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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An estimate of the Hopf degree of fractional Sobolev mappings
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by Armin Schikorra and Jean Van Schaftingen PDF
Proc. Amer. Math. Soc. 148 (2020), 2877-2891 Request permission

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 $s \in [1 - \frac {1}{4n}, 1]$ \begin{equation*} \left |\deg _H(f)\right | \lesssim [f]_{W^{s,\frac {4n-1}{s}}}^{\frac {4n}{s}}. \end{equation*} 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
  • MR Author ID: 880438
  • 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
  • MR Author ID: 730276
  • ORCID: 0000-0002-5797-9358
  • Email: Jean.VanSchaftingen@uclouvain.be
  • 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
  • © Copyright 2020 American Mathematical Society
  • 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
  • MathSciNet review: 4099776