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Volume 2008, Article ID 239414, 5 pagesdoi:10.1155/2008/239414 Research Article John-Nirenberg Type Inequalities for the Morrey-Campanato Spaces Wenming Li College of Mathematics and Inf

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 Rn.

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 ∈ L1

locRn  and a cube Q on R n , let f Q denote the average of f on Q, f Q 

1/|Q|Q f xdx We say that f has bounded mean oscillation if there is a constant C such that for any cube Q,

1

|Q|



Q

The space of functions with this property is denoted by BMO For f ∈ BMO, define the norm

on BMO by

fBMO  sup

Q

1

|Q|



Q

John and Nirenberg 1 obtained the following well-known John-Nirenberg inequality for BMO

Theorem 1 Let f ∈ BMO and fBMO/  0 Then there exist positive constants C1and C2, depending only on the dimension, such that for all cube Q and any λ > 0,

x ∈ Q :f x − f Q> λ  ≤ C1e−C2λ/ fBMO|Q|. 3

Suppose f is a locally integrable function on Rn , Q is a cube, and s is a nonnegative integer; let P Q f x be the unique polynomial of degree at most s such that



Q



f x − P Q fxx α dx 0 4 for all 0≤ |α| ≤ s Moreover, for any x ∈ Q,

P Q fx ≤ A |Q|

Q

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

Q

|Q| −β

1

|Q|



Q

f x − P Q fxq

dx

1/q

where Q is a cube Then if f − g is a polynomial of degree at most s, g also satisfies 6 and

divided by such equivalence classes will be denoted by Lβ, q, s, and 6 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, to2 4 Recently, Deng et al 5 and Duong and Yan6 gave several new characterizations for the Morrey-Campanato spaces

As noted in2, for β  0 and 1 ≤ q ≤ ∞, these spaces are variants of the BMO space For β > 0 and s ≥ nβ, the spaces Lβ, q, s are variants of the homogeneous Lipschitz spaces

˙

ΛnβRn which are duals of certain Hardy spaces See also 1

In7, we proved a John-Nirenberg-type inequality for homogeneous Lipschitz spaces

˙

ΛαRn , 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 , where β 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

that for all cube Q and any λ > 0,

x ∈ Q : |Q| −βf x − P Q fx> λ  ≤ C1e −C2λ/ f L β,1,s |Q|. 7

be determined later Applying the Calderon-Zygmund decomposition to the function

|Q| −β |fx − P Q fx| at height λ0to obtain a family of subcubes{Q j } of Q with disjoint

inte-riors such that

|Q| −βf x − P Q fx ≤ λ0 a.e Q\∞

j1

λ0< Q1j



Q j

|Q| −βf x − P Q fxdx≤ 2n λ0 for any j, 9

∞

j1

Q j ≤ 1

λ0



Q

|Q| −βf x − P Q fxdx. 10

By5, for any x ∈ Q j, we get

P Q fx − P Q j fx  P Q j



P Q f − P Q j fx

≤ Q A j



Q j

P Q fy − P Q j fy|dy. 11

Thus for any x ∈ Q j, by9 we have

|Q| −βP Q fx − P Q j fx ≤ A

Q j



Q j

|Q| −βP Q fy − P Q j fydy

≤ A

|Q j|



Q j

|Q| −βf y − P Q fydy A

Q j



Q j

|Q j|−βf y − P Q j fydy

≤ A2 n λ0 Af L β,1,s

12

Denote b  A2 n λ0 Af L β,1,s > λ0 For any x ∈ Q j, we have

|Q| −βf x − P Q fx ≤ |Q| −βP Q fx − P Q j fx Q j−βf x − P Q j fx

≤ b Q j−βf x − P Q j fx. 13

Then for any λ > 0, we have



x ∈ Q : |Q| −βf x − P Q fx> λ b

⊂x ∈ Q : |Q| −βf x − P Q fx> λ0

⊂∞

j1

Q j

14

By13 and 14,



x ∈ Q : |Q| −βf x − P Q fx> λ b

⊂∞

j1



x ∈ Q j :|Q| −βf x − P Q fx> λ b

⊂∞

j1



x ∈ Q j :Q j−βf x − P Q

j fx> λ

.

15

For any λ > 0, we set

F f λ  sup

Q

1

|Q|x ∈ Q : |Q| −βf x − P Q fx> λ. 16

Clearly, F f λ is a decreasing function on 0, ∞ and F f0 ≤ 1 Using 10, we have

1

|Q|x ∈ Q : |Q| −βf x − P Q fx> λ b ≤ F f λ 1

|Q|

∞

j1

Q j

≤ F f λ 1

λ0|Q|



Q

|Q| −βf x − P Q fxdx. 17

So for any λ ≥ 0, we get F f λ b ≤ λ−1

A2n e 1f L β,1,s is also a fixed positive number and for any λ≥ 0,

F f λ b ≤ 1

By induction argument for any k≥ 1, we get

F f



Thus, for λ ∈ kb, k 1b, we have

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, due to F f λ ≤ F f 0  1 ≤ ee −λ/b Thus,

for any λ≥ 0, we have

1

|Q|x ∈ Q : |Q| −βf x − 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

sup

Q

1

|Q|



Q



|Q| −βf x−P Q fxq

dx

1/q

≈ sup

Q

1

|Q|



Q

|Q| −βf x − P Q fxdx. 22

Proof It will su ffice to prove that f L β,q,s ≤ C q f L β,1,s for any 1 < q <∞ In fact, by 7,



Q



|Q| −βf x − P Q fxq

∞

0

λ q−1x ∈ Q : |Q| −βf x − P Q fx> λdλ

≤ C1q|Q|

∞

0

λ q−1e −C2λ/ f L β,1,s dλ

23

make the change of variables μ  C2λ/f L β,1,s, then we get

1

|Q|



Q



|Q| −βf x − P Q fxq

C2

q∞

0

μ q−1e −μ dμ

 C1qC−q2 Γqf L β,1,sq

24

which yields the desired inequality

As a consequence of the proof ofCorollary 1, we get two additional results

Corollary 2 Given β ≥ 0, s ≥ 0, 1 ≤ q < ∞, if f ∈ Lβ, q, s, then there exists λ > 0 such that for

any cube Q,

1

|Q|



Q

Corollary 3 Given β ≥ 0, s ≥ 0, 1 ≤ q < ∞, f ∈ L1

locRn , suppose there exist constants C1, C2, and

K such that for any cube Q and λ > 0,

|{x ∈ Q : |Q| −β |fx − P Q fx| > λ}| ≤ C1e−C2λ/K |Q|. 26

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 Lp,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.

By5, for any x ∈ Q j, we get

P... fxdx. 17

So for any λ ≥ 0, we get F f λ b ≤ λ−1

A2n... −λ/b 21 This concludes the proof of the theorem

Corollary Given β ≥ 0, s ≥ For all q ∈ 1, ∞, the spaces Lβ, q, s coincide, and the norms

sup

Q