Weighted Hardy and Rellich Types Inequalities on the Heisenberg Group with Sharp Constants

Abstract

This paper aims at deriving some weighted Hardy type and Rellich type inequalities with sharp constants on the Heisenberg group. The improved versions of these inequalities are established as well. The technique adopted involve the application of some elementary vectorial inequalities and some properties of Heisenberg group.

1 Preliminaries

1.1 Introduction

The main goal of this paper is to provide simple and more direct proofs of Hardy type and Rellich type inequalities on the Heisenberg group with respect to homogeneous norm, horizontal gradient and Kohn sub-Laplace operators. This class of inequalities is well known in the classical Euclidean spaces [6, 20, 28], homogeneous spaces [15, 23, 29, 31] and Riemannian manifolds [5, 13, 21, 24, 25] with numerous applications in mathematical analysis, physics, quantum mechanics, theory of linear and nonlinear partial differential equations and so on [7, 10, 16, 17]. A classical Hardy inequality measures a smooth function by its derivative, while the Rellich inequality is a generalization of the Hardy inequality to the higher order derivatives. Consider a subdomain $\Omega \subset {\mathbb {R}}^n$, the classical Hardy inequality states that

$$\begin{aligned} \int _\Omega |\nabla \phi |^pdx \ge \left( \frac{n-p}{p}\right) ^p\int _\Omega \frac{|\phi |^p}{|x|^p} dx, \ \ \ \ \ \phi \in C^\infty _0(\Omega ), \end{aligned}$$

(1.1)

where $\nabla$ is the Euclidean gradient, for $1\le p<n$, which can also be extended to the whole of ${\mathbb {R}}^n\setminus \{0\}$ for $n<p<\infty$. It is well known that the constant $\left( \frac{n-p}{p}\right) ^p$ is sharp but never achieved by a nontrivial function $\phi$. On the other hand, generalization of (1.1) to higher order derivative is the following Rellich inequality

$$\begin{aligned} \int _{{\mathbb {R}}^n} \Phi |\Delta \phi |^pdx \ge C\int _{{\mathbb {R}}^n} \Psi |\phi |^p dx, \ \ \ \ \ \phi \in C^\infty _0({\mathbb {R}}^n\setminus \{0\}), \end{aligned}$$

(1.2)

where $\Delta$ is the Euclidean Laplacian, with suitable positive weight functions $\Phi , \Psi$ and positive constant C. This was originally obtained by Rellich [30] for $p=2$ and $n\ge 5$ in the sharp form

$$\begin{aligned} \int _{{\mathbb {R}}^n} |\Delta \phi |^2 dx \ge \frac{n^2(n-4)^2}{16} \int _{{\mathbb {R}}^n} \frac{|\phi |^2}{|x|^{4}} dx, \ \ \ \ \ \phi \in C^\infty _0({\mathbb {R}}^n\setminus \{0\}). \end{aligned}$$

(1.3)

A generalization of (1.3) was obtained by Davies and Hinz [9] as follows

$$\begin{aligned} \int _{{\mathbb {R}}^n} |\Delta \phi |^pdx \ge \left( \frac{n(n-2p)(p-1)}{p^2}\right) ^p\int _{{\mathbb {R}}^n} \frac{|\phi |^p}{|x|^{2p}} dx, \ \ \ \ \ \phi \in C^\infty _0({\mathbb {R}}^n\setminus \{0\}), \end{aligned}$$

(1.4)

where $n>2p$ and the constant is sharp. Recently, there has been considerable development of the concepts of Hardy and Rellich type inequalities in sub-elliptic context. For instance, Garofalo and Lanconelli [16], Niu et al. [27], Xi and Dou [33], and D’Ambrosio [8] have obtained Hardy inequalities on the Heisenberg group with sharp constants (see also the book [31] for several related results).

Let ${\mathbb {H}}$ be the Heisenberg group of homogeneous dimension Q (detail description is given below) and $\varrho _{cc}$ be the Carnot-Carathéodory norm on ${\mathbb {H}}$. Motivated by the above quoted works, Yang [34] proved analogous Hardy type and Rellich type inequalities on the Heisenberg group with $\varrho _{cc}$ being the weight function. Precisely, they proved via a representation formula associated with $\varrho _{cc}$ that for $1<p<Q$, then

$$\begin{aligned} \int _{\mathbb {H}}|\nabla _{\mathbb {H}}\phi |^p \ge \left( \frac{Q-p}{p}\right) ^p\int _{\mathbb {H}}\frac{|\phi |^p}{\varrho _{cc}^p}, \ \ \ \ \ \phi \in C^\infty _0({\mathbb {H}}), \end{aligned}$$

(1.5)

and for $Q\ge 4$, then

$$\begin{aligned} \int _{\mathbb {H}}|\Delta _{\mathbb {H}}\phi |^2 \ge \frac{Q^2(Q-4)^2}{16}\int _{\mathbb {H}}\frac{|\phi |^2}{\varrho _{cc}^4}, \ \ \ \ \ \phi \in C^\infty _0({\mathbb {H}}\setminus \{0\}). \end{aligned}$$

(1.6)

The motivation for studying this class of inequalities on sub-Riemannian setting with noncommutative vector fields satisfying Hörmander’s condition is due to the link between Hardy and Rellich inequalities on one hand and their numerous applications as highlighted above on the other hand. From mathematical point of view, the Heisenberg group is the simplest and most important model in this setting. It is an important model to study many significant problems such as existence, continuation, eigenvalue problems, therefore, obtaining vital integral inequalities and their improvements such as those discussed in this paper become essential, and hence the theme of this paper. Our results generalize and improve the above quoted results with weight involving Koranyi-Folland homogeneous norm. In fact this paper complements some other existing results [1, 3, 4, 11, 12, 18, 19, 32].

1.2 Preliminaries

Basic facts on the Heisenberg group are summarily recalled here. Detail description can be found in Refs. [15, 31]. Heisenberg group arises from many fields such as quantum physics, differential geometry, harmonic analysis. It is a unimodular, connected and simply connected step two (Carnot) Lie group.

Let $\xi = (x,y,t)\in {\mathbb {R}}^n\times {\mathbb {R}}^n\times {\mathbb {R}}$, $n\ge 1$. The Heisenberg group ${\mathbb {H}}$ is the set ${\mathbb {H}}_n={\mathbb {R}}^{2n}\times {\mathbb {R}}$ equipped with the non-commutative group law, $\circ$, having the structure

$$\begin{aligned} \xi \circ \xi '=\left( x+x',y+y', t+t'-2\sum _{j=1}^n(x_j\cdot y'_j-x'_j\cdot y_j)\right) , \end{aligned}$$

(1.7)

where $x\cdot y$ is the usual Euclidean inner product in ${\mathbb {R}}^n$. The distance between points $\xi$ and $\xi '$ in ${\mathbb {H}}$ is defined by

$$\begin{aligned} d_{{\mathbb {H}}}(\xi ,\xi ')=|\xi '^{-1}\circ \xi |_{{\mathbb {H}}}, \end{aligned}$$

(1.8)

where $\xi '^{-1}$ is the inverse of $\xi '$ given by $\xi '^{-1}=-\xi '$ with respect to the group law $\circ$. Let $z=(x,y)$, $\xi =(z,t)$ and $\xi '=(z',t)$. By (1.7) and (1.8), one can obtain the explicit expression for $d_{{\mathbb {H}}_n}(\xi ,\xi ')$. Therefore the Heisenberg distance function (homogeneous norm) is given by

$$\begin{aligned} d:=d(\xi )=d_{{\mathbb {H}}}(\xi ,0)=(|z|^4+t^2)^{\frac{1}{4}}, \end{aligned}$$

where $|z|^2=x^2+y^2$ and 0 is the centre. A basis for the Lie algebra of the left invariant vector fields on ${\mathbb {H}}^n$ is given by ($1\le j \le n$)

$$\begin{aligned} X_j=\frac{\partial }{\partial x_j}+2y_j\frac{\partial }{\partial t}, \ \ Y_j=\frac{\partial }{\partial y_j}-2x_j\frac{\partial }{\partial t}, \ \ T= \frac{\partial }{\partial t}. \end{aligned}$$

By the Lie bracket definition one can easily check that

$$\begin{aligned}{}[X_j,X_k]=[Y_j,Y_k]=[X_j,T]=[Y_j,T]=0, \ j,k=1,2 \cdots n, \end{aligned}$$

and

$$\begin{aligned}{}[X_j,Y_k]=-4T\delta _{jk}, \end{aligned}$$

which constitute Heisenberg canonical commutation relation of quantum mechanics for position and momentum, whence the name Heisenberg group.

