# Weighted $$L^{p}$$-Hardy and $$L^{p}$$-Rellich inequalities with boundary terms on stratified Lie groups

## Abstract

In this paper, generalised weighted $$L^p$$-Hardy, $$L^p$$-Caffarelli–Kohn–Nirenberg, and $$L^p$$-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg–Pauli–Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted $$L^2$$-Rellich type inequality with the boundary term is obtained.

## Introduction

Let $${\mathbb {G}}$$ be a stratified Lie group (or a homogeneous Carnot group), with dilation structure $$\delta _{\lambda }$$ and Jacobian generators $$X_{1},\ldots ,X_{N}$$, so that N is the dimension of the first stratum of $${\mathbb {G}}$$. We refer to , or to the recent books  or  for extensive discussions of stratified Lie groups and their properties. Let Q be the homogeneous dimension of $$\mathbb {G}$$. The sub-Laplacian on $${\mathbb {G}}$$ is given by

\begin{aligned} {\mathcal {L}}=\sum _{k=1}^{N}X_{k}^{2}. \end{aligned}
(1.1)

It was shown by Folland  that the sub-Laplacian has a unique fundamental solution $$\varepsilon$$,

\begin{aligned} {\mathcal {L}} \varepsilon = \delta , \end{aligned}

where $$\delta$$ denotes the Dirac distribution with singularity at the neutral element 0 of $$\mathbb {G}$$. The fundamental solution $$\varepsilon (x,y)=\varepsilon (y^{-1}x)$$ is homogeneous of degree $$-Q+2$$ and can be written in the form

\begin{aligned} \varepsilon (x,y) = [d(y^{-1}x)]^{2-Q}, \end{aligned}
(1.2)

for some homogeneous d which is called the $${\mathcal {L}}$$-gauge. Thus, the $${\mathcal {L}}$$-gauge is a symmetric homogeneous (quasi-) norm on the stratified group $$\mathbb {G}= ({\mathbb {R}}^n,\circ ,\delta _{\lambda })$$, that is,

• $$d(x) >0$$ if and only if $$x \ne 0$$,

• $$d(\delta _{\lambda }(x))=\lambda d(x)$$ for all $$\lambda >0$$ and $$x \in \mathbb {G}$$,

• $$d(x^{-1})=d(x)$$ for all $$x \in \mathbb {G}$$.

We also recall that the standard Lebesque measure dx on $${\mathbb {R}}^{n}$$ is the Haar measure for $${\mathbb {G}}$$ (see, e.g. [9, Proposition 1.6.6]). The left invariant vector field $$X_{j}$$ has an explicit form and satisfies the divergence theorem, see e.g.  for the derivation of the exact formula: more precisely, we can write

\begin{aligned} X_{k}=\frac{\partial }{\partial x'_{k}}+ \sum _{l=2}^{r}\sum _{m=1}^{N_{l}}a_{k,m}^{(l)}(x',\ldots ,x^{(l-1)}) \frac{\partial }{\partial x_{m}^{(l)}}, \end{aligned}
(1.3)

with $$x=(x',x^{(2)},\ldots ,x^{(r)})$$, where r is the step of $$\mathbb {G}$$ and $$x^{(l)}=(x^{(l)}_1,\ldots ,x^{(l)}_{N_l})$$ are the variables in the $$l^{th}$$ stratum, see also [9, Section 3.1.5] for a general presentation. The horizontal gradient is given by

\begin{aligned} \nabla _{\mathbb {G}}:=(X_{1},\ldots , X_{N}), \end{aligned}

and the horizontal divergence is defined by

\begin{aligned} \mathrm{div}_{\mathbb {G}} v:=\nabla _{\mathbb {G}}\cdot v. \end{aligned}

The horizontal p-sub-Laplacian is defined by

\begin{aligned} {\mathcal {L}}_{p}f:=\mathrm{div}_{\mathbb {G}}(|\nabla _{\mathbb {G}}f|^{p-2}\nabla _{\mathbb {G}}f),\quad 1<p<\infty , \end{aligned}
(1.4)

and we will write

\begin{aligned} |x'|=\sqrt{x'^{2}_{1}+\ldots +x'^{2}_{N}} \end{aligned}

for the Euclidean norm on $${\mathbb {R}}^{N}.$$

Throughout this paper $$\Omega \subset {\mathbb {G}}$$ will be an admissible domain, that is, an open set $$\Omega \subset {\mathbb {G}}$$ is called an admissible domain if it is bounded and if its boundary $$\partial \Omega$$ is piecewise smooth and simple i.e., it has no self-intersections. The condition for the boundary to be simple amounts to $$\partial \Omega$$ being orientable.

We now recall the divergence formula in the form of [19, Proposition 3.1]. Let $$f_{k}\in C^{1}(\Omega )\bigcap C({\overline{\Omega }}),\,k=1,\ldots ,N$$. Then for each $$k=1,\ldots ,N,$$ we have

\begin{aligned} \int _{\Omega }X_{k}f_{k}dz= \int _{\partial \Omega }f_{k} \langle X_{k},dz\rangle . \end{aligned}
(1.5)

Consequently, we also have

\begin{aligned} \int _{\Omega }\sum _{k=1}^{N}X_{k}f_{k}dz= \int _{\partial \Omega } \sum _{k=1}^{N} f_{k}\langle X_{k},dz\rangle . \end{aligned}
(1.6)

Using the divergence formula analogues of Green’s formulae were obtained in  for general Carnot groups and in  for more abstract settings (without the group structure), for another formulation see also .

The analogue of Green’s first formula for the sub-Laplacian was given in  in the following form: if $$v \in C^1(\Omega )\cap C({\overline{\Omega }})$$ and $$u \in C^2(\Omega )\cap C^1({\overline{\Omega }})$$, then

\begin{aligned} \int _{\Omega }\left( (\mathcal {{\widetilde{\nabla }}}v) u +v{\mathcal {L}}u\right) dz=\int _{\partial \Omega }v\langle \mathcal {{\widetilde{\nabla }} }u,dz\rangle , \end{aligned}
(1.7)

where

\begin{aligned} {\widetilde{\nabla }} u = \sum _{k=1}^{N} (X_k u)X_k, \end{aligned}

and

\begin{aligned} \int _{\partial \Omega } \sum _{k=1}^{N} \langle v X_k u X_k,dz \rangle = \int _{\partial \Omega } v \langle {\widetilde{\nabla }} u, dz \rangle . \end{aligned}

Rewriting (1.7) we have

\begin{aligned} \int _{\Omega }\left( (\mathcal {{\widetilde{\nabla }} }u) v+u{\mathcal {L}}v\right) dz= & {} \int _{\partial \Omega }u\langle \mathcal {{\widetilde{\nabla }} }v,dz\rangle ,\\ \int _{\Omega }\left( (\mathcal {{\widetilde{\nabla }} }v) u+v{\mathcal {L}}u\right) dz= & {} \int _{\partial \Omega }v\langle \mathcal {{\widetilde{\nabla }} }u,dz\rangle . \end{aligned}

By using $$(\mathcal {{\widetilde{\nabla }} }u) v=(\mathcal {{\widetilde{\nabla }} }v) u$$ and subtracting one identity for the other we get Green’s second formula for the sub-Laplacian:

\begin{aligned} \int _{\Omega }(u{\mathcal {L}}v-v{\mathcal {L}}u)dz =\int _{\partial \Omega }(u\langle {\widetilde{\nabla }} v,dz\rangle -v\langle {\widetilde{\nabla }} u,dz\rangle ). \end{aligned}
(1.8)

It is important to note that the above Green’s formulae also hold for the fundamental solution of the sub-Laplacian as in the case of the fundamental solution of the (Euclidean) Laplacian since both have the same behaviour near the singularity $$z=0$$ (see [1, Proposition 4.3]).

