- Research Article
- Open Access

# Notes on Interpolation Inequalities

- Jiu-Gang Dong
^{1}and - Ti-Jun Xiao
^{2}Email author

**2011**:913403

https://doi.org/10.1155/2011/913403

© J.-G. Dong and T.-J. Xiao. 2011

**Received:**3 October 2010**Accepted:**16 November 2010**Published:**1 December 2010

## 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.

## Keywords

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

## 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.

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).

Proof.

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.

- (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:

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

Theorem 2.3.

Proof.

Hence, the proof is complete.

## Declarations

### 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).

## Authors’ Affiliations

## References

- Gilbarg D, Trudinger NS:
*Elliptic Partial Differential Equations of Second Order, Classics in Mathematics*. Springer, Berlin, Germany; 2001:xiv+517.MATHGoogle Scholar - Han YS:
*Mordern Harmonic Analysis Methods and Applications*. Science Press, Beijing, China; 1997.Google Scholar - Triebel H:
*Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library*.*Volume 18*. North-Holland, Amsterdam, The Netherlands; 1978:528.Google Scholar - Xiao T-J, Liang J:
*The Cauchy Problem for Higher-Order Abstract Differential Equations, Lecture Notes in Mathematics*.*Volume 1701*. Springer, Berlin, Germany; 1998:xii+301.View ArticleGoogle Scholar - Xiao T-J, Liang J:
**Complete second order differential equations in Banach spaces with dynamic boundary conditions.***Journal of Differential Equations*2004,**200**(1):105-136. 10.1016/j.jde.2004.01.011MathSciNetView ArticleMATHGoogle Scholar - Xiao T-J, Liang J:
**Second order parabolic equations in Banach spaces with dynamic boundary conditions.***Transactions of the American Mathematical Society*2004,**356**(12):4787-4809. 10.1090/S0002-9947-04-03704-3MathSciNetView ArticleMATHGoogle Scholar - George C:
*Exercises in Integration*. Springer, New York, NY, USA; 1984:x+550.View ArticleGoogle Scholar - John F, Nirenberg L:
**On functions of bounded mean oscillation.***Communications on Pure and Applied Mathematics*1961,**14:**415-426. 10.1002/cpa.3160140317MathSciNetView ArticleMATHGoogle Scholar - Stein EliasM:
*Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series*.*Volume 43*. Princeton University Press, Princeton, NJ, NY, USA; 1993:xiv+695.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.