A family of dilation on ${\mathbb {H}}$ is defined by $\delta _\lambda (z,t)=(\lambda z, \lambda ^2t), \lambda >0$. It is also easy to check that $|\cdot |_{{\mathbb {H}}}, X_j, Y_j, (j=1,2 \cdots n)$ are homogeneous of degree 1 with respect to dilations (where $|\cdot |_{{\mathbb {H}}}$ is as defined by (1.8)). The anisotropic dilation on ${\mathbb {H}}$ induces a homogeneous norm

$$\begin{aligned} d= d(\xi )=(|z|^4+t^2)^{\frac{1}{4}}, \end{aligned}$$

also known as Korányi-Folland non-isotropic gauge, while the homogeneous dimension with respect to dilations is $Q=2n+2$.

The horizontal gradient and divergence on ${\mathbb {H}}$ are respectively defined by

$$\begin{aligned} \nabla _{{\mathbb {H}}}f := (X_1f, \cdots , X_nf, Y_1f, \cdots , Y_nf), \end{aligned}$$

and

$$\begin{aligned} \text {div}_{{\mathbb {H}}} w:=\nabla _{{\mathbb {H}}} \cdot (w_1,w_2,\cdots , w_{2n}). \end{aligned}$$

Hence, sub-Laplacian on ${\mathbb {H}}$ (Kohn Laplacian) is defined by

$$\begin{aligned} \Delta _{{\mathbb {H}}}:=\sum _{j=1}^n\left( X_j^2+Y_j^2\right) = \text {div}_{{\mathbb {H}}}\nabla _{{\mathbb {H}}}. \end{aligned}$$

The Heisenberg group belongs to a class of homogeneous Carnot groups. It was shown by Folland [14] (see also Ref. [15]) that the sub-Laplacian on a general stratified Lie group has a unique fundamental solution $u_\varepsilon$ in distributional sense, which in the Heisenberg setting means

$$\begin{aligned} - \Delta _{\mathbb {H}} u_\varepsilon = \delta _0, \end{aligned}$$

where $\delta _0$ is the Dirac delta-distribution at the neutral element 0 of ${\mathbb {H}}^n$. The $\Delta _{\mathbb {H}}$-gauge $d(\xi )$ (i.e. the homogeneous quasi-norm on ${\mathbb {H}}$) can be written in term of the fundamental solution as

$$\begin{aligned} d = u_\varepsilon ^{\frac{1}{2-Q}}, \ \ \text {for} \ \xi \ne 0,.\\ \ \end{aligned}$$

as a continuous function smooth away from the origin. The action of $\Delta _{\mathbb {H}}$ on d is given as

$$\begin{aligned} \Delta _{\mathbb {H}}d = (Q-1) \frac{|\nabla _{\mathbb {H}} d|^2}{d} \ \ \text {in} \ {\mathbb {H}}^n\setminus \{0\}. \end{aligned}$$

A simple computation therefore gives the following identities ([27, 31])

$$\begin{aligned} |\nabla _{{\mathbb {H}}} d|&=|z|d^{-1}, \end{aligned}$$

(1.9)

$$\begin{aligned} \Delta _{{\mathbb {H}}} d&= (Q-1)|z|^2d^{-3}, \end{aligned}$$

(1.10)

$$\begin{aligned} \nabla _{{\mathbb {H}}}(|z|^{p-2})\nabla _{{\mathbb {H}}}d&=(p-2)|z|^pd^{-3}. \end{aligned}$$

(1.11)

Similarly, the following identities can be established by direct computation.

Lemma 1.1

Let d be as defined above. Then we have

$$\begin{aligned} \nabla _{{\mathbb {H}}}(|\nabla _{{\mathbb {H}}}d|^{p-2})\nabla _{{\mathbb {H}}}d&=0, \end{aligned}$$

(1.12)

$$\begin{aligned} \nabla _{\mathbb {H}}(d^{1-Q}\nabla _{{\mathbb {H}}}d)&=0, \end{aligned}$$

(1.13)

$$\begin{aligned} \nabla _{{\mathbb {H}}}(d^{1-Q}|\nabla _{{\mathbb {H}}}d|^{p-2}\nabla _{{\mathbb {H}}}d)&=0. \end{aligned}$$

(1.14)

Proof

   

$$\begin{aligned} \nabla _{{\mathbb {H}}}(|\nabla _{{\mathbb {H}}}d|^{p-2})\nabla _{{\mathbb {H}}}d&= \nabla _{\mathbb {H}}\left( \frac{|z|^{p-2}}{d^{p-2}}\right) \nabla _{\mathbb {H}}d\\&= \frac{\nabla _{{\mathbb {H}}}(|z|^{p-2})\nabla _{{\mathbb {H}}}d }{d^{p-2}} - (p-2)\frac{|\nabla _{\mathbb {H}}d|^2}{d^{p-3}}|z|^{p-2}\\&= 0 \end{aligned}$$

by applying (1.9) and (1.11).

$$\begin{aligned} \nabla _{\mathbb {H}}(d^{1-Q}\nabla _{{\mathbb {H}}}d)&= \nabla _{\mathbb {H}}(d^{1-Q}) \nabla _{\mathbb {H}}d + d^{1-Q}\Delta _{\mathbb {H}}d \\&=(1-Q)d^{1-Q}\frac{|\nabla _{\mathbb {H}}d|^2}{d} + (Q-1)d^{1-Q} \frac{|z|^2}{d^3}\\&= 0 \end{aligned}$$

by applying (1.9) and (1.10).

$$\begin{aligned}&\nabla _{{\mathbb {H}}}(d^{1-Q} |\nabla _{{\mathbb {H}}}d|^{p-2}\nabla _{{\mathbb {H}}}d) \\&\quad =\nabla _{\mathbb {H}}(d^{1-Q}\nabla _{{\mathbb {H}}}d) |\nabla _{{\mathbb {H}}}d|^{p-2} + d^{1-Q}\nabla _{\mathbb {H}}( |\nabla _{{\mathbb {H}}}d|^{p-2} ) \nabla _{\mathbb {H}}d\\&\quad = 0 \end{aligned}$$

by applying (1.12) and (1.13). $\square$

Note that the standard Lebesgue measure on ${\mathbb {R}}^n$ is the Haar measure for the homogeneous group, so we write $d\xi =dzdt$ as the Lebesgue measure on ${\mathbb {H}}$. Throughout, we denote by $B_1:=B(0,1)=\{\xi =(z,t) \in {\mathbb {H}}: d(\xi )\le 1\}$ the unit ball with respect to the homogeneous quasi norm d.

This section is concluded with the following elementary vectorial inequalities that will be of great applications to the proof of some of the results in the next sections.

Lemma 1.2

[26] There exists some constant $c(p)>0$ such that for all $u,v \in {\mathbb {R}}^n$, there holds the following inequalities

$$\begin{aligned} |u+v|^p&\ge |u|^p+p|u|^{p-2}u\cdot v +c(p)\frac{|v|^2}{(|u|+|v|)^{2-p}}, \ \ 1<p<2. \end{aligned}$$

(1.15)

$$\begin{aligned} |u+v|^p&\ge |u|^p+p|u|^{p-2}u\cdot v +c(p)|v|^p, \ \ \ \ \ \ \ \ \ 2\le p< \infty . \end{aligned}$$

(1.16)

We note that if $p=2$, (1.16) become equality and $c(2)=1$.

The rest of the paper is structured as follows: Sect. 2 is devoted to the proof of weighted $L^p$-Hardy type inequalities, ($p\ne Q+\alpha ,\ Q\ge 3$ and $\alpha \in {\mathbb {R}}$), and their improvements on the Heisenberg group. The claims for the sharp constant will also be established. While Sect. 3 is devoted to the proofs of sharp weighted Rellich type inequalities and thier improvements.

2 Weighted Hardy Type Inequalities

This section is concerned with sharp weighted Hardy type inequalities and their improvements.

