Energy Bounds for Kantorovich Transport Distances with Convex Cost Functions

TL;DR

The paper extends Ledoux's energy bounds for Kantorovich transport distances from power-type costs to general convex costs c(x,y)=L(x−y) under a two-sided Δ2 condition. It develops an infimum-convolution framework with Q_t and a Hamilton–Jacobi structure to relate transport costs to dual Luxemburg–Orlicz norms and dual Sobolev norms, constructing a bound of the form T_L(μ,ν) ≤ A Φ_L(||ν−μ||_{H^{-1,L}(μ)}) with constants determined by Φ_L and its derivatives. Key innovations include a boundary-value problem on the associated Young function and a robust smoothing argument to transfer bounds from smooth to general measures, plus a thorough treatment of vector-valued Luxemburg/Orlicz norms and their continuity under convolution. Together these results generalize Ledoux’s energy estimates to broad convex costs, linking transport inequalities with Orlicz-type functional-analytic tools and enabling energy-type control in more flexible cost settings.

Abstract

Energy bounds for Kantorovich transport distances are developed for convex cost functions. The main results extend estimates due to M. Ledoux for the Kantorovich distances $W_p$.