We estimate the Hopf degree for smooth maps
f
f
from
S
4
n
−
1
\mathbb {S}^{4n-1}
to
S
2
n
\mathbb {S}^{2n}
in the fractional Sobolev space. Namely we show that for
s
∈
[
1
−
1
4
n
,
1
]
s \in [1 - \frac {1}{4n}, 1]
|
deg
H
(
f
)
|
≲
[
f
]
W
s
,
4
n
−
1
s
4
n
s
.
\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.