Quantitative uniqueness for unbounded target measures under common costs

Investigate whether quantitative uniqueness bounds analogous to Theorem 4 can be established when the target measure μ is unbounded for standard costs, specifically for p-costs c(x,y)=||x−y||^p.

Background

The quantitative results in the paper assume the target support Y is bounded (compact), which facilitates uniform control of cost gradients and the geometry of optimal plans. The authors provide L∞ and L^q bounds under these assumptions, and extend to some unbounded-source settings with additional conditions.

They explicitly raise the question of whether comparable quantitative uniqueness phenomena persist when the target measure μ is unbounded for commonly used costs such as p-costs.