Notes on Interpolation Inequalities

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., [1–6]). 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, 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 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) As we have seen, the recursive estimation (2.8) justifies the desired exponential decay of .

• (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.