# 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 $ℍ n$ via Bessel function.

Mathematics Subject Classification (2000):

Primary 26D10

## 1 Introduction

Hardy inequality in N reads, for all $u∈ C 0 ∞ ( ℝ N )$ and N ≥ 3,

$∫ ℝ N | ∇ u | 2 d x ≥ ( N - 2 ) 2 4 ∫ ℝ N u 2 | x | 2 d x$
(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  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.() 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 () 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 () 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 (), in this note, we shall prove similar improved Hardy inequality on the Heisenberg group $ℍ n$. Recall that the Heisenberg group $ℍ n$ 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 $ℍ n$. The horizontal gradient on $ℍ n$ is the (2n) -dimensional vector given by

$∇ ℍ = ( X 1 , … , X 2 n ) = ∇ x + 2 Λ x ∂ ∂ t ,$

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 $ℍ n$, we refer to . To this end we have:

### Theorem 1.1

Let B R = {x 2n: |x| < R} and Ω H = B R × $ℍ n$. 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$u∈ C 0 ∞ ( Ω H )$

$∫ Ω H | ∇ ℍ u | 2 d x d t ≥ ( n - 1 ) 2 ∫ Ω H u 2 | x | 2 d x d t + ∫ Ω H u 2 | x | 2 V ( | x | ) d x d t$
(1.6)

and the constant (n - 1)2in (1.6) is sharp in the sense of

$( n - 1 ) 2 = inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 d x d t ∫ Ω H u 2 | x | 2 d x d t .$

### Corollary 1.2

There holds, for$u∈ C 0 ∞ ( Ω H )$,

$∫ Ω H | ∇ ℍ u | 2 ≥ ( n - 1 ) 2 ∫ Ω H u 2 | x | 2 + 1 4 ∑ j = 1 k ∫ Ω H u 2 | x | 2 ∏ i = 1 j log ( i ) R | x | - 2$
(1.7)

and the constant 1/4 is sharp in the sense of

$1 4 = inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 - ( n - 1 ) 2 ∫ Ω H u 2 | x | 2 - 1 4 ∑ j = 1 k - 1 ∫ Ω H u 2 | x | 2 ∏ i = 1 j log ( i ) R | x | - 2 ∫ Ω H u 2 | x | 2 ∏ i = 1 k log ( i ) R | x | - 2 .$

### Corollary 1.3

There holds, for $u∈ C 0 ∞ ( Ω H )$ and D ≥ R,

$∫ Ω H | ∇ ℍ u | 2 ≥ ( n - 1 ) 2 ∫ Ω H u 2 | x | 2 + 1 4 ∑ k = 1 ∞ ∫ Ω H u 2 | x | 2 X 1 2 | x | D ⋅ ⋯ ⋅ X k 2 | x | D ,$
(1.8)

and the constant 1/4 is sharp in the sense of

$1 4 = inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 - ( n - 1 ) 2 ∫ Ω H u 2 | x | 2 - 1 4 ∑ j = 1 k - 1 ∫ Ω H u 2 | x | 2 X 1 2 | x | D ⋯ X j 2 | x | D ∫ Ω H u 2 | x | 2 X 1 2 | x | D ⋅ ⋯ ⋅ X k 2 | x | D .$

## 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 $f∈ C 0 ∞ ( ℝ 2 n )$. Decomposing f into spherical harmonics we get (see e.g., )

$f = ∑ k = 0 ∞ f k ( r ) ϕ k ( σ ) ,$

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 $ℍ n$ is the (2n)-dimensional vector given by

$∇ ℍ = ( X 1 , … , X 2 n ) = ∇ x + 2 Λ x ∂ ∂ t ,$

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 $ϕ∈ C 0 ∞ ( ℍ n )$,

$⟨ x , ∇ ℍ ϕ ⟩ = ⟨ x , ∇ x ϕ ⟩ + 2 ⟨ x , Λ x ⟩ ∂ ϕ ∂ t (1) = ⟨ x , ∇ x ϕ ⟩ . (2) (3)$
(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

