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

TL;DR

This paper identifies the optimal Besov space for the Hessian determinant to act continuously as a distribution, extending previous work on the Jacobian determinant and characterizing the space via Sobolev traces.

Contribution

It establishes the optimal Besov space for the Hessian determinant's continuity and links it to Sobolev space traces, providing a comprehensive understanding of the distributional action.

Findings

Hessian determinant is continuous on the Besov space B(2−2/N,N).

All continuity results in this Besov scale follow from this main result.

Counterexamples show failure of continuity in B(2−2/N,p) for p>N.

Abstract

We establish optimal continuity results for the action of the Hessian determinant on spaces of Besov type into the space of distributions on $\mathbb{R}^N$. 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 $B(2-\frac{2}{N},N)$, 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 $B(2-\frac{2}{N},N)$ as the space of traces of functions in the Sobolev space $W^{2,N}(\mathbb{R}^{N+2})$ on the subspace $\mathbb{R}^N$ of codimension $2$. The most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in $B(2-\frac{2}{N},p)$ with $p>N$.