 Yanping Chen 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. Let μ Ω , μ S ,andμ ∗, λ denote the Marcinkiewicz integral, the parameterized area integral, and the parameterized Littlewood-Paley g ∗ λ function, respectively. In this paper, the authors give a characterization of BMO space by the boundedness of the commutators of μ Ω , μ S ,andμ ∗, λ on the generalized Morrey space L p,ϕ R n . 1. Introduction Let S n−1 {x ∈ R n : |x| 1} be the unit sphere in R n equipped with the Lebesgue measure dσ. Suppose that Ω satisfies the following conditions. aΩis the homogeneous function of degree zero on R n \{0},thatis, Ω μx Ω x , for any μ>0,x∈ R n \ { 0 } . 1.1 bΩhas mean zero on S n−1 ,thatis, S n−1 Ω x dσ x 0. 1.2 2 Journal of Inequalities and Applications cΩ∈ LipS n−1 ,thatis, Ω x − Ω y ≤ x − y , for any x ,y ∈ S n−1 . 1.3 In 1958, Stein 1 defined the Marcinkiewicz integral of higher dimension μ Ω as μ Ω f x ∞ 0 | F Ω,t x | 2 dt t 3 1/2 , 1.4 where F Ω,t x |x−y|≤t Ω x − y x − y n−1 f y dy. 1.5 We refer to see 1, 2 for the properties of μ Ω . Let 0 <<nand λ>1. The parameterized area integral μ S and the parameterized Littlewood-Paley g ∗ λ function μ ∗, λ are defined by μ S f x ⎛ ⎝ Γ x 1 t |y−z|<t Ω y − z y − z n− f z dz 2 dydt t n1 ⎞ ⎠ 1/2 , 1.6 where Γx{y,t ∈ R n1 : |x − y| <t}, and μ ∗, λ f x ⎛ ⎝ R n1 t t x − y λn 1 t |y−z|<t Ω y − z y − z n− f z dz 2 dydt t n1 ⎞ ⎠ 1/2 , 1.7 respectively. μ S and μ ∗, λ play very important roles in harmonic analysis and PDE e.g., see 3–8. Before stating our result, let us recall some definitions. For b ∈ L loc R n , the commutator b, μ Ω formed by b and the Marcinkiewicz integral μ Ω are defined by b, μ Ω f x ⎛ ⎝ ∞ 0 |x−y|≤t Ω x − y x − y n−1 b x − b y f y dy 2 dt t 3 ⎞ ⎠ 1/2 . 1.8 Journal of Inequalities and Applications 3 Let 0 <<nand λ>1. The commutator b, μ S of μ S and the commutator b, μ ∗, λ of μ ∗, λ are defined, respectively, by b, μ S f x ⎛ ⎝ Γ x 1 t |y−z|≤t Ω y − z y − z n− b x − b z f z dz 2 dydt t n1 ⎞ ⎠ 1/2 , 1.9 b, μ ∗, λ f x ⎛ ⎝ R n1 t t x − y λn × 1 t |y−z|≤t Ω y − z y − z n− b x − b z f z dz 2 dydt t n1 ⎞ ⎠ 1/2 . 1.10 Let b ∈ L loc R n .Itissaidthatb ∈ BMOR n if b ∗ : sup B⊂R n M b, B < ∞, 1.11 where B Bx, r denotes the ball in R n centered at x and with radius r, M b, B 1 | B | B | b x − b B | dx, 1.12 and b B 1/|B| B bydy. There are some results about the boundedness of the commutators formed by BMO functions with μ Ω , μ S ,andμ ∗, λ see 7, 9, 10. Many important operators gave a characterization of BMO space. In 1976, Coifman et al. 11 gave a characterization of BMO space by the commutator of Riesz transform; in 1982, Chanillo 12 studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space. The purpose of this paper is to give a characterization of BMO space by the boundedness of the commutators of μ Ω , μ S ,andμ ∗, λ on the generalized Morrey space L p,ϕ R n . Definition 1.1. Let 1 <p<∞. Suppose that ϕ : 0, ∞ → 0, ∞ be such that ϕt is nonincreasing and t 1/p ϕt is nondecreasing. The generalized Morrey space L p,ϕ is defined by L p,ϕ R n f ∈ L loc R n : f L p,ϕ < ∞ , 1.13 where f L p,ϕ sup x∈R n r>0 1 ϕ | B x, r | 1 | B x, r | Bx,r f y p dy 1/p . 