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Poincaré inequalities for the Heisenberg group target. (English) Zbl 1075.22004

Let \(\Omega\) be a bounded and connected Lipschitz domain in \({\mathbb R}^n\) (\(n \geq 2)\). Poincaré inequality in the classical case: Let \(p > 1\). Then there exists a constant \(C\) depending only on \(\Omega\), \(n\) and \(p\), such that for every function \(u \in W^{1,p}(\Omega, {\mathbb R}^n)\), the Sobolev space, we have \[ \int_{\Omega}|u(x)-\lambda_{u}|^p\,dx \leq C \int_{\Omega}|\nabla u|^p \,dx \] where \(\lambda_u = \frac{1}{|\Omega|}\int_{\Omega}u(x)dx\). In this paper some Poincaré type inequalities are obtained for the maps of the Heisenberg group target. Let \({\mathbb H}^{m} = {\mathbb R}^{2m+1}\) be a Heisenberg group with group law \[ (x_1, y_1, t_1)\cdot(x_2, y_2, t_2)= (x_1+x_2, y_1+y_2, t_1+t_2+2(y_2x_1-x_2y_1)) \] for every \(u_1=(x_1, y_1, t_1)\), \(u_2=(x_2, y_2, t_2) \in {\mathbb R}^{2m+1}\). For every \(u_1=(x_1, y_1, t_1)\), \(u_2=(x_2, y_2, t_2) \in {\mathbb H}^{m}\), the metric \(d(u_1, u_2)\) in the Heisenberg group \({\mathbb H}^m\) is defined as \[ d(u_1, u_2) =|u_2 u_1^{-1}|=[((x_2-x_1)^2+(y_2-y_1)^2)^2 + (t_2-t_1+2(x_2y_1-x_1y_2))^2]^{1/4}. \] Let \(2 \leq p <\infty\). A function \(u=(z, t): \Omega \rightarrow {\mathbb H}^m\) is in \(L^p(\Omega, {\mathbb H}^m)\) if for some \(h_0 \in \Omega\), one has \( \int_{\Omega}(d(u(h), u(h_0)))^p \,dh <\infty \). A function \(u=(z,t): \Omega \rightarrow {\mathbb H}^m\) is in the Sobolev space \(W^{1,p}(\Omega, {\mathbb H}^m)\) if \(u \in L^p(\Omega, {\mathbb H}^m)\) and \( \int_{\Omega}|\nabla z|^{p}(q) dq <\infty\).
Theorem (Poincaré type inequality). Let \(2 \leq p< \infty\). Then there exists a constant \(C\) depending only on \(\Omega\), \(n\), \(m\) and \(p\), such that for every function \(u=(z, t)= (x, y, t) \in W^{1,p}(\Omega, {\mathbb H}^m)\), \( \int_{\Omega}d(u(q), \lambda_{u})^p \,dq \leq C_{\Omega}\int_{\Omega} |\nabla z|^{p}(q) \,dq \). Here \(\lambda_{u}=(\lambda_x, \lambda_y, \lambda_t)\) and \(\lambda_f = \frac{1}{|\Omega|}\int_{\Omega}f(q) \,dq\).

MSC:

22E20 General properties and structure of other Lie groups
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D15 Inequalities for sums, series and integrals
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