Weighted Hardy and Rellich inequalities in different related contexts have been recently considered in  and . For the general importance of such inequalities we can refer to . Some boundary terms have appeared in . For these inequalities in the setting of general homogeneous groups we refer to .

The main aim of this paper is to give the generalised weighted $$L^p$$-Hardy and $$L^p$$-Rellich type inequalities on stratified groups. In Sect. 2, we present a weighted $$L^p$$-Caffarelli–Kohn–Nirenberg type inequality with boundary term on stratified group $$\mathbb {G}$$, which implies, in particular, the weighted $$L^p$$-Hardy type inequality. As consequences of those inequalities, we recover most of the known Hardy type inequalities and Heisenberg–Pauli–Weyl type uncertainty principles on stratified group $$\mathbb {G}$$ (see  for discussions in this direction). In Sect. 3, a weighted $$L^p$$-Rellich type inequality is investigated. Moreover, a weighted $$L^2$$-Rellich type inequality with the boundary term is obtained together with its consequences.

Usually, unless we state explicitly otherwise, the functions u entering all the inequalities are complex-valued.

## Weighted $$L^p$$-Hardy type inequalities with boundary terms and their consequences

In this section we derive several versions of the $$L^p$$ weighted Hardy inequalities.

### Weighted $$L^p$$-Cafferelli-Kohn-Nirenberg type inequalities with boundary terms

We first present the following weighted $$L^p$$-Cafferelli–Kohn–Nirenberg type inequalities with boundary terms on the stratified Lie group $$\mathbb {G}$$ and then discuss their consequences. The proof of Theorem 2.1 is analogous to the proof of Davies and Hinz , but is now carried out in the case of the stratified Lie group $$\mathbb {G}$$. The boundary terms also give new addition to the Euclidean results in . The classical Caffarelli–Kohn–Nirenberg inequalities in the Euclidean setting were obtained in .

Let $$\mathbb {G}$$ be a stratified group with N being the dimension of the first stratum, and let V be a real-valued function in $$L_{loc}^1(\Omega )$$ with partial derivatives of order up to 2 in $$L_{loc}^1(\Omega )$$, and such that $${\mathcal {L}} V$$ is of one sign. Then we have:

### Theorem 2.1

Let $$\Omega$$ be an admissible domain in the stratified group $${\mathbb {G}}$$, and let V be a real-valued function such that $${\mathcal {L}} V<0$$ holds a.e. in $$\Omega$$. Then for any complex-valued $$u \in C^2(\Omega )\cap C^1({\overline{\Omega }})$$, and all $$1<p<\infty$$, we have the inequality

\begin{aligned} \left\| |{\mathcal {L}} V|^{\frac{1}{p}} u\right\| _{L^p(\Omega )}^p \le p \left\| \frac{|\nabla _{\mathbb {G}} V|}{|{\mathcal {L}} V|^{\frac{p-1}{p}}}|\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )} \left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^{p-1} -\int _{\partial \Omega } |u|^p\langle {\widetilde{\nabla }} V,dx\rangle . \end{aligned}
(2.1)

Note that if u vanishes on the boundary $$\partial \Omega$$, then (2.1) extends the Davies and Hinz result  to the weighted $$L^p$$-Hardy type inequality on stratified groups:

\begin{aligned} \left\| |{\mathcal {L}} V|^{\frac{1}{p}} u\right\| _{L^p(\Omega )} \le p \left\| \frac{|\nabla _{\mathbb {G}} V|}{|{\mathcal {L}} V|^{\frac{p-1}{p}}}|\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )}, \quad 1<p<\infty . \end{aligned}
(2.2)

### Proof of Theorem 2.1

Let $$\upsilon _{\epsilon }:=(|u|^2 + \epsilon ^2)^{\frac{1}{2}}-\epsilon$$. Then $$\upsilon _{\epsilon }^p \in C^2({\Omega }) \cap C^1({{\overline{\Omega }}})$$ and using Green’s first formula (1.7) and the fact that $${\mathcal {L}} V < 0$$ we get

\begin{aligned} \int _{\Omega }|{\mathcal {L}} V|\upsilon _{\epsilon }^p dx&= - \int _{\Omega } {\mathcal {L}} V \upsilon _{\epsilon }^p dx \\&= \int _{\Omega } ({\widetilde{\nabla }}V) \upsilon _{\epsilon }^p dx - \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle \\&= \int _{\Omega } \nabla _{\mathbb {G}} V \cdot \nabla _{\mathbb {G}} \upsilon _{\epsilon }^p dx - \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle \\&\le \int _{\Omega } |\nabla _{\mathbb {G}} V| |\nabla _{\mathbb {G}} \upsilon _{\epsilon }^p| dx - \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle \\&= p \int _{\Omega } \left( \frac{|\nabla _{\mathbb {G}} V|}{|{\mathcal {L}} V|^{\frac{p-1}{p}}}\right) |{\mathcal {L}} V|^{\frac{p-1}{p}} \upsilon _{\epsilon }^{p-1}|\nabla _{\mathbb {G}} \upsilon _{\epsilon }|dx- \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle , \end{aligned}

where $$({\widetilde{\nabla }}u)v = \nabla _{\mathbb {G}} u\cdot \nabla _{\mathbb {G}} v$$. We have

\begin{aligned} \nabla _{\mathbb {G}} \upsilon _{\epsilon } = (|u|^2 +\epsilon ^2)^{-\frac{1}{2}} |u|\nabla _{\mathbb {G}} |u|, \end{aligned}

since $$0 \le \upsilon _{\epsilon } \le |u|$$. Thus,

\begin{aligned} \upsilon _{\epsilon }^{p-1} |\nabla _{\mathbb {G}} \upsilon _{\epsilon }| \le |u|^{p-1}|\nabla _{\mathbb {G}}|u||. \end{aligned}

On the other hand, let $$u(x)=R(x)+iI(x)$$, where R(x) and I(x) denote the real and imaginary parts of u. We can restrict to the set where $$u\ne 0$$. Then we have

\begin{aligned} (\nabla _{\mathbb {G}} |u|)(x) = \frac{1}{|u|} (R(x)\nabla _{\mathbb {G}} R(x) + I(x)\nabla _{\mathbb {G}} I(x)) \quad \text {if} \quad u \ne 0. \end{aligned}
(2.3)

Since

\begin{aligned} \left| \frac{1}{|u|} (R\nabla _{\mathbb {G}} R+I \nabla _{\mathbb {G}} I)\right| ^2 \le |\nabla _{\mathbb {G}} R|^2 + |\nabla _{\mathbb {G}} I|^2, \end{aligned}
(2.4)

we get that $$|\nabla _{\mathbb {G}} |u||\le |\nabla _{\mathbb {G}} u|$$ a.e. in $$\Omega$$. Therefore,

\begin{aligned} \int _{\Omega }|{\mathcal {L}} V|\upsilon _{\epsilon }^p dx\le & {} p \int _{\Omega } \left( \frac{|\nabla _{\mathbb {G}} V|}{|{\mathcal {L}} V|^{\frac{p-1}{p}}} |\nabla _{\mathbb {G}} u|\right) |{\mathcal {L}} V|^{\frac{p-1}{p}}|u|^{p-1} dx- \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle \\\le & {} p \left( \int _{\Omega } \left( \frac{|\nabla _{\mathbb {G}} V|^p}{|{\mathcal {L}} V|^{(p-1)}} |\nabla _{\mathbb {G}} u|^p\right) dx \right) ^{\frac{1}{p}} \left( \int _{\Omega }|{\mathcal {L}} V||u|^p dx\right) ^{\frac{p-1}{p}}\\&- \int _{\partial \Omega }\upsilon _{\epsilon }^p \langle {\widetilde{\nabla }} V,dx\rangle , \end{aligned}

where we have used Hölder’s inequality in the last line. Thus, when $$\epsilon \rightarrow 0$$, we obtain (2.1). $$\square$$

