Classifying Globally Lipschitz Transport Settings

Definitions and mathematical statement

A general definition of a globally Lipschitz transport map can be given as follows; the supplied excerpt does not specify whether this is the precise framework intended. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces equipped with Borel probability measures $\mu$ and $\nu$, respectively. A Borel measurable map $T:X\to Y$ transports $\mu$ to $\nu$ if $T_{\#}\mu=\nu$, meaning that \[ \nu(B)=\mu(T^{-1}(B))\qquad\text{for every Borel set }B\subseteq Y. \] The map $T$ is globally Lipschitz if there exists a finite constant $L\geq 0$ such that \[ d_Y(T(x),T(x'))\leq L\,d_X(x,x')\qquad\text{for all }x,x'\in X. \] If the intended transport map is defined only almost everywhere, the corresponding global property requires specifying a representative and the set on which this inequality must hold.

Source question. Find additional settings in which the transport map under discussion is globally Lipschitz.

The preceding definitions explain the terminology but do not specify the missing setting. The supplied source excerpt identifies neither the source and target measures or spaces, nor the construction or selection rule for "the map," nor the previously established settings relative to which "additional" is meant. Consequently, it does not determine whether the question concerns the existence of some Lipschitz transport, the regularity of a particular transport construction, or a property of every map in a specified class. No particular domain, optimality condition, boundary condition, or uniform dependence of $L$ can be inferred. The excerpt also states that further results have been obtained, citing its reference [2], which is not identified in the supplied record. A more specific quantified problem requires this missing context.

Short English statement

The question asks for further settings where the transport map being discussed satisfies a single finite Lipschitz bound throughout its domain. The supplied excerpt does not identify that map, its domain, its measures, or the settings already known, so the intended mathematical problem remains underdetermined.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005262$, source problem code catalogue entry, title: Classifying Globally Lipschitz Transport Settings. The registry supplies the status open and the statement label corrected_verified; these are supplied curation labels, not independent mathematical verification.
• Supplied report citation: Heat Kernels, Stochastic Processes and Functional Inequalities (2023), Oberwolfach Reports, pp. 2811--2893, DOI 10.4171/owr/2022/49. https://doi.org/10.4171/owr/2022/49. The extracted passage carries page number 2845 and the contribution title Globally Lipschitz transport maps, attributed to Max Fathi, joint work with Dan Mikulincer and Yair Shenfeld. The registry records proposed year 2022 and no separately identified proposer.
• Supplied literature source: Oberwolfach Report 49. https://ems.press/content/serial-article-files/46987.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes the search for further settings as a natural question, and states that the source gives no general classification. This is supplied dated curation, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The original excerpt mentions further results and cites its reference [2]. No bibliographic identification of [2] is supplied.