$| ∂ r u | = | ⟨ x , ∇ x u ⟩ | | x | = | ⟨ x , ∇ ℍ u ⟩ | | x | ≤ | ∇ ℍ u | .$

Therefore, we obtain, by (2.6)

$( n - 1 ) 2 ∫ Ω H u 2 | x | 2 d x d t + ∫ Ω H u 2 | x | 2 V ( | x | ) d x d t ≤ ∫ Ω H | ∇ ℍ u | 2 d x d t .$
(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 $w ( t ) ∈ C 0 ∞ ( ℝ )$. Since ϕ is radial, we have

$| ∇ ℍ u ( x , t ) | 2 = 〈 w ( t ) ∇ x ϕ ( | x | ) + 2 ϕ ( | x | ) Λ x w ′ ( t ) , w ( t ) ∇ x ϕ ( | x | ) + 2 ϕ ( | x | ) Λ x w ′ ( t ) 〉 = | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | Λ x | 2 ϕ 2 ( w ′ ( t ) ) 2 + 4 〈 ∇ x ϕ ( | x | ) , Λ x 〉 ϕ ( | x | ) w ′ ( t ) = | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | Λ x | 2 ϕ 2 ( w ′ ( t ) ) 2 + 4 ϕ ′ ( | x | ) 〈 x | x | , Λ x 〉 ϕ ( | x | ) w ′ ( t ) = | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 .$

Here we use the fact |Λx| = |x| since Λ is a orthogonal matrix. Therefore,

$∫ Ω H | ∇ ℍ u | 2 d x d t ∫ Ω H u 2 | x | 2 d x d t = ∫ Ω H | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) ∫ Ω H ϕ ( | x | ) 2 w ( t ) 2 | x | 2 + 4 ∫ Ω H | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 ∫ Ω H ϕ ( | x | ) 2 w ( t ) 2 | x | 2 = ∫ B R | ∇ x ϕ ( | x | ) | 2 d x ⋅ ∫ − ∞ + ∞ w 2 ( t ) d t ∫ B R ϕ ( | x | ) 2 | x | 2 d x ⋅ ∫ − ∞ + ∞ w 2 ( t ) d t + 4 ∫ B R | x | 2 ϕ 2 ( | x | ) d x ⋅ ∫ − ∞ + ∞ ( w ′ ( t ) ) 2 ∫ B R ϕ ( | x | ) 2 | x | 2 d x ⋅ ∫ − ∞ + ∞ w 2 ( t ) d t = ∫ B R | ∇ x ϕ ( | x | ) | 2 d x ∫ B R ϕ ( | x | ) 2 | x | 2 d x + 4 ∫ B R | x | 2 ϕ 2 ( | x | ) d x ∫ B R ϕ ( | x | ) 2 | x | 2 d x ⋅ ∫ − ∞ + ∞ ( w ′ ( t ) ) 2 ∫ − ∞ + ∞ w 2 ( t ) d t$

Since

$inf w ( t ) ∈ C 0 ∞ ( ℝ ) \ { 0 } ∫ ℝ | w ′ ( t ) | 2 d t ∫ ℝ | w ( t ) | 2 d t = 0,$

we obtain

$inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 d x d t ∫ Ω H u 2 | x | 2 d x d t ≤ inf ϕ ∈ C 0 ∞ ( B R ) \ { 0 } ∫ B R | ∇ x ϕ ( | x | ) | 2 d x ∫ B R ϕ ( | x | ) 2 | x | 2 d x = ( n − 1 ) 2 .$

The proof of Theorem 1.1 is completed.