### Consequences of theorem 2.1

As consequences of Theorem 2.1, we can derive the horizontal $$L^p$$-Caffarelli–Kohn–Nirenberg type inequality with the boundary term on the stratified group $$\mathbb {G}$$ which also gives another proof of $$L^p$$-Hardy type inequality, and also yet another proof of the Badiale-Tarantello conjecture  (for another proof see e.g.  and references therein).

### Corollary 2.2

Let $$\Omega$$ be an admissible domain in a stratified group $$\mathbb {G}$$ with $$N\ge 3$$ being dimension of the first stratum, and let $$\alpha , \beta \in {\mathbb {R}}$$. Then for all $$u \in C^2(\Omega \backslash \{x'=0\})\cap C^1({\overline{\Omega }} \backslash \{x'=0\})$$, and any $$1<p<\infty$$, we have

\begin{aligned}&\frac{|N-\gamma |}{p} \left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| ^p_{L^p(\Omega )} \le \left\| \frac{\nabla _{\mathbb {G}} u}{|x'|^{\alpha }} \right\| _{L^p(\Omega )} \left\| \frac{u}{|x'|^{\frac{\beta }{p-1}}} \right\| ^{p-1}_{L^p(\Omega )} \nonumber \\&\quad -\frac{1}{p} \int _{\partial \Omega } |u|^p\langle {\widetilde{\nabla }} |x'|^{2-\gamma },dx\rangle , \end{aligned}
(2.5)

for $$2<\gamma <N$$ with $$\gamma = \alpha +\beta +1,$$ and where $$|\cdot |$$ is the Euclidean norm on $${\mathbb {R}}^{N}$$. In particular, if u vanishes on the boundary $$\partial \Omega$$, we have

\begin{aligned} \frac{|N-\gamma |}{p} \left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| ^p_{L^p(\Omega )} \le \left\| \frac{\nabla _{\mathbb {G}} u}{|x'|^{\alpha }} \right\| _{L^p(\Omega )} \left\| \frac{u}{|x'|^{\frac{\beta }{p-1}}} \right\| ^{p-1}_{L^p(\Omega )}. \end{aligned}
(2.6)

### Proof of Corollary 2.2

To obtain (2.5) from (2.1) , we take $$V= |x'|^{2-\gamma }$$. Then

\begin{aligned} |\nabla _{\mathbb {G}} V| = |2-\gamma ||x'|^{1-\gamma },\qquad |{\mathcal {L}} V| =|(2-\gamma )(N-\gamma )| |x'|^{-\gamma }, \end{aligned}

and observe that $${\mathcal {L}} V = (2-\gamma )(N-\gamma ) |x'|^{-\gamma }<0.$$ To use (2.1) we calculate

\begin{aligned} \left\| |{\mathcal {L}} V|^{\frac{1}{p}} u\right\| _{L^p(\Omega )}^p= & {} |(2-\gamma )(N-\gamma )|\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}} \right\| _{L^p(\Omega )}^p,\\ \left\| \frac{|\nabla _{\mathbb {G}} V|}{|{\mathcal {L}} V|^{\frac{p-1}{p}}} \nabla _{\mathbb {G}} u \right\| _{L^p(\Omega )}= & {} \frac{|2-\gamma |}{|(2-\gamma )(N-\gamma )|^{\frac{p-1}{p}}} \left\| \frac{|\nabla _{\mathbb {G}} u|}{|x'|^{\frac{\gamma -p}{p}}} \right\| _{L^p(\Omega )},\\ \left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^{p-1}= & {} |(2-\gamma )(N-\gamma )|^{\frac{p-1}{p}}\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| _{L^p(\Omega )}^{p-1}. \end{aligned}

Thus, (2.1) implies

\begin{aligned} \frac{|N-\gamma |}{p}\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}} \right\| _{L^p(\Omega )}^p \le \left\| \frac{\nabla _{\mathbb {G}} u}{|x'|^{\frac{\gamma -p}{p}}} \right\| _{L^p(\Omega )}\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| _{L^p(\Omega )}^{p-1} - \frac{1}{p}\int _{\partial \Omega } |u|^p\langle {\widetilde{\nabla }} |x'|^{2-\gamma },dx\rangle . \end{aligned}

If we denote $$\alpha = \frac{\gamma -p}{p}$$ and $$\frac{\beta }{p-1}=\frac{\gamma }{p}$$, we get (2.5). $$\square$$

#### Badiale–Tarantello conjecture

Theorem 2.1 also gives a new proof of the generalised Badiale-Tarantello conjecture  (see, also ) on the optimal constant in Hardy inequalities in $${\mathbb {R}}^n$$ with weights taken with respect to a subspace.

### Proposition 2.3

Let $$x=(x',x'') \in {\mathbb {R}}^N \times {\mathbb {R}}^{n-N}$$, $$1\le N \le n$$, $$2<\gamma <N$$ and $$\alpha , \beta \in {\mathbb {R}}$$. Then for any $$u \in C_0^{\infty }({\mathbb {R}}^n \backslash \{x'=0\} )$$ and all $$1<p<\infty$$, we have

\begin{aligned} \frac{|N-\gamma |}{p} \left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| ^p_{L^p({\mathbb {R}}^n)} \le \left\| \frac{\nabla u}{|x'|^{\alpha }} \right\| _{L^p({\mathbb {R}}^n)} \left\| \frac{u}{|x'|^{\frac{\beta }{p-1}}} \right\| ^{p-1}_{L^p({\mathbb {R}}^n)}, \end{aligned}
(2.7)

where $$\gamma = \alpha +\beta +1$$ and $$|x'|$$ is the Euclidean norm $${\mathbb {R}}^N$$. If $$\gamma \ne N$$ then the constant $$\frac{|N-\gamma |}{p}$$ is sharp.

The proof of Proposition 2.3 is similar to Corollary 2.2, so we sketch it only very briefly.

