Lifting of fractional Sobolev mappings to noncompact covering space

TL;DR

This paper studies the conditions under which fractional Sobolev maps between compact Riemannian manifolds can be lifted to noncompact covering spaces, providing characterizations and optimal estimates in certain parameter regimes.

Contribution

It characterizes liftings of fractional Sobolev maps to noncompact coverings and derives optimal nonlinear fractional Sobolev estimates when specific conditions are met.

Findings

Characterization of liftings for fractional Sobolev maps when sp > 1.

Optimal nonlinear fractional Sobolev estimates for sp ≥ dimension of the domain.

Nonlinear description of the sum of certain fractional Sobolev spaces.

Abstract

Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$, a Riemannian covering $\pi : \smash{\widetilde{\mathcal{N}}} \to \mathcal{N}$ by a noncompact covering space $\smash{\widetilde{\mathcal{N}}}$, $1 < p < \infty$ and $0 < s < 1$, the space of liftings of fractional Sobolev maps in $\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathcal{N})$ is characterized when $sp > 1$ and an optimal nonlinear fractional Sobolev estimate is obtained when moreover $sp \ge \dim \mathcal{M}$. A nonlinear characterization of the sum of spaces $\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathbb{R}) + \smash{\dot{W}^{1, sp}} (\mathcal{M}, \mathbb{R})$ is also provided.