1.14 4 Journal of Inequalities and Applications We refer to see 13, 14 for the known results of the generalized Morrey space L p,ϕ for some suitable ϕ.Notingthatϕt ≡ t −1/p , we get the Lebesque space L p R n . For ϕt t λ/n−1/p 0 <λ<n, L p,ϕ R n coincides with the Morrey space L p,λ R n . The main result in this paper is as follows. Theorem 1.2. Assume that ϕt is nonincreasing and t 1/p ϕt is nondecreasing. Suppose that b, μ Ω is defined as 1.8, Ω satisfies 1.1, 1.2, and Ω x − Ω y ≤ C 1 log2/ x − y γ ,C 1 > 0,γ>1,x ,y ∈ S n−1 . 1.15 If b, μ Ω is bounded on L p,ϕ R n for some p 1 <p<∞,thenb ∈ BMOR n . Theorem 1.3. Let 0 <<nand 1 <p<∞. Assume that ϕt is nonincreasing and t 1/p ϕt is nondecreasing. Suppose that b, μ S is defined as 1.9, Ω satisfies 1.1, 1.2, and 1.15.Ifb, μ S is a bounded operator on L p,ϕ R n for some p 1 <p<∞,thenb ∈ BMOR n . Theorem 1.4. Let 0 <<n, λ>1, and 1 <p<∞. Assume that ϕt is nonincreasing and t 1/p ϕt is nondecreasing. Suppose that b, μ ∗,ϕ λ is defined as 1.10, Ω satisfies 1.1, 1.2, and 1.15.Ifb, μ ∗, λ is on L p,ϕ R n for some p 1 <p<∞,thenb ∈ BMOR n . Remark 1.5. It is easy to check that b, μ S fx ≤ 2 λn b, μ ∗, λ fxsee, e.g., the proof of 19 in 15, page 89, we therefore give only the proofs of Theorem 1.2 for b, μ Ω and Theorem 1.3 for b, μ S . Remark 1.6. It is easy to see that the condition 1.15 is weaker than Lip β S n−1 for 0 <β≤ 1. In the proof of Theorems 1.2 and 1.3, we will use some ideas in 16. However, because Marcinkiewicz integral and the parameterized Littlewood-Paley operators are neither the convolution operator nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof. 2. Proof of Theorem 1.2 Let us begin with recalling some known conclusion. Similar to the proof of 17, we can easily get the following. Lemma 2.1. If Ω satisfies conditions 1.1, 1.2, and 1.15,letβ>0, then for |x| > 2|y|, we have Ω x − y x − y β − Ω x | x | β ≤ C | x | β log | x | / y γ . 2.1 Now let us return to the proof of Theorem 1.2. Suppose that b, μ Ω is a bounded operator on L p,ϕ R n , we are going to prove that b ∈ BMOR n . Journal of Inequalities and Applications 5 We may assume that b, μ Ω L p,ϕ →L p,ϕ 1. We want to prove that, for any x 0 ∈ R n and r ∈ R , the inequality N 1 | B x 0 ,r | Bx 0 ,r b y − a 0 dy ≤ A p, Ω,n,γ 2.2 holds, where a 0 |Bx 0 ,r| −1 Bx 0 ,r bydy. Since b − a 0 ,μ Ω b, μ Ω , we may assume that a 0 0. Let f y sgn b y − c 0 χ Bx 0 ,r y , 2.3 where c 0 1/|Bx 0 ,r| Bx 0 ,r sgnbydy. Since 1/|Bx 0 ,r| Bx 0 ,r bydy a 0 0, we can easily get |c 0 | < 1. Then, f has the following properties: f ∞ ≤ 2, 2.4 supp f ⊂ B x 0 ,r , 2.5 R n f y