In this talk, we describe a program of work studying continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$, obtained in collaboration with D. Jerison. A key ingredient in the analysis is the characterization of certain Besov spaces as the space of traces of functions in Sobolev spaces. We will also discuss some related results on quantitative aspects of the classical weak stability of the Hessian determinant mapping.