Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 913403, 6 pages
doi:10.1155/2011/913403
Research Article
Notes on Interpolation Inequalities
Jiu-Gang Dong1 and Ti-Jun Xiao2
1
2
Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences,
Fudan University, Shanghai 200433, China
Correspondence should be addressed to Ti-Jun Xiao, xiaotj@ustc.edu.cn
Received 3 October 2010; Accepted 16 November 2010
Academic Editor: Toka Diagana
Copyright q 2011 J.-G. Dong and T.-J. Xiao. 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.
An easy proof of the John-Nirenberg inequality is provided by merely using the CalderónZygmund decomposition. Moreover, an interpolation inequality is presented with the help of the
John-Nirenberg inequality.
1. Introduction
It is well known that various interpolation inequalities play an important role in the study of
operational equations, partial differential equations, and variation problems see, e.g., 1–6.
So, it is an issue worthy of deep investigation.
Let Q0 be either Rn or a fixed cube in Rn . For f ∈ L1loc Q0 , write
1
f : sup
BMO
|Q|
Q⊂Q0
f − fQ dx,
1.1
Q
where the supremum is taken over all cubes Q ⊂ Q0 and fQ : 1/|Q| Q fdx.
Recall that BMOQ0 is the set consisting of all locally integrable functions on Q0
such that fBMO < ∞, which is a Banach space endowed with the norm · BMO . It is clear
that any bounded function on Q0 is in BMOQ0 , but the converse is not true. On the other
hand, the BMO space is regarded as a natural substitute for L∞ in many studies. One of the
important features of the space is the John-Nirenberg inequality. There are several versions
of its proof; see, for example, 2, 7–9. Stimulated by these works, we give, in this paper, an
easy proof of the John-Nirenberg inequality by using the Calderón-Zygmund decomposition
only. Moreover, with the help of this inequality, an interpolation inequality is showed for Lp
and BMO norms.
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2. Results and Proofs
Lemma 2.1 John-Nirenberg inequality. If f ∈ BMOQ0 , then there exist positive constants c1 ,
c2 such that, for each cube Q ⊂ Q0 ,
x ∈ Q : fx − fQ > t ≤ c1 exp − c2 t |Q|,
f BMO
t > 0.
2.1
Proof. Without loss of generality, we can and do assume that fBMO 1.
For each t > 0, let Ft denote the least number for which we have
x ∈ Q : fx − fQ > t ≤ Ft|Q|,
2.2
for any cube Q ⊂ Q0 . It is easy to see that Ft ≤ 1t > 0 and Ft is decreasing.
Fix a cube Q ⊂ Q0 . Applying the Calderón-Zygmund decomposition cf., e.g., 2, 9
to |fx − fQ | on Q, with 2n as the separating number, we get a sequence of disjoint cubes
{Qj } and E such that
⎞
Q ⎝ Qj ⎠ ∪ E,
⎛
2.3
j
fx − fQ ≤ 2n ,
1
2n < Qj for a.e. x ∈ E,
f − fQ dx ≤ 4n .
Qj
2.4
2.5
Using 2.5, we have
Qj < 1 |Q|.
2n
j
2.6
From 2.3, 2.4, and 2.6, we deduce that for t > 4n ,
x ∈ Q : fx − fQ > t x ∈ Qj : fx − fQ > t j
x ∈ Qj : fx − fQ > t − 4n ≤
j
1 Qj x ∈ Qj : fx − fQ > t − 4n Qj j
≤
1
Ft − 4n |Q|.
2n
2.7
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This yields that
Ft ≤
1
Ft − 4n ,
2n
2.8
t > 4n .
Let γ t − 14−n t > 4n , μ 1 γ4n . Then 0 < μ ≤ t. By iterating, we get
−n
Ft ≤ F μ F 1 γ4n ≤ 2−nγ ≤ 2−nt−14 −1
−n
2n14 exp − log 2 n4−n t , t > 4n .
2.9
Thus, letting
−n
c1 2n14 ,
c2 log 2 n4−n
2.10
gives that
Ft ≤ c1 e−c2 t ,
t > 0,
2.11
0 < t ≤ 4n .
2.12
since
Ft ≤ 1 ≤ c1 e−c2 t ,
This completes the proof.
Remark 2.2. 1 As we have seen, the recursive estimation 2.8 justifies the desired
exponential decay of Ft.