### Proof of Proposition 2.3

Let us take $$V= |x'|^{2-\gamma }$$. We observe that $$\Delta V = (2-\gamma )(N-\gamma ) |x'|^{-\gamma }<0,$$ as well as $$|\nabla V| = |2-\gamma ||x'|^{(1-\gamma )}$$ and $$|\Delta V| = |(2-\gamma )(N-\gamma )| |x'|^{-\gamma }$$. Then (2.1) with

\begin{aligned} \left\| |\Delta V|^{\frac{1}{p}} u\right\| _{L^p({\mathbb {R}}^n)}^p= & {} |(2-\gamma )(N-\gamma )|\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}} \right\| _{L^p({\mathbb {R}}^n)}^p,\\ \left\| \frac{|\nabla V|}{|\Delta V|^{\frac{p-1}{p}}} \nabla u \right\| _{L^p({\mathbb {R}}^n)}= & {} \frac{|2-\gamma |}{|(2-\gamma )(N-\gamma )|^{\frac{p-1}{p}}} \left\| \frac{\nabla u}{|x'|^{\frac{\gamma -p}{p}}} \right\| _{L^p({\mathbb {R}}^n)},\\ \left\| |\Delta V|^{\frac{1}{p}} u \right\| _{L^p({\mathbb {R}}^n)}^{p-1}= & {} |(2-\gamma )(N-\gamma )|^{\frac{p-1}{p}}\left\| \frac{u}{|x'|^{\frac{\gamma }{p}}}\right\| _{L^p({\mathbb {R}}^n)}^{p-1}, \end{aligned}

and denoting $$\alpha = \frac{\gamma -p}{p}$$ and $$\frac{\beta }{p-1}=\frac{\gamma }{p}$$, implies (2.7). $$\square$$

In particular, if we take $$\beta =(\alpha +1)(p-1)$$ and $$\gamma =p(\alpha +1)$$, then (2.7) implies

\begin{aligned} \frac{|N-p(\alpha +1) |}{p} \left\| \frac{u}{|x'|^{\alpha +1}}\right\| _{L^p({\mathbb {R}}^n)} \le \left\| \frac{\nabla u}{|x'|^{\alpha }} \right\| _{L^p({\mathbb {R}}^n)}, \end{aligned}
(2.8)

where $$1<p< \infty$$, for all $$u \in C_0^{\infty }({\mathbb {R}}^n \backslash \{x'=0\})$$, $$\alpha \in {\mathbb {R}}$$, with sharp constant. When $$\alpha =0$$, $$1<p<N$$ and $$2\le N\le n$$, the inequality (2.8) implies that

\begin{aligned} \left\| \frac{u}{|x'|} \right\| _{L^p({\mathbb {R}}^n)} \le \frac{p}{N-p} \left\| \nabla u \right\| _{L^p({\mathbb {R}}^n)}, \end{aligned}
(2.9)

which given another proof of the Badiale-Tarantello conjecture from [3, Remark 2.3].

#### The local Hardy type inequality on $$\mathbb {G}$$.

As another consequence of Theorem 2.1 we obtain the local Hardy type inequality with the boundary term, with d being the $${\mathcal {L}}$$-gauge as in (1.2).

### Corollary 2.4

Let $$\Omega \subset \mathbb {G}$$ with $$0 \notin \partial \Omega$$ be an admissible domain in a stratified group $$\mathbb {G}$$ of homogeneous dimension $$Q\ge 3.$$ Let $$0>\alpha > 2-Q$$. Let $$u \in C^{1}(\Omega \backslash \{0\})\cap C({\overline{\Omega }}\backslash \{0\})$$. Then we have

\begin{aligned} \frac{|Q+\alpha -2|}{p} \left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}} d|^{\frac{2}{p}} u \right\| _{L^p(\Omega )}\le & {} \left\| d^{\frac{p+\alpha -2}{p}} |\nabla _{\mathbb {G}}d|^{\frac{2-p}{p}} |\nabla _{\mathbb {G}}u| \right\| _{L^p(\Omega )} \nonumber \\&\quad -\,\frac{1}{p}\left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}} d|^{\frac{2}{p}} u\right\| ^{1-p}_{L^p(\Omega )} \nonumber \\&\quad \times \,\int _{\partial \Omega } d^{\alpha -1} |u|^p \langle {\widetilde{\nabla }} d,dx\rangle . \end{aligned}
(2.10)

This extends the local Hardy type inequality that was obtained in  for $$p=2$$:

\begin{aligned} \frac{|Q+\alpha -2|}{2} \left\| d^{\frac{\alpha -2}{2}} |\nabla _{\mathbb {G}} d| u \right\| _{L^2(\Omega )}\le & {} \left\| d^{\frac{\alpha }{2}} |\nabla _{\mathbb {G}}u| \right\| _{L^2(\Omega )} \nonumber \\&\quad -\,\frac{1}{2} \left\| d^{\frac{\alpha -2}{2}} |\nabla _{\mathbb {G}} d| u\right\| _{L^2(\Omega )}^{-1}\nonumber \\&\quad \times \,\int _{\partial \Omega } d^{\alpha -1} |u|^2 \langle {\widetilde{\nabla }} d,dx\rangle . \end{aligned}
(2.11)

### Proof of Corollary 2.4

First, we can multiply both sides of the inequality (2.1) by $$\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^{1-p}$$, so that we have

\begin{aligned} \left\| |{\mathcal {L}}V|^{\frac{1}{p}}u \right\| _{L^p(\Omega )} \le p \left\| \frac{|\nabla _{\mathbb {G}}V|}{|{\mathcal {L}}V|^{\frac{p-1}{p}}}|\nabla _{\mathbb {G}} u|\right\| _{L^p(\Omega )} - \left\| |{\mathcal {L}} V|^{\frac{1}{p}} u\right\| _{L^p(\Omega )}^{1-p} \int _{\partial \Omega }|u|^p \langle {\widetilde{\nabla }}V,dx\rangle .\nonumber \\ \end{aligned}
(2.12)

Now, let us take $$V=d^{\alpha }$$. We have

\begin{aligned} {\mathcal {L}} d^{\alpha } = \nabla _{\mathbb {G}}(\nabla _{\mathbb {G}} \varepsilon ^{\frac{\alpha }{2-Q}})= & {} \nabla _{\mathbb {G}}\left( \frac{\alpha }{2-Q}\varepsilon ^{\frac{\alpha +Q-2}{2-Q}}\nabla _{\mathbb {G}}\varepsilon \right) \\= & {} \frac{\alpha (\alpha +Q-2)}{(2-Q)^2} \varepsilon ^{\frac{\alpha -4+2Q}{2-Q}}|\nabla _{\mathbb {G}} \varepsilon |^2 + \frac{\alpha }{2-Q}\varepsilon ^{\frac{\alpha +Q-2}{2-Q}} {\mathcal {L}} \varepsilon . \end{aligned}

Since $$\varepsilon$$ is the fundamental solution of $${\mathcal {L}}$$, we have

\begin{aligned} {\mathcal {L}} d^{\alpha } = \frac{\alpha (\alpha +Q-2)}{(2-Q)^2} \varepsilon ^{\frac{\alpha -4+2Q}{2-Q}}|\nabla _{\mathbb {G}} \varepsilon |^2= \alpha (\alpha +Q-2)d^{\alpha -2}|\nabla _{\mathbb {G}}d|^2. \end{aligned}

We can observe that $${\mathcal {L}}d^{\alpha } <0$$, and also the identities

\begin{aligned}&\left\| |{\mathcal {L}}d^{\alpha }|^{\frac{1}{p}} u \right\| _{L^p(\Omega )} = \alpha ^{\frac{1}{p}} |Q+\alpha -2|^{\frac{1}{p}} \left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}}d|^{\frac{2}{p}} u \right\| _{L^p(\Omega )},\\&\left\| \frac{|\nabla _{\mathbb {G}}d^{\alpha }|}{|{\mathcal {L}}d^{\alpha }|^{\frac{p-1}{p}}} |\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )} = \alpha ^{\frac{1}{p}}|Q+\alpha -2|^{\frac{1-p}{p}} \left\| d^{\frac{\alpha -2+p}{p}} |\nabla _{\mathbb {G}}d|^{\frac{2-p}{p}} |\nabla _{\mathbb {G}}u| \right\| _{L^p(\Omega )},\\&\left\| |{\mathcal {L}}d^{\alpha }|^{\frac{1}{p}} u \right\| ^{1-p}_{L^p(\Omega )} \int _{\partial \Omega } |u|^p\langle {\widetilde{\nabla }} d^{\alpha },dx\rangle = \alpha ^{\frac{1}{p}} |Q+\alpha -2|^{\frac{1-p}{p}} \left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}}d|^{\frac{2}{p}} u \right\| ^{1-p}_{L^p(\Omega )}\\&\quad \int _{\partial \Omega } d^{\alpha -1}|u|^p\langle {\widetilde{\nabla }} d,dx\rangle . \end{aligned}

