# 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 $u\in {C}_{0}^{\infty }\left({ℝ}^{N}\right)$ and N ≥ 3,

$\underset{{ℝ}^{N}}{\int }|\nabla u{|}^{2}\mathsf{\text{d}}x\ge \frac{{\left(N-2\right)}^{2}}{4}\underset{{ℝ}^{N}}{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x$
(1.1)

and $\frac{{\left(N-2\right)}^{2}}{4}$ 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 $u\in {C}_{0}^{\infty }\left(\Omega \right)$,

$\underset{\Omega }{\int }|\nabla u{|}^{2}\mathsf{\text{d}}x\ge \frac{{\left(N-2\right)}^{2}}{4}\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+{z}_{0}^{2}{\left(\frac{{\omega }_{N}}{|\Omega |}\right)}^{\frac{2}{N}}\underset{\Omega }{\int }{u}^{2}\mathsf{\text{d}}x,$
(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 $u\in {C}_{0}^{\infty }\left(\Omega \right)$, there exists a constant C n,k such that

$\underset{\Omega }{\int }|\nabla u{|}^{2}\mathsf{\text{d}}x\ge \frac{{\left(N-2\right)}^{2}}{4}\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+{C}_{n,k}\sum _{j=1}^{k}\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}{\left(\prod _{i=1}^{j}{log}^{\left(i\right)}\frac{\rho }{|x|}\right)}^{-2}\mathsf{\text{d}}x,$
(1.3)

where

$\rho =\left(\underset{x\in \Omega }{sup}|x|\right)\left({e}^{{e}^{{.}^{{.}^{e\left(k-times\right)}}}}\right),$

log(1)(.) = log(.) and log(k)(.) = log(log(k-1)(.)) for k ≥ 2. Filippas and Tertikas () proved that, for $u\in {C}_{0}^{\infty }\left(\Omega \right)$, there holds

$\underset{\Omega }{\int }|\nabla u{|}^{2}\ge \frac{{\left(N-2\right)}^{2}}{4}\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}+\frac{1}{4}\sum _{k=1}^{\infty }\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdot \cdots \cdot {X}_{k}^{2}\left(\frac{|x|}{D}\right),$
(1.4)

where D ≥ sup xΩ|x|,

${X}_{1}\left(s\right)={\left(1-\mathrm{ln}s\right)}^{-1},\phantom{\rule{0.25em}{0ex}}{X}_{k}\left(s\right)={X}_{1}\left({X}_{k-1}\left(t\right)\right)$

for k ≥ 2 and $\frac{1}{4}$ 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 $u\in {C}_{0}^{\infty }\left({B}_{\rho }\right)$, where B ρ = {x n :|x| < ρ},

$\underset{\Omega }{\int }|\nabla u{|}^{2}\mathsf{\text{d}}x\ge \frac{{\left(N-2\right)}^{2}}{4}\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{\Omega }{\int }\frac{{u}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x$
(1.5)

if and only if the ordinary differential equation

${y}^{″}\left(r\right)+\frac{{y}^{\prime }\left(r\right)}{r}+V\left(r\right)y\left(r\right)=0$

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

$\left(x,t\right)\circ \left({x}^{\prime },{t}^{\prime }\right)=\left(x+{x}^{\prime },t+{t}^{\prime }+2\sum _{j=1}^{n}\left({x}_{2j}{{x}^{\prime }}_{2j-1}-{x}_{2j-1}{{x}^{\prime }}_{2j}\right)\right).$

The vector fields

$\begin{array}{c}{X}_{2j-1}=\frac{\partial }{\partial {x}_{2j-1}}+2{x}_{2j}\frac{\partial }{\partial t},\\ {X}_{2j}=\frac{\partial }{\partial {x}_{2j}}-2{x}_{2j-1}\frac{\partial }{\partial t},\end{array}$

(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

${\nabla }_{ℍ}=\left({X}_{1},\dots ,{X}_{2n}\right)={\nabla }_{x}+2\Lambda x\frac{\partial }{\partial t},$

where ${\nabla }_{x}=\left(\frac{\partial }{\partial {x}_{1}},\dots ,\frac{\partial }{\partial {x}_{2n}}\right),\Lambda$ is a skew symmetric and orthogonal matrix given by