2.1 Weighted $L^p$-Hardy Inequalities

Theorem 2.1

Let ${\mathbb {H}}$ be the Heisenberg group of homogeneous dimension $Q \ge 3$. Let $\alpha \in {\mathbb {R}}$ and $p\ge 1$ such that $p\ne Q+\alpha$. Then the following inequality

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^pd\xi \ge \left| \frac{Q+\alpha -p}{p}\right| ^p \int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }} d\xi \end{aligned}$$

(2.1)

holds for all $f\in C^\infty _0({\mathbb {H}}\setminus \{0\})$ and the constant $\left| \frac{Q+\alpha -p}{p}\right| ^p$ is sharp.

Proof

Let $f\in C^\infty _0({\mathbb {H}}\setminus \{0\})$ and define a function $g=d^{-\gamma }f$, where $\gamma \ne 0$ will be chosen later. Direct computation implies

$$\begin{aligned} |\nabla _{\mathbb {H}}f|^p=|\gamma d^{\gamma -1}g \nabla _{\mathbb {H}}d+ d^\gamma \nabla _{\mathbb {H}}g|^p. \end{aligned}$$

Clearly by Lemma 1.2 we have for $u,v \in {\mathbb {R}}^n$ and $p\ge 1$

$$\begin{aligned} |u+v|^p \ge |u|^p+p|u|^{p-2}u\cdot v. \end{aligned}$$

Thus, one can write

$$\begin{aligned} |\nabla _{\mathbb {H}}f|^p \ge |\gamma |^pd^{p(\gamma -1)}|g|^p|\nabla _{\mathbb {H}}d|^p + p|\gamma |^{p-2}\gamma d^{\gamma p-p+1}|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d |g|^{p-2}g\nabla _{\mathbb {H}}g. \end{aligned}$$

Multiplying the last inequality by $d^\alpha$ and integrating by parts over ${\mathbb {H}}$ gives

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^p d\xi&\ge |\gamma |^p\int _{\mathbb {H}}d^{\alpha +\gamma p-p}|g|^p|\nabla _{\mathbb {H}}d|^p d\xi \nonumber \\&\quad + |\gamma |^{p-2}\gamma \int _{\mathbb {H}}d^{\alpha +\gamma p-p+1}|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d \nabla _{\mathbb {H}}(|g|^p) d\xi \nonumber \\&= |\gamma |^p\int _{\mathbb {H}}d^{\alpha +\gamma p-p}|g|^p|\nabla _{\mathbb {H}}d|^p d\xi \\&\quad - |\gamma |^{p-2}\gamma \int _{\mathbb {H}}\nabla _{\mathbb {H}}(d^{\alpha +\gamma p-p+1}|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d) |g|^p d\xi . \nonumber \end{aligned}$$

(2.2)

Choosing $\gamma = \frac{p-Q-\alpha }{p}\ne 0$, we have by (1.12) that

$$\begin{aligned} \nabla _{\mathbb {H}}(d^{\alpha +\gamma p-p+1}|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d) = \nabla _{{\mathbb {H}}}\cdot (d^{1-Q}|\nabla _{{\mathbb {H}}}d|^{p-2}\nabla _{{\mathbb {H}}}d) =0. \end{aligned}$$

Therefore the last integral in (2.2) vanishes and we thus arrive at

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^pd\xi \ge \Big |\frac{Q+\alpha -p}{p}\Big |^p \int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }} d\xi \end{aligned}$$

(2.3)

which implies the required inequality (2.1).

In the next we prove the claim that the constant ${\mathcal {C}}_{{\mathbb {H}}} = \left| \frac{Q+\alpha -p}{p}\right| ^p$ is sharp in the sense that

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {H}}} = \inf _{0\ne f \in C^\infty _0({\mathbb {H}}_n\setminus \{0\})} \frac{\int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^pd\xi }{\int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }} d\xi } =: {\mathcal {A}}_{{\mathbb {H}}} \end{aligned}$$

must hold. In other words, we need to show that ${\mathcal {C}}_{{\mathbb {H}}}={\mathcal {A}}_{{\mathbb {H}}}$. We use the idea from Ref. [8] (see also Refs. [22, 23, 35] for similar methods). Obviously, by the inequality (2.1) and by passing to the infimum in the last equation

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {H}}} \le {\mathcal {A}}_{{\mathbb {H}}} \end{aligned}$$

holds for all $f \in C^\infty _0({\mathbb {H}}\setminus \{0\})$. Thus, we need to show that

$$\begin{aligned} {\mathcal {C}}_{{\mathbb {H}}} \ge {\mathcal {A}}_{{\mathbb {H}}} \end{aligned}$$

and for this purpose, we define the following radial function which can be approximated by some smooth function with compact support in ${\mathbb {H}}$

$$\begin{aligned} f_\epsilon (d):= \left\{ \begin{array}{ll} d^{\frac{Q+\alpha -p}{p}+\epsilon }, \ \ \ \ 0\le d \le 1\\ \ \\ d^{\frac{p-\alpha -Q}{p}-\epsilon }, \ \ \ \ d>1 \end{array} \right. \end{aligned}$$

for $\epsilon >0$. Therefore

$$\begin{aligned} \int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^p d^{\alpha -p}|f_\epsilon |^p d\xi = \int _{B_1} |\nabla _{\mathbb {H}}d|^p d^{Q+2\alpha -2p+\epsilon p} d\xi + \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^p d^{-Q -\epsilon p} d\xi . \end{aligned}$$

Since the integrands on the right hand side (of the last equation) are integrable, both integrals on that side are finite and so also the integral on the left hand side.

Also

$$\begin{aligned} d^{\alpha }| \nabla _{\mathbb {H}}f_\epsilon |^p&= \Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p |\nabla _{\mathbb {H}}d|^p d^{Q+2\alpha -2p+\epsilon p}, \ \ \ \ \ 0\le d \le 1 \\ d^{\alpha }| \nabla _{\mathbb {H}}f_\epsilon |^p&= \Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p |\nabla _{\mathbb {H}}d|^p d^{-Q-\epsilon p}, \ \ \ \ \ \ \ d > 1. \end{aligned}$$

Thus we have

$$\begin{aligned}&\frac{\Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p}{{\mathcal {A}}_{{\mathbb {H}}}}\int _{\mathbb {H}}d^{\alpha }| \nabla _{\mathbb {H}}f_\epsilon |^pd\xi \ge \Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p \int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^p d^{\alpha -p}|f_\epsilon |^p d\xi \\&\quad = \Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p \left[ \int _{B_1} |\nabla _{\mathbb {H}}d|^p d^{Q+2\alpha -2p+\epsilon p} d\xi + \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^p d^{-Q -\epsilon p} d\xi \right] \\&\quad = \int _{\mathbb {H}}d^{\alpha }| \nabla _{\mathbb {H}}f_\epsilon |^p d\xi \end{aligned}$$

implying that $\Big |\frac{Q+\alpha -p}{p}+\epsilon \Big |^p \ge {\mathcal {A}}_{{\mathbb {H}}}$. Then, ${\mathcal {C}}_{{\mathbb {H}}} \ge {\mathcal {A}}_{{\mathbb {H}}}$ by letting $\epsilon \rightarrow 0$. Therefore ${\mathcal {C}}_{{\mathbb {H}}} = {\mathcal {A}}_{{\mathbb {H}}}$. This concludes the proof. $\square$

Remark 2.2

   

• (1) There are many special cases of (2.1) that are present in the literature, for examples, the cases $\alpha =0$, $\alpha =p$ and $\alpha =-p$ (see D’Ambrossio [8], Niu-Zhang-Wang [27], Abolarinwa-Ruzhansky [2] for these and more examples). The case $p=2$ is well known.

• (2) Setting $\alpha =0$ and replacing Korányi-Folland distance norm d with Carnot-Carathéodory norm $\varrho _{cc}$ the inequality (2.1) reduces to (1.5).

• (3) The inequality (2.1) is strict except for $f\equiv 0$. It is therefore natural to expect additional term on the right hand side of (2.1). (See Theorems 2.3 and  2.5).

• (4) The proof of sharpness of the constant is well known by different methods. (See Refs. [8, 22, 23, 27, 35] for examples).

• (5) Theorem 2.1 above can also be compared with [33, Theorem 2.1], which has been derived using a different method for $1<p<\infty$. Indeed, if $\beta$ and $\alpha - p$ in [33, eqn. (2.5)] are respectively replaced by p and $-\alpha$, then [33, Theorem 2.2] coincides with our Theorem 2.1.

