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An improved Hardy type inequality on Heisenberg group

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

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 foru 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, foru 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 forf 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(rk ) 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.

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

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YX designed and performed all the steps of proof in this research and also wrote the paper. All authors read and approved the final manuscript.

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Xiao, Y. An improved Hardy type inequality on Heisenberg group. J Inequal Appl 2011, 38 (2011). https://doi.org/10.1186/1029-242X-2011-38

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Keywords

  • Hardy inequality
  • Heisenberg group