Open Access

An improved Hardy type inequality on Heisenberg group

Journal of Inequalities and Applications20112011:38

https://doi.org/10.1186/1029-242X-2011-38

Received: 2 April 2011

Accepted: 25 August 2011

Published: 25 August 2011

Abstract

Motivated by the work of Ghoussoub and Moradifam, we prove some improved Hardy inequalities on the Heisenberg group n via Bessel function.

Mathematics Subject Classification (2000):

Primary 26D10

Keywords

Hardy inequalityHeisenberg group

1 Introduction

Hardy inequality in N reads, for all u C 0 ( N ) and N ≥ 3,
N | u | 2 d x ( N - 2 ) 2 4 N u 2 | x | 2 d x
(1.1)
and ( N - 2 ) 2 4 is the best constant in (1.1) and is never achieved. A similar inequality with the same best constant holds in N is replaced by an arbitrary domain Ω N and Ω contains the origin. Moreover, in case Ω N is a bounded domain, Brezis and Vázquez [1] have improved it by establishing that for u C 0 ( Ω ) ,
Ω | u | 2 d x ( N - 2 ) 2 4 Ω u 2 | x | 2 d x + z 0 2 ω N | Ω | 2 N Ω u 2 d x ,
(1.2)
where ω N and |Ω| denote the volume of the unit ball and Ω, respectively, and z0 = 2.4048... denotes the first zero of the Bessel function J0(z). Inequality (1.2) is optimal in case Ω is a ball centered at zero. Triggered by the work of Brezis and Vázquez (1.2), several Hardy inequalities have been established in recent years. In particular, Adimurthi et al.([2]) proved that, for u C 0 ( Ω ) , there exists a constant C n,k such that
Ω | u | 2 d x ( N - 2 ) 2 4 Ω u 2 | x | 2 d x + C n , k j = 1 k Ω u 2 | x | 2 i = 1 j log ( i ) ρ | x | - 2 d x ,
(1.3)
where
ρ = sup x Ω | x | e e . . e ( k - t i m e s ) ,
log(1)(.) = log(.) and log(k)(.) = log(log(k-1)(.)) for k ≥ 2. Filippas and Tertikas ([3]) proved that, for u C 0 ( Ω ) , there holds
Ω | u | 2 ( N - 2 ) 2 4 Ω u 2 | x | 2 + 1 4 k = 1 Ω u 2 | x | 2 X 1 2 | x | D X k 2 | x | D ,
(1.4)
where D ≥ sup xΩ|x|,
X 1 ( s ) = ( 1 ln s ) 1 , X k ( s ) = X 1 ( X k 1 ( t ) )
for k ≥ 2 and 1 4 is the best constant in (1.4) and is never achieved. More recently, Ghoussoub and Moradifam ([4]) give a necessary and sufficient condition on a radially symmetric potential V(|x|) on Ω that makes it an admissible candidate for an improved Hardy inequality. It states that the following improved Hardy inequality holds for u C 0 ( B ρ ) , where B ρ = {x n :|x| < ρ},
Ω | u | 2 d x ( N - 2 ) 2 4 Ω u 2 | x | 2 d x + Ω u 2 | x | 2 V ( | x | ) d x
(1.5)
if and only if the ordinary differential equation
y ( r ) + y ( r ) r + V ( r ) y ( r ) = 0

has a positive solution on (0, ρ]. These include inequalities (1.2)-(1.4).

Motivated by the work of Ghoussoub and Moradifam ([4]), in this note, we shall prove similar improved Hardy inequality on the Heisenberg group n . Recall that the Heisenberg group n is the Carnot group of step two whose group structure is given by
( x , t ) ( x , t ) = x + x , t + t + 2 j = 1 n x 2 j x 2 j - 1 - x 2 j - 1 x 2 j .
The vector fields
X 2 j - 1 = x 2 j - 1 + 2 x 2 j t , X 2 j = x 2 j - 2 x 2 j - 1 t ,
(j = 1,..., n) are left invariant and generate the Lie algebra of n . The horizontal gradient on n is the (2n) -dimensional vector given by
= ( X 1 , , X 2 n ) = x + 2 Λ x t ,
where x = x 1 , , x 2 n , Λ is a skew symmetric and orthogonal matrix given by
Λ = diag ( J 1 , , J n ) , J 1 = = J n = 0 1 - 1 0 .