2.2 Improved Hardy Type Inequalities

Theorem 2.3

Let ${\mathbb {H}}$ be the Heisenberg group of homogeneous dimension $Q \ge 3$ and $Q+\alpha -2>0$, $\alpha \in {\mathbb {R}}$. Let $\Omega \subset {\mathbb {H}}$ be a bounded domain with smooth boundary, $0 \in \Omega$ and $\sup _\Omega d< R<\infty$. Then the following inequality

$$\begin{aligned} \int _\Omega d^\alpha |\nabla _{\mathbb {H}}f|^2d\xi \ge {\mathscr {C}}_Q\int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{2-\alpha }} d\xi +\frac{1}{4}\int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{2-\alpha }(\ln R/d)^2} d\xi , \end{aligned}$$

(2.4)

where ${\mathscr {C}}_Q=\Big (\frac{Q+\alpha -2}{2}\Big )^2$, holds for all compactly supported smooth function $f\in C^\infty _0(\Omega )$.

Proof

Define $\phi :=d^{-\lambda }f$, $\lambda <0$ for any $f \in C^\infty _0(\Omega )$. Then

$$\begin{aligned} |\nabla _{\mathbb {H}}f|^2 =\lambda ^2d^{2(\lambda -1)}|\nabla _{\mathbb {H}}d|^2 \phi ^2 + 2 \lambda d^{2\lambda -1}\nabla _{\mathbb {H}}d \phi \nabla _{\mathbb {H}}\phi + d^{2\lambda } |\nabla _{\mathbb {H}}\phi |^2 \end{aligned}$$

and

$$\begin{aligned} \int _\Omega d^\alpha |\nabla _{\mathbb {H}}f|^2 d\xi&=\lambda ^2 \int _\Omega d^{\alpha +2\lambda -2}|\nabla _{\mathbb {H}}d|^2 \phi ^2d\xi + \lambda \int _\Omega d^{\alpha + 2\lambda -1}\nabla _{\mathbb {H}}d \nabla _{\mathbb {H}}\phi ^2 d\xi \nonumber \\&\quad + \int _\Omega d^{\alpha +2\lambda } |\nabla _{\mathbb {H}}\phi |^2 d\xi \nonumber \\&=\lambda ^2 \int _\Omega d^{\alpha +2\lambda -2}|\nabla _{\mathbb {H}}d|^2 \phi ^2d\xi - \lambda \int _\Omega \nabla _{\mathbb {H}}\cdot (d^{\alpha + 2\lambda -1}\nabla _{\mathbb {H}}d)\phi ^2 d\xi \nonumber \\&\quad + \int _\Omega d^{\alpha +2\lambda } |\nabla _{\mathbb {H}}\phi |^2 d\xi . \end{aligned}$$

(2.5)

The integrand of the middle term on the right hand side of (2.5) can be simplified further by using (1.9) and (1.10) as follows

$$\begin{aligned} \nabla _{\mathbb {H}}\cdot (d^{\alpha + 2\lambda -1}\nabla _{\mathbb {H}}d)\phi ^2&= (\alpha +2\lambda -1)|\nabla _{\mathbb {H}}d|^2 d^{\alpha +2\lambda -2} + d^{\alpha +2\lambda -1}\Delta _{\mathbb {H}}d\\&= (\alpha +2\lambda +Q-2)\frac{|z|^2}{d^2}d^{\alpha +2\lambda -2}. \end{aligned}$$

Substituting the last expression into (2.5) yields

$$\begin{aligned} \int _\Omega d^\alpha |\nabla _{\mathbb {H}}f|^2 d\xi&= [\lambda ^2 -\lambda (\alpha +Q-2)] \int _\Omega \frac{|z|^2}{d^2}d^{\alpha +2\lambda -2}\phi ^2 d\xi + \int _\Omega d^{\alpha +2\lambda } |\nabla _{\mathbb {H}}\phi |^2 d\xi . \end{aligned}$$

By the maximization procedure the function $\lambda \mapsto \lambda ^2 -\lambda (\alpha +Q-2)$ attains its maximum for $\lambda =\frac{2-Q-\alpha }{2}$ with the maximum being $(\frac{Q+\alpha -2}{2})^2$. Therefore

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^2 d\xi&= \Big (\frac{Q+\alpha -2}{2}\Big )^2\int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^2}{d^{2-\alpha }}d\xi + \int _{\mathbb {H}}d^{2-Q} |\nabla _{\mathbb {H}}\phi |^2 d\xi . \end{aligned}$$

(2.6)

Now define a smooth function $\psi$ by $\psi :=(\ln \frac{R}{d})^{-\frac{1}{2}}\phi$. Then we have

$$\begin{aligned} |\nabla _{\mathbb {H}}\phi |^2= \frac{1}{4}(\ln R/d)^{-1}\frac{|\nabla _{\mathbb {H}}d|^2}{d^2}\psi ^2 + (\ln R/d)|\nabla _{\mathbb {H}}\psi |^2 - \frac{1}{2}\frac{\nabla _{\mathbb {H}}d}{d}\nabla _{\mathbb {H}}\psi ^2 \end{aligned}$$

and then

$$\begin{aligned} \int _\Omega d^{2-Q} |\nabla _{\mathbb {H}}\phi |^2 d\xi =&\frac{1}{4}\int _\Omega d^{-Q}|\nabla _{\mathbb {H}}d|^2 (\ln R/d)^{-1}\psi ^2 d\xi + \int _\Omega d^{2-Q}(\ln R/d)|\nabla _{\mathbb {H}}\psi |^2d\xi \nonumber \\&- \frac{1}{2} \int _\Omega d^{1-Q}\nabla _{\mathbb {H}}d\nabla \psi ^2d\xi . \end{aligned}$$

(2.7)

The last integral in (2.7) vanishes due to integration by parts and identities in (1.13). This can be shown as follows:

$$\begin{aligned} \frac{1}{2} \int _\Omega d^{1-Q}\nabla _{\mathbb {H}}d\nabla \psi ^2d\xi&= - \frac{1}{2} \int _\Omega \nabla _{\mathbb {H}}(d^{1-Q}\nabla _{\mathbb {H}}d)\psi ^2d\xi = 0. \end{aligned}$$

Therefore (2.7) implies

$$\begin{aligned} \int _\Omega d^{2-Q} |\nabla _{\mathbb {H}}\phi |^2 d\xi&= \frac{1}{4}\int _\Omega d^{-Q}|\nabla _{\mathbb {H}}d|^2 (\ln R/d)^{-1}\psi ^2 d\xi + \int _\Omega d^{2-Q}(\ln R/d)|\nabla _{\mathbb {H}}\psi |^2d\xi \nonumber \\&\ge \frac{1}{4}\int _\Omega d^{-Q}|\nabla _{\mathbb {H}}d|^2 \frac{\phi ^2}{(\ln R/d)^2} d\xi \nonumber \\&= \frac{1}{4}\int _\Omega d^{\alpha -2}\frac{|z|^2}{d^2} \frac{|f|^2}{(\ln R/d)^2} d\xi , \end{aligned}$$

(2.8)

where we have used $\psi =(\ln \frac{R}{d})^{-\frac{1}{2}}\phi$ and $\phi =d^{-\alpha }f$ with $\lambda =\frac{2-Q-\alpha }{2}$. Putting (2.8) into (2.6) we arrive at (2.4). This completes the proof. $\square$

Remark 2.4

Heuristically, if one chooses $\lambda =\frac{2-Q-\alpha }{2}$ in (2.5), obiviously, the middle integral on the right hand side of the equation vanishes at once. Then one can conclude at that point using (2.8) without resulting to the maximization of coefficient.