Using (2.12) we arrive at

\begin{aligned}&\frac{|Q+\alpha -2|}{p} \left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}} d|^{\frac{2}{p}} u \right\| _{L^p(\Omega )} \le \left\| d^{\frac{p+\alpha -2}{p}} |\nabla _{\mathbb {G}}d|^{\frac{2-p}{p}} |\nabla _{\mathbb {G}}u| \right\| _{L^p{\Omega }} \\&\quad -\frac{1}{p} \left\| d^{\frac{\alpha -2}{p}} |\nabla _{\mathbb {G}} d|^{\frac{2}{p}} u\right\| ^{1-p}_{L^p(\Omega )} \int _{\partial \Omega } d^{\alpha -1} |u|^p \langle {\widetilde{\nabla }} d,dx\rangle , \end{aligned}

which implies (2.10). $$\square$$

### Uncertainty type principles

The inequality (2.12) implies the following Heisenberg-Pauli-Weyl type uncertainty principle on stratified groups.

### Corollary 2.5

Let $$\Omega \subset \mathbb {G}$$ be admissible domain in a stratified group $$\mathbb {G}$$ and let $$V \in C^2(\Omega )$$ be real-valued. Then for any complex-valued function $$u \in C^2(\Omega )\cap C^1({\overline{\Omega }})$$ we have

\begin{aligned}&\left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| \frac{|\nabla _{\mathbb {G}}V|}{|{\mathcal {L}}V|^{\frac{p-1}{p}}} |\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )} \nonumber \\&\ge \frac{1}{p} \left\| u \right\| ^2_{L^p(\Omega )} + \frac{1}{p}\left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| ^{1-p}_{L^p(\Omega )}\int _{\partial \Omega } |u|^p \langle {\widetilde{\nabla }} V,dx\rangle .\nonumber \\ \end{aligned}
(2.13)

In particular, if u vanishes on the boundary $$\partial \Omega$$, then we have

\begin{aligned} \left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| \frac{|\nabla _{\mathbb {G}}V|}{|{\mathcal {L}}V|^{\frac{p-1}{p}}} |\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )} \ge \frac{1}{p} \left\| u \right\| ^2_{L^p(\Omega )}. \end{aligned}
(2.14)

### Proof of Corollary 2.5

By using the extended Hölder inequality and (2.12) we have

\begin{aligned}&\left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| \frac{|\nabla _{\mathbb {G}}V|}{|{\mathcal {L}}V|^{\frac{p-1}{p}}} |\nabla _{\mathbb {G}} u| \right\| _{L^p(\Omega )} \\&\quad \ge \frac{1}{p} \left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )} \left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )} \\&\qquad +\,\frac{1}{p} \left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| ^{1-p}_{L^p(\Omega )}\int _{\partial \Omega } |u|^p \langle {\widetilde{\nabla }} V,dx\rangle ,\\&\quad \ge \frac{1}{p} \left\| |u|^2 \right\| _{L^{\frac{p}{2}}(\Omega )} + \frac{1}{p}\left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| ^{1-p}_{L^p(\Omega )}\int _{\partial \Omega } |u|^p \langle {\widetilde{\nabla }} V,dx\rangle .\\&\quad = \frac{1}{p} \left\| u \right\| ^2_{L^p(\Omega )} + \frac{1}{p}\left\| |{\mathcal {L}}V|^{-\frac{1}{p}} u \right\| _{L^p(\Omega )}\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| ^{1-p}_{L^p(\Omega )}\int _{\partial \Omega } |u|^p \langle {\widetilde{\nabla }} V,dx\rangle , \end{aligned}

proving (2.13). $$\square$$

By setting $$V =|x'|^{\alpha }$$ in the inequality (2.14), we recover the Heisenberg–Pauli–Weyl type uncertainty principle on stratified groups as in  and :

\begin{aligned} \left( \int _{\Omega } |x'|^{2-\alpha } |u|^p dx \right) \left( \int _{\Omega } |x'|^{\alpha +p-2} |\nabla _{\mathbb {G}} u|^p dx \right) \ge \left( \frac{N+\alpha -2}{p} \right) ^p \left( \int _{\Omega } |u|^p dx \right) ^2. \end{aligned}

In the abelian case $$\mathbb {G}=({\mathbb {R}}^n,+)$$, taking $$N=n\ge 3$$, for $$\alpha =0$$ and $$p=2$$ this implies the classical Heisenberg–Pauli–Weyl uncertainty principle for all $$u \in C^{\infty }_0({\mathbb {R}}^n \backslash \{0\})$$:

\begin{aligned} \left( \int _{{\mathbb {R}}^n} |x|^2 |u(x)|^2 dx\right) \left( \int _{{\mathbb {R}}^n} |\nabla u(x)|^2 dx \right) \ge \left( \frac{n-2}{2} \right) ^2 \left( \int _{{\mathbb {R}}^n} |u(x)|^2 dx \right) ^2. \end{aligned}

By setting $$V =d^{\alpha }$$ in the inequality (2.14), we obtain another uncertainty type principle:

\begin{aligned}&\left( \int _{\Omega } \frac{|u|^p}{d^{\alpha -2}|\nabla _{\mathbb {G}}d|^2} dx \right) \left( \int _{\Omega } d^{\alpha +p-2}|\nabla _{\mathbb {G}}d|^{2-p}|\nabla _{\mathbb {G}}u|^p dx\right) \\&\quad \ge \left( \frac{Q+\alpha -2}{p} \right) ^p \left( \int _{\Omega } |u|^p dx\right) ^2; \end{aligned}

taking $$p=2$$ and $$\alpha =0$$ this yields

\begin{aligned} \left( \int _{\Omega } \frac{d^{2}}{|\nabla _{\mathbb {G}}d|^2}|u|^2 dx \right) \left( \int _{\Omega } |\nabla _{\mathbb {G}}u|^2 dx\right) \ge \left( \frac{Q-2}{2} \right) ^2 \left( \int _{\Omega } |u|^2 dx\right) ^2. \end{aligned}

## Weighted $$L^p$$-Rellich type inequalities

In this section we establish weighted Rellich inequalities with boundary terms. We consider first the $$L^2$$ and then the $$L^p$$ cases. The analogous $$L^2$$-Rellich inequality on $${\mathbb {R}}^n$$ was proved by Schmincke  (and generalised by Bennett ).