For more information about n , we refer to [58]. To this end we have:

Theorem 1.1

Let B R = {x 2n: |x| < R} and Ω H = B R × n . Let V(|x|) be a radially symmetric decreasing nonnegative function on B R . If the ordinary differential equation
y ( r ) + y ( r ) r + V ( r ) y ( r ) = 0
has a positive solution on (0, R], then the following improved Hardy inequality holds for u C 0 ( Ω H )
Ω H | u | 2 d x d t ( n - 1 ) 2 Ω H u 2 | x | 2 d x d t + Ω H u 2 | x | 2 V ( | x | ) d x d t
(1.6)
and the constant (n - 1)2in (1.6) is sharp in the sense of
( n - 1 ) 2 = inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 d x d t Ω H u 2 | x | 2 d x d t .

Corollary 1.2

There holds, for u C 0 ( Ω H ) ,
Ω H | u | 2 ( n - 1 ) 2 Ω H u 2 | x | 2 + 1 4 j = 1 k Ω H u 2 | x | 2 i = 1 j log ( i ) R | x | - 2
(1.7)
and the constant 1/4 is sharp in the sense of
1 4 = inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 - ( n - 1 ) 2 Ω H u 2 | x | 2 - 1 4 j = 1 k - 1 Ω H u 2 | x | 2 i = 1 j log ( i ) R | x | - 2 Ω H u 2 | x | 2 i = 1 k log ( i ) R | x | - 2 .

Corollary 1.3

There holds, for u C 0 ( Ω H ) and D ≥ R,
Ω H | u | 2 ( n - 1 ) 2 Ω H u 2 | x | 2 + 1 4 k = 1 Ω H u 2 | x | 2 X 1 2 | x | D X k 2 | x | D ,
(1.8)
and the constant 1/4 is sharp in the sense of
1 4 = inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 - ( n - 1 ) 2 Ω H u 2 | x | 2 - 1 4 j = 1 k - 1 Ω H u 2 | x | 2 X 1 2 | x | D X j 2 | x | D Ω H u 2 | x | 2 X 1 2 | x | D X k 2 | x | D .

2 Proof

To prove the main result, we first need the following preliminary result.

Lemma 2.1

Let B R = {x 2n: |x| < R} and V(|x|) be a radially symmetric decreasing nonnegative function on B R . If the ordinary differential equation
y ( r ) + y ( r ) r + V ( r ) y ( r ) = 0
has a positive solution on (0, R], then the following improved Hardy inequality holds for f C 0 ( B R ) ,
B R | r f | 2 d x ( n - 1 ) 2 B R f 2 | x | 2 d x + B R f 2 | x | 2 V ( | x | ) d x ,
(2.1)

where r = |x| and r = x , | x | is the radial derivation.

Proof

Observe that if f is radial, i.e., f(x) = f(r), then | f| = |∂ r f|. By inequality (1.5), inequality (2.1) holds.

