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

2014/11/19 by Eric Baer, David Jerison, Baer, Eric +1
Economics, Econometrics and Finance · Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Navier-Stokes equation solutions #Stochastic processes and financial applications #math.AP #math.CA #math.FA

paper · pdf · doi:10.48550/arxiv.1411.5303

26 pages

arxiv created 2014/11/19 · openalex publication_date 2014/11/19 · arxiv updated 2014/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^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^2,N(ℝ^N+2) on the subspace ℝ^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.

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