2.3 Improved $L^p$-Hardy Type Inequalities

Theorem 2.5

Suppose all the conditions of Theorem  2.1 hold for $p\ge 2$. Then there exists a constant ${\mathcal {D}}_p>0$ such that

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^pd\xi \ge \Big |\frac{Q+\alpha -p}{p}\Big |^p \int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }} d\xi + {\mathcal {D}}_p \int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }(\ln \frac{R}{d})^p} d\xi , \end{aligned}$$

(2.9)

holds for all $f\in C^\infty _0({\mathbb {H}}\setminus \{0\})$, where $\sup _{\mathbb {H}}d<R<\infty .$

Proof

For $f\in C^\infty _0({\mathbb {H}}\setminus \{0\})$, define $g=d^{-\gamma }f$, $\gamma \ne 0$ as in the proof of Theorem 2.1. Using (1.16) of Lemma 1.2 we have

$$\begin{aligned} |\nabla _{\mathbb {H}}f|^p&\ge |\gamma |^pd^{p(\gamma -1)}|g|^p|\nabla _{\mathbb {H}}d|^p + p|\gamma |^{p-2}\gamma d^{\gamma p-p+1}|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d |g|^{p-2}g\nabla _{\mathbb {H}}g \\&\quad +c(p)d^{\gamma p}|\nabla _{\mathbb {H}}g|^p. \end{aligned}$$

Following the same steps as in the proof of Theorem 2.1 we arrive at (2.1) but with an extra term which will be analyzed further, that is

$$\begin{aligned} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}f|^pd\xi \ge \Big |\frac{Q+\alpha -p}{p}\Big |^p \int _{\mathbb {H}}\frac{|z|^p}{d^p}\frac{|f|^p}{d^{p-\alpha }} d\xi + c(p) \int _{\mathbb {H}}d^{\alpha +\gamma p} |\nabla _{\mathbb {H}}g|^p d\xi . \end{aligned}$$

(2.10)

Define a function $\psi :=\varphi ^{-\frac{1}{p}}g$ and compute using Lemma 1.2 for $p>1$

$$\begin{aligned} |\nabla _{\mathbb {H}}g|^p&= \Big |\frac{1}{p}\psi \varphi ^{\frac{1}{p}-1}\nabla _{\mathbb {H}}\varphi + \varphi ^{\frac{1}{p}}\nabla _{\mathbb {H}}\psi \Big |^p \\&\ge \frac{1}{p^p}|\psi |^p|\varphi |^{1-p}|\nabla _{\mathbb {H}}\varphi |^p +p^{1-p}\frac{|\nabla _{\mathbb {H}}\varphi |^{p-2}}{|\varphi |^{p-2}} \nabla _{\mathbb {H}}\varphi \nabla _{\mathbb {H}}|\psi |^p. \end{aligned}$$

Therefore,

$$\begin{aligned} c(p) \int _{\mathbb {H}}d^{\alpha +\gamma p} |\nabla _{\mathbb {H}}g|^p d\xi \ge&\frac{c(p)}{p^p}\int _{\mathbb {H}}d^{\alpha +\gamma p} |\psi |^p \frac{|\nabla _{\mathbb {H}}\varphi |^p}{|\varphi |^{p-1}}d\xi \nonumber \\&+{\widetilde{c}}(p)\int _{\mathbb {H}}d^{\alpha +\gamma p} \frac{|\nabla _{\mathbb {H}}\varphi |^{p-2}}{|\varphi |^{p-2}} \nabla _{\mathbb {H}}\varphi \nabla _{\mathbb {H}}( |\psi |^p)d\xi , \end{aligned}$$

(2.11)

where ${\widetilde{c}}(p)=c(p)p^{1-p}$.

Setting $\varphi :=\ln \frac{R}{d}$ we have

$$\begin{aligned} \nabla _{\mathbb {H}}\varphi = -\frac{\nabla _{\mathbb {H}}d}{d} \ \ \ \text {and} \ \ \ |\nabla _{\mathbb {H}}\varphi |^p = \frac{|\nabla _{\mathbb {H}}d|^p}{d^p}. \end{aligned}$$

(2.12)

Then referring to the definitions of $\psi$ and g, (i.e., $\psi :=\varphi ^{-\frac{1}{p}}g$ and $g=d^{-\gamma }f$), the first integral on the right hand side of (2.11) implies

$$\begin{aligned} \frac{c(p)}{p^p}\int _{\mathbb {H}}d^{\alpha +\gamma p} |\psi |^p \frac{|\nabla _{\mathbb {H}}\varphi |^p}{|\varphi |^{p-1}}d\xi = \frac{c(p)}{p^p}\int _{\mathbb {H}}d^\alpha \frac{|\nabla _{\mathbb {H}}d|^p}{d^p} \frac{|f|^p}{(\ln \frac{R}{d})^p}d\xi . \end{aligned}$$

(2.13)

Using $\gamma =\frac{p-Q-\alpha }{p}$ and $\nabla _{\mathbb {H}}\varphi = -\frac{\nabla _{\mathbb {H}}d}{d}$ into the second integral on the right hand side of (2.11), then applying integration by parts and (2.10) gives

$$\begin{aligned}&{\widetilde{c}}(p)\int _{\mathbb {H}}d^{\alpha +\gamma p} \frac{|\nabla _{\mathbb {H}}\varphi |^{p-2}}{|\varphi |^{p-2}} \nabla _{\mathbb {H}}\varphi \nabla _{\mathbb {H}}|\psi |^p d\xi \nonumber \\&\quad = - {\widetilde{c}}(p)\int _{\mathbb {H}}d^{1-Q} \frac{|\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d}{|\varphi |^{p-2}} \nabla _{\mathbb {H}}|\psi |^p d\xi \nonumber \\&\quad ={\widetilde{c}}(p)\int _{\mathbb {H}}\nabla _{\mathbb {H}}\Big (d^{1-Q} \frac{|\nabla _{\mathbb {H}}d|^{p-2}}{|\varphi |^{p-2}} \nabla _{\mathbb {H}}d\Big ) |\psi |^p d\xi \nonumber \\&\quad = - {\widetilde{c}}(p)\int _{\mathbb {H}}\frac{ d^{1-Q} |\nabla _{\mathbb {H}}d|^{p-2}\nabla _{\mathbb {H}}d}{|\varphi |^{2(p-2)}} \nabla _{\mathbb {H}}(|\varphi |^{p-2}) |\psi |^p d\xi \nonumber \\&\quad = (p-2){\widetilde{c}}(p)\int _{\mathbb {H}}d^{\alpha -p}|\nabla _{\mathbb {H}}d|^p \frac{|f|^p}{(\ln \frac{R}{d})^p}d\xi \end{aligned}$$

(2.14)

by using (1.12) and (2.12). Combining (2.11), (2.13) and (2.14) with (2.10) yields the expected inequality (2.9) with ${\mathcal {D}}_p=\frac{c(p)}{p^p}(p-1)^2>0$. $\square$

Remark 2.6

If $p=2$, we can see from Lemma 1.2 that $c(p)=1$. Thus, the explicit value of ${\mathcal {D}}_p$ can be computed as ${\mathcal {D}}_p=\frac{c(p)}{p^p}(p-1)^2 = \frac{1}{4}$ in the case $p=2$ as in Theorem 2.3.

3 Rellich Type Inequalities

In this section, we are concerned with Rellich type inequalities with weights given in terms of the homogeneous quasi norm (Heisenberg distance function). First, we start with an $L^2$-Rellich inequalities and their improved versions and we conclude with $L^p$-Rellich inequalities and their improved versions

3.1 $L^2$-Rellich Type Inequalities

Theorem 3.1

Let ${\mathbb {H}}$ be the Heisenberg group of homogeneous dimension $Q \ge 3$ and $Q+\alpha -4> 0$ for $\alpha \in {\mathbb {R}}$. Then for all $f \in C^\infty _0({\mathbb {H}}\setminus \{0\})$

$$\begin{aligned} \int _{\mathbb {H}}\frac{d^{\alpha +2}}{|z|^2} |\Delta _{\mathbb {H}}f|^2d\xi \ge \frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16} \int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }} d\xi \end{aligned}$$

(3.1)

and the constant $\frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16}$ is sharp.

Moreover, if $4-Q<\alpha <Q$ and $\sup _\Omega d<R<\infty$, where $\Omega \in {\mathbb {H}}$, then

$$\begin{aligned} \int _\Omega \frac{d^{\alpha +2}}{|z|^2}|\Delta _{\mathbb {H}}f|^2d\xi \ge {\mathscr {A}}_{Q,\alpha }\int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }} d\xi + {\mathcal {B}}_{Q,\alpha } \int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }(\ln R/d)^2} d\xi , \end{aligned}$$

(3.2)

where ${\mathscr {A}}_{Q,\alpha }:=\frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16}$ and ${\mathcal {B}}_{Q,\alpha }:=\frac{(Q+\alpha -4)(Q-\alpha )}{8}$, holds for all compactly support function $f\in C^\infty _0(\Omega )$.