Now let f C 0 ( B R ) . If we extend f as zero outside B R , we may consider f C 0 ( 2 n ) . Decomposing f into spherical harmonics we get (see e.g., [9])
f = k = 0 f k ( r ) ϕ k ( σ ) ,
where ϕ k (σ) are the orthonormal eigenfunctions of the Laplace-Beltrami operator with responding eigenvalues
c k = k ( N + k - 2 ) , k 0 .
The functions f k (r) belong to C 0 ( B R ) , satisfying f k (r) = O(r k ) and f k ( r ) = O ( r k - 1 ) as r → 0. So
B R | r f | 2 d x = k = 0 B R | f k | 2 d x
(2.2)
and
( n - 1 ) 2 B R f 2 | x | 2 d x + B R f 2 | x | 2 V ( | x | ) d x = k = 0 ( n - 1 ) 2 B R f k 2 | x | 2 d x + B R f k 2 | x | 2 V ( | x | ) d x .
(2.3)
Note that if f is radial, then inequality (2.1) holds. We have, since f k ( r ) C 0 ( B R ) ,
B R | f k | 2 d x ( n - 1 ) 2 B R f k 2 | x | 2 d x + B R f k 2 | x | 2 V ( | x | ) d x .
Therefore, by (2.2) and (2.3),
B R | r f | 2 d x = k = 0 B R | f k ' | 2 d x k = 0 ( ( n 1 ) 2 B R f k 2 | x | 2 d x + B R f k 2 | x | 2 V ( | x | ) d x ) = ( n 1 ) 2 B R f 2 | x | 2 d x + B R f 2 | x | 2 V ( | x | ) d x .

This completes the proof of lemma 2.1.

Proof of Theorem 1.1

Recall that the horizontal gradient on n is the (2n)-dimensional vector given by
= ( X 1 , , X 2 n ) = x + 2 Λ x t ,
where x = ( x 1 , , x 2 n ) , Λ is a skew symmetric and orthogonal matrix given by
Λ = diag ( J 1 , , J n ) , J 1 = = J n = 0 1 - 1 0 .
Therefore, for any ϕ C 0 ( n ) ,
x , ϕ = x , x ϕ + 2 x , Λ x ϕ t (1) = x , x ϕ . (2) (3)
(2.4)

Here we use the fact x, Λx = 0 since Λ is a skew symmetric matrix.

Since u C 0 ( Ω H ) , for every t , u ( , t ) C 0 ( B R ) . By Lemma 2.1,
B R | r u | 2 d x ( n - 1 ) 2 B R u 2 | x | 2 d x + B R u 2 | x | 2 V ( | x | ) d x
(2.5)
Integrating both sides of the inequality (2.5) with respect to t, we have,
Ω H | r u | 2 d x d t ( n - 1 ) 2 Ω H u 2 | x | 2 d x d t + Ω H u 2 | x | 2 V ( | x | ) d x d t
(2.6)
By (2.4) and the pointwise Schwartz inequality, we have
| r u | = | x , x u | | x | = | x , u | | x | | u | .
Therefore, we obtain, by (2.6)
( n - 1 ) 2 Ω H u 2 | x | 2 d x d t + Ω H u 2 | x | 2 V ( | x | ) d x d t Ω H | u | 2 d x d t .
(2.7)
To see the constant (n - 1)2 is sharp, we choose u(x, t) = ϕ(|x|)w(t) with ϕ ( | x | ) C 0 ( B R ) and w ( t ) C 0 ( ) . Since ϕ is radial, we have
| u ( x , t ) | 2 = w ( t ) x ϕ ( | x | ) + 2 ϕ ( | x | ) Λ x w ( t ) , w ( t ) x ϕ ( | x | ) + 2 ϕ ( | x | ) Λ x w ( t ) = | x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | Λ x | 2 ϕ 2 ( w ( t ) ) 2 + 4 x ϕ ( | x | ) , Λ x ϕ ( | x | ) w ( t ) = | x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | Λ x | 2 ϕ 2 ( w ( t ) ) 2 + 4 ϕ ( | x | ) x | x | , Λ x ϕ ( | x | ) w ( t ) = | x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 .
Here we use the fact |Λx| = |x| since Λ is a orthogonal matrix. Therefore,
Ω H | u | 2 d x d t Ω H u 2 | x | 2 d x d t = Ω H | x ϕ ( | x | ) | 2 w 2 ( t ) Ω H ϕ ( | x | ) 2 w ( t ) 2 | x | 2 + 4 Ω H | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 Ω H ϕ ( | x | ) 2 w ( t ) 2 | x | 2 = B R | x ϕ ( | x | ) | 2 d x + w 2 ( t ) d t B R ϕ ( | x | ) 2 | x | 2 d x + w 2 ( t ) d t + 4 B R | x | 2 ϕ 2 ( | x | ) d x + ( w ( t ) ) 2 B R ϕ ( | x | ) 2 | x | 2 d x + w 2 ( t ) d t = B R | x ϕ ( | x | ) | 2 d x B R ϕ ( | x | ) 2 | x | 2 d x + 4 B R | x | 2 ϕ 2 ( | x | ) d x B R ϕ ( | x | ) 2 | x | 2 d x + ( w ( t ) ) 2 + w 2 ( t ) d t
Since
inf w ( t ) C 0 ( ) \ { 0 } | w ( t ) | 2 d t | w ( t ) | 2 d t = 0,
we obtain
inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 d x d t Ω H u 2 | x | 2 d x d t inf ϕ C 0 ( B R ) \ { 0 } B R | x ϕ ( | x | ) | 2 d x B R ϕ ( | x | ) 2 | x | 2 d x = ( n 1 ) 2 .

