Skip to content

Advertisement

  • Research
  • Open Access

\(L^{p}\) Hardy type inequality in the half-space on the H-type group

Journal of Inequalities and Applications20162016:129

https://doi.org/10.1186/s13660-016-1070-8

  • Received: 25 November 2015
  • Accepted: 19 April 2016
  • Published:

Abstract

In the current work we studied Hardy type and \(L^{p}\) Hardy type inequalities in the half-space on the H-type group, where the Hardy inequality in the upper half-space \(\mathbf{R}_{+}^{n}\) was proved by Tidblom in (J. Funct. Anal. 221:482-495, 2005).

Keywords

  • H-type group
  • Hardy type inequality
  • Green’s function

1 Introduction

In recent years a lot of authors studied the Hardy inequalities (see [15]). They are the extensions of the original inequality by Hardy [6]. The Heisenberg group, denoted by \(\mathbf{H}_{n}\), is also very popular in mathematics (see [711]). By \(\mathbf{H}_{n,+}=\{(z,t)\in\mathbf {H}_{n}| z\in C^{n}, t>0\}\) is denoted the half-space on the Heisenberg group. A Hardy type inequality on \(\mathbf{H}_{n,+}\) in [4] is stated as follows. For \(u\in C_{0}^{\infty}(\mathbf{H}_{n,+})\), we have
$$\begin{aligned} \int_{\mathbf{H}_{n,+}}\vert \nabla_{\mathbf{H}_{n}}u\vert ^{2} \,dz\,dt \geq \int_{\mathbf{H}_{n,+}}\frac{\vert z\vert ^{2}}{t^{2}}\vert u\vert ^{2}\,dz \,dt +\frac{(Q+2)(Q-2)}{4} \int_{\mathbf{H}_{n,+}}\rho ^{-4}\vert z\vert ^{2} \vert u\vert ^{2}\,dz\,dt, \end{aligned}$$
where \(\rho=(\vert z\vert ^{4}+t^{2})^{\frac{1}{4}}\) and \(Q=2n+2\) is the homogeneous dimension of the Heisenberg group. We know that the H-type group, denoted by \(\mathbf{H}=\{(z,t)\in\mathbf {H}| z\in C^{n}, t\in\mathbf{R}^{m}\}\), is the nilpotent Lie group introduced by Kaplan (see [12]). We also know that \(\mathbf {H}_{n}\) is a nilpotent Lie group with homogeneous dimension \(2n+2\). The homogeneous dimension of H is \(2n+2m\). Kaplan introduced the H-type group as a direct generalization of the Heisenberg group, which motivates us to study the H-type group.
In this paper we prove the Hardy type inequality in the half-space on the H-type group (see Theorem 2.1). The half-space on the H-type group is given by \(\mathbf{H}^{+}=\{(z,t)\in\mathbf{H}| t_{m}>0\}\). For \(u\in C_{0}^{\infty}(\mathbf{H}^{+})\), we have
$$\begin{aligned} \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{2} \,dz\,dt \geq& \frac{1}{16} \int_{\mathbf{H}^{+}}\frac {\vert z\vert ^{2}}{t_{m}^{2}}\vert u\vert ^{2}\,dz \,dt \\ &{}+\frac{(Q-2)(Q+2)}{4} \int_{\mathbf {H}^{+}}d(z,t)^{-4}\vert z\vert ^{2} \vert u\vert ^{2}\,dz\,dt \\ &{}-(Q+2) \int_{\mathbf{H}^{+}}d(z,t)^{-4}\sum_{k=1}^{m-1} \bigl\langle U^{(k)}z, U^{(m)}z \bigr\rangle \frac{t_{k}\vert u\vert ^{2}}{t_{m}} \,dz\,dt, \end{aligned}$$
where \(d(z,t)=(\vert z\vert ^{4}+16\vert t\vert ^{2})^{\frac{1}{4}}\) and \(Q=2n+2m\) is the homogeneous dimension of the H-type group.

In [1], the \(L^{p}\) Hardy inequalities in the upper half-space \(\mathbf{R}_{+}^{n}\) were studied. So we are also interested in the \(L^{p}\) Hardy type inequalities in the half-space on the H-type group.

In the remainder of this section we give a basic concept of H-type group and a useful theorem.