### Theorem 3.1

Let $$\Omega$$ be an admissible domain in a stratified group $$\mathbb {G}$$ with $$N \ge 2$$ being the dimension of the first stratum. If a real-valued function $$V\in C^2(\Omega )$$ satisfies $${\mathcal {L}} V(x) < 0$$ for all $$x \in \Omega$$, then for every $$\epsilon >0$$ we have

\begin{aligned} \left\| \frac{|V|}{|{\mathcal {L}} V|^{\frac{1}{2}}} {\mathcal {L}} u \right\| ^2_{L^2(\Omega )}&\ge 2\epsilon \left\| V^{\frac{1}{2}} |\nabla _{\mathbb {G}} u| \right\| ^2_{L^2(\Omega )} + \epsilon (1-\epsilon ) \left\| |{\mathcal {L}}V|^{\frac{1}{2}} u\right\| ^2_{L^2(\Omega )}\nonumber \\&\quad - \epsilon \int _{\partial \Omega } (|u|^2\langle {\widetilde{\nabla }}V,dx\rangle - V\langle {\widetilde{\nabla }} |u|^2,dx\rangle ), \end{aligned}
(3.1)

for all complex-valued functions $$u \in C^2(\Omega )\cap C^1({\overline{\Omega }})$$. In particular, if u vanishes on the boundary $$\partial \Omega$$, we have

\begin{aligned} \left\| \frac{|V|}{|{\mathcal {L}} V|^{\frac{1}{2}}} {\mathcal {L}} u \right\| ^2_{L^2(\Omega )} \ge 2\epsilon \left\| V^{\frac{1}{2}} |\nabla _{\mathbb {G}} u| \right\| ^2_{L^2(\Omega )} + \epsilon (1-\epsilon ) \left\| |{\mathcal {L}}V|^{\frac{1}{2}} u\right\| ^2_{L^2(\Omega )}. \end{aligned}

### Proof of Theorem 3.1

Using Green’s second identity (1.8) and that $${\mathcal {L}} V(x) <0$$ in $$\Omega$$, we obtain

\begin{aligned} \int _{\Omega } |{\mathcal {L}} V||u|^2dx&= -\int _{\Omega }V{\mathcal {L}}|u|^2 dx - \int _{\partial \Omega } (|u|^2\langle {\widetilde{\nabla }}V,dx\rangle - V\langle {\widetilde{\nabla }} |u|^2,dx\rangle ) \\&=-\,2\int _{\Omega }V\left( \mathrm{Re}({\overline{u}}{\mathcal {L}} u)+|\nabla _{\mathbb {G}} u|^2\right) dx \\&\quad -\,\int _{\partial \Omega } (|u|^2\langle {\widetilde{\nabla }}V,dx\rangle - V\langle {\widetilde{\nabla }} |u|^2,dx\rangle ). \end{aligned}

Using the Cauchy–Schwartz inequality we get

\begin{aligned} \int _{\Omega } |{\mathcal {L}} V||u|^2 dx&\le 2\left( \frac{1}{\epsilon } \int _{\Omega } \frac{|V|^2}{|{\mathcal {L}} V|}|{\mathcal {L}} u|^2 dx\right) ^{\frac{1}{2}} \left( \epsilon \int _{\Omega }|{\mathcal {L}} V||u|^2 dx\right) ^{\frac{1}{2}}\\&\quad -\,2 \int _{\Omega }V|\nabla _{\mathbb {G}} u|^2 dx - \int _{\partial \Omega } (|u|^2\langle {\widetilde{\nabla }}V,dx\rangle - V\langle {\widetilde{\nabla }} |u|^2,dx\rangle ) \\&\le \frac{1}{\epsilon } \int _{\Omega } \frac{|V|^2}{|{\mathcal {L}} V|} |{\mathcal {L}} u|^2 dx + \epsilon \int _{\Omega }|{\mathcal {L}} V||u|^2dx \\&\quad - 2\int _{\Omega } V |\nabla _{\mathbb {G}} u|^2 dx- \int _{\partial \Omega } (|u|^2\langle {\widetilde{\nabla }}V,dx\rangle - V\langle {\widetilde{\nabla }} |u|^2,dx\rangle ), \end{aligned}

yielding (3.1). $$\square$$

### Corollary 3.2

Let $$\mathbb {G}$$ be a stratified group with N being the dimension of the first stratum. If $$\alpha >-2$$ and $$N>\alpha +4$$ then for all $$u \in C_0^{\infty }(\mathbb {G}\backslash \{x'=0\})$$ we have

\begin{aligned} \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|{\mathcal {L}} u|^2}{|x'|^{\alpha }} dx \ge \frac{(N+\alpha )^2(N-\alpha -4)^2}{16} \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|u|^2}{|x'|^{\alpha +4}}dx. \end{aligned}
(3.2)

### Proof of Corollary 3.2

Let us take $$V(x)=|x'|^{-(\alpha +2)}$$ in Theorem 3.1, which can be applied since $$x'=0$$ is not in the support of u. Then we have

\begin{aligned} \nabla _{\mathbb {G}} V = -(\alpha +2)|x'|^{-\alpha -4}x',\qquad {\mathcal {L}} V = - (\alpha +2)(N-\alpha -4)|x'|^{-(\alpha +4)}. \end{aligned}

Let us set $$C_{N,\alpha }:=(\alpha +2)(N-\alpha -4)$$. Observing that

\begin{aligned} {\mathcal {L}}V = -C_{N,\alpha }|x'|^{-(\alpha +4)}<0 , \end{aligned}

for $$|x'| \ne 0$$, it follows from (3.1) that

\begin{aligned} \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|{\mathcal {L}} u|^2}{|x'|^{\alpha }} dx&\ge 2 C_{N,\alpha } \epsilon \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|\nabla _{\mathbb {G}} u|^2}{|x'|^{\alpha +2}}dx \nonumber \\&\quad +C_{N,\alpha }^2 \epsilon (1-\epsilon ) \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|u|^2}{|x'|^{\alpha +4}}dx. \end{aligned}
(3.3)

To obtain (3.2), let us apply the $$L^p$$-Hardy type inequality (2.2) by taking $$V(x)=|x'|^{\alpha +2}$$ for $$\alpha \in (-2,N-4)$$, so that

\begin{aligned} \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|\nabla _{\mathbb {G}} u|^2}{|x'|^{\alpha +2}} dx \ge \frac{(N-\alpha -4)^2}{4} \int _{\mathbb {G}\backslash \{x'=0\}} \frac{|u|^2}{|x'|^{\alpha +4}}dx, \end{aligned}

and then choosing $$\epsilon =(N+\alpha )/4(\alpha +2)$$ for (3.3), which is the choice of $$\epsilon$$ that gives the maximum right-hand side. $$\square$$

We can now formulate the $$L^p$$-version of weighted $$L^p$$-Rellich type inequalities.

### Theorem 3.3

Let $$\Omega$$ be an admissible domain in a stratified group $$\mathbb {G}$$. If $$0<V \in C(\Omega )$$, $${\mathcal {L}} V <0$$, and $${\mathcal {L}}(V^{\sigma })\le 0$$ on $$\Omega$$ for some $$\sigma >1$$, then for all $$u \in C_0^{\infty }(\Omega )$$ we have

\begin{aligned} \left\| |{\mathcal {L}} V|^{\frac{1}{p}}u \right\| _{L^p(\Omega )} \le \frac{p^2}{(p-1)\sigma +1} \left\| \frac{V}{|{\mathcal {L}}V|^{\frac{p-1}{p}}}{\mathcal {L}}u \right\| _{L^p(\Omega )}, \quad 1\le p < \infty . \end{aligned}
(3.4)

Theorem 3.3 will follow by Lemma 3.5, by putting $$C =\frac{(p-1)(\sigma -1)}{p}$$ in Lemma 3.4.