$\Lambda =\mathsf{\text{diag}}\left({J}_{1},\dots ,{J}_{n}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{J}_{1}=\cdots ={J}_{n}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right).$

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

${y}^{″}\left(r\right)+\frac{{y}^{\prime }\left(r\right)}{r}+V\left(r\right)y\left(r\right)=0$

has a positive solution on (0, R], then the following improved Hardy inequality holds for$u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)$

$\underset{{\Omega }_{H}}{\int }|{\nabla }_{ℍ}u{|}^{2}\mathsf{\text{d}}x\mathsf{\text{d}}t\ge {\left(n-1\right)}^{2}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x\mathsf{\text{d}}t+\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x\mathsf{\text{d}}t$
(1.6)

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

${\left(n-1\right)}^{2}=\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{inf}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}\mathsf{\text{d}}x\mathsf{\text{d}}t}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x\mathsf{\text{d}}t}.$

### Corollary 1.2

There holds, for$u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)$,

$\underset{{\Omega }_{H}}{\int }|{\nabla }_{ℍ}u{|}^{2}\ge {\left(n-1\right)}^{2}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}+\frac{1}{4}\sum _{j=1}^{k}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}{\left(\prod _{i=1}^{j}{log}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}$
(1.7)

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

$\frac{1}{4}=\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{inf}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{log}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{log}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}.$

### Corollary 1.3

There holds, for $u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)$ and D ≥ R,

$\underset{{\Omega }_{H}}{\int }|{\nabla }_{ℍ}u{|}^{2}\ge {\left(n-1\right)}^{2}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}+\frac{1}{4}\sum _{k=1}^{\infty }\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdot \cdots \cdot {X}_{k}^{2}\left(\frac{|x|}{D}\right),$
(1.8)

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

$\frac{1}{4}=\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{inf}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-\phantom{\rule{2.77695pt}{0ex}}{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdot \cdots \cdot {X}_{k}^{2}\left(\frac{|x|}{D}\right)}.$

## 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

${y}^{″}\left(r\right)+\frac{{y}^{\prime }\left(r\right)}{r}+V\left(r\right)y\left(r\right)=0$

has a positive solution on (0, R], then the following improved Hardy inequality holds for$f\in {C}_{0}^{\infty }\left({B}_{R}\right)$,

$\underset{{B}_{R}}{\int }|{\partial }_{r}f{|}^{2}\mathsf{\text{d}}x\ge {\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{{B}_{R}}{\int }\frac{{f}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x,$
(2.1)

where r = |x| and${\partial }_{r}=\frac{⟨x,\nabla ⟩}{|x|}$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 $f\in {C}_{0}^{\infty }\left({B}_{R}\right)$. If we extend f as zero outside B R , we may consider $f\in {C}_{0}^{\infty }\left({ℝ}^{2n}\right)$. Decomposing f into spherical harmonics we get (see e.g., )

$f=\sum _{k=0}^{\infty }{f}_{k}\left(r\right){\varphi }_{k}\left(\sigma \right),$

where ϕ k (σ) are the orthonormal eigenfunctions of the Laplace-Beltrami operator with responding eigenvalues

${c}_{k}=k\left(N+k-2\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}k\ge 0.$

The functions f k (r) belong to ${C}_{0}^{\infty }\left({B}_{R}\right)$, satisfying f k (r) = O(rk ) and ${f}_{k}^{\prime }\left(r\right)=O\left({r}^{k-1}\right)$ as r → 0. So

$\underset{{B}_{R}}{\int }|{\partial }_{r}f{|}^{2}\mathsf{\text{d}}x=\sum _{k=0}^{\infty }\underset{{B}_{R}}{\int }|{f}_{k}^{\prime }{|}^{2}\mathsf{\text{d}}x$
(2.2)

and

${\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{{B}_{R}}{\int }\frac{{f}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x=\sum _{k=0}^{\infty }\left({\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x\right).$
(2.3)

Note that if f is radial, then inequality (2.1) holds. We have, since ${f}_{k}\left(r\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)$,

