Vol. 14, No. 6, 2021

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Lifting in compact covering spaces for fractional Sobolev mappings

Petru Mironescu and Jean Van Schaftingen

Vol. 14 (2021), No. 6, 1851–1871
Abstract

Let π : 𝒩˜ 𝒩 be a Riemannian covering, with 𝒩, 𝒩˜ smooth compact connected Riemannian manifolds. If is an m-dimensional compact simply connected Riemannian manifold, 0 < s < 1 and 2 sp < m, we prove that every mapping u Ws,p(,𝒩 ) has a lifting in Ws,p; i.e., we have u = π ũ for some mapping ũ Ws,p(, 𝒩˜). 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 Ws,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 m, provided there is no topological obstruction on u; i.e., u = π ũ holds in this range provided u is in the strong closure of C(,𝒩 ).

However, when 0 < s < 1, sp = 1 and m 2, we show that an (analytical) obstruction still arises, even in the absence of topological obstructions. More specifically, we construct some map u Ws,p(,𝒩 ) in the strong closure of C(,𝒩 ) such that u = π ũ does not hold for any ũ Ws,p(, 𝒩˜).

Keywords
analytical obstruction, finite-sheeted covering, Riemannian covering, fractional Sobolev spaces of mappings
Mathematical Subject Classification 2010
Primary: 46E35
Secondary: 58D15
Milestones
Received: 23 July 2019
Revised: 19 November 2019
Accepted: 19 March 2020
Published: 7 September 2021
Authors
Petru Mironescu
CNRS UMR 5208
Institut Camille Jordan
Université Claude Bernard Lyon 1
Villeurbanne
France
Simion Stoilow Institute of Mathematics of the Romanian Academy
Bucureşti
România
Jean Van Schaftingen
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Louvain-la-Neuve
Belgium