2 There exists a gap in the proof of the John-Nirenberg inequality given in 2.
Actually, for a decreasing function Gt : 0, ∞ → 0, 1, the following estimate:
G2 · 2n α ≤
1
G2n α,
α
2.13
α>1
does not generally imply such a property, that is, the existence of constants c1 , c2 > 0 such
that
Gt ≤ c1 e−c2 t ,
2.14
t > 0.
We present the following function as a counter example:
5 −1 2
Gt exp − log
log t 1 ,
3
t > 0.
2.15
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In fact, it is easy to see that there are no constants c1 , c2 > 0 such that 2.14 holds. On the
other hand, we have
G t
Gt
5 −1 logt 1
− log
,
2
3
t1
t > 0.
2.16
Integrating both sides of the above equation from 2n α to 2 · 2n α, we obtain
n
5 −1 2·2 α logt 1
G2 · 2 α exp −2 log
dt G2n α
3
t
1
n
2 α
n
5 −1 2
2 n
n
exp − log
log 2 · 2 α 1 − log 2 α 1 G2n α
3
5 −1
2 · 2n α 1
n
n
G2n α
exp − log
log2 · 2 α 12 α 1 · log
2.17
3
2n α 1
≤ exp − log2 · 2n α 12n α 1 G2n α
≤
2 ·
2n α
1
G2n α
12n α 1
1
G2n α,
α
where the fact that
2 · 2n α 1 5
>
2n α 1
3
α > 1
2.18
is used to get the first inequality above. This means that
G2 · 2n α ≤
1
G2n α,
α
α > 1.
2.19
Next, we make use of the John-Nirenberg inequality to obtain an interpolation
inequality for Lp and BMO norms.
Theorem 2.3. Suppose that 1 ≤ p < r < ∞ and f ∈ Lp Q0 ∩ BMOQ0 . Then we have
1−p/r
f r ≤ constf p/r
f Lp
BMO .
L
2.20
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0. In view of the
Proof. If fBMO 0, the proof is trivial; so we assume that fBMO /
Calderón-Zygmund decomposition theorem, there exists a sequence of disjoint cubes {Qj }
and E such that
⎞
Q0 ⎝ Qj ⎠ ∪ E,
⎛
2.21
j
fxp ≤ f p
BMO
p
1
f BMO < Qj for a.e. x ∈ E,
2.22
fxp dx ≤ 2n f p
BMO .
Qj
2.23
From 2.23, we get
Qj < 1
f p
j
BMO
f Qj
1
Qj fxdx ≤
Qj
p
f p
fx dx Lp ,
f p
Q0
BMO
1
Qj fxp dx
Qj
1/p
2.24
≤ 2n/p f BMO .
Using 2.21–2.24, together with Lemma 2.1, yields that, for λ > 2n/p fBMO ,
x ∈ Q0 : fx > λ x ∈ Qj : fx > λ j
≤
x ∈ Qj : fx − fQj > λ − fQj j
≤
1 Qj x ∈ Qj : fx − fQ > λ − 2n/p f j
BMO Qj j
c2 n/p f BMO Qj λ−2
≤
c1 exp − f j
BMO
c2
≤ c1 exp − f BMO
f p p
f BMO
p L .
f BMO
n/p λ−2
2.25
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From 2.25, we obtain
r
f r r
L
r
∞
λr−1 x ∈ Q0 : fx > λ dλ
0
2n/p fBMO
λr−1 x ∈ Q0 : fx > λ dλ
0
r
≤r
∞
2n/p
f BMO
2n/p fBMO
λr−1 x ∈ Q0 : fx > λ dλ
r−1
λ
0
r
2.26
∞
2n/p f BMO
p
f p
L
dλ
λp
r−1
λ
c2
c1 exp − f BMO
f f p p
f BMO
p L dλ
f BMO
n/p λ−2
r n/pr−p f r−p f p p rc1 2n/pr−1 f r−p f p p
2
L
L
BMO
BMO
r−p
c2
r−p p
≤ constf BMO f Lp .
Hence, the proof is complete.
Acknowledgments
The authors would like to thank the referees for helpful comments and suggestions. The work
was supported partly by the NSF of China 11071042 and the Research Fund for Shanghai
Key Laboratory for Contemporary Applied Mathematics 08DZ2271900.
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