$\underset{{B}_{R}}{\int }|{f}_{k}^{\prime }{|}^{2}\mathsf{\text{d}}x\ge {\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x.$

Therefore, by (2.2) and (2.3),

$\begin{array}{l}\underset{{B}_{R}}{\int }{|{\partial }_{r}f|}^{2}\text{d}x=\sum _{k=0}^{\infty }\underset{{B}_{R}}{\int }{|{f}_{k}^{\text{'}}|}^{2}\text{d}x\\ \ge \sum _{k=0}^{\infty }\left({\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{{|x|}^{2}}\text{d}x+\underset{{B}_{R}}{\int }\frac{{f}_{k}^{2}}{{|x|}^{2}}V\left(|x|\right)\text{d}x\right)\\ {=\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{f}^{2}}{{|x|}^{2}}\text{d}x+\underset{{B}_{R}}{\int }\frac{{f}^{2}}{{|x|}^{2}}V\left(|x|\right)\text{d}x.\end{array}$

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

${\nabla }_{ℍ}=\left({X}_{1},\dots ,{X}_{2n}\right)={\nabla }_{x}+2\Lambda x\frac{\partial }{\partial t},$

where ${\nabla }_{x}=\left(\frac{\partial }{\partial {x}_{1}},\dots ,\frac{\partial }{\partial {x}_{2n}}\right),\Lambda$ is a skew symmetric and orthogonal matrix given by

$\Lambda =\mathsf{\text{diag}}\left({J}_{1},\dots ,{J}_{n}\right),\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{J}_{1}=\cdots ={J}_{n}=\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill \end{array}\right).$

Therefore, for any $\varphi \in {C}_{0}^{\infty }\left({ℍ}^{n}\right)$,

$\begin{array}{lll}\hfill ⟨x,{\nabla }_{ℍ}\varphi ⟩& =⟨x,{\nabla }_{x}\varphi ⟩+2⟨x,\Lambda x⟩\frac{\partial \varphi }{\partial t}\phantom{\rule{2em}{0ex}}& \hfill \text{(1)}\\ =⟨x,{\nabla }_{x}\varphi ⟩.\phantom{\rule{2em}{0ex}}& \hfill \text{(2)}\\ \hfill \text{(3)}\end{array}$
(2.4)

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

Since $u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)$, for every t ,$u\left(\cdot ,t\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)$. By Lemma 2.1,

$\underset{{B}_{R}}{\int }|{\partial }_{r}u{|}^{2}\mathsf{\text{d}}x\ge {\left(n-1\right)}^{2}\underset{{B}_{R}}{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x+\underset{{B}_{R}}{\int }\frac{{u}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x$
(2.5)

Integrating both sides of the inequality (2.5) with respect to t, we have,

$\underset{{\Omega }_{H}}{\int }|{\partial }_{r}u{|}^{2}\mathsf{\text{d}}x\mathsf{\text{d}}t\ge {\left(n-1\right)}^{2}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x\mathsf{\text{d}}t+\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x\mathsf{\text{d}}t$
(2.6)

By (2.4) and the pointwise Schwartz inequality, we have

$|{\partial }_{r}u|=\frac{|⟨x,{\nabla }_{x}u⟩|}{|x|}=\frac{|⟨x,{\nabla }_{ℍ}u⟩|}{|x|}\le \phantom{\rule{2.77695pt}{0ex}}|{\nabla }_{ℍ}u|.$

Therefore, we obtain, by (2.6)

${\left(n-1\right)}^{2}\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}\mathsf{\text{d}}x\mathsf{\text{d}}t+\underset{{\Omega }_{H}}{\int }\frac{{u}^{2}}{|x{|}^{2}}V\left(|x|\right)\mathsf{\text{d}}x\mathsf{\text{d}}t\le \underset{{\Omega }_{H}}{\int }|{\nabla }_{ℍ}u{|}^{2}\mathsf{\text{d}}x\mathsf{\text{d}}t.$
(2.7)

To see the constant (n - 1)2 is sharp, we choose u(x, t) = ϕ(|x|)w(t) with $\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)$ and $w\left(t\right)\in {C}_{0}^{\infty }\left(ℝ\right)$. Since ϕ is radial, we have

