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 [10], or to the recent books [4] or [9] 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 [10] 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. [9] 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 [19] for general Carnot groups and in [20] for more abstract settings (without the group structure), for another formulation see also [11].

The analogue of Green’s first formula for the sub-Laplacian was given in [19] 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 [15] and [13]. For the general importance of such inequalities we can refer to [2]. Some boundary terms have appeared in [24]. For these inequalities in the setting of general homogeneous groups we refer to [22].

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 [21] 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 [8], 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 [8]. The classical Caffarelli–Kohn–Nirenberg inequalities in the Euclidean setting were obtained in [6].

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 [8] 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 [3] (for another proof see e.g. [18] and references therein).

Horizontal $L^p$-Caffarelli–Kohn–Nirenberg inequalities with the boundary term

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 [3] (see, also [18]) 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 [19] 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 [17] and [20]:

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

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 [23] (and generalised by Bennett [5]).

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 [14], 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 [16].