Let \((z,t), (z',t')\in\mathbf{H}\), \(U^{(j)}\) is a \(2n\times2n\) skew-symmetric orthogonal matrix and \(U^{(j)}\) satisfy \(U^{(i)}U^{(j)}+U^{(j)}U^{(i)}=0\), \(i, j=1,2,\ldots,m\) with \(i\neq j\). The group law is given by
$$\begin{aligned} (z,t) \bigl(z',t'\bigr)=\biggl(z+z',t+t'+ \frac{1}{2}\Im\bigl(zz'\bigr)\biggr), \end{aligned}$$
where \((\Im(zz'))_{j}=\langle z,U^{(j)}z'\rangle\), \(\langle z,U^{(j)}z'\rangle\) is the inner product of z and \(U^{(j)}z'\) on \(\mathbf{R}^{2n}\).
The left invariant vector fields are given by
$$\begin{aligned}& X_{j}=\frac{\partial}{\partial x_{j}}+\frac{1}{2}\sum _{k=1}^{m}\Biggl(\sum_{i=1}^{2n}s_{i}U_{i,j}^{(k)} \Biggr)\frac{\partial}{\partial t_{k}}, \quad j=1,2,\ldots,n, \\& Y_{j}=\frac{\partial}{\partial y_{j}}+\frac{1}{2}\sum _{k=1}^{m}\Biggl(\sum_{i=1}^{2n}s_{i}U_{i,j+n}^{(k)} \Biggr)\frac{\partial}{\partial t_{k}}, \quad j=1,2,\ldots,n, \\& T_{k}=\frac{\partial}{\partial t_{k}}, \quad k=1,2,\ldots,m, \end{aligned}$$
where \(s_{i}=x_{i}\) for \(i=1,2,\ldots,n\) and \(s_{i}=y_{i-n}\) for \(i=n+1,n+2,\ldots,2n\). The sub Laplacian \(\mathcal{L}\) is defined by
$$\begin{aligned} \mathcal{L}=-\sum_{j=1}^{n} \bigl(X_{j}^{2}+Y_{j}^{2}\bigr). \end{aligned}$$
We write \(\nabla_{\mathbf{H}}=(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n})\) and
$$\begin{aligned} \operatorname {div}_{\mathbf{H}}(f_{1},f_{2},\ldots,f_{2n})= \sum_{j=1}^{n}(X_{j}f_{j}+Y_{j}f_{j+n}). \end{aligned}$$
We define the Kohn Laplacian \(\Delta_{\mathbf{H}}\) by
$$\begin{aligned} \Delta_{\mathbf{H}}=\sum_{j=1}^{n} \bigl(X_{j}^{2}+Y_{j}^{2} \bigr). \end{aligned}$$
(1)
On the Heisenberg group, a fundamental solution for the sub Laplacian was studied in [13]. Similarly, we give a fundamental solution for \(\Delta_{\mathbf{H}}\) below. For \(0< r<\infty\) and \((z,t)\in\mathbf {H}\), we define \(\delta_{r}(z,t)=(rz,r^{2}t)\).

Theorem 1.1

A fundamental solution for \(\Delta_{\mathbf{H}}\) with source at 0 is given by \(c_{n,m}d(z,t)^{-Q+2}\), where
$$\begin{aligned} c_{n,m}^{-1}=4(n+m+1) (1-n-m) \int_{\mathbf{H}}\vert z\vert ^{2} \bigl(d(z,t)^{4}+1 \bigr)^{\frac{-n-m-3}{2}}\,dz\,dt. \end{aligned}$$
For \(u(z,t)\in C_{0}^{\infty}(\mathbf{H})\), we have
$$\bigl\langle \Delta_{\mathbf{H}}u(z,t), c_{n,m}d(z,t)^{-Q+2} \bigr\rangle _{L^{2}(\mathbf{H})}=u(0,0). $$

Proof

For \(\varepsilon>0\), let \(d_{\varepsilon}(z,t)=(d(z,t)^{4}+\varepsilon ^{4})^{\frac{1}{4}}\), similar to [13], by equation (1) and a direct calculation, we have
$$\begin{aligned} \Delta_{\mathbf{H}}d_{\varepsilon}(z,t)^{-Q+2}=\varepsilon^{-Q} \phi \bigl(\delta_{\frac{1}{\varepsilon}}(z,t)\bigr), \end{aligned}$$
(2)
where
$$\begin{aligned} \phi(z,t)=4(n+m+1) (1-n-m)\vert z\vert ^{2} \bigl(d(z,t)^{4}+1 \bigr)^{\frac {-n-m-3}{2}}. \end{aligned}$$
From this, it follows that, for all \(u(z,t)\in C_{0}^{\infty}(\mathbf{H})\),
$$\begin{aligned} \bigl\langle \Delta_{\mathbf{H}}u(z,t), c_{n,m}d(z,t)^{-Q+2} \bigr\rangle _{L^{2}(\mathbf{H})} =&\lim_{\varepsilon\rightarrow0}\bigl\langle \Delta_{\mathbf{H}}u(z,t), c_{n,m}d_{\varepsilon}(z,t)^{-Q+2} \bigr\rangle _{L^{2}(\mathbf{H})} \\ =&\lim_{\varepsilon\rightarrow0}\bigl\langle u(z,t), c_{n,m}\Delta _{\mathbf{H}}d_{\varepsilon}(z,t)^{-Q+2}\bigr\rangle _{L^{2}(\mathbf {H})} \\ =&u(0,0). \end{aligned}$$
 □
For \(\varepsilon>0\), the Green’s function on the half-space on the H-type group is given by
$$\begin{aligned} G(z,t,\varepsilon) =&\frac{1}{ (\vert z\vert ^{4}+16\sum_{j=1}^{m-1}t_{j}^{2}+16(t_{m}-\varepsilon)^{2} )^{\frac {Q-2}{4}}} -\frac{1}{ (\vert z\vert ^{4}+16\sum_{j=1}^{m-1}t_{j}^{2}+16(t_{m}+\varepsilon)^{2} )^{\frac {Q-2}{4}}}. \end{aligned}$$

2 Result

We give the main results of this paper in this section.