$\begin{array}{l}{|{\nabla }_{ℍ}u\left(x,t\right)|}^{2}\phantom{\rule{0.25em}{0ex}}=〈w\left(t\right){\nabla }_{x}\varphi \left(|x|\right)+2\varphi \left(|x|\right)\Lambda x{w}^{\prime }\left(t\right),w\left(t\right){\nabla }_{x}\varphi \left(|x|\right)+2\varphi \left(|x|\right)\Lambda x{w}^{\prime }\left(t\right)〉\\ =\phantom{\rule{0.25em}{0ex}}{|{\nabla }_{x}\varphi \left(|x|\right)|}^{2}{w}^{2}\left(t\right)+4{|\Lambda x|}^{2}{\varphi }^{2}{\left({w}^{\prime }\left(t\right)\right)}^{2}+4〈{\nabla }_{x}\varphi \left(|x|\right),\Lambda x〉\varphi \left(|x|\right){w}^{\prime }\left(t\right)\\ =\phantom{\rule{0.25em}{0ex}}{|{\nabla }_{x}\varphi \left(|x|\right)|}^{2}{w}^{2}\left(t\right)+4|\Lambda x{|}^{2}{\varphi }^{2}{\left({w}^{\prime }\left(t\right)\right)}^{2}+4{\varphi }^{\prime }\left(|x|\right)〈\frac{x}{|x|},\Lambda x〉\varphi \left(|x|\right){w}^{\prime }\left(t\right)\\ =\phantom{\rule{0.25em}{0ex}}{|{\nabla }_{x}\varphi \left(|x|\right)|}^{2}{w}^{2}\left(t\right)+4{|x|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t\right){\right)}^{2}.\end{array}$

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

$\begin{array}{c}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}\text{d}x\text{d}t}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}\text{d}x\text{d}t}=\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}{w}^{2}\left(t\right)}{{\int }_{{\Omega }_{H}}\frac{\varphi {\left(|x|\right)}^{2}w{\left(t\right)}^{2}}{|x{|}^{2}}}+4\frac{{\int }_{{\Omega }_{H}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t{\right)\right)}^{2}}{{\int }_{{\Omega }_{H}}\frac{\varphi {\left(|x|\right)}^{2}w{\left(t\right)}^{2}}{|x{|}^{2}}}\\ =\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}\text{d}x\cdot {\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}{{\int }_{{B}_{R}}\frac{\varphi {\left(|x|\right)}^{2}}{|x{|}^{2}}\text{d}x\cdot {\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}+4\frac{{\int }_{{B}_{R}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)\text{d}x\cdot {\int }_{-\infty }^{+\infty }{\left({w}^{\prime }\left(t\right)\right)}^{2}}{{\int }_{{B}_{R}}\frac{\varphi {\left(|x|\right)}^{2}}{|x{|}^{2}}\text{d}x\cdot {\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}\\ =\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}\text{d}x}{{\int }_{{B}_{R}}\frac{\varphi {\left(|x|\right)}^{2}}{|x{|}^{2}}\text{d}x}+4\frac{{\int }_{{B}_{R}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)\text{d}x}{{\int }_{{B}_{R}}\frac{\varphi {\left(|x|\right)}^{2}}{|x{|}^{2}}\text{d}x}\cdot \frac{{\int }_{-\infty }^{+\infty }{\left({w}^{\prime }\left(t\right)\right)}^{2}}{{\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}\end{array}$

Since

$\underset{w\left(t\right)\in {C}_{0}^{\infty }\left(ℝ\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{ℝ}|{w}^{\prime }\left(t{\right)|}^{2}\text{d}t}{{\int }_{ℝ}|w\left(t{\right)|}^{2}\text{d}t}=0,$

we obtain

$\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}\text{d}x\text{d}t}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}\text{d}x\text{d}t}\le \underset{\varphi \in {C}_{0}^{\infty }\left({B}_{R}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}\text{d}x}{{\int }_{{B}_{R}}\frac{\varphi {\left(|x|\right)}^{2}}{|x{|}^{2}}\text{d}x}=\left(n-{1\right)}^{2}.$

