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Notes on Interpolation Inequalities
Advances in Difference Equations volumeÂ 2011, ArticleÂ number:Â 913403 (2011)
Abstract
An easy proof of the JohnNirenberg inequality is provided by merely using the CalderÃ³nZygmund decomposition. Moreover, an interpolation inequality is presented with the help of the JohnNirenberg 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., [1â€“6]). 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 JohnNirenberg inequality. There are several versions of its proof; see, for example, [2, 7â€“9]. Stimulated by these works, we give, in this paper, an easy proof of the JohnNirenberg inequality by using the CalderÃ³nZygmund 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 (JohnNirenberg inequality).
If , then there exist positive constants , such that, for each cube ,
Proof.
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 CalderÃ³nZygmund 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
since
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 JohnNirenberg 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
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 JohnNirenberg inequality to obtain an interpolation inequality for and BMO norms.
Theorem 2.3.
Suppose that and . Then we have
Proof.
If , the proof is trivial; so we assume that . In view of the CalderÃ³nZygmund decomposition theorem, there exists a sequence of disjoint cubes and such that
From (2.23), we get
Using (2.21)â€“(2.24), together with Lemma 2.1, yields that, for ,
From (2.25), we obtain
Hence, the proof is complete.
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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).
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Dong, JG., Xiao, TJ. Notes on Interpolation Inequalities. Adv Differ Equ 2011, 913403 (2011). https://doi.org/10.1155/2011/913403
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DOI: https://doi.org/10.1155/2011/913403