Volume 2011, Article ID 913403, 6 pagesdoi:10.1155/2011/913403 Research Article Notes on Interpolation Inequalities Jiu-Gang Dong1 and Ti-Jun Xiao2 1 Department of Mathematics, Universit
Trang 1Volume 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 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
2 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
Copyrightq 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 Calder ´on-Zygmund 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 Q0be either R n or a fixed cube in R n For f ∈ L1
locQ0, write
f
BMO: sup
Q⊂Q0
1
|Q|
Q
where the supremum is taken over all cubes Q ⊂ Q0and fQ: 1/|Q|Q fdx.
Recall that BMOQ0 is the set consisting of all locally integrable functions on Q0
such thatfBMO < ∞, which is a Banach space endowed with the norm · BMO It is clear
that any bounded function on Q0is 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 Calder ´on-Zygmund decomposition
only Moreover, with the help of this inequality, an interpolation inequality is showed for L p
and BMO norms
Trang 22 Results and Proofs
Lemma 2.1 John-Nirenberg inequality If f ∈ BMOQ0, then there exist positive constants c1,
c2such that, for each cube Q ⊂ Q0,
x ∈ Q :f x − fQ> t ≤ c1exp
− c2
f
BMO
t
|Q|, 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 :f x − 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 Calder ´on-Zygmund decompositioncf., e.g., 2,9
to|fx − fQ | on Q, with 2 n as the separating number, we get a sequence of disjoint cubes
{Qj} and E such that
Q
⎛
⎝
j
Q j
⎞
f x − fQ ≤ 2 n , for a.e x ∈ E, 2.4
2n < Q1j
Q j
Using2.5, we have
j
Q j< 1
From2.3, 2.4, and 2.6, we deduce that for t > 4 n,
x ∈ Q :f x − fQ> t
j
x ∈ Q j:f x − fQ> t
≤
j
x ∈ Q j :f x − fQ> t − 4 n
j
Q j 1
Q jx ∈ Q j:f x − fQ> t − 4 n
≤ 1
2n F t − 4 n |Q|.
2.7
Trang 3This yields that
F t ≤ 1
2n F t − 4 n , t > 4 n 2.8
Let γ t − 14 −n t > 4 n , μ 1 γ4 n Then 0 < μ ≤ t By iterating, we get
F t ≤ Fμ
F1 γ4 n
≤ 2−nγ ≤ 2−nt−14 −n−1
2n14 −nexp
−log 2
n4 −n t
, t > 4 n
2.9
Thus, letting
c1 2n14 −n, c2log 2
n4 −n 2.10 gives that
F t ≤ c1e −c2t , t > 0, 2.11 since
F t ≤ 1 ≤ c1e −c2t , 0 < t ≤ 4 n 2.12 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
α G2n α , α > 1 2.13
does not generally imply such a property, that is, the existence of constants c1, c2 > 0 such
that
G t ≤ c1e −c2t , t > 0. 2.14
We present the following function as a counter example:
G t exp
−
log5 3
−1 log2t 1
Trang 4
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
G t
−log5 3
−1
2logt 1
t 1
Integrating both sides of the above equation from 2n α to 2 · 2 n α, we obtain
G2 · 2n α exp
−2
log5 3
−12·2n α
2n α
logt 1
t 1 dt
G2n α
exp
−
log5 3
−1 log22 · 2n α 1 − log22n α 1 G2n α
exp
−log5 3
−1 log2 · 2n α 12n α 1 · log
2· 2n α 1
2n α 1
G2n α
≤ exp− log2 · 2n α 12n α 1G2n α
2 · 2n 1
α 12n α 1G2n α
≤ 1
α G2n α ,
2.17
where the fact that
2· 2n α 1
2n α 1 >
5
3 α > 1 2.18
is used to get the first inequality above This means that
G2 · 2n α ≤ 1
α G2n α , α > 1. 2.19
Next, we make use of the John-Nirenberg inequality to obtain an interpolation
inequality for L pand BMO norms
Theorem 2.3 Suppose that 1 ≤ p < r < ∞ and f ∈ L p Q0 ∩ BMOQ0 Then we have
f
L r ≤ constfp/r
L p f1−p/r
Trang 5Proof If fBMO 0, the proof is trivial; so we assume that fBMO/ 0 In view of the Calder ´on-Zygmund decomposition theorem, there exists a sequence of disjoint cubes{Qj}
and E such that
Q0
⎛
⎝
j
Q j
⎞
fxp≤fp
BMO for a.e x ∈ E, 2.22
fp
BMO< Q1j
Q j
f xp
dx ≤ 2 nfp
BMO. 2.23
From2.23, we get
j
Q j< 1
fp
BMO
Q0
fxp
dx
fp
L p
fp
BMO
,
f
Q j Q1j
Q j
f xdx ≤ 1
Q j
Q j
fxp
dx
1/p
≤ 2n/pf
BMO.
2.24
Using2.21–2.24, together withLemma 2.1, yields that, for λ > 2 n/p fBMO,
x ∈ Q0:f x> λ
j
x ∈ Q j:f x> λ
≤
j
x ∈ Q j: fx − fQ
j
> λ −fQ
j
≤
j
Q j 1
Q jx ∈ Q j:fx − fQ
j
> λ − 2 n/pf
BMO
≤
j
c1exp
− c2
f
BMO
λ − 2 n/pf
BMO
Qj
≤ c1exp
− c2
f
BMO
λ − 2 n/pf
BMO
fp
L p
fp
BMO
.
2.25
Trang 6From2.25, we obtain
fr
L r r
∞
0
λ r−1x ∈ Q0:f x> λdλ
r
2n/p fBMO 0
λ r−1x ∈ Q0:f x> λdλ
r
∞
2n/pfBMOλ
r−1x ∈ Q0:f x> λdλ
≤ r
2n/p fBMO 0
λ r−1fp
L p
λ p dλ
r
∞
2n/pfBMOλ
r−1 c1exp
− c2
f
BMO
λ − 2 n/pf
BMO
ffp
L p
fp
BMO
dλ
r
r − p2
n/pr−pfr−p
BMOfp
L prc1
c2
2n/pr−1fr−p
BMOfp
L p
≤ constfr−p
BMOfp
L p
2.26
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 China11071042 and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics08DZ2271900
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