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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 239414, 5 pages doi:10.1155/2008/239414 Research Article John-Nirenberg Type Inequalities for the Morrey-Campanato Spaces Wenming Li College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016, Hebei, China Correspondence should be addressed to Wenming Li, lwmingg@sina.com Received 17 April 2007; Accepted 3 December 2007 Recommended by Y. Giga We give John-Nirenberg type inequalities for the Morrey-Campanato spaces on R n . Copyright q 2008 Wenming Li. 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. Given a function f ∈ L 1 loc R n and a cube Q on R n ,letf Q denote the average of f on Q, f Q 1/|Q| Q fxdx. We say that f has bounded mean oscillation if there is a constant C such that for any cube Q, 1 |Q| Q fx − f Q dx ≤ C. 1 The space of functions with this property is denoted by BMO. For f ∈ BMO, define the norm on BMO by f BMO sup Q 1 |Q| Q fx − f Q dx. 2 John and Nirenberg 1 obtained the following well-known John-Nirenberg inequality for BMO. Theorem 1. Let f ∈ BMO and f BMO / 0. Then there exist positive constants C 1 and C 2 , depending only on the dimension, such that for all cube Q and any λ>0, x ∈ Q : fx − f Q >λ ≤ C 1 e −C 2 λ/f BMO |Q|. 3 2 Journal of Inequalities and Applications Suppose f is a locally integrable function on R n , Q is a cube, and s is a nonnegative integer; let P Q fx be the unique polynomial of degree at most s such that Q fx − P Q fx x α dx 0 4 for all 0 ≤|α|≤s. Moreover, for any x ∈ Q, P Q fx ≤ A |Q| Q fx dx, 5 where the constant A is independent of f and Q. Clearly, A ≥ 1. For β ≥ 0,s≥ 0, 1 ≤ q<∞, we will say that a locally integrable function fx belongs to the Morrey-Campanato spaces Lβ, q,s if f Lβ,q,s sup Q |Q| −β 1 |Q| Q fx − P Q fx q dx 1/q < ∞, 6 where Q is a cube. Then if f − g is a polynomial of degree at most s, g also satisfies 6 and f Lβ,q,s g Lβ,q,s . If this is the case we say that f and g are equivalent, the quotient space divided by such equivalence classes will be denoted by Lβ, q,s,and6 defines its norm. These spaces played an important role in the study of partial differential equations and they were studied extensively. Reader is referred, in particular, to 2–4. Recently, Deng et al. 5 and Duong and Yan 6 gave several new characterizations for the Morrey-Campanato spaces. As noted in 2,forβ 0and1≤ q ≤∞, these spaces are variants of the BMO space. For β>0ands ≥ nβ, the spaces Lβ, q, s are variants of the homogeneous Lipschitz spaces ˙ Λ nβ R n which are duals of certain Hardy spaces. See also 1. In 7, we proved a John-Nirenberg-type inequality for homogeneous Lipschitz spaces ˙ Λ α R n , 0 <α<1. In this note, we will show that a similar inequality is also true for the Morrey-Campanato spaces Lβ, q,s on R n ,whereβ is nonnegative, 1 ≤ q ≤∞, and the integer s ≥ 0. Our main result can be stated as follows. Theorem 2 John-Nirenberg-type inequality. Given β ≥ 0 and s ≥ 0, let f ∈ Lβ, 1,s and f Lβ,1,s / 0. Then there exist positive constants C 1 and C 2 , depending only on the dimension, such that for all cube Q and any λ>0, x ∈ Q : |Q| −β fx − P Q fx >λ ≤ C 1 e −C 2 λ/f Lβ,1,s |Q|. 7 Proof. Let Q be a fixed cube and let λ 0 be some positive real number which will be determined later. Applying the Calderon-Zygmund decomposition to the function |Q| −β |fx − P Q fx| at height λ 0 to obtain a family of subcubes {Q j } of Q with disjoint inte- riors such that |Q| −β fx − P Q fx ≤ λ 0 a.e.Q\ ∞ j1 Q j , 8 λ 0 < 1 Q j Q j |Q| −β fx − P Q fx dx ≤ 2 n λ 0 for any j, 9 ∞ j1 Q j ≤ 1 λ 0 Q |Q| −β fx − P Q fx dx. 10 Wenming Li 3 By 5, for any x ∈ Q j ,weget P Q fx − P Q j fx P Q j P Q f − P Q j f x ≤ A Q j Q j P Q fy − P Q j fy|dy. 11 Thus for any x ∈ Q j ,by9 we have |Q| −β P Q fx − P Q j fx ≤ A Q j Q j |Q| −β P Q fy − P Q j fy dy ≤ A |Q j | Q j |Q| −β fy − P Q fy dy A Q j Q j |Q j | −β fy − P Q j fy dy ≤ A2 n λ 0 Af Lβ,1,s . 12 Denote b A2 n λ 0 Af Lβ,1,s >λ 0 . For any x ∈ Q j ,wehave |Q| −β fx − P Q fx ≤|Q| −β P Q fx − P Q j fx Q j −β fx − P Q j fx ≤ b Q j −β fx − P Q j fx . 