Theorem 2.1

For \(u\in C_{0}^{\infty}(\mathbf{H}^{+})\), we have
$$\begin{aligned} \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{2} \,dz\,dt \geq& \frac{1}{16} \int_{\mathbf{H}^{+}}\frac {\vert z\vert ^{2}}{t_{m}^{2}}\vert u\vert ^{2}\,dz \,dt \\ &{}+\frac{(Q-2)(Q+2)}{4} \int_{\mathbf {H}^{+}}d(z,t)^{-4}\vert z\vert ^{2} \vert u\vert ^{2}\,dz\,dt \\ &{}-(Q+2) \int_{\mathbf{H}^{+}}d(z,t)^{-4}\sum_{k=1}^{m-1} \bigl\langle U^{(k)}z, U^{(m)}z \bigr\rangle \frac{t_{k}\vert u\vert ^{2}}{t_{m}} \,dz\,dt. \end{aligned}$$

The theorems below show us the \(L^{p}\) Hardy type inequalities in the half-space on the H-type group.

Theorem 2.2

Let \(u\in C_{0}^{\infty}(\mathbf{H}^{+})\) and \(1< p<\infty\), then
$$\begin{aligned} \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac {p-1}{p}\biggr)^{p}\frac{p}{4} \int_{\mathbf{H}^{+}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{t_{m}^{p}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p}\frac{p-1}{2^{\frac{p}{p-1}}} \int_{\mathbf {H}^{+}}\frac{1}{t_{m}^{p}}\vert z\vert ^{\frac{p}{p-1}} \vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(3)

Theorem 2.3

Let \(u\in C_{0}^{\infty}(\mathbf{H}^{+})\) and \(1< p<\infty\), then
$$\begin{aligned} \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac {p-1}{p}\biggr)^{p}\frac{p}{4} \int_{\mathbf{H}^{+}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{t_{m}^{p}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p}\frac{p-1}{2^{\frac{p}{p-1}}} \int_{\mathbf {H}^{+}}\frac{1}{t_{m}^{p}}\bigl\vert 1-t_{m}^{p-1} \bigr\vert ^{\frac{p}{p-1}}\vert z\vert ^{\frac {p}{p-1}}\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(4)

We also study the \(L^{p}\) Hardy type inequalities in the H-type group.

Theorem 2.4

Let \(u\in C_{0}^{\infty}(\mathbf{H})\) and \(1< p<\infty\), then
$$\begin{aligned} \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac{p-1}{p}\biggr)^{p}p(Q-2)^{2} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{d^{(-Q+2)p+2Q}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p}(p-1) (Q-2)^{\frac{p}{p-1}} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert z\vert ^{\frac{p}{p-1}}}{d^{(-Q+2)p+\frac{p}{p-1}Q}}\,dz\,dt. \end{aligned}$$
(5)

Theorem 2.5

Let \(u\in C_{0}^{\infty}(\mathbf{H})\) and \(1< p<\infty\). Then
$$\begin{aligned} & \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \\ &\quad \geq\biggl(\frac{p-1}{p}\biggr)^{p-1}c_{n,m}^{-1} \bigl\vert u(0)\bigr\vert ^{p}+ \biggl(\frac {p-1}{p} \biggr)^{p}p(Q-2)^{2} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{d^{(-Q+2)p+2Q}}\,dz\,dt \\ &\qquad {}-\biggl(\frac{p-1}{p}\biggr)^{p}(p-1) (Q-2)^{\frac{p}{p-1}} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert 1-d^{(-Q+2)(p-1)}\vert ^{\frac{p}{p-1}}\vert z\vert ^{\frac {p}{p-1}}}{d^{(-Q+2)p+\frac{p}{p-1}Q}}\,dz\,dt. \end{aligned}$$
(6)

3 Hardy type inequality

This section is to show the Hardy type inequality in \(\mathbf{H}^{+}\).

