  In this paper we introduce the notion of distributional $k$-Hessian
associated with Besov type functions in Euclidean $n$-space, $k=2,\ldots,n$.
Particularly, inspired by recent work of Baer and Jerison on distributional
Hessian determinant, we show that the distributional $k$-Hessian is weak
continuous on the Besov space $B(2-\frac{2}{k},k)$, and the result is optimal
in the framework of the space $B(s,p)$, i.e., the distributional $k$-Hessian is
well defined in $B(s,p)$ if and only if $B(s,p)\subset
B_{loc}(2-\frac{2}{k},k)$.