The proof of Theorem 1.1 is completed.

Proof of Corollary 1.2

By Theorem 1.1 and following [4], it is enough to show the constant 1/4 is sharp. Choose u(x, t) = ϕ(|x|)w(t) with ϕ ( | x | ) C 0 ( B R ) and w ( t ) C 0 ( ) . By the proof of Theorem 1.1,
| u ( x , t ) | 2 = | x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 .
Therefore,
Ω H | u | 2 ( n 1 ) 2 Ω H u 2 | x | 2 1 4 j = 1 k 1 Ω H u 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 Ω H u 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 = Ω H | x ϕ ( | x | ) | 2 w 2 ( t ) ( n 1 ) 2 Ω H ϕ 2 w 2 ( t ) | x | 2 1 4 j = 1 k 1 Ω H ϕ 2 w 2 ( t ) | x | 2 ( i = 1 j log ( i ) R | x | ) 2 Ω H ϕ 2 w 2 ( t ) | x | 2 ( i = 1 k log ( i ) R | x | ) 2 + 4 Ω H | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 Ω H ϕ 2 w 2 ( t ) | x | 2 ( i = 1 k log ( i ) R | x | ) 2 = B R | x ϕ ( | x | ) | 2 ( n 1 ) 2 B R ϕ 2 | x | 2 1 4 j = 1 k 1 B R ϕ 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 B R ϕ 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 + 4 B R | x | 2 ϕ 2 ( | x | ) B R ϕ 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 + ( w ( t ) ) 2 + w 2 ( t ) d t .
Since
inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 ( n 1 ) 2 Ω H u 2 | x | 2 1 4 j = 1 k 1 Ω H u 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 Ω H u 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 inf ϕ ( | x | ) C 0 ( B R ) \ { 0 } B R | x ϕ ( | x | ) | 2 ( n 1 ) 2 B R ϕ 2 | x | 2 1 4 j = 1 k 1 B R ϕ 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 B R ϕ 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 = 1 4 .
we have
inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 ( n 1 ) 2 Ω H u 2 | x | 2 1 4 j = 1 k 1 Ω H u 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 Ω H u 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 inf ϕ ( | x | ) C 0 ( B R ) \ { 0 } B R | x ϕ ( | x | ) | 2 ( n 1 ) 2 B R ϕ 2 | x | 2 1 4 j = 1 k 1 B R ϕ 2 | x | 2 ( i = 1 j log ( i ) R | x | ) 2 B R ϕ 2 | x | 2 ( i = 1 k log ( i ) R | x | ) 2 = 1 4 .

Here we use the fact that the sharp constant in inequality (1.3) is 1/4 (see [4]). This completes the proof of Corollary 1.2.