Proof of Theorem 2.1

Let \(v(z,t)=G(z,t,\varepsilon)^{-\frac{1}{2}}u(z,t)\). Write \(t^{\varepsilon }=(0,\ldots,0,\varepsilon)\). We know that \(G(0,t^{\varepsilon },\varepsilon)=\infty\), so we have \(v(0,t^{\varepsilon})=0\) and \(u(z,t)=G(z,t,\varepsilon)^{\frac{1}{2}}v(z,t)\). Then we obtain
$$\begin{aligned} \nabla_{\mathbf{H}}u=\biggl(\frac{1}{2}\frac{\nabla_{\mathbf{H}}G}{G}+ \frac {\nabla_{\mathbf{H}}v}{v}\biggr)u \end{aligned}$$
and
$$\begin{aligned} & \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{2} \,dz\,dt \\ &\quad =\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt+ \int_{\mathbf{H}^{+}}\frac{\langle\nabla _{\mathbf{H}}G, \nabla_{\mathbf{H}}v\rangle}{Gv} \vert u\vert ^{2}\,dz \,dt+ \int _{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}v\vert ^{2}}{v^{2}}\vert u\vert ^{2}\,dz \,dt \\ &\quad =\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt+ \int_{\mathbf{H}^{+}}v\langle\nabla _{\mathbf{H}}G, \nabla_{\mathbf{H}}v \rangle \,dz\,dt+ \int_{\mathbf {H}^{+}}\vert \nabla_{\mathbf{H}}v\vert ^{2}G \,dz\,dt \\ &\quad =\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt+\frac{1}{2} \int_{\mathbf{H}^{+}}\bigl\langle \nabla_{\mathbf{H}}G, \nabla_{\mathbf{H}}v^{2}\bigr\rangle \,dz\,dt+ \int_{\mathbf {H}^{+}}\vert \nabla_{\mathbf{H}}v\vert ^{2}G \,dz\,dt \\ &\quad =\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt+\frac {1}{2}c_{n,m}^{-1}v^{2} \bigl(0,t^{\varepsilon}\bigr)+ \int_{\mathbf{H}^{+}}\vert \nabla _{\mathbf{H}}v\vert ^{2}G \,dz\,dt \\ &\quad =\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt+ \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf {H}}v\vert ^{2}G \,dz\,dt \\ &\quad \geq\frac{1}{4} \int_{\mathbf{H}^{+}}\frac{\vert \nabla_{\mathbf {H}}G\vert ^{2}}{G^{2}}\vert u\vert ^{2}\,dz \,dt. \end{aligned}$$
Using L’Hospital’s rule, we also have
$$\begin{aligned} \lim_{\varepsilon\rightarrow0^{+}}\frac{G(z,t,\varepsilon)}{\varepsilon }=16(Q-2)t_{m}d(z,t)^{-Q-2} \end{aligned}$$
and
$$\begin{aligned} \lim_{\varepsilon\rightarrow0^{+}}\biggl\vert \frac{\nabla_{\mathbf {H}}G(z,t,\varepsilon)}{\varepsilon}\biggr\vert ^{2} =&\bigl(16(Q-2)\bigr)^{2} \bigl(t_{m}^{2} \bigl\vert \nabla_{\mathbf{H}}d(z,t)^{-Q-2}\bigr\vert ^{2} \\ &{}+2d(z,t)^{-Q-2}t_{m}\bigl\langle \nabla_{\mathbf{H}}d(z,t)^{-Q-2}, \nabla _{\mathbf{H}}t_{m} \bigr\rangle \\ &{}+\bigl(d(z,t)^{-Q-2}\bigr)^{2}\vert \nabla_{\mathbf{H}}t_{m} \vert ^{2} \bigr). \end{aligned}$$
Because
$$\begin{aligned} \nabla_{\mathbf{H}}t_{m}=\Biggl(\frac{1}{2}\sum _{i=1}^{2n}s_{i}U_{i,1}^{(m)}, \ldots,\frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,n}^{(m)}, \frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,1+n}^{(m)}, \ldots,\frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,2n}^{(m)} \Biggr), \end{aligned}$$
from this we can see that
$$\begin{aligned} \vert \nabla_{\mathbf{H}}t_{m}\vert =& \Biggl(\Biggl( \frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,1}^{(m)} \Biggr)^{2}+\cdots +\Biggl(\frac{1}{2}\sum _{i=1}^{2n}s_{i}U_{i,n}^{(m)} \Biggr)^{2} \\ &{}+\Biggl(\frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,1+n}^{(m)} \Biggr)^{2}+\cdots+\Biggl(\frac {1}{2}\sum _{i=1}^{2n}s_{i}U_{i,2n}^{(m)} \Biggr)^{2} \Biggr)^{\frac {1}{2}} \\ =&\frac{1}{2}\vert z\vert . \end{aligned}$$
By a direct calculation, we get
$$\begin{aligned} \bigl\vert \nabla_{\mathbf{H}}d(z,t)\bigr\vert ^{2}= \frac{\vert z\vert ^{2}}{d(z,t)^{2}}. \end{aligned}$$
Thus we have
$$\begin{aligned}& \bigl\vert \nabla_{\mathbf {H}}d(z,t)^{-Q-2}\bigr\vert ^{2}=(Q+2)^{2}d(z,t)^{-2Q-8}\vert z\vert ^{2}, \\& \vert \nabla_{\mathbf{H}}t_{m}\vert ^{2}= \frac{1}{4}\vert z\vert ^{2} , \end{aligned}$$
(7)
and
$$\begin{aligned} \bigl\langle \nabla_{\mathbf{H}}d(z,t)^{-Q-2},\nabla_{\mathbf{H}}t_{m} \bigr\rangle =2(-Q-2)d(z,t)^{-Q-6}\Biggl(\sum _{k=1}^{m}\bigl\langle U^{(k)}z, U^{(m)}z \bigr\rangle t_{k}\Biggr). \end{aligned}$$
Consequently, we have
$$\begin{aligned} \lim_{\varepsilon\rightarrow0^{+}}\biggl\vert \frac{\nabla_{\mathbf {H}}G(z,t,\varepsilon)}{\varepsilon }\biggr\vert ^{2} =&\bigl(16(Q-2)\bigr)^{2}\Biggl((Q+2)^{2}d(z,t)^{-2Q-8} \vert z\vert ^{2}t_{m}^{2} \\ &{}+4(-Q-2) d(z,t)^{-2Q-8}\bigl\langle U^{(m)}z, U^{(m)}z \bigr\rangle t_{m}^{2} +\frac{1}{4}d(z,t)^{-2Q-4}\vert z\vert ^{2} \\ &{}+4(-Q-2)d(z,t)^{-2Q-8}\sum_{k=1}^{m-1} \bigl\langle U^{(k)}z, U^{(m)}z \bigr\rangle t_{m}t_{k}\Biggr). \end{aligned}$$
This finishes the proof of the theorem. □

4 \(L^{p}\) Hardy type inequality

In this section, we are going to consider the \(L^{p}\) Hardy type inequalities in \(\mathbf{H}^{+}\) and H, respectively. Let Ω be a domain in H. We write \(\varrho(z,t)=\operatorname {dist}((z,t),\partial\Omega)\). Similar to [1], we have the lemma below.

