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.