Optimal function spaces for the weak continuity of the distributional $k$-Hessian

TL;DR

This paper defines distributional $k$-Hessian for Besov functions and proves its weak continuity on an optimal Besov space, extending previous work on the Hessian determinant.

Contribution

It introduces the distributional $k$-Hessian for Besov functions and establishes the optimal function space for its weak continuity.

Findings

Distributional $k$-Hessian is weakly continuous on $B(2-rac{2}{k},k)$.

The result is optimal within the Besov space framework.

The paper generalizes recent work on the Hessian determinant.

Abstract

In this paper we introduce the notion of distributional $k$-Hessian associated with Besov type functions in Euclidean $n$-space, $k=2,\ldots,n$. Particularly, inspired by recent work of Baer and Jerison on distributional Hessian determinant, we show that the distributional $k$-Hessian is weak continuous on the Besov space $B(2-\frac{2}{k},k)$, and the result is optimal in the framework of the space $B(s,p)$, i.e., the distributional $k$-Hessian is well defined in $B(s,p)$ if and only if $B(s,p)\subset B_{loc}(2-\frac{2}{k},k)$.