Proof of Corollary 1.3

The proof is similar to that of Corollary 1.2 and it is enough to show the constant 1/4 is sharp. Choose u(x, t) = ϕ(|x|)w(t) with ϕ ( | x | ) C 0 ( B R ) and w ( t ) C 0 ( ) . Then
| u ( x , t ) | 2 = | x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 .
Therefore,
Ω H | u | 2 ( n 1 ) 2 Ω H u 2 | x | 2 1 4 j = 1 k 1 Ω H u 2 | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) Ω H u 2 | x | 2 X 1 2 ( | x | D ) X k 2 ( | x | D ) = Ω H | x ϕ ( | x | ) | 2 w 2 ( t ) ( n 1 ) 2 Ω H ϕ 2 w 2 ( t ) | x | 2 1 4 j = 1 k 1 Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) + 4 Ω H | x | 2 ϕ 2 ( | x | ) ( w ( t ) ) 2 Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) = B R | x ϕ ( | x | ) | 2 ( n 1 ) 2 B R ϕ 2 | x | 2 1 4 j = 1 k 1 B R ϕ 2 | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) B R ϕ 2 | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) + 4 B R | x | 2 ϕ 2 ( | x | ) B R ϕ 2 | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) + ( w ( t ) ) 2 + w 2 ( t ) d t .
Thus
inf u C 0 ( Ω H ) \ { 0 } Ω H | u | 2 ( n 1 ) 2 Ω H u 2 | x | 2 1 4 j = 1 k 1 Ω H u 2 | x | 2 X 1 2 ( | x | D ) X j 2 ( | x | D ) Ω H u 2 | x | 2 X 1 2 ( | x | D ) X k 2 ( | x | D ) inf ϕ ( | x | ) C 0 ( B R ) \ { 0 } B R | x ϕ ( | x | ) | 2 ( n 1 ) 2 B R ϕ 2 | x | 2 1 4 j = 1 k 1 B R ϕ 2 | x | 2 ( | x | D ) X j 2 ( | x | D ) B R ϕ 2 | x | 2 ( | x | D ) X j 2 ( | x | D ) = 1 4 .
since
inf w ( t ) C 0 ( ) \ { 0 } | w ( t ) | 2 d t | w ( t ) | 2 d t = 0.

This completes the proof of Corollary 1.3.

Declarations

Acknowledgements

The authors thanks the referee for his/her careful reading and very useful comments which improved the final version of this paper.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Xiaogan University

References

  1. Brezis H, Vázquez JL: Blowup solutions of some nonlinear elliptic problems. Rev Mat Univ Comp Madrid 1997, 10: 443–469.MATHGoogle Scholar
  2. Adimurthi N, Chaudhuri N, Ramaswamy N: An improved Hardy Sobolev inequality and its applications. Proc Amer Math Soc 2002, 130: 489–505. 10.1090/S0002-9939-01-06132-9MATHMathSciNetView ArticleGoogle Scholar
  3. Filippas S, Tertikas A: Optimizing improved Hardy inequalities. J Funct Anal 2002,192(1):186–233. 10.1006/jfan.2001.3900MATHMathSciNetView ArticleGoogle Scholar
  4. Ghoussoub N, Moradifam A: On the best possible remaining term in the improved Hardy inequality. Proc Nat Acad Sci 2008,105(37):13746–13751. 10.1073/pnas.0803703105MATHMathSciNetView ArticleGoogle Scholar
  5. Garofalo N, Lanconelli E: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann Inst Fourier(Grenoble) 1990, 40: 313–356.MATHMathSciNetView ArticleGoogle Scholar
  6. Luan J, Yang Q: A Hardy type inequality in the half-space on n and Heisenberg group. J Math Anal Appl 2008, 347: 645–651. 10.1016/j.jmaa.2008.06.048MATHMathSciNetView ArticleGoogle Scholar
  7. Niu P, Zhang H, Wang Y: Hardy type and Rellich type inequalities on the Heisenberg group. Proc Amer Math Soc 2001, 129: 3623–3630. 10.1090/S0002-9939-01-06011-7MATHMathSciNetView ArticleGoogle Scholar
  8. Yang Q: Best constants in the Hardy-Rellich type inequalities on the Heisenberg group. J Math Anal Appl 2008, 342: 423–431. 10.1016/j.jmaa.2007.12.014MATHMathSciNetView ArticleGoogle Scholar
  9. Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Adv Math 2007, 209: 407–459. 10.1016/j.aim.2006.05.011MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Xiao; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.