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Notes on Interpolation Inequalities


An easy proof of the John-Nirenberg inequality is provided by merely using the Calderón-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., [16]). So, it is an issue worthy of deep investigation.

Let be either or a fixed cube in . For , write


where the supremum is taken over all cubes and .

Recall that is the set consisting of all locally integrable functions on such that , which is a Banach space endowed with the norm . It is clear that any bounded function on is in , but the converse is not true. On the other hand, the BMO space is regarded as a natural substitute for 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, 79]. 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 and BMO norms.

2. Results and Proofs

Lemma 2.1 (John-Nirenberg inequality).

If , then there exist positive constants , such that, for each cube ,



Without loss of generality, we can and do assume that .

For each , let denote the least number for which we have


for any cube . It is easy to see that and is decreasing.

Fix a cube . Applying the Calderón-Zygmund decomposition (cf., e.g., [2, 9]) to on , with as the separating number, we get a sequence of disjoint cubes and such that


Using (2.5), we have


From (2.3), (2.4), and (2.6), we deduce that for ,


This yields that


Let , . Then . By iterating, we get


Thus, letting


gives that




This completes the proof.

Remark 2.2.

  1. (1)

    As we have seen, the recursive estimation (2.8) justifies the desired exponential decay of .

  2. (2)

    There exists a gap in the proof of the John-Nirenberg inequality given in [2]. Actually, for a decreasing function , the following estimate:


does not generally imply such a property, that is, the existence of constants such that


We present the following function as a counter example:


In fact, it is easy to see that there are no constants such that (2.14) holds. On the other hand, we have


Integrating both sides of the above equation from to , we obtain


where the fact that


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


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

Theorem 2.3.

Suppose that and . Then we have



If , the proof is trivial; so we assume that . In view of the Calderón-Zygmund decomposition theorem, there exists a sequence of disjoint cubes and such that


From (2.23), we get


Using (2.21)–(2.24), together with Lemma 2.1, yields that, for ,


From (2.25), we obtain


Hence, the proof is complete.


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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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Correspondence to Ti-Jun Xiao.

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Dong, JG., Xiao, TJ. Notes on Interpolation Inequalities. Adv Differ Equ 2011, 913403 (2011).

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