Optimal function spaces for continuity of the Hessian determinant as a distribution
Author(s)Baer, Eric; Jerison, David S
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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.
Journal of Functional Analysis
Baer, Eric, and David Jerison. “Optimal Function Spaces for Continuity of the Hessian Determinant as a Distribution.” Journal of Functional Analysis, vol. 269, no. 5, Sept. 2015, pp. 1482–514.