dy 0, 2.6 f y b y > 0,y∈ B x 0 ,r , 2.7 1 | B x 0 ,r | R n f y b y dy N. 2.8 In this proof for j 1, ,15, A j is a positive constant depending only on Ω,p, n, γ,and A i 1 ≤ i<j. Since Ω satisfies 1.2, then there exists an A 1 such that 0 <A 1 < 1and σ x ∈ S n−1 : Ω x ≥ 2C 1 log2/A 1 γ > 0, 2.9 where σ is the measure on S n−1 which is induced from the Lebesgue measure on R n .Bythe condition 1.15,itiseasytoseethat Λ : x ∈ S n−1 : Ω x ≥ 2C 1 log2/A 1 γ 2.10 is a closed set. We claim that if x ∈ Λ and y ∈ S n−1 , satisfying x − y ≤ A 1 , then Ω y ≥ C 1 log2/A 1 γ . 2.11 6 Journal of Inequalities and Applications In fact, since |Ωx − Ωy |≤C 1 /log2/|x − y | γ ≤ C 1 /log2/A 1 γ , note that Ωx ≥ 2C 1 /log2/A 1 γ , we can get Ωy ≥ C 1 /log2/A 1 γ . Taking A 2 > 3/A 1 ,let G x ∈ R n : | x − x 0 | ≥ A 2 r, x − x 0 ∈ Λ . 2.12 For x ∈ G, we have b, μ Ω f x ≥ μ Ω bf x − | b x | μ Ω f x ⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0 |x−y|≤t Ω x − y x − y n−1 b y f y dy 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 − | b x | ⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0 |x−y|≤t Ω x − y x − y n−1 f y dy 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 : I 1 − I 2 . 2.13 For I 1 , noting that if y ∈ Bx 0 ,r, then |x − x 0 | >A 2 |y − x 0 | for x ∈ G. Thus, we have x − y − x − x 0 ≤ 2 y − x 0 | x − x 0 | ≤ 2 A 2 <A 1 . 2.14 Using 2.11,wegetΩx − y ≥ C 1 /log2/A 1 γ . Noting that |x − x 0 ||x − y|, it follows from 2.5, 2.7, 2.8,andH ¨ older’s inequality that I 1 ≥ ⎧ ⎪ ⎨ ⎪ ⎩ ∞ | x−x 0 | ⎛ ⎜ ⎝ Bx 0 ,r Ω x − y b y f y x − y n−1 χ {|x−y|≤t} y dy ⎞ ⎟ ⎠ 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 ≥ ⎛ ⎜ ⎝ ∞ | x−x 0 | Bx 0 ,r Ω x − y b y f y x − y n−1 χ {|x−y|≤t} dy dt t 3 ⎞ ⎟ ⎠ ∞ | x−x 0 | dt t 3 −1/2 ≥ C 1 log2/A 1 γ | x − x 0 | Bx 0 ,r x − y −n1 b y f y |x−x 0 |≤t |x−y|≤t dt t 3 dy ≥ C log2/A 1 γ | x − x 0 | −n Bx 0 ,r b y f y dy A 3 Nr n | x − x 0 | −n . 2.15 Journal of Inequalities and Applications 7 For x ∈ G,byΩ ∈ L ∞ S n−1 , 2.4, 2.5, 2.6, the Minkowski inequality, and Lemma 2.1,we obtain I 2 | b x | ⎧ ⎨ ⎩ ∞ 0 R n f y Ω x − y x − y n−1 χ {|x−y|≤t} − Ω x − x 0 | x − x 0 | n−1 χ {|x−x 0 |≤t} dy 2 dt t 3 ⎫ ⎬ ⎭ 1/2 ≤ | b x | ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎝ ∞ 0 |x−y|≤t<|x−x 0 | Ω x − y x − y n−1 f y dy 2 dt t 3 ⎞ ⎠ 1/2 ⎛ ⎝ ∞ 0 |x−x 0 |≤t<|x−y| | Ω x − x 0 | | x − x 0 | n−1 f y dy 2 dt t 3 ⎞ ⎠ 1/2 ⎛ ⎜ ⎝ ∞ 0 ⎛ ⎝ |x−x 0 |≤t |x−y|≤t Ω x − y x − y n−1 − Ω x − x 0 | x − x 0 | n−1 f y dy ⎞ ⎠ 2 dt t 3 ⎞ ⎟ ⎠ 1/2 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ≤ | b x | ⎧ ⎨ ⎩ Bx 0 ,r Ω x − y x − y n−1 f y |x−y|≤t<|x−x 0 | dt t 3 1/2 dy Bx 0 ,r | Ω x − x 0 | | x − x 0 | n−1 f y |x−y|>t≥|x−x 0 | dt t 3 1/2 dy Bx 0 ,r f y Ω x − y x − y n−1 − Ω x − x 0 | x − x 0 | n−1 ⎛ ⎝ |x−y|≤t |x−x 0 |≤t dt t 3 ⎞ ⎠ 1/2 dy ⎫ ⎪ ⎬ ⎪ ⎭ ≤ C | b x | r 1/2 Bx 0 ,r f y | x − x 0 | n1/2 dy Bx 0 ,r f y | x − x 0 | n log | x − x 0 | /r γ dy ≤ A 4 | b x | r n | x − x 0 | −n log | x − x 0 | r −γ . 2.16 Let F x ∈ G : | b x | > A 3 N 2A 4 log | x − x 0 | r γ , | x − x 0 | <N 1/n r . 2.17 Without loss of generality, we may assume that N>A 2 > 1, otherwise, we get the desired 8 Journal of Inequalities and Applications result. Since ϕt is nonincreasing, it follows that ϕ|Bx 0 ,N 1/n r| ≤ ϕ|Bx 0 ,r|ϕr n .By 2.13, 2.15,and2.16, we have f p L p,ϕ ≥ b, μ Ω f p L p,ϕ ≥ 1 ϕ B x 0 ,N 1/n r p B x 0 ,N 1/n r |x−x 0 |<N 1/n r b, μ Ω f x p dx ≥ 1 ϕ r n p Nr n G\F∩{|x−x 0 |<N 1/n r} 1 2 A 3 Nr n | x − x 0 | −n p dx ≥ 1 ϕ r n p Nr n {A 5 |F|A 2 r n 1/n <|x−x 0 |<N 1/n r}∩G 1 2 A 3 Nr n | x − x 0 | −n p dx ω n−1 ϕ r n p Nr n A 3 Nr n 2 p N 1/n r A 5 | F | A 2 r n 1/n t −pnn−1 dt ≥ ω n−1 ϕ r n p Nr n p−1 A 3 /2 p n − np N 1−p r n1−p − A 1−pn 5 | F | A 2 r n 1−p . 2.18 Thus, | F | A 2 r n 1−p ≤ A 6 N 1−p r n1−p 1 ϕ r n p f p L p,ϕ . 2.19 Now, we claim that f L p,ϕ ≤ C ϕ r n , 2.20 where C is independent of r. In fact, f L p,ϕ sup x∈R n t>0 1 ϕ | B x, t | 1 | B x, t | Bx,t f y p dy 1/p . 2.21 Now, we consider the L p,ϕ norm of f in the following two cases. Case 1 t>r. Since s 1/p ϕs is nondecreasing in s, then 1 ϕ | B x, t | 1 | B x, t | 1/p ≤ 1 ϕ r n 1 r n/p . 2.22 Journal of Inequalities and Applications 9 Thus, f L p,ϕ ≤ sup x∈R n t>0 1 ϕ r n 1 r n/p Bx,t f y p dy 1/p sup x∈R n t>0 1 ϕ r n 1 r n/p Bx,t∩Bx 0 ,r f y p dy 1/p ≤ C ϕ r n . 2.23 Case 2 t ≤ r. Since ϕs is nonincreasing in s, then 1 ϕ | B x, t | ≤ 1 ϕ r n . 2.24 Thus, f L p,ϕ ≤ sup x∈R n t>0 1 ϕ r n 1 | B x, t | Bx,t f y p dy 1/p ≤ C ϕ r n . 2.25 Now, 2.20 is established. Then, by 2.19 and 2.20,weget | F | A 2 r n ≥ A 7 Nr n . 2.26 If N ≤ 2A −1 7 A n 2 , then Theorem 1.2 is proved. If N>2A −1 7 A n 2 , then | F | ≥ A 7 2 Nr n . 2.27 Let gyχ Bx 0 ,r y. For x ∈ F, we have b, μ Ω g x ≥ | b x | ⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0 |x−y|≤t Ω x − y x − y n−1 g y dy 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 − ⎧ ⎪ ⎨ ⎪ ⎩ ∞ 0 |x−y|≤t Ω x − y x − y n−1 b y g y dy 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 : K 1 − K 2 . 2.28 10 Journal of Inequalities and Applications Noting that if y ∈ Bx 0 ,r and x ∈ F,weget|x − y − x − x 0 |≤A 1 . Applying 2.11,we have Ωx − y ≥ C 1 /log2/A 1 γ . Since |x − y||x − x 0 | when y ∈ Bx 0 ,r and x ∈ F,it follows that K 1 ≥ | b x | ⎧ ⎪ ⎨ ⎪ ⎩ ∞ | x−x 0 | ⎛ ⎜ ⎝ Bx 0 ,r Ω x − y x − y n−1 χ {|x−y|≤t} y dy ⎞ ⎟ ⎠ 2 dt t 3 ⎫ ⎪ ⎬ ⎪ ⎭ 1/2 ≥ | b x | ⎛ ⎜ ⎝ ∞ | x−x 0 | Bx 0 ,r Ω x − y x − y n−1 χ {|x−y|≤t} dy dt t 3 ⎞ ⎟ ⎠ ∞ | x−x 0 | dt t 3 −1/2 ≥ C 1 | b x | log2/A 1 γ | x − x 0 | Bx 0 ,r x − y −n1 |x−x 0 |≤t |x−y|≤t dt t 3 dy ≥ A 8 | b x || x − x 0 | −n Bx 0 ,r dy A 8 r n | b x || x − x 0 | −n . 2.29 By Ω ∈ L ∞ S n−1 , |x− x 0 ||x −y| when y ∈ Bx 0 ,r and x ∈ F and the Minkowski inequality, we have K 2 ≤ C Bx 0 ,r b y x − y n dy ≤ A 9 | x − x 0 | −n Bx 0 ,r b y dy A 9 Nr n | x − x 0 | −n . 