Lemma 4.1

Let \(u\in C_{0}^{\infty}(\Omega)\), \(l\in\{1,2,3,\ldots\}\), \(1< p<\infty \), \(s\in(-\infty, lp-1)\), \(F_{j}\in C^{1}(\Omega)\), \(j=1,2,\ldots,2n\), \(F=(F_{1},F_{2},\ldots,F_{2n})\) and \(w\in C^{1}(\Omega)\) be a nonnegative weight function. We write \(C(p,l,s)=(\frac {lp-s-1}{p})^{p}\), then we have
$$\begin{aligned} \int_{\Omega}\frac{\vert \nabla_{\mathbf{H}}u\vert ^{p}w}{\varrho ^{(l-1)p-s}}\,dz\,dt \geq& C(p,l,s) \int_{\Omega}\frac{p\vert u\vert ^{p}\vert \nabla _{\mathbf{H}}\varrho \vert ^{2}w}{\varrho^{lp-s}}\,dz\,dt \\ &{}-C(p,l,s) \int_{\Omega}\frac{p\vert u\vert ^{p}\Delta_{\mathbf{H}}\varrho w}{(lp-s-1)\varrho^{lp-s-1}}\,dz\,dt \\ &{}+C(p,l,s) \int_{\Omega}\frac{p\operatorname {div}_{\mathbf{H}}F\vert u\vert ^{p} w}{lp-s-1}\,dz\,dt \\ &{}-C(p,l,s) \int_{\Omega}\frac{p-1}{\varrho^{lp-s}}\bigl\vert \nabla_{\mathbf {H}} \varrho-\varrho^{lp-s-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \\ &{}+\biggl(\frac{lp-s-1}{p}\biggr)^{p-1} \int_{\Omega}\nabla_{\mathbf{H}}w\biggl(F-\frac {\nabla_{\mathbf{H}}\varrho}{\varrho^{lp-s-1}} \biggr)\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(8)

