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 $${\rho }_{U,p}=Z_{U,p}^{-1}\exp (-{\mathrm{\Theta }}_p(U)),{\mathrm{\Theta }}_p(t)=p\log (1+\frac{t}{p}),p\in [n,+\mathrm{\infty }],$$ with finite normalization constant $Z_{U,p}$, and with the convention ${\mathrm{\Theta }}_{\mathrm{\infty }}(t)=t$. We allow different parameters for source and target, $d,D\in [n,+\mathrm{\infty }]$, with $d\le D$. Our global estimate is uniform in $n,d,D$, and in the case $d=D<+\mathrm{\infty }$, 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 $d,D\to +\mathrm{\infty }$ and they allow us to recover Caffarelli’s celebrated contraction theorem with sharp constants.