2.30 Thus, by 2.28, 2.29,and2.30,weget,forx ∈ F, b, μ Ω g x ≥ A 8 r n | b x || x − x 0 | −n − A 9 Nr n | x − x 0 | −n . 2.31 Similar to the proof of 2.20, we can easily get g L p,ϕ ≤ C/ϕr n .Thus,by2.31, [...]... 13 T Mizuhara, Commutators of singular integral operators on Morrey spaces with general growth functions,” Surikaisekikenkyusho K¯ kyuroku, no 1102, pp 49–63, 1999, Proceedings of the Coference on o ¯ ¯ ¯ Harmonic Analysis and Nonlinear Partial Differential Equations, Kyoto, Japan, 1998 14 Y Komori and T Mizuhara, “Factorization of functions in H 1 Rn and generalized Morrey spaces,” Mathematische Nachrichten,... “A note on the Marcinkiewicz integral,” Colloquium Mathematicum, vol 60-61, no 1, pp 235–243, 1990 8 E M Stein, The development of square functions in the work of A Zygmund,” Bulletin of the American Mathematical Society, vol 7, no 2, pp 359–376, 1982 9 Y Ding, S Lu, and K Yabuta, On commutators of Marcinkiewicz integrals with rough kernel,” Journal of Mathematical Analysis and Applications, vol... authors wish to express their gratitude to the referee for his/her valuable comments and suggestions The research was supported by NSF of China Grant nos.: 10901017, 10931001 , SRFDP of China Grant no.: 20090003110018 , and NSF of Zhenjiang Grant no.: Y7080325 References 1 E M Stein, On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Transactions of the American Mathematical Society, vol... be as 2.3 , then 2.4 – 2.8 hold In this proof for j 1, , 13, Bj is a positive constant depending only on Ω, p, n, , and Bi 1 ≤ i < j Since Ω satisfies 1.2 , then there exists a B1 such that 0 < B1 < 1 and σ x ∈ Sn−1 : Ω x ≥ 2C1 log 2/B1 γ > 0, 3.2 Journal of Inequalities and Applications 13 where σ is the measure on Sn−1 which is induced from the Lebesgue measure on Rn By the condition 1.15 , it... 2006 20 Journal of Inequalities and Applications 15 E M Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no 30, Princeton University Press, Princeton, NJ, USA, 1970 16 A Uchiyama, On the compactness of operators of Hankel type,” Tˆ hoku Mathematical Journal, vol o 30, no 1, pp 163–171, 1978 17 Y Ding, “A note on end properties of Marcinkiewicz... 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Corporation Journal of Inequalities and Applications Volume 2010, Article ID 961502, 20 pages doi:10.1155/2010/961502 Research Article Commutators of Littlewood-Paley Operators on the Generalized Morrey. nor the linear operators, hence, we need new ideas and nontrivial estimates in the proof. 2. Proof of Theorem 1.2 Let us begin with recalling some known conclusion. Similar to the proof of 17,. paper is to give a characterization of BMO space by the boundedness of the commutators of μ Ω , μ S ,andμ ∗, λ on the generalized Morrey space L p,ϕ R n . Definition 1.1. Let 1 <p<∞. Suppose
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