### Lemma 3.4

Let $$\Omega$$ an admissible domain in a stratified group $$\mathbb {G}$$. If $$V \ge 0$$, $${\mathcal {L}} V <0$$, and there exists a constant $$C\ge 0$$ such that

\begin{aligned} C \left\| |{\mathcal {L}} V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^p \le p(p-1) \left\| V^{\frac{1}{p}} |u|^{\frac{p-2}{p}}|\nabla _{\mathbb {G}} u|^{\frac{2}{p}}\right\| _{L^p(\Omega )}^p, \quad 1<p<\infty , \end{aligned}
(3.5)

for all $$u \in C_0^{\infty }(\Omega )$$, then we have

\begin{aligned} (1+C)\left\| |{\mathcal {L}} V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )} \le p \left\| \frac{V}{|{\mathcal {L}}V|^{\frac{p-1}{p}}} {\mathcal {L}} u \right\| _{L^p(\Omega )}, \end{aligned}
(3.6)

for all $$u \in C_0^{\infty }(\Omega )$$. If $$p=1$$ then the statement holds for $$C=0$$.

### Proof of Lemma 3.4

We can assume that u is real-valued by using the following identity (see [7, p. 176]):

\begin{aligned} \forall z \in {\mathbb {C}}: |z|^p= \left( \int _{-\pi }^{\pi } |\cos {\vartheta }|^p d\vartheta \right) ^{-1} \int _{\pi }^{-\pi } |\mathrm{Re}(z)\cos {\vartheta }+\mathrm{Im}(z)\sin {\vartheta }|^{p} d \vartheta , \end{aligned}

which can be proved by writing $$z=r(\cos {\phi }+i\sin {\phi })$$ and simplifying.

Let $$\epsilon >0$$ and set $$u_{\epsilon }:=(|u|^2+\epsilon ^2)^{p/2}-\epsilon ^p$$. Then $$0\le u_{\epsilon } \in C_0^{\infty }$$ and

\begin{aligned} \int _{\Omega }|{\mathcal {L}}V|u_{\epsilon } dx = - \int _{\Omega } ({\mathcal {L}}V)u_{\epsilon } dx = - \int _{\Omega }V {\mathcal {L}}u_{\epsilon } dx, \end{aligned}

where

\begin{aligned} {\mathcal {L}}u_{\epsilon }&= {\mathcal {L}} \left( (|u|^2+\epsilon ^2)^{\frac{p}{2}}-\epsilon ^p \right) = \nabla _{\mathbb {G}} \cdot (\nabla _{\mathbb {G}}((|u|^2+\epsilon ^2)^{\frac{p}{2}}-\epsilon ^p))\\&= \nabla _{\mathbb {G}}(p(|u|^2+\epsilon ^2)^{\frac{p-2}{2}} u \nabla _{\mathbb {G}}u)\\&= p(p-2)(|u|^2+\epsilon ^2)^{\frac{p-4}{2}}u^2|\nabla _{\mathbb {G}}u|^2 +\,p(|u|^2+\epsilon ^2)^{\frac{p-2}{2}}|\nabla _{\mathbb {G}}u|^2 \\&\quad +\, p(|u|^2+\epsilon ^2)^{\frac{p-2}{2}}u{\mathcal {L}}u. \end{aligned}

Then

\begin{aligned} \int _{\Omega }|{\mathcal {L}}V|u_{\epsilon } dx&= -\,\int _{\Omega }\left( p(p-2)u^2(u^2+\epsilon ^2)^{\frac{p-4}{2}}+p(u^2+\epsilon ^2)^{\frac{p-2}{2}}\right) V|\nabla _{\mathbb {G}}u|^2dx \\&\quad - p\int _{\Omega } V u(u^2+\epsilon ^2)^{\frac{p-2}{2}}{\mathcal {L}}udx. \end{aligned}

Hence

\begin{aligned}&\int _{\Omega } |{\mathcal {L}}V|u_{\epsilon } + \left( p(p-2)u^2(u^2+\epsilon ^2)^{\frac{p-4}{2}}+p(u^2+\epsilon ^2)^{\frac{p-2}{2}}\right) V|\nabla _{\mathbb {G}}u|^2 dx \\&\quad \le p\int _{\Omega } V|u|(u^2+\epsilon ^2)^{\frac{p-2}{2}}|{\mathcal {L}}u|dx. \end{aligned}

When $$\epsilon \rightarrow 0$$, the integrand on the right is bounded by $$V(\max |u|^2+1)^{(p-1)/2}\max |{\mathcal {L}}u|$$ and it is integrable because $$u \in C_0^{\infty }(\Omega )$$, and so the integral tends to $$\int _{\Omega }V|u|^{p-1}|{\mathcal {L}}u|dx$$ by the dominated convergence theorem. The integrand on the left is non-negative and tends to $$|{\mathcal {L}} V||u|^p + p(p-1)V|u|^{p-2}|\nabla _{\mathbb {G}}u|^2$$ pointwise, only for $$u \ne 0$$ when $$p<2$$, otherwise for any x. It then follows by Fatou’s lemma that

\begin{aligned} \left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^p + p(p-1)\left\| V^{\frac{1}{p}}|u|^{\frac{p-2}{p}}|\nabla _{\mathbb {G}}u|^{\frac{2}{p}} \right\| _{L^p(\Omega )}^p \le p \left\| V^{\frac{1}{p}} |u|^{\frac{p-1}{p}} |{\mathcal {L}}u|^{\frac{1}{p}}\right\| _{L^p(\Omega )}^p. \end{aligned}

By using (3.5), followed by the Hölder inequality, we obtain

\begin{aligned} (1+C)\left\| |{\mathcal {L}}V|^{\frac{1}{p}} u\right\| _{L^p(\Omega )}^p&\le p \left\| |{\mathcal {L}}V|^{(p-1)}V^{\frac{1}{p}}|u|^{\frac{p-1}{p}}|{\mathcal {L}}V|^{-(p-1)} |{\mathcal {L}} u|^{\frac{1}{p}} \right\| ^p \\&\le p \left\| |{\mathcal {L}}V|^{\frac{1}{p}} u \right\| _{L^p(\Omega )}^{p-1} \left\| \frac{|V|}{|{\mathcal {L}}V|^{\frac{p-1}{p}}} {\mathcal {L}} u \right\| _{L^p(\Omega )}. \end{aligned}

This implies (3.6). $$\square$$

### Lemma 3.5

Let $$\Omega$$ be an admissible domain in a stratified group $$\mathbb {G}$$. If $$0<V\in C(\Omega )$$, $${\mathcal {L}} V <0$$, and $${\mathcal {L}}V^{\sigma } \le 0$$ on $$\Omega$$ for some $$\sigma >1$$, then we have

\begin{aligned} (\sigma -1) \int _{\Omega } |{\mathcal {L}}V||u|^p dx \le p^2 \int _{\{x\in \Omega , u(x) \ne 0\}} V|u|^{p-2}|\nabla _{\mathbb {G}} u|^2 dx< \infty , \quad 1<p<\infty , \end{aligned}
(3.7)

for all $$u \in C_0^{\infty }(\Omega )$$.