Proof

A direct computation shows that

$$\begin{aligned} \Delta _{\mathbb {H}}d^{\alpha -2}&= (\alpha -2)(\alpha -3)d^{\alpha -4}|\nabla _{\mathbb {H}}d|^2 + (\alpha -2)d^{\alpha -3}\Delta _{\mathbb {H}}d\\&= (\alpha -2)(Q+\alpha -4)d^{\alpha -4}\frac{|z|^2}{d^2}. \end{aligned}$$

On the other hand, using integration by parts we get (f is assumed to be real-valued without loss of generality)

$$\begin{aligned} \int _{\mathbb {H}}|f|^2\Delta _{\mathbb {H}}d^{\alpha -2} d\xi = \int _{\mathbb {H}}d^{\alpha -2} (2f \Delta _{\mathbb {H}}f + 2|\nabla _{\mathbb {H}}f|^2)d\xi . \end{aligned}$$

Since $Q+\alpha -4> 0$, we have

$$\begin{aligned} (\alpha -2)(Q+\alpha -4)\int _{\mathbb {H}}d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi&= \int _{\mathbb {H}}|f|^2\Delta _{\mathbb {H}}d^{\alpha -2} d\xi \\&= 2 \int _{\mathbb {H}}d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi + 2 \int _{\mathbb {H}}d^{\alpha -2} |\nabla _{\mathbb {H}}f|^2d\xi . \end{aligned}$$

Rearranging we have

$$\begin{aligned} (\alpha -2)(Q+\alpha -4)\int _{\mathbb {H}}d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi - 2 \int _{\mathbb {H}}d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi = 2 \int _{\mathbb {H}}d^{\alpha -2} |\nabla _{\mathbb {H}}f|^2d\xi . \end{aligned}$$

(3.3)

Applying the weighted Hardy inequality (2.1) replacing $\alpha$ by $\alpha -2$ and $p=2$ gives

$$\begin{aligned}&(\alpha -2)(Q+\alpha -4)\int _{\mathbb {H}}d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi - 2 \int _{\mathbb {H}}d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi \\&\quad \ge 2 \Big (\frac{Q+\alpha -4}{2}\Big )^2 \int _{\mathbb {H}}d^{\alpha -4} \frac{|z|^2}{d^2}|f|^2 d\xi . \end{aligned}$$

The last inequality implies after rearranging and adding like terms

$$\begin{aligned} \frac{(Q-\alpha )(Q+\alpha -4)}{4}\int _{\mathbb {H}}d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi \le - \int _{\mathbb {H}}d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi . \end{aligned}$$

(3.4)

Using Cauchy-Schwarz inequality

$$\begin{aligned} -\int _{\mathbb {H}}d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi&\le \int _{\mathbb {H}}d^{\alpha -2} |f| |\Delta _{\mathbb {H}}f| d\xi \nonumber \\&\le \Big (\int _{\mathbb {H}}d^{\alpha -4}|\nabla _{\mathbb {H}}d|^2|f|^2d\xi \Big )^{\frac{1}{2}}\Big (\int _{\mathbb {H}}d^{\alpha }\frac{|\Delta _{\mathbb {H}}f|^2}{|\nabla _{\mathbb {H}}d|^2}d\xi \Big )^{\frac{1}{2}}. \end{aligned}$$

(3.5)

Combining (3.4) and (3.5) proves (3.1).

To prove the inequality (3.2), we apply the improved weighted Hardy type inequality (2.4) to the right hand side of (3.3) and obtain

$$\begin{aligned}&(\alpha -2)(Q+\alpha -4)\int _\Omega d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi - 2 \int _\Omega d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi \\&\quad = 2 \int _\Omega d^{\alpha -2} |\nabla _{\mathbb {H}}f|^2d\xi \\&\quad \ge 2 \Big [\Big (\frac{Q+\alpha -4}{2}\Big )^2 \int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }} d\xi + \frac{1}{4}\int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }(\ln R/d)^2} d\xi \Big ]. \end{aligned}$$

Rearranging

$$\begin{aligned}&\frac{(Q-\alpha )(Q+\alpha -4)}{4}\int _\Omega d^{\alpha -4}\frac{|z|^2}{d^2}|f|^2d\xi + \frac{1}{4}\int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }(\ln R/d)^2} d\xi \nonumber \\&\quad \le - \int _\Omega d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi . \end{aligned}$$

(3.6)

Applying the Young’s inequality for $\beta >0$ ($\beta$ to be determined later)

$$\begin{aligned} -\int _\Omega d^{\alpha -2} f \Delta _{\mathbb {H}}f d\xi&\le \beta \int _\Omega d^{\alpha -4}|\nabla _{\mathbb {H}}d|^2|f|^2d\xi + \frac{1}{4\beta }\int _\Omega d^{\alpha }\frac{|\Delta _{\mathbb {H}}f|^2}{|\nabla _{\mathbb {H}}d|^2}d\xi . \end{aligned}$$

(3.7)

Putting (3.7) and (3.6) together

$$\begin{aligned} \int _\Omega d^{\alpha }\frac{|\Delta _{\mathbb {H}}f|^2}{|\nabla _{\mathbb {H}}d|^2}d\xi \ge&[(Q-\alpha )(Q+\alpha -4)\beta -4\beta ^2]\int _\Omega \frac{|z|^2}{d^2} \frac{|f|^2}{d^{4-\alpha }} d\xi \\&+\beta \int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }(\ln R/d)^2} d\xi . \end{aligned}$$

Let $\Psi (\beta ):= \beta [(Q-\alpha )(Q+\alpha -4) -4\beta ]$. Obviously, the maximum of the $\Psi (\beta )$ is

$$\begin{aligned} \Psi (\beta _\star )= \Big [\frac{(Q-\alpha )(Q+\alpha -4)}{4}\Big ]^2 \end{aligned}$$

achieved at the point $\beta _\star =\frac{(Q-\alpha )(Q+\alpha -4)}{8}$. Hence we arrive at

$$\begin{aligned} \int _\Omega d^{\alpha }\frac{|\Delta _{\mathbb {H}}f|^2}{|\nabla _{\mathbb {H}}d|^2}d\xi \ge&\Psi (\beta _\star ) \int _\Omega \frac{|z|^2}{d^2} \frac{|f|^2}{d^{4-\alpha }} d\xi + \beta _\star \int _\Omega \frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }(\ln R/d)^2} d\xi \end{aligned}$$

which is the required (3.2).

In the next we prove that the constant $\frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16}$ appearing in (3.1) is sharp in the sense that

$$\begin{aligned} \frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16} = \inf _{0\ne f \in C^\infty _0({\mathbb {H}}\setminus \{0\})} \frac{\int _{\mathbb {H}}\frac{d^{\alpha +2}}{|z|^2}|\Delta _{\mathbb {H}}f|^2d\xi }{\int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^2}{d^{4-\alpha }} d\xi } =: {\mathcal {D}}_{{\mathbb {H}}} \end{aligned}$$

(3.8)

must hold.

Thus, we are required to prove that $D_{Q,\alpha }:=\frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16} = {\mathcal {D}}_{{\mathbb {H}}_n}$. Clearly by (3.1) and passing to the infimum in (3.8) we have $D_{Q,\alpha } \le {\mathcal {D}}_{{\mathbb {H}}}$. Hence, it only remains to prove that $D_{Q,\alpha } \ge {\mathcal {D}}_{{\mathbb {H}}}$.

For $\eta >0$, we define the following radial function which can be approximated by smooth function with compact support in ${\mathbb {H}}$

$$\begin{aligned} f_\eta (d):= \left\{ \begin{array}{ll} (\frac{4-\alpha -Q}{2}-\eta )(d-1)+1, &{} \ 0\le d \le 1,\\ \ \\ d^{\frac{4-\alpha -Q}{2}-\eta }, &{} d>1. \end{array} \right. \end{aligned}$$

