Let $\pi :\widetilde{\mathcal{𝒩}}\to \mathcal{𝒩}$ be a Riemannian covering, with $\mathcal{𝒩}$, $\widetilde{\mathcal{𝒩}}$ smooth compact connected Riemannian manifolds. If $\mathcal{ℳ}$ is an $m$-dimensional compact simply connected Riemannian manifold, $0<s<1$ and $2\le sp<m$, we prove that every mapping $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ has a lifting in $W^{s,p}$; i.e., we have $u=\pi \circ ũ$ for some mapping $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result settles completely the question of the lifting in Sobolev spaces over covering spaces.

The proof relies on an a priori estimate of the oscillations of $W^{s,p}$ maps with $0<s<1$ and $sp>1$, in dimension $1$. Our argument also leads to the existence of a lifting when $0<s<1$ and $1<sp<2\le m$, provided there is no topological obstruction on $u$; i.e., $u=\pi \circ ũ$ holds in this range provided $u$ is in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$.

However, when $0<s<1$, $sp=1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in the absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ such that $u=\pi \circ ũ$ does not hold for any $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$.

让 $\pi ：\stackrel{～}{\mathcal{𝒩}}\to \mathcal{𝒩}$ 是黎曼覆盖，具有 $\mathcal{𝒩}$, $\stackrel{～}{\mathcal{𝒩}}$光滑紧凑连通黎曼流形。如果$\mathcal{ℳ}$ 是一个 $米$-维紧单连通黎曼流形， $0<秒<1$ 和 $2\le 秒磷<米$，我们证明每一个映射 $你\in {宽}^{秒,磷}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ 有一个提升 ${宽}^{秒,磷}$; 即，我们有$你=\pi \circ ㄧ$ 对于一些映射 $ㄧ\in {宽}^{秒,磷}\left(\mathcal{ℳ},\stackrel{～}{\mathcal{𝒩}}\right)$. 结合 Bourgain、Brezis 和 Mironescu 以及 Bethuel 和 Chiron 之前的贡献，我们的结果完全解决了 Sobolev 空间中覆盖空间的提升问题。

证明依赖于对振荡的先验估计 ${宽}^{秒,磷}$ 映射与 $0<秒<1$ 和 $秒磷>1$,在 维度上$1$. 我们的论点也导致了提升的存在，当$0<秒<1$ 和 $1<秒磷<2\le 米$，只要没有拓扑障碍 $你$; IE，$你=\pi \circ ㄧ$ 保持在这个范围内提供 $你$ 处于强封闭状态 $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$.

然而，当 $0<秒<1$, $秒磷=1$ 和 $米\ge 2$，我们表明，即使没有拓扑障碍，（分析）障碍仍然存在。更具体地说，我们构造了一些地图$你\in {宽}^{秒,磷}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ 在强大的关闭 $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ 以至于 $你=\pi \circ ㄧ$ 不适合任何 $ㄧ\in {宽}^{秒,磷}\left(\mathcal{ℳ},\stackrel{～}{\mathcal{𝒩}}\right)$.