The Hölder continuities of the maps in \(W^{1,p}(\Omega,\mathbb H^{n})\).
The aim of this paper is to establish the Hölder continuity of maps in the Sobolev space \(W^{1,p}(\Omega, {\mathbb H}^{n})\), which includes a Campanato type theorem and a Morrey type theorem. The Heisenberg group \({\mathbb H}^{n}\) is the Lie group whose underlying space is \({\mathbb C}^{n} \times {\mathbb R}\), endowed with the group law \[ (z, t)\cdot (z', t')=(z+z', t+t'+2 \text{ Im} z\cdot z'), \] for \(z, z' \in {\mathbb C}^{n}\) and \(t, t' \in {\mathbb R}\). There is a metric \(d(u,v)\) on \({\mathbb H}^{ n}\) defined by \[ d(u,v)=|vu^{-1}|= [((x_{v}-x_{u})^{2}+ (y_{v}- y_{u})^{2})^{2}+t_{v}-t_{u}+ 2(x_{v}y_{u}- x_{u}y_{v })^{2}]^{1/ 4} \] for \(u=(x_{u}, y_{u}, t_{u}), v=(x_{v}, y_{v}, t_{v}) \in {\mathbb H}^{n}\) and a homogeneous gauge defined as \[ |u|_{ {\mathbb H}^{n}} = [(x^{2}+y^{2})^{2} +t^{2}]^{1/4} \] for \(u=(x,y,t) \in {\mathbb H}^{n}\). Let \(1 < p< \infty\), \(\Omega\) be a bounded domain in \({\mathbb R}^{m}\). A function \(u: \Omega \to {\mathbb H}^{n}\) is in \(L^{p}(\Omega, {\mathbb H}^{n})\) if for some \(h_{0} \in \Omega\), \[ \int_{\Omega}d(u(h), u(h_{0}) )^{p}\,dh< \infty. \] A function \(u : \Omega \to {\mathbb H}^{n}\) is in the Sobolev space \(W^{1,p}(\Omega, {\mathbb H}^{n})\) if \(u \in L^{p}( \Omega, {\mathbb H}^{n})\) and \[ E_{p,\Omega}(u)= \sup_{f \in C_{C}(\Omega), 0 \leq f \leq 1} \limsup_{\varepsilon \to 0} \int_{\Omega}f(h)e_{u,\varepsilon}(h)\,dh < \infty \] where \[ e_{u,\varepsilon}(h_{0})=\int_{|h-h_{0}|=\varepsilon} \left(\frac {d(u(h_{0}), u(h))}{\varepsilon}\right)^{p} \frac{d \sigma_{ \varepsilon} (h)}{\varepsilon^{m-1}}. \] Let \(\alpha > 0\) be given. The map \(u : \Omega \to {\mathbb H}^{n}\) is said to be Hölder continuous of order \(\alpha\) with a positive constant \(K\) if \[ d(u(h_{1}), u(h_{2})) \leq K|h_{1}-h_{2}|^{\alpha} \] for all \(h_{1},h_{2} \in \Omega\), and is denoted as \(u \in C^{\alpha}(\Omega,{\mathbb H}^{n})\); when \(\alpha=1\), we call \(u\) Lipschitz continuous. Theorem A. Let \(\Omega\) be a bounded domain of \({\mathbb R}^{m}\). If \(u\) is Lipschitz continuous with a positive constant \(K\) on \(\Omega\), then \(u \in W^{1,p}(\Omega, {\mathbb H}^{n})\). Moreover, we have \(E_{p, \Omega}(u) \leq K^{p}|\Omega|\). The authors prove the following Campanato type theorem: Theorem B. Let \(\Omega\) be a bounded open domain of \({\mathbb R}^{m}\). Suppose that \(u \in L^{p}(\Omega, {\mathbb H}^{n})\) satisfies \[ \int_{B_{r}(h_{0})}d(u(h), u_{h_{0},r})^{p}\,dh \leq M^{p}r^{m+p\alpha} \] for any ball \(B_{r}(h_{0}) \subset \Omega\) and \(0<\alpha <1\), where \(u_{h_{0}, r}=(1/|B_{r}(h_{0})|)\int_{B_{r}(h_{0})}u(h)\,dh\). Then \(u \in C^{\alpha}(\Omega, {\mathbb H}^{n})\) for \(p \geq 2\). Furthermore, for \(p \geq 4\) and any \(\Omega' \subset \subset \Omega\), the following more precise estimate holds: \[ \sup_{\Omega'} |u|_{{\mathbb H}^{n}}+ \sup_{h_{1},h_{2} \in \Omega', h_{1} \neq h_{2}} \frac{d(u(h_{1}), u(h_{2}))} {|h_{1}-h_{2} |^{\alpha}} \leq C(m,p,\alpha, \Omega', \Omega) (M+\| u\|_{L^{p}(\Omega,{\mathbb H}^{n})}). \] The authors prove the following Morrey type theorem. Theorem C. Let \(\Omega\) be a bounded open domain of \({\mathbb R}^{m}\). Suppose \(u=(z,t) \in W^{1,p}(\Omega, {\mathbb H}^{n})\) satisfies \[ \int_{B_{r}(h_{0})}|\nabla z|^{p}\,dh \leq M^{p}r^{m-p+p\alpha} \] for any \(B_{r}(h_{0}) \subset \Omega\) and some \(\alpha \in \)\textrm{(0,1)}. Then \(u \in C^{\alpha}(\Omega, {\mathbb H}^{n})\) for \(p \geq 2\). Furthermore, for \(p \geq 4\) and any \(\Omega' \subset \subset \Omega\), the following more precise estimate holds: \[ \sup_{\Omega'}|u|_{{\mathbb H}^{n}}+\sup_{h_{1},h_{2} \in \Omega' , h_{1}\neq h_{2}}\frac{d(u(h_{1}),u(h_{2}))}{|h_{1}-h_{2}|^{\alpha}}\leq C(m, p, \alpha, \Omega', \Omega)(M+||u||_{L^{p}(\Omega, {\mathbb H}^{n})}). \]
scientific article; zbMATH DE number 2056092
scientific article