Hessian measures II

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

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…

旋转对称下Christoffel-Minkowski问题和Hessian方程的显式解

Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries

本文在旋转对称性下，给出了凸旋转体Christoffel-Minkowski问题的显式解，并研究了混合面积测度的存在性问题，通过构造显式的凸解来解决混合Monge-Ampère方程，特别地处理了R^n上的k-Hessian方程的Dirichlet问题。

On uniform continuity of convex bodies with respect to measures in Banach spaces

Let $μ$ be a probability measure on a separable Banach space $X$. A subset $U\subset X$ is $μ$-continuous if $μ(\partial U)=0$. In the paper the $μ$-continuity and uniform $μ$-continuity of convex bodies in $X$, especially of balls and half-spaces, is considered. The $μ$-continuity is interesting for study of the Glivenko-Cantelli theorem in Banach spaces. Answer to a question of F. Topsøe is given.

On the (dis)continuity of the Fourier transform of measures

In this paper, we will study the continuity of the Fourier transform of measures with respect to the vague topology. We show that the Fourier transform is vaguely discontinuous on R, but becomes continuous when restricting to a class of Fourier transformable measures such that either the measures, or their Fourier transforms are equi-translation bounded. We discuss continuity of the Fourier transform in the product and norm topology. We show that vague convergence of positive definite measures implies the equi translation boundedness of the Fourier transforms, which explains the continuity of …

通过测度值导子与BV-瓦瑟斯坦曲线建立度量空间上的连续性方程

Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves

本文引入度量空间上的连续性方程概念，基于测度值导子刻画配备1-瓦瑟斯坦距离的概率测度空间中的BV-曲线，其公式与Almi等人的相关工作一致，还扩展了此前的概率表示研究。

度量空间中曲线的速度测度与绝对连续性

The speed measure and absolute continuity for curves in metric spaces

本文研究了度量空间中曲线的速度测度及其绝对连续性，通过速度测度与勒贝格测度的导数关系，推广了巴纳赫-扎雷斯基定理。