An estimate of the Hopf degree of fractional Sobolev mappings

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 $s\in [1-\frac{1}{4n},1]$ \[ \left |{\rm deg}_H(f)\right | \lesssim [f]_{W^{s,\frac{4n-1}{s}}}^{\frac{4n}{s}}. \] Our argument is based on the Whitehead integral formula and commutator estimates for Jacobian-type expressions.