A dimension-free interpolation of Caffarelli's contraction theorem

Technical Summary: A Dimension-Free Interpolation of Caffarelli's Contraction Theorem

Problem Statement
The paper addresses the regularity of the Brenier map (the optimal transport map for quadratic cost) between two probability measures on $\mathbb{R}^n$. Specifically, it seeks global Lipschitz estimates for the gradient of the transport potential $\nabla \phi$ when the source and target densities belong to a specific family of measures that interpolate between polynomial-type densities and log-concave densities.

The densities are defined as:
$$\rho_{U,p} = Z_{U,p}^{-1} \exp(-\Theta_p(U)), \quad \Theta_p(t) = p \log\left(1 + \frac{t}{p}\right)$$
where $p \in [n, +\infty]$. The case $p = +\infty$ corresponds to the log-concave densities ($\Theta_\infty(t) = t$) studied in Caffarelli's celebrated contraction theorem. The authors consider a transport map pushing a source measure $\rho_{V,d}$ forward to a target measure $\rho_{W,D}$, where the parameters $d$ and $D$ satisfy $n \le d \le D \le +\infty$.

The central challenge is to establish dimension-free bounds on the Hessian of the transport potential ($\|D^2\phi\|_{L^\infty}$) that hold uniformly across the dimension $n$ and the interpolation parameters $d$ and $D$. Previous work by the authors and others (specifically [5]) provided estimates for the polynomial-to-polynomial case ($d=D < \infty$) but suffered from an exponential dependence on the dimension. Caffarelli's theorem ($d=D=\infty$) provided a sharp, dimension-free bound, but extending this sharpness to the intermediate polynomial regime with uniform constants remained an open problem.

Methodology
The authors combine two distinct analytical approaches found in the literature:

1. Maximum Principle Argument: Following the approach in [5], the authors utilize the Monge-Ampère equation and a maximum principle applied to the second derivatives of the potential. This involves analyzing the equation at a point where the second directional derivative attains its maximum.
2. Monotonicity and Growth Estimates: Following the approach in [9] (inspired by Fathi), the authors employ monotonicity properties of the optimal transport map and concentration inequalities to derive quantitative growth bounds on the transport map itself (i.e., bounds on $|\nabla \phi|$).

The core technical innovation is the integration of these two viewpoints. The authors first derive a "two-parameter maximum principle" (Lemma 2.1) that relates the Hessian of the potential to the Hessians of the potentials $V$ and $W$ and the gradients of the map. They then insert explicit, finite-parameter growth estimates for $|\nabla \phi|$ (derived via Fathi's argument in Propositions 2.2 and 2.3) into this maximum principle inequality.

A critical aspect of the methodology is the careful handling of limits. To recover the sharp Caffarelli bound in the limit $d, D \to \infty$, the authors demonstrate that the order of limits is crucial: one must first fix a radius $R$, let $D \to \infty$ and $d \to \infty$, and then let $R \to \infty$. This ordering ensures that the constants remain stable and converge to the sharp values.

Key Contributions and Results

1. Global Dimension-Free Hessian Estimate (Theorem 1.2):
   The paper establishes a global Lipschitz bound for the Brenier map between $\rho_{V,d}$ and $\rho_{W,D}$ where $d \le D$. The bound is uniform in the dimension $n$ and the parameters $d, D$.

   • Improvement over [5]: In the case $d=D < \infty$, this result improves upon the bounds in [5] by removing the exponential dependence on the dimension $d$.
   • Structure: The constant $C_{glob}$ depends on structural quantities of the potentials $V$ and $W$ (specifically their Hessian bounds $C_V^{(2)}, c_W^{(2)}$ and gradient bounds) but is independent of $n$.

2. Localized Hessian Estimate (Theorem 1.3):
   The authors prove a refined estimate for the Hessian of the potential inside a fixed ball $B_R(0)$. This localized bound involves constants ($R_{d,D}(R)$ and $G_{d,D}(R)$) that control the growth of the transport map within that ball.

   • Stability: These localized constants are stable under the limits $d, D \to +\infty$.
   • Mechanism: The proof relies on defining a "Fathi radius" $R_{d,D}(R)$ which bounds the image of the ball $B_R(0)$ under the transport map, ensuring the maximum principle can be applied effectively within a controlled domain.

3. Recovery of Caffarelli's Theorem (Corollaries 1.4 and 1.5):
   By taking the limit $D \to +\infty$ (polynomial-to-log-concave case) and subsequently $d \to +\infty$, the authors recover Caffarelli's contraction theorem with sharp constants.

   • Corollary 1.4: Provides a bound for the case $D=+\infty$ and finite $d$, showing the growth factor disappears.
   • Corollary 1.5: By taking the limits in the specific order ($D \to \infty$, then $d \to \infty$, then $R \to \infty$), the authors rigorously recover the classical result: if $f=e^{-V}$ and $g=e^{-W}$ with $D^2V \le C_V^{(2)}I$ and $D^2W \ge c_W^{(2)}I$, then $\|DT\|_{L^\infty} \le \sqrt{C_V^{(2)}/c_W^{(2)}}$.

Significance and Claims
The paper claims to provide a unified framework that interpolates between polynomial-type densities and log-concave densities while maintaining dimension-free control.

• Uniformity: The primary significance is the uniformity of the estimates. The bounds do not deteriorate as the dimension $n$ increases, nor do they depend exponentially on the interpolation parameters.
• Sharpness: The methodology allows for the recovery of the sharp constants in Caffarelli's theorem, which previous polynomial-to-polynomial estimates could not achieve due to their dimensional dependence.
• Robustness: The approach is described as "extremely robust," handling different source and target parameters ($d \le D$) and recovering known results as special cases (e.g., the polynomial-to-polynomial case of [5] and the log-concave case of Caffarelli).

The authors note that the restriction $d \le D$ is natural; if $d > D$, the transport map may exhibit superlinear growth at infinity, preventing a global Lipschitz bound. The paper concludes by standard regularization arguments to extend the smooth maximum principle results to the general $C^{1,1}_{loc}$ setting required by the hypotheses.