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Notes on Interpolation Inequalities

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Abstract

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

(1.1)

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 ,

(2.1)

Proof.

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

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

(2.2)

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

(2.3)
(2.4)
(2.5)

Using (2.5), we have

(2.6)

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

(2.7)

This yields that

(2.8)

Let , . Then . By iterating, we get

(2.9)

Thus, letting

(2.10)

gives that

(2.11)

since

(2.12)

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:

    (2.13)

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

(2.14)

We present the following function as a counter example:

(2.15)

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

(2.16)

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

(2.17)

where the fact that

(2.18)

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

(2.19)

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

(2.20)

Proof.

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

(2.21)
(2.22)
(2.23)

From (2.23), we get

(2.24)

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

(2.25)

From (2.25), we obtain

(2.26)

Hence, the proof is complete.

References

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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).

Author information

Correspondence to Ti-Jun Xiao.

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Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Positive Constant