### Proof of Lemma 3.5

We shall use that

\begin{aligned} 0 \ge {\mathcal {L}}(V^{\sigma }) = \sigma V^{\sigma -2}\left( (\sigma -1)|\nabla _{\mathbb {G}} V|^2+V{\mathcal {L}}V\right) , \end{aligned}
(3.8)

and hence

\begin{aligned} (\sigma -1)|\nabla _{\mathbb {G}}V|^2 \le V |{\mathcal {L}}V|. \end{aligned}

Now we use the inequality (2.2) for $$p=2$$ to get

\begin{aligned} (\sigma -1)\int _{\Omega } |{\mathcal {L}} V||u|^2 dx&\le 4(\sigma -1)\int _{\Omega } \frac{|\nabla _{\mathbb {G}} V|^2}{|{\mathcal {L}} V|} |\nabla _{\mathbb {G}}u|^2 dx \nonumber \\&\le 4 \int _{\Omega } V |\nabla _{\mathbb {G}}u|^2 dx = 4 \int _{\{x\in \Omega ; u(x)\ne 0, |\nabla _{\mathbb {G}}u|\ne 0 \}} V |\nabla _{\mathbb {G}} u|^2 dx, \end{aligned}
(3.9)

the last equality valid since $$|\{x \in \Omega ; u(x)=0, |\nabla _{\mathbb {G}} u| \ne 0 \}|=0$$. This proves Lemma 3.5 for $$p=2$$.

For $$p \ne 2$$, put $$v_{\epsilon }=(u^2 +\epsilon ^2)^{p/4}-\epsilon ^{p/2}$$, and let $$\epsilon \rightarrow 0$$. Since $$0\le v_{\epsilon }\le |u|^{\frac{p}{2}}$$, the left-hand side of (3.9), with u replaced by $$v_{\epsilon }$$, tends to $$(\sigma -1)\int _{\Omega }|{\mathcal {L}}V||u|^pdx$$ by the dominated convergence theorem. If $$u \ne 0$$, then

\begin{aligned} |\nabla _{\mathbb {G}}v_{\epsilon }|^2 V =\left| \frac{p}{2}u(u^2+\epsilon ^2)^{\frac{p-4}{4}}\nabla _{\mathbb {G}}u\right| ^2 V. \end{aligned}

For $$\epsilon \rightarrow 0$$ we obtain

\begin{aligned} |\nabla _{\mathbb {G}}u|^p V = \frac{p^2}{4} |u|^{p-2}|\nabla _{\mathbb {G}}u|^2 V. \end{aligned}

It follows as in the proof of Lemma 3.4, by using Fatou’s lemma, that the right-hand side of (3.9) tends to

\begin{aligned} p^2\int _{\{x\in \Omega ; u(x) \ne 0,|\nabla _{\mathbb {G}}u|\ne 0\}} V |u|^{p-2}|\nabla _{\mathbb {G}}u|^2dx, \end{aligned}

and this completes the proof. $$\square$$

### Corollary 3.6

Let $$\mathbb {G}$$ be a stratified group with N being the dimension of the first stratum. Then for any $$2<\alpha <N$$ and all $$u \in C_0^{\infty }(\mathbb {G}\backslash \{x'=0\})$$ we have the inequality

\begin{aligned} \int _{\mathbb {G}} \frac{|u|^p}{|x'|^{\alpha }} dx \le C^p_{(N,p,\alpha )} \int _{\mathbb {G}} \frac{|{\mathcal {L}}u|^p}{|x'|^{\alpha -2p}}dx, \end{aligned}
(3.10)

where

\begin{aligned} C_{(N,p,\alpha )} = \frac{p^2}{(N-\alpha )\left( (p-1)N+\alpha -2p\right) }. \end{aligned}
(3.11)

### Proof of Corollary 3.6

Let us choose $$V=|x'|^{-(\alpha -2)}$$ in Theorem 3.3, so that

\begin{aligned} {\mathcal {L}} V = -(\alpha -2)(N-\alpha )|x'|^{-\alpha }, \end{aligned}

and we note that when $$2<\alpha <N$$, we have $${\mathcal {L}} V < 0$$ for $$|x'|\ne 0$$. Now it follows from (3.4) that

\begin{aligned} (\alpha -2)^p(N-\alpha )^p \int _{\mathbb {G}} \frac{|u|^p}{|x'|^{\alpha }} dx \le \frac{p^{2p}}{[(p-1)\sigma +1]^p} \int _{\mathbb {G}} \frac{|{\mathcal {L}}u|^p}{|x'|^{\alpha -2p}}dx. \end{aligned}
(3.12)

By taking $$\sigma =(N-2)/(\alpha -2)$$, we arrive at

\begin{aligned} \int _{\mathbb {G}} \frac{|u|^p}{|x'|^{\alpha }} dx \le \frac{p^{2p}}{(N-\alpha )^p\left( (p-1)N+\alpha -2p\right) ^p} \int _{\mathbb {G}} \frac{|{\mathcal {L}}u|^p}{|x'|^{\alpha -2p}}dx, \end{aligned}

which proves (3.10)–(3.11). $$\square$$

### Corollary 3.7

Let $$\mathbb {G}$$ be a stratified Lie group and let $$d=\varepsilon ^{\frac{1}{2-Q}}$$, where $$\varepsilon$$ is the fundamental solution of the sub-Laplacian $${\mathcal {L}}$$. Assume that $$Q\ge 3$$, $$\alpha <2$$, and $$Q+\alpha -4>0$$. Then for all $$u \in C_0^{\infty }(\mathbb {G}\backslash \{0\})$$ we have

\begin{aligned} \frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16} \int _{\mathbb {G}} d^{\alpha -4} |\nabla _{\mathbb {G}} d|^2 |u|^2 dx \le \int _{\mathbb {G}} \frac{d^{\alpha }}{|\nabla _{\mathbb {G}}d|^2} |{\mathcal {L}} u|^2 dx. \end{aligned}
(3.13)

The inequality (3.13) was obtained by Kombe , but now we get it as an immediate consequence of Theorem 3.3.

### Proof of Corollary 3.7

Let us choose $$V=d^{\alpha -2}$$ in Theorem 3.3. Then

\begin{aligned} {\mathcal {L}} V = (\alpha -2)(Q+\alpha -4)d^{\alpha -4}|\nabla _{\mathbb {G}}d|^2. \end{aligned}

Note that for $$Q+\alpha -4>0$$ and $$\alpha <2$$, we have $${\mathcal {L}}V<0$$ for all $$x\ne 0.$$ If $$p=2$$ then from (3.4) it follows that

\begin{aligned} (\alpha -2)^2(Q+\alpha -4)^2\int _{\mathbb {G}} d^{\alpha -4}|\nabla _{\mathbb {G}} d|^2 |u|^2 dx \le \frac{16}{(\sigma +1)^2}\int _{\mathbb {G}} \frac{d^{\alpha }}{|\nabla _{\mathbb {G}}d|^2} |{\mathcal {L}}u|^2 dx. \end{aligned}

By taking $$\sigma =(Q-2\alpha +2)/(\alpha -2)$$ we get

\begin{aligned} \frac{(Q+\alpha -4)^2(Q-\alpha )^2}{16} \int _{\mathbb {G}} d^{\alpha -4} |\nabla _{\mathbb {G}} d|^2 |u|^2 dx \le \int _{\mathbb {G}} \frac{d^{\alpha }}{|\nabla _{\mathbb {G}}d|^2} |{\mathcal {L}} u|^2 dx, \end{aligned}

proving inequality (3.13). $$\square$$

### Remark 3.8

In the abelian case, when $$\mathbb {G}\equiv ({\mathbb {R}}^n,+)$$ with $$d =|x|$$ being the Euclidean norm, and $$\alpha =0$$ in inequality (3.13), we recover the classical Rellich inequality .

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Correspondence to Michael Ruzhansky.

## Additional information

The first author was supported by the EPSRC Grant EP/R003025/1 and by the Leverhulme Research Grant RPG-2017-151. No new data was collected or generated during the course of this research.

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