Then

$$\begin{aligned} |\Delta _{\mathbb {H}}f_\eta |^2 := \left\{ \begin{array}{ll} (\frac{Q+\alpha -4}{2}+\eta )^2(Q-1)^2 \frac{|\nabla _{\mathbb {H}}d|^4}{{d^2}}, &{} d \le 1,\\ \ \\ (\frac{Q+\alpha -4}{2}+\eta )^2(\frac{Q-\alpha }{2}-\eta )^2 d^{-Q-\alpha -2\eta } |\nabla _{\mathbb {H}}d|^4, &{} d>1 \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {H}}\frac{d^{\alpha +2}}{|z|^2} |\Delta _{\mathbb {H}}f_\eta |^2 = \Lambda (Q,\alpha ,\eta )\int _{B_1}d^{\alpha -2}|\nabla _{\mathbb {H}}d|^2 + {\widetilde{\Lambda }}(Q,\alpha ,\eta )\int _{{\mathbb {H}}\setminus B_1} d^{-Q-2\eta }|\nabla _{\mathbb {H}}d|^2, \end{aligned}$$

where $\Lambda (Q,\alpha ,\eta ):= (\frac{Q+\alpha -4}{2}+\eta )^2(Q-1)^2$ and ${\widetilde{\Lambda }}(Q,\alpha ,\eta ):= (\frac{Q+\alpha -4}{2}+\eta )^2(\frac{Q-\alpha }{2}-\eta )^2$, and $B_1$ is the unit Heisenberg ball with respect to homogeneous quasi norm as defined in the introduction. Note that the integral $\int _{B_1}d^{\alpha -2}|\nabla _{\mathbb {H}}d|^2d\xi$ is finite since $|\nabla _{\mathbb {H}}d|$ is uniformly bounded and $Q+\alpha -4>0$. Therefore

$$\begin{aligned} \int _{\mathbb {H}}\frac{d^{\alpha +2}}{|z|^2} |\Delta _{\mathbb {H}}f_\eta |^2 = {\widetilde{\Lambda }}(Q,\alpha ,\eta )\int _{{\mathbb {H}}\setminus B_1} d^{-Q-2\eta }|\nabla _{\mathbb {H}}d|^2 + O(1), \end{aligned}$$

On the other hand,

$$\begin{aligned} \int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi = \int _{B_1} |\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi + \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi . \end{aligned}$$

Clearly, the integral $\int _{B_1} |\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi$ is finite and we get

$$\begin{aligned} \int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi&= \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^2 d^{\alpha -4}|f_\eta |^2d\xi +O(1)\\&= \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^2 d^{-Q-2\eta }d\xi +O(1). \end{aligned}$$

Sending $\eta \rightarrow 0$ and noting that $\int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^2 d^{-Q-2\eta }d\xi \rightarrow \infty$, we get that ${\mathcal {D}}_{\mathbb {H}}\le D(Q,\alpha )$. The proof is therefore complete. $\square$

The next theorem contains the $L^p$-Rellich type inequalities and improvement.

3.2 Improved $L^p$-Rellich Type Inequalities

Theorem 3.2

Let ${\mathbb {H}}$ be the Heisenberg group of homogeneous dimension $Q \ge 3$. Let $2-Q<\alpha <0$, then there holds for $f \in C^\infty _0({\mathbb {H}}\setminus \{0\})$ and $1<p<\infty$

$$\begin{aligned} {\mathscr {A}}_{Q,p} \int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^p}{d^{2-\alpha }} d\xi \le \int _{\mathbb {H}}\Big (\frac{d^2}{|z|^2}\Big )^{p-1} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}}d\xi . \end{aligned}$$

(3.9)

Moreover, the constant ${\mathscr {A}}_{Q,p} = \Big (\frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^2}\Big )^p$ is sharp.

Furthermore, we have the improved version

$$\begin{aligned} {\mathscr {A}}_{Q,p} \int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^p}{d^{2-\alpha }} d\xi&+{\mathscr {B}}_{Q,p}\int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{|f|^p}{d^{2-\alpha }(\ln R/d)^2} d\xi \le \int _{\mathbb {H}}\Big (\frac{d^2}{|z|^2}\Big )^{p-1} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}}d\xi , \end{aligned}$$

(3.10)

where

$$\begin{aligned} {\mathscr {A}}_{Q,p} = \Big (\frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^p}\Big )^p \end{aligned}$$

and

$$\begin{aligned} {\mathscr {B}}_{Q,p} = \frac{(p-1)}{p^p} \left( \frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p}\right) ^{p-1}. \end{aligned}$$

Remark 3.3

The first part of Theorem 3.2 above can also be compared with [33, Theorem 2.2], which has been derived using a different method for $1<p<\infty$. Indeed, if $\alpha$ in [33, eqn. (2.10)] is replaced by $2-\alpha$, then [33, eqn. (2.10)] coincides with (3.10) above.

Proof

For $\epsilon >0$, define $f_\epsilon :=(|f|^2+\epsilon ^2)^{\frac{p}{2}}-\epsilon ^p \in C^\infty _0({\mathbb {H}}_n\setminus \{0\})$ with same support as f. Compute

$$\begin{aligned} \Delta _{\mathbb {H}}f_\epsilon&= p (|f|^2+\epsilon ^2)^{\frac{p}{2}-1}|\nabla _{\mathbb {H}}f|^2 +p(p-2)(|f|^2+\epsilon ^2)^{\frac{p}{2}-2}f^2|\nabla _{\mathbb {H}}f|^2 \\&\quad + p (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f\\&\ge p(p-1)(|f|^2+\epsilon ^2)^{\frac{p}{2}-2}f^2|\nabla _{\mathbb {H}}f|^2 + p (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f\\&= \frac{4(p-1)}{p}|\nabla _{\mathbb {H}}h_\epsilon |^2 + p (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f, \end{aligned}$$

where we have defined $h_\epsilon := (|f|^2+\epsilon ^2)^{\frac{p}{4}}-\epsilon ^{\frac{p}{2}} \in C^\infty _0({\mathbb {H}}_n\setminus \{0\})$ and used

$$\begin{aligned} (|f|^2+\epsilon ^2)^{\frac{p}{2}-2}f^2|\nabla _{\mathbb {H}}f|^2 = \frac{4}{p^2}|\nabla _{\mathbb {H}}h_\epsilon |^2. \end{aligned}$$

Thus,

$$\begin{aligned} - p (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f \ge \frac{4(p-1)}{p}|\nabla _{\mathbb {H}}h_\epsilon |^2-\Delta _{\mathbb {H}}f_\epsilon . \end{aligned}$$

Multiplying the last inequality by $d^\alpha$ ($2-Q<\alpha <0$) and integrating over ${\mathbb {H}}$ gives

$$\begin{aligned} - p \int _{\mathbb {H}}d^\alpha (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f d\xi \ge \frac{4(p-1)}{p} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}h_\epsilon |^2 d\xi - \int _{\mathbb {H}}d^\alpha \Delta _{\mathbb {H}}f_\epsilon d\xi . \end{aligned}$$

Integration by parts gives

$$\begin{aligned} - \int _{\mathbb {H}}d^\alpha \Delta _{\mathbb {H}}f_\epsilon d\xi&= - \int _{\mathbb {H}}\Delta _{\mathbb {H}}d^\alpha f_\epsilon d\xi \\&= -\alpha (\alpha -1) \int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^2 d^{\alpha -2}f_\epsilon d\xi - \alpha \int _{\mathbb {H}}d^{\alpha -1}\Delta _{\mathbb {H}}d f_\epsilon d\xi \\&= - \alpha (Q+\alpha -2)\int _{\mathbb {H}}d^{\alpha -2}\frac{|z|^2}{d^2}f_\epsilon d\xi . \end{aligned}$$

Therefore

$$\begin{aligned} - p \int _{\mathbb {H}}d^\alpha (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f d\xi \ge&\frac{4(p-1)}{p} \int _{\mathbb {H}}d^\alpha |\nabla _{\mathbb {H}}h_\epsilon |^2 d\xi \nonumber \\&- \alpha (Q+\alpha -2)\int _{\mathbb {H}}d^{\alpha -2}\frac{|z|^2}{d^2}f_\epsilon d\xi . \end{aligned}$$

(3.11)

Applying the weighted Hardy inequality (2.1) (with $p=2$) to (3.11), we have

$$\begin{aligned}&\frac{(p-1)(Q+\alpha -2)^2}{p} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{h^2_\epsilon }{d^{2-\alpha }} d\xi - \alpha (Q+\alpha -2)\int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{f_\epsilon }{d^{2-\alpha }} d\xi \\&\quad \le p \int _{\mathbb {H}}d^\alpha (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} |f| |\Delta _{\mathbb {H}}f| d\xi . \end{aligned}$$

Letting $\epsilon \rightarrow 0$, applying the Lebesgue dominated convergence theorem and Hölder’s inequality, we have

$$\begin{aligned}&\frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^2} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi \le \int _{\mathbb {H}}d^\alpha |f|^{p-1} |\Delta _{\mathbb {H}}f| d\xi \\&\quad \le \Big (\int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi \Big )^{\frac{p-1}{p}} \Big (\int _{\mathbb {H}}\frac{d^{2(p-1})}{|z|^{2(p-1})} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}} d\xi \Big )^{\frac{1}{p}}. \end{aligned}$$

Raising both sides to the power of p and rearranging leads to the desired inequality (3.9).