Proof

Applying Hölder’s inequality, we can deduce that
$$\begin{aligned} &p^{p} \int_{\Omega}\frac{\vert \nabla_{\mathbf{H}}u\vert ^{p}w}{\varrho ^{(l-1)p-s}}\,dz\,dt \biggl( \int_{\Omega}\biggl\vert \frac{\nabla_{\mathbf{H}}\varrho }{\varrho^{l(p-1)+\frac{s}{p}-s}}-\varrho^{l-1-\frac{s}{p}}F \biggr\vert ^{\frac {p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \biggr)^{p-1} \\ &\quad \geq p^{p}\biggl\vert \int_{\Omega}\biggl(\frac{\nabla_{\mathbf{H}}\varrho w}{\varrho ^{lp-s-1}}-Fw\biggr) \bigl(\operatorname {sign}(u) \vert u\vert ^{p-1}\bigr)\nabla_{\mathbf {H}}u\,dz\,dt \biggr\vert ^{p}. \end{aligned}$$
On the other hand, by partial integration we get
$$\begin{aligned} &p^{p}\biggl\vert \int_{\Omega}\biggl(\frac{\nabla_{\mathbf{H}}\varrho w}{\varrho ^{lp-s-1}}-Fw\biggr) \bigl(\operatorname {sign}(u) \vert u\vert ^{p-1}\bigr)\nabla_{\mathbf {H}}u\,dz\,dt \biggr\vert ^{p} \\ &\quad =\biggl\vert \int_{\Omega} \biggl(\biggl(\frac{(lp-s-1)\vert \nabla_{\mathbf{H}}\varrho \vert ^{2}}{\varrho^{lp-s}}-\frac{\Delta_{\mathbf{H}}\varrho}{\varrho ^{lp-s-1}}+ \operatorname {div}_{\mathbf{H}}F\biggr)w\\ &\qquad {}+\nabla_{\mathbf{H}}w\biggl(F-\frac {\nabla_{\mathbf{H}}\varrho}{\varrho^{lp-s-1}} \biggr) \biggr)\vert u\vert ^{p}\,dz\,dt\biggr\vert ^{p}. \end{aligned}$$
Thus we obtain
$$\begin{aligned} &p^{p} \int_{\Omega}\frac{\vert \nabla_{\mathbf{H}}u\vert ^{p}w}{\varrho ^{(l-1)p-s}}\,dz\,dt \\ &\quad \geq\biggl\vert \int_{\Omega} \biggl(\biggl(\frac{(lp-s-1)\vert \nabla_{\mathbf{H}}\varrho \vert ^{2}}{\varrho^{lp-s}}-\frac{\Delta_{\mathbf{H}}\varrho}{\varrho ^{lp-s-1}}+ \operatorname {div}_{\mathbf{H}}F\biggr)w\\ &\qquad {}+\nabla_{\mathbf{H}}w\biggl(F-\frac {\nabla_{\mathbf{H}}\varrho}{\varrho^{lp-s-1}} \biggr) \biggr)\vert u\vert ^{p}\,dz\,dt\biggr\vert ^{p} \\ &\qquad {}\times\biggl( \int_{\Omega}\biggl\vert \frac{\nabla_{\mathbf{H}}\varrho}{\varrho ^{l(p-1)+\frac{s}{p}-s}}-\varrho^{l-1-\frac{s}{p}}F \biggr\vert ^{\frac {p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \biggr)^{-p+1}. \end{aligned}$$
It is clear that \(\frac{\vert a\vert ^{p}}{b^{p-1}}\geq pa-(p-1)b\) for \(b>0\). Then we have equation (8). □
For \(F=0\), we have
$$\begin{aligned} \int_{\Omega}\frac{\vert \nabla_{\mathbf{H}}u\vert ^{p}w}{\varrho ^{(l-1)p-s}}\,dz\,dt \geq& C(p,l,s) \int_{\Omega}\frac{p\vert u\vert ^{p}\vert \nabla _{\mathbf{H}}\varrho \vert ^{2}w}{\varrho^{lp-s}}\,dz\,dt \\ &{}-C(p,l,s) \int_{\Omega}\frac{p\vert u\vert ^{p}\Delta_{\mathbf{H}}\varrho w}{(lp-s-1)\varrho^{lp-s-1}}\,dz\,dt \\ &{}-C(p,l,s) \int_{\Omega}\frac{p-1}{\varrho^{lp-s}}\vert \nabla_{\mathbf {H}}\varrho \vert ^{\frac{p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \\ &{}-\biggl(\frac{lp-s-1}{p}\biggr)^{p-1} \int_{\Omega}\frac{\nabla_{\mathbf {H}}w\nabla_{\mathbf{H}}\varrho}{\varrho^{lp-s-1}}\vert u\vert ^{p}\,dz \,dt. \end{aligned}$$
Now, let us discuss the \(L^{p}\) Hardy type inequalities in \(\mathbf {H}^{+}\). Let \(l=1\), \(s=0\), and \(w=1\), we have by equation (8)
$$\begin{aligned} \int_{\Omega} \vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac {p-1}{p}\biggr)^{p} \int_{\Omega}\frac{p\vert u\vert ^{p}\vert \nabla_{\mathbf{H}}\varrho \vert ^{2}}{\varrho^{p}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\Omega}\frac{p\vert u\vert ^{p}\Delta_{\mathbf {H}}\varrho}{(p-1)\varrho^{p-1}}\,dz\,dt \\ &{}+\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\Omega}\frac{p\operatorname {div}_{\mathbf {H}}F\vert u\vert ^{p} }{p-1}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\Omega}\frac{p-1}{\varrho^{p}}\bigl\vert \nabla _{\mathbf{H}} \varrho-\varrho^{p-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(9)
For \(\Omega=\mathbf{H}^{+}\), we have \(\varrho=t_{m}\). So we get
$$\begin{aligned} \int_{\mathbf{H}^{+}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac {p-1}{p}\biggr)^{p} \int_{\mathbf{H}^{+}}\frac{p\vert u\vert ^{p}\vert \nabla_{\mathbf {H}}t_{m}\vert ^{2}}{t_{m}^{p}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}^{+}}\frac{p\vert u\vert ^{p}\Delta _{\mathbf{H}}t_{m} }{(p-1)t_{m}^{p-1}}\,dz\,dt \\ &{}+\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}^{+}}\frac{p\operatorname {div}_{\mathbf {H}}F\vert u\vert ^{p} }{p-1}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}^{+}}\frac{p-1}{t_{m}^{p}}\bigl\vert \nabla _{\mathbf{H}}t_{m}-t_{m}^{p-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(10)

Proof of Theorem 2.2

We know that
$$\begin{aligned} \vert \nabla_{\mathbf{H}}t_{m}\vert =\frac{1}{2}\vert z \vert \end{aligned}$$
and
$$\begin{aligned} \Delta_{\mathbf{H}}t_{m}=0. \end{aligned}$$
Set \(F=0\), using equation (10), then we obtain equation (3). □

Proof of Theorem 2.3

Set \(F=\nabla_{\mathbf{H}}t_{m}\). Since \(U^{(m)}\) is a \(2n\times2n\) skew-symmetric orthogonal matrix, we have
$$\begin{aligned} \operatorname {div}_{\mathbf{H}}F =&\sum_{j=1}^{n}X_{j} \frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,j}^{(m)}+ \sum_{j=1}^{n}Y_{j} \frac{1}{2}\sum_{i=1}^{2n}s_{i}U_{i,j+n}^{(m)} \\ =&\sum_{j=1}^{n}\frac{1}{2}U_{j,j}^{(m)}+ \sum_{j=1}^{n}\frac {1}{2}U_{j+n,j+n}^{(m)} \\ =&0. \end{aligned}$$
Using equation (10), we have equation (4). □

Now we are going to deal with the \(L^{p}\) Hardy type inequalities in H.

