DC Calculus

TL;DR

This paper extends DC Calculus to Alexandrov spaces with curvature bounds, enabling the definition of Hessian and Laplacian as measure-valued objects, bridging the gap between nonsmooth and smooth geometric analysis.

Contribution

It introduces a framework for defining Hessian and Laplacian of DC functions on Alexandrov spaces, generalizing smooth manifold concepts to nonsmooth settings.

Findings

Hessian of DC functions is a measure-valued tensor.

Laplacian of DC functions is a Radon measure.

Properties of these objects mirror those on smooth manifolds.

Abstract

In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a Radon measure respectively. We show that these objects share various properties with their analogues on smooth Riemannian manifolds.