In order to prove (3.10), we apply the improved weighted Hardy inequality (2.4) (or (2.9) for $p=2$) to (3.11) to obtain

$$\begin{aligned}&- p \int _{\mathbb {H}}d^\alpha (|f|^2+\epsilon ^2)^{\frac{p}{2}-1} f \Delta _{\mathbb {H}}f d\xi \ge \frac{(p-1)(Q+\alpha -2)^2}{p} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{h^2_\epsilon }{d^{2-\alpha }}d\xi \nonumber \\&\quad +\frac{p-1}{p} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{h^2_\epsilon }{d^{2-\alpha }(\ln \frac{R}{d})^2} d\xi - \alpha (Q+\alpha -2)\int _{\mathbb {H}}\frac{|z|^2}{d^2}\frac{f_\epsilon }{d^{2-\alpha }} d\xi . \end{aligned}$$

Letting $\epsilon \rightarrow 0$, using Lebesgue dominated convergence theorem, rearranging and then applying Hölder’s and Young’s inequalities (for $\varepsilon >0$)

$$\begin{aligned}&\frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^2} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi +\frac{p-1}{p^2} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }(\ln \frac{R}{d})^2} d\xi \\&\quad \le \int _{\mathbb {H}}d^\alpha |f|^{p-1} |\Delta _{\mathbb {H}}f| d\xi \\&\quad \le \left( \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi \right) ^{\frac{p-1}{p}} \left( \int _{\mathbb {H}}\frac{d^{2(p-1})}{|z|^{2(p-1})} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}} d\xi \right) ^{\frac{1}{p}}\\&\quad \le \frac{p-1}{p}\varepsilon ^{-\frac{p}{p-1}} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi + \frac{\varepsilon ^{p}}{ p} \int _{\mathbb {H}}\frac{d^{2(p-1})}{|z|^{2(p-1})} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}}. \end{aligned}$$

The Young’s inequality used here can be described as follows: Let

$$\begin{aligned} A:= \int _{\mathbb {H}}\frac{d^{2(p-1})}{|z|^{2(p-1})} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}} d\xi \ \text {and}\ \ B:= \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi . \end{aligned}$$

Then for any $\varepsilon >0$, Young’s inequality implies

$$\begin{aligned} AB = \varepsilon A \frac{B}{\varepsilon } \le \frac{1}{p}\left( \varepsilon A\right) ^p + \frac{1}{q}\left( \frac{B}{\varepsilon }\right) ^q, \ \text {with} \ \ q =\frac{p}{p-1} \ \text {being conjugate to}\ p. \end{aligned}$$

Therefore

$$\begin{aligned} \int _{\mathbb {H}}\frac{d^{2(p-1})}{|z|^{2(p-1})} \frac{|\Delta _{\mathbb {H}}f|^p}{d^{2-\alpha -2p}} \ge \Phi (\varepsilon )\int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }} d\xi +\frac{p-1}{p}\varepsilon ^{-p} \int _{\mathbb {H}}\frac{|z|^2}{d^2} \frac{|f|^p}{d^{2-\alpha }(\ln \frac{R}{d})^2} d\xi , \end{aligned}$$

(3.12)

where $\Phi (\varepsilon )$ is a scalar function given by

$$\begin{aligned} \Phi (\varepsilon ):=\frac{\varepsilon ^{-p} }{p}\left[ (Q+\alpha -2)[(p-1)(Q-2)-\alpha ]-p(p-1)\varepsilon ^{-\frac{p}{p-1}}\right] . \end{aligned}$$

Obviously, the maximum of $\varepsilon \mapsto \Phi (\varepsilon )$ is

$$\begin{aligned} \max _{\varepsilon >0}\Phi (\varepsilon )= \Phi (\varepsilon _\star )= \left( \frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^2}\right) ^p \end{aligned}$$

achieved at the point $\varepsilon _\star$ given by

$$\begin{aligned} \varepsilon _\star ^{\frac{p}{p-1}} =\frac{p^2}{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}. \end{aligned}$$

Replacing $\Phi (\varepsilon )$ and $\varepsilon$ in (3.12) with $\Phi (\varepsilon _\star )$ and $\varepsilon _\star$, respectively, we obtain (3.10) at once.

In conclusion we show that the constant ${\mathscr {A}}_{Q,p} = \Big (\frac{(Q+\alpha -2)[(p-1)(Q-2)-\alpha ]}{p^2}\Big )^p$ appearing in (3.9) is sharp. Following the same procedure as before, we consider the function

$$\begin{aligned} f_\eta (d):= \left\{ \begin{array}{ll} C_\eta , \ \ \ \ 0\le d \le 1\\ \ \\ d^{\frac{2-\alpha -Q}{p}-\eta }, \ \ \ \ d>1 \end{array} \right. \end{aligned}$$

for $\eta >0$ and $C_\eta$ is a constant function.

A simple computation gives (for $d>1$)

$$\begin{aligned} | \Delta _{\mathbb {H}}f_\eta |^p = \Big (\frac{Q+\alpha -2}{p}-\eta \Big )^p \Big (\frac{(p-1)(Q-2)-\alpha }{p}-\eta \Big )^p |\nabla _{\mathbb {H}}d|^{2p} d^{2-\alpha -Q-2p-\eta p}. \end{aligned}$$

and

$$\begin{aligned}&\int _\Omega \left( \frac{d^2}{|z|^2}\right) ^{p-1}\frac{|\Delta _{\mathbb {H}}f_\eta |^p}{d^{2-\alpha -2p}}d\xi \\&\quad = \Big (\frac{Q+\alpha -2}{p}-\eta \Big )^p \Big (\frac{(p-1)(Q-2)-\alpha }{p}-\eta \Big )^p \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^{p} d^{-Q-\eta p}d\xi \\&\quad = \Big (\frac{Q+\alpha -2}{p}-\eta \Big )^p \Big (\frac{(p-1)(Q-2)-\alpha }{p}-\eta \Big )^p \int _{{\mathbb {H}}\setminus B_1} |\nabla _{\mathbb {H}}d|^{p} d^{\alpha -2}|f_\eta |^pd\xi . \end{aligned}$$

Passing to the limit $\eta \rightarrow 0$ one gets at once that the constant ${\mathscr {A}}_{Q,p}$ is sharp. $\square$

Remark 3.4

Setting $\alpha =0$, we have (3.1) reduces to $Q\ge 5$ and

$$\begin{aligned} \int _{\mathbb {H}}\frac{ |\Delta _{\mathbb {H}}f|^2}{|\nabla _{\mathbb {H}}d|^2}d\xi \ge \frac{Q^2(Q-4)^2}{16}\int _{\mathbb {H}}|\nabla _{\mathbb {H}}d|^2 \frac{|f|^2}{d^4} d\xi . \end{aligned}$$

(3.13)

Replacing d with Carnot-Carathéodory distance norm $\varrho _{cc}$ we obtain (1.6) since $\varrho _{cc}$ is Lipschitz and $|\nabla _{\mathbb {H}}\varrho _{cc}|^2 =1$, (we can do this because $\varrho _{cc}$ is equivalent to any other norm on ${\mathbb {H}}$). In a similar spirit, setting $\alpha =-2$, $p=2$ and $d\equiv \varrho _{cc}$ in (3.10) (Theorem 3.2) we also obtain (1.6).

4 Conclusion

In this paper we have derived some weighted Hardy type and Rellich type inequalities with sharp constants and their improved versions on the Heisenberg group. Improvement of this class of inequalities is being a hot research topic since few decades ago, see Refs. [2, 5, 8, 12, 13, 19, 24, 25, 31, 34] for instance. The book [31] contains detail discussion on this class of inequalities on homogeneous groups. Some more general results and some improvement of Hardy and Rellich inequalities were established in Ref. [33] (this reference was pointed to us by one of the referees at the review stage). The present paper therefore provides simple and more direct proofs of improved Hardy type and Rellich type inequalities on the Heisenberg group with respect to homogeneous norm, horizontal gradient and Kohn sub-Laplace operators. The results obtained generalize and improved several results in literature as reported in the paper. From mathematical point of view, the Heisenberg group is the simplest and most important model in sub-Riemannian setting with noncommutative vector fields satisfying Hörmander’s condition. Hence, it would be interesting to extend these results to some other examples of homogeneous groups.