Lemma 4.2

Let \(u\in C_{0}^{\infty}(\mathbf{H})\), \(l\in\{1,2,3,\ldots\}\), \(1< p<\infty\), \(s\in(-\infty, lp-1)\), \(F_{j}\in C^{1}(\mathbf{H})\), \(j=1,2,\ldots,2n\), \(F=(F_{1},F_{2},\ldots,F_{2n})\) and \(w\in C^{1}(\mathbf{H})\) be a nonnegative weight function. Then we have
$$\begin{aligned} & \int_{\mathbf{H}}\frac{\vert \nabla_{\mathbf {H}}u\vert ^{p}w}{(d^{-Q+2})^{(l-1)p-s}}\,dz\,dt \\ &\quad \geq C(p,l,s) \int_{\mathbf{H}}\frac{p\vert u\vert ^{p}\vert \nabla_{\mathbf {H}}d^{-Q+2}\vert ^{2}w}{(d^{-Q+2})^{lp-s}}\,dz\,dt +C(p,l,s) \int_{\mathbf{H}}\frac{p\operatorname {div}_{\mathbf{H}}F\vert u\vert ^{p} w}{lp-s-1}\,dz\,dt \\ & \qquad {}-C(p,l,s) \int_{\mathbf{H}}\frac{p-1}{(d^{-Q+2})^{lp-s}}\bigl\vert \nabla _{\mathbf{H}}d^{-Q+2}-\bigl(d^{-Q+2}\bigr)^{lp-s-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \\ &\qquad {}+\biggl(\frac{lp-s-1}{p}\biggr)^{p-1} \int_{\mathbf{H}}\nabla_{\mathbf {H}}w\biggl(F-\frac{\nabla_{\mathbf{H}}d^{-Q+2}}{(d^{-Q+2})^{lp-s-1}} \biggr)\vert u\vert ^{p}\,dz\,dt, \end{aligned}$$
(11)
where \(C(p,l,s)=(\frac{lp-s-1}{p})^{p}\).

Proof

Similar to Lemma 4.1, we have
$$\begin{aligned} & \int_{\mathbf{H}}\frac{\vert \nabla_{\mathbf {H}}u\vert ^{p}w}{(d^{-Q+2})^{(l-1)p-s}}\,dz\,dt \\ &\quad \geq C(p,l,s) \int_{\mathbf{H}}\frac{p\vert u\vert ^{p}\vert \nabla_{\mathbf {H}}d^{-Q+2}\vert ^{2}w}{(d^{-Q+2})^{lp-s}}\,dz\,dt \\ & \qquad {}-C(p,l,s) \int_{\mathbf{H}}\frac{p\vert u\vert ^{p}\Delta_{\mathbf{H}}d^{-Q+2} w}{(lp-s-1)(d^{-Q+2})^{lp-s-1}}\,dz\,dt \\ & \qquad {}+C(p,l,s) \int_{\mathbf{H}}\frac{p\operatorname {div}_{\mathbf{H}}F\vert u\vert ^{p} w}{lp-s-1}\,dz\,dt \\ & \qquad {}-C(p,l,s) \int_{\mathbf{H}}\frac{p-1}{(d^{-Q+2})^{lp-s}}\bigl\vert \nabla _{\mathbf{H}}d^{-Q+2}-\bigl(d^{-Q+2}\bigr)^{lp-s-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u\vert ^{p} w\,dz\,dt \\ & \qquad {}+\biggl(\frac{lp-s-1}{p}\biggr)^{p-1} \int_{\mathbf{H}}\nabla_{\mathbf {H}}w\biggl(F-\frac{\nabla_{\mathbf{H}}d^{-Q+2}}{(d^{-Q+2})^{lp-s-1}} \biggr)\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(12)
We know that \(c_{n,m}d(z,t)^{-Q+2}\) is a fundamental solution for \(\Delta_{\mathbf{H}}\). So we have
$$\begin{aligned} \int_{\mathbf{H}}\frac{\vert u\vert ^{p}\Delta_{\mathbf{H}}d^{-Q+2} w}{(d^{-Q+2})^{lp-s-1}}\,dz\,dt=c_{n,m}^{-1} \bigl\vert u(0)\bigr\vert ^{p}w(0)d(0)^{(Q-2)(lp-s-1)}=0. \end{aligned}$$
 □
For \(l=1\), \(s=0\), and \(w=1\), we have
$$\begin{aligned} & \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \\ &\quad \geq \biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac{p\vert u\vert ^{p}\vert \nabla _{\mathbf{H}}d^{-Q+2}\vert ^{2}}{(d^{-Q+2})^{p}}\,dz\,dt \\ & \qquad {}+\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac{p\operatorname {div}_{\mathbf {H}}F\vert u\vert ^{p} }{p-1}\,dz\,dt \\ & \qquad {}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac {p-1}{(d^{-Q+2})^{p}}\bigl\vert \nabla_{\mathbf {H}}d^{-Q+2}- \bigl(d^{-Q+2}\bigr)^{p-1}F\bigr\vert ^{\frac{p}{p-1}}\vert u \vert ^{p}\,dz\,dt. \end{aligned}$$
(13)
Set \(F=0\), then we get
$$\begin{aligned} \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \geq& \biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac{p\vert u\vert ^{p}\vert \nabla _{\mathbf{H}}d^{-Q+2}\vert ^{2}}{(d^{-Q+2})^{p}}\,dz\,dt \\ &{}-\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac {p-1}{(d^{-Q+2})^{p}}\bigl\vert \nabla_{\mathbf{H}}d^{-Q+2} \bigr\vert ^{\frac {p}{p-1}}\vert u\vert ^{p}\,dz\,dt. \end{aligned}$$
(14)

