Notes on Interpolation Inequalities

doi:10.1155/2011/913403

Research Article

Notes on Interpolation Inequalities

Jiu-Gang Dong

_{and Ti-Jun Xiao}

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,[email protected]

Received 3 October 2010; Accepted 16 November 2010

Academic Editor: Toka Diagana

Copyrightq2011 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 problemssee, e.g.,1–6. So, it is an issue worthy of deep investigation.

LetQ0be eitherRnor a fixed cube inRn. Forf∈L1_locQ0, write

_f

BMO: sup

Q⊂Q0

1 |Q|

Q

_{f−fQdx, 1.1}

where the supremum is taken over all cubesQ⊂Q0andfQ: 1/|Q|

Qfdx.

Recall that BMOQ0 is the set consisting of all locally integrable functions on Q0

such thatf_BMO < ∞, which is a Banach space endowed with the norm · BMO. It is clear

that any bounded function onQ0is in BMOQ0, but the converse is not true. On the other

hand, the BMO space is regarded as a natural substitute forL∞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 forLp

2. Results and Proofs

Lemma 2.1John-Nirenberg inequality. Iff ∈BMOQ0, then there exist positive constantsc1, c2such that, for each cubeQ⊂Q0,

_{x∈Q:fx−fQ> t≤c1exp−} c2 _f

BMO

t |Q|, t >0. 2.1

Proof. Without loss of generality, we can and do assume thatf_BMO1. For eacht >0, letFtdenote the least number for which we have

_x∈Q:fx−fQ> t≤Ft|Q|, 2.2

for any cubeQ⊂Q0. It is easy to see thatFt≤1t >0andFtis decreasing.

Fix a cubeQ ⊂Q0. Applying the Calder ´on-Zygmund decompositioncf., e.g.,2,9

to|fx−fQ|onQ, with 2n as the separating number, we get a sequence of disjoint cubes

{Qj}andEsuch that

Q ⎛ ⎝

j

Qj

⎞

⎠_{∪E, 2.3}

_fx−fQ≤2n_{, for a.e. x∈E, 2.4}

2n< 1 Qj

Qj

_f−fQdx≤4n_.

2.5

Using2.5, we have

j

_Qj< 1

2n|Q|. 2.6

From2.3,2.4, and2.6, we deduce that fort >4n,

_{x∈Q:fx−fQ> t} j

x∈Qj:fx−fQ> t

≤

j

_x∈Qj :fx−fQ> t−4n

j

_Qj 1

_Qjx∈Qj:fx−fQ> t−4n

≤ 1 2nFt−4

n_|Q|.

This yields that

Ft≤ 1

2nFt−4

n_{, t >4}n_{. 2.8}

Letγ t−14−n _{t >4}n_,μ1γ4n_{. Then 0< μ≤t. By iterating, we get}

Ft≤FμF1γ4n≤2−nγ≤2−nt−14−n−1

2n14−nexp−log 2n4−nt, t >4n.

2.9

Thus, letting

c1 2n14 −n

, c2

log 2n4−n 2.10

gives that

Ft≤c1e−c2t, t >0, 2.11

since

Ft≤1≤c1e−c2t, 0< t≤4n. 2.12

This completes the proof.

Remark 2.2. 1 As we have seen, the recursive estimation 2.8 justifies the desired exponential decay ofFt.

2 There exists a gap in the proof of the John-Nirenberg inequality given in 2. Actually, for a decreasing functionGt:0,∞ → 0,1, the following estimate:

G2·2nα≤ 1 αG2

n_{α, α >1 2.13}

does not generally imply such a property, that is, the existence of constantsc1, c2 > 0 such

that

Gt≤c1e−c2t, t >0. 2.14

We present the following function as a counter example:

Gt exp

−

log5 3

−1

In fact, it is easy to see that there are no constantsc1, c2 > 0 such that2.14holds. On the

other hand, we have

Gt Gt

−log5 3

−1

2logt1

t1 , t >0. 2.16

Integrating both sides of the above equation from 2n_{αto 2·2}n_{α, we obtain}

G2·2nα exp

−2

log5 3

₋12·2n_α

2n_α

logt1

t1 dt G2

n_α

exp

−

log5 3

₋1

log22·2nα1−log22nα1 G2nα

exp

−log5 3

−1

log2·2nα12nα1·log

2·2nα1 2n_α1 G2

n_α

≤exp−log2·2nα12nα1G2nα

1

2·2n_α12n_α1G2 n_α

≤ 1

αG2

n_α,

2.17

where the fact that

2·2n_α1

2n_α1 >

5

3 α >1 2.18

is used to get the first inequality above. This means that

G2·2nα≤ 1 αG2

n_{α, α >1. 2.19}

Next, we make use of the John-Nirenberg inequality to obtain an interpolation inequality forLp_{and BMO norms.}

Theorem 2.3. Suppose that1≤p < r <∞andf∈Lp_Q

0∩BMOQ0. Then we have

_f

Lr ≤constf

p/r Lp f

1−p/r

Proof. If f_BMO 0, the proof is trivial; so we assume that f_BMO/0. In view of the Calder ´on-Zygmund decomposition theorem, there exists a sequence of disjoint cubes{Qj}

andEsuch that

From2.25, we obtain

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 China11071042and the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics08DZ2271900.

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