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

Abstract

We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on R[superscript 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(2-2/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(2-2/N,N) as the space of traces of functions in the Sobolev space W[superscript 2,N](R[superscript N+2]) on the subspace R[superscript N] 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.