Proof of Theorem 2.4

It is obvious that
$$\begin{aligned} \vert \nabla_{\mathbf{H}}d\vert ^{2}=\frac{\vert z\vert ^{2}}{d^{2}}. \end{aligned}$$
So we have
$$\begin{aligned} \bigl\vert \nabla_{\mathbf{H}}d^{-Q+2}\bigr\vert ^{2}=(Q-2)^{2}d^{2(-Q+1)}\frac {\vert z\vert ^{2}}{d^{2}}. \end{aligned}$$
(15)
From this together with (14), we get equation (5). □

Proof of Theorem 2.5

Let \(F=\nabla_{\mathbf{H}}d^{-Q+2}\). Then we have \(\operatorname {div}_{\mathbf{H}}F=\operatorname {div}_{\mathbf{H}}\nabla _{\mathbf{H}}d^{-Q+2}=\Delta_{\mathbf{H}}d^{-Q+2}\). From equations (13) and (15), it follows that
$$\begin{aligned} & \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \\ &\quad \geq \biggl(\frac{p-1}{p}\biggr)^{p}p(Q-2)^{2} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{d^{(-Q+2)p+2Q}}\,dz\,dt \\ & \qquad {}+\biggl(\frac{p-1}{p}\biggr)^{p} \int_{\mathbf{H}}\frac{p\Delta_{\mathbf {H}}d^{-Q+2}\vert u\vert ^{p} }{p-1}\,dz\,dt \\ &\qquad {}-\biggl(\frac{p-1}{p}\biggr)^{p}(p-1) (Q-2)^{\frac{p}{p-1}} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert 1-d^{(-Q+2)(p-1)}\vert ^{\frac{p}{p-1}}\vert z\vert ^{\frac {p}{p-1}}}{d^{(-Q+2)p+\frac{p}{p-1}Q}}\,dz\,dt, \end{aligned}$$
which implies that
$$\begin{aligned} & \int_{\mathbf{H}}\vert \nabla_{\mathbf{H}}u\vert ^{p} \,dz\,dt \\ &\quad \geq\biggl(\frac{p-1}{p}\biggr)^{p-1}c_{n,m}^{-1} \bigl\vert u(0)\bigr\vert ^{p}+ \biggl(\frac {p-1}{p} \biggr)^{p}p(Q-2)^{2} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert z\vert ^{2}}{d^{(-Q+2)p+2Q}}\,dz\,dt \\ & \qquad {}-\biggl(\frac{p-1}{p}\biggr)^{p}(p-1) (Q-2)^{\frac{p}{p-1}} \int_{\mathbf{H}}\frac {\vert u\vert ^{p}\vert 1-d^{(-Q+2)(p-1)}\vert ^{\frac{p}{p-1}}\vert z\vert ^{\frac {p}{p-1}}}{d^{(-Q+2)p+\frac{p}{p-1}Q}}\,dz\,dt. \end{aligned}$$
So we have equation (6). □

Declarations

Acknowledgements

The work for this paper is supported by the National Natural Science Foundation of China (No. 11271091, 11471040).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, P.R. China

References

  1. Tidblm, J: A Hardy inequality in the half-space. J. Funct. Anal. 221, 482-495 (2005) MathSciNetView ArticleGoogle Scholar
  2. Barbatis, G, Filippas, S, Tertikas, A: A unified approach to improved \(L^{p}\) Hardy inequalities with best constants. Trans. Am. Math. Soc. 356, 2169-2196 (2004) MathSciNetView ArticleMATHGoogle Scholar
  3. Hoffmann-Ostenhof, M, Hoffmann-Ostenhof, T, Laptev, A: A geometrical version of Hardy’s inequality. J. Funct. Anal. 189, 539-548 (2002) MathSciNetView ArticleMATHGoogle Scholar
  4. Luan, JW, Yang, QH: A Hardy type inequality in the half-space on \(\mathbf{R}^{n}\) and Heisenberg group. J. Math. Anal. Appl. 347, 645-651 (2008) MathSciNetView ArticleMATHGoogle Scholar
  5. Tidblm, J: \(L^{p}\) Hardy inequalities in general domains. http://www.math.su.se/reports/2003/4/ (2003)
  6. Hardy, G: Note on a theorem of Hilbert. Math. Z. 6, 314-317 (1920) MathSciNetView ArticleMATHGoogle Scholar
  7. Folland, GB: Harmonic Analysis in the Phase Space. Princeton University Press, Princeton (1989) MATHGoogle Scholar
  8. Geller, D: Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 26, 205-254 (1980) MathSciNetView ArticleMATHGoogle Scholar
  9. Geller, D, Stein, EM: Singular convolution operators on the Heisenberg group. Bull. Am. Math. Soc. 6, 99-103 (1982) MathSciNetView ArticleMATHGoogle Scholar
  10. Stein, EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) MATHGoogle Scholar
  11. Thangavelu, S: Harmonic Analysis on the Heisenberg Group. Birkhäuser, Boston (1998) View ArticleMATHGoogle Scholar
  12. Kaplan, A: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147-153 (1980) MathSciNetView ArticleMATHGoogle Scholar
  13. Folland, GB: A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79, 373-376 (1973) MathSciNetView ArticleMATHGoogle Scholar

Copyright

© He and Yin 2016

Advertisement