Weighted interpolation and stability of power-cost transport maps from a Gaussian source

Fix 1 < p < 2 and the standard Gaussian source on R^d. For arbitrary target probabilities in a fixed ball, we derive a quantitative L^2 stability bound for the optimal maps associated with |x-y|^p/p. We first prove a Gaussian weighted interpolation inequality at exponent (p-1)/(p+1) by one-sided derivative increments, local finite differences, and summation over enlarged cubes. A compact potential estimate is localized with controlled constants. Subtraction of the common radial cost yields a uniform Lipschitz bound and an explicit logarithmic loss. The map exponent is (p-1)^2/(p(p+1)). No sharpness is claimed. The compact potential argument is credited to Mischler-Trevisan and reproduced with the dependence needed here. The quadratic case is excluded from the claimed contribution.