The Hölder continuities of the maps in \(W^{1,p}(\Omega,\mathbb H^{n})\).
type: Item
instance of: scholarly article
title: The Hölder continuities of the maps in \(W^{1,p}(\Omega,\mathbb H^{n})\).
zbMATH Open document ID: 1042.46017
DOI: 10.1016/S0096-3003(03)00157-7
author: Gao Jia
author: Xiao-Ping Yang
author: Jinde Cao
published in: Applied Mathematics and Computation
publication date: 2004-03-14T00:00:00Z
review text: 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\…
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 22E30
zbMATH DE Number: 2056092
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Hölder continuity
zbMATH Keywords: Campanato type estimate
zbMATH Keywords: Morrey type estimate
zbMATH Keywords: Sobolev space
reviewed by: Koichi Saka
MaRDI profile type: Publication
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full work available at URL: s0096-3003(03)00157-7
OpenAlex ID: W1877263118
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description: scientific article; zbMATH DE number 2056092