Families of extensions of the Kantorovich-Rubinstein and Lipschitz norms

TL;DR

This paper introduces parameterized families of extensions for the Kantorovich-Rubinstein and Lipschitz norms, establishing duality relations between these extended spaces on compact metric spaces.

Contribution

It extends these norms from specific measure and function spaces to broader classes, parameterized by p and q, and proves duality when p and q are conjugates.

Findings

Extended norms to all countably additive measures and Lipschitz functions.

Established isometric isomorphism between dual spaces for conjugate parameters.

Provided a unified framework linking Kantorovich-Rubinstein and Lipschitz norms.

Abstract

We propose a family of extensions of the Kantorovich-Rubinstein norm from the space of zero-charge countably additive measures on a compact metric space to the space of all countably additive measures, and a family of extensions of the Lipschitz norm from the quotient space of Lipschitz functions on a compact metric space to the space of all Lipschitz functions. These families are parameterized by $p,q \in [1,\infty]$, and if $p,q$ are H\"older conjugates, then the dual of the resulting $p$-Kantorovich space is isometrically isomorphic to the resulting $q$-Lipschitz space.