An estimate of the Hopf degree of fractional Sobolev mappings

Schikorra, Armin; Van Schaftingen, Jean

doi:10.1090/proc/15026

An estimate of the Hopf degree of fractional Sobolev mappings
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by Armin Schikorra and Jean Van Schaftingen

Proc. Amer. Math. Soc. 148 (2020), 2877-2891

DOI: https://doi.org/10.1090/proc/15026

Published electronically: March 25, 2020

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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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Proceedings of the American Mathematical Society

An estimate of the Hopf degree of fractional Sobolev mappings
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Abstract: