Hardy type and Rellich type inequalities on the Heisenberg group (Q2750859)

Hardy's inequality in \(\mathbb{R}^1\) is the following:NEWLINENEWLINENEWLINEIf \(p>1\), \(f(x) \geq 0,\) and \(F(x)=\int_0^x f(t) dt\), then NEWLINE\[NEWLINE \int_0^{\infty} \left(\frac{F}{x} \right) dx < \left(\frac{p}{p-1} \right)^p \int_0^{\infty}f^p dx, NEWLINE\]NEWLINE unless \(f \equiv 0\). The constant is the best possible.NEWLINENEWLINENEWLINEThe authors show that a similar inequality holds on the nilpotent Lie group, in particular, on the Heisenberg group \(H_n\). Let \(d=d(x,y,t)=(|z|^4+t^2)^{1/4}\), \(|z|^2=x^2+y^2\), \(z=(x,y) \in \mathbb{R}^n \times \mathbb{R}^n\), \(t \in \mathbb{R}\), \(Q\) the homogeneous dimension, \(\nabla_{H_n}\Phi=(X_1 \Phi,\cdots, X_n \Phi,Y_1 \Phi, \cdots,Y_n \Phi)\), \(\{X_j,y_j \}_{j=1}^{n}\) the basis of left invariant vector fields on \(H_n\), \(x_j=\frac{\partial}{\partial x_j}+2y_j \frac{\partial}{\partial t}\), and \(Y_j=\frac{\partial}{\partial y_j}-2x_j\frac{\partial}{\partial t}\). Let \(\Phi \in C_0^{\infty}(H_0 \setminus \{0 \})\), \(1 < p <Q\). Then it follows the Hardy type inequalities NEWLINE\[NEWLINE \int_{H_n}|\nabla_{H_n} \Phi|^p \geq \left(\frac{Q-p}{p} \right)^p \int_{H_n} \left( \frac{|z|}{d} \right)^p \frac{|\Phi|^p}{d^p} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \int_{H_n}|\nabla_{H_n} \Phi|^p \geq \left(\frac{Q-p}{p} \right)^p \int_{H_n} \left( \frac{|z|}{d} \right)^p \frac{|\Phi|^p}{(1+d)^p}. NEWLINE\]NEWLINE Furthermore they prove a Rellich type inequality for the unique continuation of the sub-Laplacian \(\triangle_{H_n}=\sum_{j=1}^{n}(X_j^2+Y_j^2).\)NEWLINENEWLINENEWLINETheorem. Let \(\Phi \in C_0^\infty(H_0\setminus\{0\})\), \(1<p<Q\). Then it follows the Rellich type inequality NEWLINE\[NEWLINE \int_{H_n}|\triangle_{H_n} \Phi|^p +C_0\int_{H_n} \frac{|z|^{2(p-2)}}{d^{4(p-1)}}|\Phi|^p \geq C_1 \left( \frac{|z|^{2p}}{d^{4p}} \right)^p,NEWLINE\]NEWLINE where \(C_0\) and \(C_1\) only depend on \(Q\) and \(p\).NEWLINENEWLINENEWLINEIn the proof they establish a Picone's type identity for the operators \(\nabla_{H_n}\) as well as the \(p\)-Laplacian \(\triangle_{H_n}\) and lead estimates for these operators.