  We establish optimal continuity results for the action of the Hessian
determinant on spaces of Besov type into the space of distributions on
$\mathbb{R}^N$. In particular, inspired by recent work of Brezis and Nguyen on
the distributional Jacobian determinant, we show that the action is continuous
on the Besov space of fractional order $B(2-\frac{2}{N},N)$, and that all
continuity results in this scale of Besov spaces are consequences of this
result.
  A key ingredient in the argument is the characterization of
$B(2-\frac{2}{N},N)$ as the space of traces of functions in the Sobolev space
$W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ of codimension $2$.
The most delicate and elaborate part of the analysis is the construction of a
counterexample to continuity in $B(2-\frac{2}{N},p)$ with $p>N$.
