Strong density results in trace spaces of maps between manifolds (Q1014798)

Let \({\mathcal X}\) and \({\mathcal Y}\) be smooth, connected, compact Riemannian manifolds without boundary that are isometrically embedded in \(\mathbb R^ l\) and \(\mathbb R^ N\), respectively, and equipped with metrics induced by the Euclidean norms of the ambient spaces. \(W^{1/p}({\mathcal X})=W^{1-1/p,p}({\mathcal X})\) is called the fractional Sobolev space if it is the Banach space of \(L^P\)-functions \(u:{\mathcal X}\to\mathbb R\) which have finite \(W^{1-1/p,p}\)-seminorm \(|u|_{1/p,{\mathcal X}}\) endowed with the norm \(\|u\|_{1/p,{\mathcal X}}\). If \(W^{1,p}({\mathcal X},\mathbb R^N)\) is the space of vector-valued maps \(u=(u^1,u^,\dots,u^N)\) such that \(u^j\in W^{1/p}({\mathcal X})\), then \(W^{1/p}({\mathcal X},{\mathcal Y})\) is the class of maps \(u\in W^{1/p}({\mathcal X},\mathbb R^N)\) such that \(u(x)\in{\mathcal Y}\) for \(x\in{\mathcal X}\). In this paper, the author deals with with strong density results of the class \(W^{1/p}({\mathcal X},{\mathcal Y})\) of smooth maps between \({\mathcal X}\) and \({\mathcal Y}\) in fractional Sobolev spaces \(W^{1/p}({\mathcal X})\) given by the traces of Sobolev maps in \(W^{1,p}\).