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 $\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)$ and $w\left(t\right)\in {C}_{0}^{\infty }\left(ℝ\right)$. By the proof of Theorem 1.1,

$|{\nabla }_{ℍ}u\left(x,t{\right)|}^{2}=|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}{w}^{2}\left(t\right)+4|x{|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t{\right)\right)}^{2}.$

Therefore,

$\begin{array}{l}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ =\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}{w}^{2}\left(t\right)-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ \phantom{\rule{0.25em}{0ex}}+4\frac{{\int }_{{\Omega }_{H}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t{\right)\right)}^{2}}{{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ =\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}-{\left(n-1\right)}^{2}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ \phantom{\rule{0.25em}{0ex}}+4\frac{{\int }_{{B}_{R}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\cdot \frac{{\int }_{-\infty }^{+\infty }{\left({w}^{\prime }\left(t\right)\right)}^{2}}{{\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}.\end{array}$

Since

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ \le \underset{\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}-{\left(n-1\right)}^{2}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ =\frac{1}{4}.\end{array}$

we have

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ \le \underset{\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}-{\left(n-1\right)}^{2}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{j}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{\left({\prod }_{i=1}^{k}{\mathrm{log}}^{\left(i\right)}\frac{R}{|x|}\right)}^{-2}}\\ =\frac{1}{4}.\end{array}$

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 $\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)$ and $w\left(t\right)\in {C}_{0}^{\infty }\left(ℝ\right)$. Then

$|{\nabla }_{ℍ}u\left(x,t{\right)|}^{2}=|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}{w}^{2}\left(t\right)+4|x{|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t{\right)\right)}^{2}.$

Therefore,

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdot \cdots \cdot {X}_{k}^{2}\left(\frac{|x|}{D}\right)}\\ =\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}{w}^{2}\left(t\right)-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}\\ \phantom{\rule{0.25em}{0ex}}+4\frac{{\int }_{{\Omega }_{H}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)\left({w}^{\prime }\left(t{\right)\right)}^{2}}{{\int }_{{\Omega }_{H}}\frac{{\varphi }^{2}{w}^{2}\left(t\right)}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}\\ =\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}-{\left(n-1\right)}^{2}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}\\ \phantom{\rule{0.25em}{0ex}}+4\frac{{\int }_{{B}_{R}}|x{|}^{2}{\varphi }^{2}\left(|x|\right)}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}\cdot \frac{{\int }_{-\infty }^{+\infty }{\left({w}^{\prime }\left(t\right)\right)}^{2}}{{\int }_{-\infty }^{+\infty }{w}^{2}\left(t\right)\text{d}t}.\end{array}$

Thus

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}\underset{u\in {C}_{0}^{\infty }\left({\Omega }_{H}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{\Omega }_{H}}|{\nabla }_{ℍ}u{|}^{2}-{\left(n-1\right)}^{2}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{\Omega }_{H}}\frac{{u}^{2}}{|x{|}^{2}}{X}_{1}^{2}\left(\frac{|x|}{D}\right)\cdot \cdots \cdot {X}_{k}^{2}\left(\frac{|x|}{D}\right)}\\ \le \underset{\varphi \left(|x|\right)\in {C}_{0}^{\infty }\left({B}_{R}\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{{B}_{R}}|{\nabla }_{x}\varphi \left(|x{|\right)|}^{2}-{\left(n-1\right)}^{2}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}-\frac{1}{4}{\sum }_{j=1}^{k-1}{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}{{\int }_{{B}_{R}}\frac{{\varphi }^{2}}{|x{|}^{2}}\left(\frac{|x|}{D}\right)\cdots {X}_{j}^{2}\left(\frac{|x|}{D}\right)}\\ =\frac{1}{4}.\end{array}$

since

$\underset{w\left(t\right)\in {C}_{0}^{\infty }\left(ℝ\right)\\left\{0\right\}}{\mathrm{inf}}\frac{{\int }_{ℝ}|{w}^{\prime }\left(t{\right)|}^{2}\text{d}t}{{\int }_{ℝ}|w\left(t{\right)|}^{2}\text{d}t}=0.$

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