- Research Article
- Open Access
Notes on Interpolation Inequalities
© J.-G. Dong and T.-J. Xiao. 2011
- Received: 3 October 2010
- Accepted: 16 November 2010
- Published: 1 December 2010
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.
- Differential Equation
- Partial Differential Equation
- Ordinary Differential Equation
- Functional Analysis
- Positive Constant
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.
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.
Lemma 2.1 (John-Nirenberg inequality).
Without loss of generality, we can and do assume that .
for any cube . It is easy to see that and is decreasing.
This completes the proof.
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 . Actually, for a decreasing function , the following estimate:
Next, we make use of the John-Nirenberg inequality to obtain an interpolation inequality for and BMO norms.
Hence, the proof is complete.
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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