13 Then for any λ>0, we have x ∈ Q : |Q| −β fx − P Q fx >λ b ⊂ x ∈ Q : |Q| −β fx − P Q fx >λ 0 ⊂ ∞ j1 Q j . 14 By 13 and 14, x ∈ Q : |Q| −β fx − P Q fx >λ b ⊂ ∞ j1 x ∈ Q j : |Q| −β fx − P Q fx >λ b ⊂ ∞ j1 x ∈ Q j : Q j −β fx − P Q j fx >λ . 15 For any λ>0, we set F f λsup Q 1 |Q| x ∈ Q : |Q| −β fx − P Q fx >λ . 16 Clearly, F f λ is a decreasing function on 0, ∞ and F f 0 ≤ 1. Using 10,wehave 1 |Q| x ∈ Q : |Q| −β fx − P Q fx >λ b ≤ F f λ 1 |Q| ∞ j1 Q j ≤ F f λ 1 λ 0 |Q| Q |Q| −β fx − P Q fx dx. 17 4 Journal of Inequalities and Applications So for any λ ≥ 0, we get F f λ b ≤ λ −1 0 f Lβ,1,s F f λ. Taking λ 0 ef Lβ,1,s ,thenb A2 n e 1f Lβ,1,s is also a fixed positive number and for any λ ≥ 0, F f λ b ≤ 1 e F f λ. 18 By induction argument for any k ≥ 1, we get F f k 1b ≤ e −k F f b. 19 Thus, for λ ∈ kb,k 1b,wehave F f λ ≤ F f kb ≤ e −k F f 0 ≤ ee −λ/b . 20 Notice that this inequality is also true for λ ∈ 0,b,duetoF f λ ≤ F f 01 ≤ ee −λ/b . Thus, for any λ ≥ 0, we have 1 |Q| x ∈ Q : |Q| −β fx − P Q fx >λ ≤ ee −λ/b . 21 This concludes the proof of the theorem. Corollary 1. Given β ≥ 0,s≥ 0. For all q ∈ 1, ∞, the spaces Lβ, q, s coincide, and the norms · Lβ,q,s are equivalent, namely, sup Q 1 |Q| Q |Q| −β fx−P Q fx q dx 1/q ≈ sup Q 1 |Q| Q |Q| −β fx − P Q fx dx. 22 Proof. It will suffice to prove that f Lβ,q,s ≤ C q f Lβ,1,s for any 1 <q<∞.Infact,by7, Q |Q| −β fx − P Q fx q dx ≤ q ∞ 0 λ q−1 x ∈ Q : |Q| −β fx − P Q fx >λ dλ ≤ C 1 q|Q| ∞ 0 λ q−1 e −C 2 λ/f Lβ,1,s dλ 23 make the change o f variables μ C 2 λ/f Lβ,1,s ,thenweget 1 |Q| Q |Q| −β fx − P Q fx q dx ≤ C 1 q f Lβ,1,s C 2 q ∞ 0 μ q−1 e −μ dμ C 1 qC −q 2 Γq f Lβ,1,s q 24 which yields the desired inequality. As a consequence of the proof of Corollary 1, we get two additional results. Corollary 2. Given β ≥ 0,s≥ 0, 1 ≤ q<∞,iff ∈ Lβ, q,s, then there exists λ>0 such that for any cube Q, 1 |Q| Q e λ|Q| −β |fx−P Q fx| dx < ∞. 25 Wenming Li 5 Corollary 3. Given β ≥ 0,s≥ 0, 1 ≤ q<∞, f ∈ L 1 loc R n , suppose there exist constants C 1 , C 2 ,and K such that for any cube Q and λ>0, |{x ∈ Q : |Q| −β |fx − P Q fx| >λ}| ≤ C 1 e −C 2 λ/K |Q|. 26 Then f ∈ Lβ, q, s. Acknowledgments The author would like to express his deep thanks to the referee for several valuable remarks and suggestions. This work was supported by National Natural Science Foundation of China nos. 10771049 and 60773174. References 1 F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Communications on Pure and Applied Mathematics, vol. 14, no. 3, pp. 415–426, 1961. 2 S. Janson, M. Taibleson, and G. Weiss, “Elementary characterizations of the Morrey-Campanato spaces,” in Harmonic Analysis (Cortona, 1982), vol. 992 of Lecture Notes in Mathematics, pp. 101–114, Springer, Berlin, Germany, 1983. 3 J. Peetre, “On the theory of L p,λ spaces,” Journal of Functional Analysis, vol. 4, no. 1, pp. 71–87, 1969. 4 M. H. Taibleson and G. Weiss, “The molecular characterization of certain Hardy spaces,” Ast ´ erisque, vol. 77, pp. 67–149, 1980. 5 D. Deng, X. T. Duong, and L. Yan, “A characterization of the Morrey-Campanato spaces,” Mathematische Zeitschrift, vol. 250, no. 3, pp. 641–655, 2005. 6 X. T. Duong and L. Yan, “New function spaces of BMO type, the John-Nirenberg inequality, interpola- tion, and applications,” Communications on Pure and Applied Mathematics, vol. 58, no. 10, pp. 1375–1420, 2005. 7 W. Li, “John-Nirenberg inequality and self-improving properties,” Journal of Mathematical Research and Exposition, vol. 25, no. 1, pp. 42–46, 2005. . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 239414, 5 pages doi:10.1155/2008/239414 Research Article John-Nirenberg Type Inequalities for the Morrey-Campanato Spaces Wenming. characterizations for the Morrey-Campanato spaces. As noted in 2 ,for 0and1≤ q ≤∞, these spaces are variants of the BMO space. For β>0ands ≥ nβ, the spaces Lβ, q, s are variants of the homogeneous. by Y. Giga We give John-Nirenberg type inequalities for the Morrey-Campanato spaces on R n . Copyright q 2008 Wenming Li. This is an open access article distributed under the Creative Commons
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