Caffarelli Estimates under Lipschitz Perturbations

Source abstract

We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let $d\ge1$, $L\ge0$, and let $B:\mathbb{R}^d\to\mathbb{R}$ be globally $L$-Lipschitz. We prove that the Brenier map from the standard Gaussian measure $γ_d$ to the probability measure proportional to $\mathrm{e}^{-B}γ_d$ has a globally Lipschitz representative whose bound depends only on $L$. The Brenier potential also belongs to $C^{1,1}(\mathbb{R}^d)$ and satisfies dimension-free two-sided Hessian bounds. Let $\mathfrak{C}(0,L)$ denote the upper-bound constant, then $$\log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \qquad(L\to\infty),$$ and a one-dimensional example with Brenier map $T_L$ satisfies $\log\operatorname{Lip}(T_L)\ge L^2/2$. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let $V,W:\mathbb{R}^d\to\mathbb{R}$ be potentials, and let $Q,P$ be positive-definite matrices. Using the distributional curvature bounds $D^2V\preceq Q$ and $D^2W\succeq P$, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.