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Proceedings of the American Mathematical Society

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An estimate of the Hopf degree of fractional Sobolev mappings


Authors: Armin Schikorra and Jean Van Schaftingen
Journal: Proc. Amer. Math. Soc. 148 (2020), 2877-2891
MSC (2010): Primary 46E35, 55Q25; Secondary 55M25, 55P99, 58A12
DOI: https://doi.org/10.1090/proc/15026
Published electronically: March 25, 2020
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Abstract | References | Similar Articles | Additional Information

Abstract: We estimate the Hopf degree for smooth maps $ f$ from $ \mathbb{S}^{4n-1}$ to $ \mathbb{S}^{2n}$ in the fractional Sobolev space. Namely we show that for $ s \in [1 - \frac {1}{4n}, 1]$

$\displaystyle \left \vert\deg _H(f)\right \vert \lesssim [f]_{W^{s,\frac {4n-1}{s}}}^{\frac {4n}{s}}.$    

Our argument is based on the Whitehead integral formula and commutator estimates for Jacobian-type expressions.

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Additional Information

Armin Schikorra
Affiliation: Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, Pennsylvania 15260
Email: armin@pitt.edu

Jean Van Schaftingen
Affiliation: Institut de Recherche en Mathématique et Physique (IRMP), Université catholique de Louvain (UCLouvain), Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Email: Jean.VanSchaftingen@uclouvain.be

DOI: https://doi.org/10.1090/proc/15026
Received by editor(s): May 6, 2019
Published electronically: March 25, 2020
Additional Notes: The first author gratefully acknowledges support by the Simons foundation, grant no 579261.
The second author gratefully acknowledges support by Fonds de la Recherche Scientifique–FNRS, Mandat d’Impulsion Scientifique F.4523.17, “Topological singularities of Sobolev maps”.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2020 American Mathematical Society