Necessity of moment bounds for unbounded source measures in L^q quantitative uniqueness

Ascertain whether the uniform q/2-moment assumption on the source measure ρ in Theorem 5 (which provides L^q diameter bounds for quadratic cost with unbounded supp ρ and star-shaped Ω) is necessary to obtain quantitative uniqueness for unbounded source measures.

Background

Theorem 5 establishes an L^q bound on the diameter of the set of Kantorovich potentials for quadratic cost when Ω is star-shaped and the source ρ has a finite q/2-moment, with Y compact. The proof integrates the L∞ control from a curve-based bound and uses the moment assumption to control the integral.

The authors are unsure whether the moment assumption is an artifact of the proof technique or a genuine necessity for quantitative uniqueness in the unbounded-source setting.