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 RN . 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 − 2 N , N) and, furthermore, 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 − 2 N , N) as the space of traces of functions in the Sobolev space W 2,N (RN+2) on the subspace RN of codimension 2. The most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in B(2− 2 N , p) with p > N .