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\(L^{p}\) Hardy type inequality in the half-space on the H-type group
Journal of Inequalities and Applications volume 2016, Article number: 129 (2016)
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).
1 Introduction
In recent years a lot of authors studied the Hardy inequalities (see [1–5]). 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 [7–11]). 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
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
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
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
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
We write \(\nabla_{\mathbf{H}}=(X_{1},\ldots,X_{n},Y_{1},\ldots,Y_{n})\) and
We define the Kohn Laplacian \(\Delta_{\mathbf{H}}\) by
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
For \(u(z,t)\in C_{0}^{\infty}(\mathbf{H})\), we have
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
where
From this, it follows that, for all \(u(z,t)\in C_{0}^{\infty}(\mathbf{H})\),
□
For \(\varepsilon>0\), the Green’s function on the half-space on the H-type group is given by
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
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
Theorem 2.3
Let \(u\in C_{0}^{\infty}(\mathbf{H}^{+})\) and \(1< p<\infty\), then
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
Theorem 2.5
Let \(u\in C_{0}^{\infty}(\mathbf{H})\) and \(1< p<\infty\). Then
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
and
Using L’Hospital’s rule, we also have
and
Because
from this we can see that
By a direct calculation, we get
Thus we have
and
Consequently, we have
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
Proof
Applying Hölder’s inequality, we can deduce that
On the other hand, by partial integration we get
Thus we obtain
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
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)
For \(\Omega=\mathbf{H}^{+}\), we have \(\varrho=t_{m}\). So we get
Proof of Theorem 2.2
We know that
and
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
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
where \(C(p,l,s)=(\frac{lp-s-1}{p})^{p}\).
Proof
Similar to Lemma 4.1, we have
We know that \(c_{n,m}d(z,t)^{-Q+2}\) is a fundamental solution for \(\Delta_{\mathbf{H}}\). So we have
□
For \(l=1\), \(s=0\), and \(w=1\), we have
Set \(F=0\), then we get
Proof of Theorem 2.4
It is obvious that
So we have
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
which implies that
So we have equation (6). □
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Acknowledgements
The work for this paper is supported by the National Natural Science Foundation of China (No. 11271091, 11471040).
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He, J., Yin, M. \(L^{p}\) Hardy type inequality in the half-space on the H-type group. J Inequal Appl 2016, 129 (2016). https://doi.org/10.1186/s13660-016-1070-8
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DOI: https://doi.org/10.1186/s13660-016-1070-8