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 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 afunction f ∈ L1loc Rn and a cube Q on Rn , let fQ denote the average of f on Q, fQ 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
fx − fQ dx ≤ C.
1
|Q| Q
The space of functions with this property is denoted by BMO. For f ∈ BMO, define the norm
on BMO by
1
fx − fQ dx.
fBMO sup
2
|Q|
Q
Q
John and Nirenberg 1 obtained the following well-known John-Nirenberg inequality
for BMO.
0. Then there exist positive constants C1 and C2 , depending
Theorem 1. Let f ∈ BMO and fBMO /
only on the dimension, such that for all cube Q and any λ > 0,
x ∈ Q : fx − fQ > λ ≤ C1 e−C2 λ/fBMO |Q|.
3
2
Journal of Inequalities and Applications
Suppose f is a locally integrable function on Rn , Q is a cube, and s is a nonnegative
integer; let PQ fx be the unique polynomial of degree at most s such that
4
fx − PQ fx xα dx 0
Q
for all 0 ≤ |α| ≤ s. Moreover, for any x ∈ Q,
PQ fx ≤ A
|Q|
fxdx,
5
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
1/q
1
fx − PQ fxq dx
fLβ,q,s sup |Q|−β
< ∞,
6
|Q| Q
Q
where Q is a cube. Then if f − g is a polynomial of degree at most s, g also satisfies 6 and
fLβ,q,s gLβ,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, 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, 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, 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
Λ̇nβ Rn which are duals of certain Hardy spaces. See also 1.
In 7, we proved a John-Nirenberg-type inequality for homogeneous Lipschitz spaces
Λ̇α Rn , 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 Rn , 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
fLβ,1,s /
0. Then there exist positive constants C1 and C2 , depending only on the dimension, such
that for all cube Q and any λ > 0,
x ∈ Q : |Q|−β fx − PQ fx > λ ≤ C1 e−C2 λ/fLβ,1,s |Q|.
7
Proof. Let Q be a fixed cube and let λ0 be some positive real number which will
be determined later. Applying the Calderon-Zygmund decomposition to the function
|Q|−β |fx − PQ fx| at height λ0 to obtain a family of subcubes {Qj } of Q with disjoint interiors such that
∞
8
|Q|−β fx − PQ fx ≤ λ0 a.e. Q \ Qj ,
1
λ0 < Qj j1
|Q|−β fx − PQ fxdx ≤ 2n λ0
for any j,
9
Qj
∞ Qj ≤ 1
|Q|−β fx − PQ fxdx.
λ0 Q
j1
10
Wenming Li
3
By 5, for any x ∈ Qj , we get
PQ fx − PQ fx PQ PQ f − PQ f x
j
j
j
A
P fy − PQ fy|dy.
≤ j
Qj Q Q
11
j
Thus for any x ∈ Qj , by 9 we have
A
|Q|−β PQ fx − PQj fx ≤ Qj ≤
A
|Qj |
|Q|−β PQ fy − PQj fydy
Qj
A
|Q|−β fy − PQ fydy Qj Qj
|Qj |−β fy − PQj fydy
12
Qj
≤ A2n λ0 AfLβ,1,s .
Denote b A2n λ0 AfLβ,1,s > λ0 . For any x ∈ Qj , we have
−β |Q|−β fx − PQ fx ≤ |Q|−β PQ fx − PQj fx Qj fx − PQj fx
13
−β ≤ b Qj fx − PQj fx.
Then for any λ > 0, we have
∞
Qj .
x ∈ Q : |Q|−β fx − PQ fx > λ b ⊂ x ∈ Q : |Q|−β fx − PQ fx > λ0 ⊂
j1
14
By 13 and 14,
∞
x ∈ Qj : |Q|−β fx − PQ fx > λ b
x ∈ Q : |Q|−β fx − PQ fx > λ b ⊂
j1
∞
−β ⊂
x ∈ Qj : Qj fx − PQj fx > λ .
15
j1
For any λ > 0, we set
Ff λ sup
Q
1 x ∈ Q : |Q|−β fx − PQ fx > λ .
|Q|
16
Clearly, Ff λ is a decreasing function on 0, ∞ and Ff 0 ≤ 1. Using 10, we have
∞ 1 1 Qj x ∈ Q : |Q|−β fx − PQ fx > λ b ≤ Ff λ
|Q|
|Q| j1
1
≤ Ff λ
|Q|−β fx − PQ fxdx.
λ0 |Q| Q
17
4
Journal of Inequalities and Applications
So for any λ ≥ 0, we get Ff λ b ≤ λ−1
0 fLβ,1,s Ff λ. Taking λ0 efLβ,1,s , then b A2n e 1fLβ,1,s is also a fixed positive number and for any λ ≥ 0,
Ff λ b ≤
1
Ff λ.
e
18
By induction argument for any k ≥ 1, we get
Ff k 1b ≤ e−k Ff b.
19
Thus, for λ ∈ kb, k 1b, we have
Ff λ ≤ Ff kb ≤ e−k Ff 0 ≤ ee−λ/b .
20
Notice that this inequality is also true for λ ∈ 0, b, due to Ff λ ≤ Ff 0 1 ≤ ee−λ/b . Thus,
for any λ ≥ 0, we have
1 x ∈ Q : |Q|−β fx − PQ fx > λ ≤ ee−λ/b .
|Q|
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
−β |Q| fx−PQ fx dx
1/q
Q
1
≈ sup
|Q|
Q
|Q|−β fx − PQ fxdx.
22
Q
Proof. It will suffice to prove that fLβ,q,s ≤ Cq fLβ,1,s for any 1 < q < ∞. In fact, by 7,
q
|Q|−β fx − PQ fx dx ≤ q
∞
λq−1 x ∈ Q : |Q|−β fx − PQ fx > λ dλ
0
∞
λq−1 e−C2 λ/fLβ,1,s dλ
≤ C1 q|Q|
Q
23
0
make the change of variables μ C2 λ/fLβ,1,s , then we get
1
|Q|
Q
q ∞
f
q
Lβ,1,s
μq−1 e−μ dμ
|Q|−β fx − PQ fx dx ≤ C1 q
C2
0
q
−q
C1 qC2 Γq fLβ,1,s
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 < ∞, if f ∈ Lβ, q, s, then there exists λ > 0 such that for
any cube Q,
1
−β
eλ|Q| |fx−PQ fx| dx < ∞.
25
|Q| Q
Wenming Li
5
Corollary 3. Given β ≥ 0, s ≥ 0, 1 ≤ q < ∞, f ∈ L1loc Rn , suppose there exist constants C1 , C2 , and
K such that for any cube Q and λ > 0,
|{x ∈ Q : |Q|−β |fx − PQ fx| > λ}| ≤ C1 e−C2 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
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