Abstract: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.
| Subjects: | Analysis of PDEs (math.AP) |
| MSC classes: | 35J96, 35B65, 35A23 |
| Cite as: | arXiv:2605.24443 [math.AP] |
| (or arXiv:2605.24443v1 [math.AP] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24443 arXiv-issued DOI via DataCite (pending registration) |
From: Bader Ammari [view email]
[v1]
Sat, 23 May 2026 07:30:20 UTC (32 KB)
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