### Proof of Corollary 1.2

By Theorem 1.1 and following , 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 $w ( t ) ∈ C 0 ∞ ( ℝ )$. By the proof of Theorem 1.1,

$| ∇ ℍ u ( x , t ) | 2 = | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 .$

Therefore,

$∫ Ω H | ∇ ℍ u | 2 − ( n − 1 ) 2 ∫ Ω H u 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 = ∫ Ω H | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) − ( n − 1 ) 2 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 + 4 ∫ Ω H | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 = ∫ B R | ∇ x ϕ ( | x | ) | 2 − ( n − 1 ) 2 ∫ B R ϕ 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 + 4 ∫ B R | x | 2 ϕ 2 ( | x | ) ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 ⋅ ∫ − ∞ + ∞ ( w ′ ( t ) ) 2 ∫ − ∞ + ∞ w 2 ( t ) d t .$

Since

$inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 − ( n − 1 ) 2 ∫ Ω H u 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 ≤ inf ϕ ( | x | ) ∈ C 0 ∞ ( B R ) \ { 0 } ∫ B R | ∇ x ϕ ( | x | ) | 2 − ( n − 1 ) 2 ∫ B R ϕ 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 = 1 4 .$

we have

$inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 − ( n − 1 ) 2 ∫ Ω H u 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ Ω H u 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 ≤ inf ϕ ( | x | ) ∈ C 0 ∞ ( B R ) \ { 0 } ∫ B R | ∇ x ϕ ( | x | ) | 2 − ( n − 1 ) 2 ∫ B R ϕ 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 j log ( i ) R | x | ) − 2 ∫ B R ϕ 2 | x | 2 ( ∏ i = 1 k log ( i ) R | x | ) − 2 = 1 4 .$

Here we use the fact that the sharp constant in inequality (1.3) is 1/4 (see ). 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 $w ( t ) ∈ C 0 ∞ ( ℝ )$. Then

$| ∇ ℍ u ( x , t ) | 2 = | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) + 4 | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 .$

Therefore,

$∫ Ω H | ∇ ℍ u | 2 − ( n − 1 ) 2 ∫ Ω H u 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H u 2 | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ∫ Ω H u 2 | x | 2 X 1 2 ( | x | D ) ⋅ ⋯ ⋅ X k 2 ( | x | D ) = ∫ Ω H | ∇ x ϕ ( | x | ) | 2 w 2 ( t ) − ( n − 1 ) 2 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) + 4 ∫ Ω H | x | 2 ϕ 2 ( | x | ) ( w ′ ( t ) ) 2 ∫ Ω H ϕ 2 w 2 ( t ) | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) = ∫ B R | ∇ x ϕ ( | x | ) | 2 − ( n − 1 ) 2 ∫ B R ϕ 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ B R ϕ 2 | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ∫ B R ϕ 2 | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) + 4 ∫ B R | x | 2 ϕ 2 ( | x | ) ∫ B R ϕ 2 | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ⋅ ∫ − ∞ + ∞ ( w ′ ( t ) ) 2 ∫ − ∞ + ∞ w 2 ( t ) d t .$

Thus

$inf u ∈ C 0 ∞ ( Ω H ) \ { 0 } ∫ Ω H | ∇ ℍ u | 2 − ( n − 1 ) 2 ∫ Ω H u 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ Ω H u 2 | x | 2 X 1 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ∫ Ω H u 2 | x | 2 X 1 2 ( | x | D ) ⋅ ⋯ ⋅ X k 2 ( | x | D ) ≤ inf ϕ ( | x | ) ∈ C 0 ∞ ( B R ) \ { 0 } ∫ B R | ∇ x ϕ ( | x | ) | 2 − ( n − 1 ) 2 ∫ B R ϕ 2 | x | 2 − 1 4 ∑ j = 1 k − 1 ∫ B R ϕ 2 | x | 2 ( | x | D ) ⋯ X j 2 ( | x | D ) ∫ B R ϕ 2 | x | 2 ( | x | D ) ⋯ X j 2 ( | x | D ) = 1 4 .$

since

$inf w ( t ) ∈ C 0 ∞ ( ℝ ) \ { 0 } ∫ ℝ | w ′ ( t ) | 2 d t ∫ ℝ | w ( t ) | 2 d t = 0.$

This completes the proof of Corollary 1.3.

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## Acknowledgements

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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Correspondence to Ying-Xiong Xiao.

### Competing interests

The author declares that they have no competing interests.

### Authors' contributions

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 