Abstract
In this note we provide simple and short proofs for a class of Hardy-Rellich type inequalities with the best constant, which extends some recent results.
MSC:26D15, 35A23.
Similar content being viewed by others
1 Introduction
It is well known that Hardy’s inequality and its generalizations play important roles in many areas of mathematics. The classical Hardy inequality is given by, for ,
where , the constant is optimal and not attained.
Recently there has been a considerable interest in studying the Hardy-type and Rellich-type inequalities. See, for example, [1–7]. In [8] Caffarelli, Kohn and Nirenberg proved a rather general interpolation inequality with weights. That is the following so-called Caffarelli-Kohn-Nirenberg inequality. For any , there exists such that
where
and
In [9] Costa proved the following -case version for a class of Caffarelli-Kohn-Nirenberg inequalities with a sharp constant by an elementary method. For all and ,
where the constant is sharp.
On the other hand, the Rellich inequality is a generalization of the Hardy inequality to second-order derivatives, and the classical Rellich inequality in states that for and ,
The constant is sharp and never achieved. In [10] Tetikas and Zographopoulos obtained a corresponding stronger versions of the Rellich inequality which reads
for all and . In [11] Costa obtained a new class of Hardy-Rellich type inequalities which contain (1.5) as a special case. If , then
where the constant is sharp.
The goal of this paper is to extend the above (1.3) and (1.6) to the general case for by a different and direct approach.
2 Main results
In this section, we will give the proof of the main theorems.
Theorem 1 For all and , one has
where and the constant is sharp.
Proof Let , and . By integration by parts and the Hölder inequality, one has
Then
It remains to show the sharpness of the constant. By the condition with equality in the Hölder inequality, we consider the following family of functions:
and
where is a positive number sequence converging to as . By direct computation and the limit process, we know the constant is sharp. □
Remark 1 When , the inequality (2.1) covers the inequality (2.4) in [9].
Remark 2 When , , the inequality (2.1) is the classical Hardy inequality:
When we take special values for a, b, the following corollary holds.
Corollary 1 (i) When , the inequality (2.1) is just the weighted Hardy inequality:
-
(ii)
When , according to the inequality (2.1), we have
(2.5) -
(iii)
When and , we obtain the inequality
(2.6)
By a similar method, we can prove the following case Hardy-Rellich type inequality.
Theorem 2 Let , . Then, for any , the following holds:
where , and is the p-Laplacian operator.
Proof Set , it is easy to see
On the other hand,
which means
Then, we can deduce from (2.8) and (2.9)
That is,
By the Hölder inequality,
note that . Thus
We mention that we do not know whether the constant in (2.7) is optimal or not. □
Corollary 2 When , we have the following inequalities:
-
(i)
when , , the inequality (2.7) is equivalent to the inequality
(2.14) -
(ii)
When , , we obtain the inequality
(2.15) -
(iii)
When , , we get
(2.16)
References
Adimurthi AS: Role of the fundamental solution in Hardy-Sobolev type inequalities. Proc. R. Soc. Edinb., Sect. A 2006, 136: 1111–1130. 10.1017/S030821050000490X
Garofalo N, Lanconelli E: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 1990, 40: 313–356. 10.5802/aif.1215
Goldstein JA, Kombe I: Nonlinear degenerate parabolic equations on the Heisenberg group. Int. J. Evol. Equ. 2005, 1: 1–22.
Goldstein JA, Zhang QS: On a degenerate heat equation with a singular potential. J. Funct. Anal. 2001, 186: 342–359. 10.1006/jfan.2001.3792
Jin Y, Han Y: Weighted Rellich inequality on H -type groups and nonisotropic Heisenberg groups. J. Inequal. Appl. 2010., 2010: Article ID 158281
Jin Y, Zhang G: Degenerate p -Laplacian operators and Hardy type inequalities on h -type groups. Can. J. Math. 2010, 62: 1116–1130. 10.4153/CJM-2010-033-9
García Azorero JP, Peral Alonso I: Hardy inequalities and some critical elliptic and parabolic problems. J. Differ. Equ. 1998, 144: 441–476. 10.1006/jdeq.1997.3375
Caffarelli L, Kohn R, Nirenberg L: First order interpolation inequalities with weights. Compos. Math. 1984, 53: 259–275.
Costa DG: Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities. J. Math. Anal. Appl. 2008, 337: 311–317. 10.1016/j.jmaa.2007.03.062
Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Adv. Math. 2007, 209: 407–459. 10.1016/j.aim.2006.05.011
Costa DG:On Hardy-Rellich type inequalities in . Appl. Math. Lett. 2009, 22: 902–905. 10.1016/j.aml.2008.02.018
Acknowledgements
This work is supported by NNSF of China (11001240), ZJNSF (LQ12A01023) and the foundation of the Zhejiang University of the Technology (20100229).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors jointly worked on the results and they read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Di, Y., Jiang, L., Shen, S. et al. A note on a class of Hardy-Rellich type inequalities. J Inequal Appl 2013, 84 (2013). https://doi.org/10.1186/1029-242X-2013-84
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-84