Optimal function spaces for continuity of the Hessian determinant as a distribution

We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on RNRN. 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(2−2N,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(2−2N,N) as the space of traces of functions in the Sobolev space W2,N(RN+2)W2,N(RN+2) on the subspace RNRN of codimension 2. The most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in B(2−2N,p) with p>Np>N.