A dimension-free interpolation of Caffarelli's contraction theorem

We prove global Lipschitz estimates for Brenier maps between probability measures on $\mathbb{R}^n$ whose densities belong to the family $$ \rho_{U,\,p}=Z_{U,\, p}^{-1}\exp(-\Theta_p(U)), \qquad \Theta_p(t)=p\log\Bigl(1+\frac{t}{p}\Bigr), \qquad p\in[n,+\infty], $$ with finite normalization constant $Z_{U,\, p}$, and with the convention $\Theta_{\infty}(t)=t$. We allow different parameters for source and target, $d,D\in[n,+\infty]$, with $d\le D$. Our global estimate is uniform in $n,d,D$, and in the case $d=D<+\infty$, it improves the bounds of arXiv:2404.05456 by removing their exponential dependence on the dimension. We also prove localized estimates inside fixed balls $B_R$ whose constants are stable under the limits $d,D\to+\infty$ and they allow us to recover Caffarelli's celebrated contraction theorem with sharp constants.