Inserted: 28 feb 2022
Last Updated: 17 nov 2022
Journal: J. Funct. An.
This paper is an answer to a conjecture in https:/arxiv.orgabs2010.12946
We prove some Lorentz-type estimates for the average in time of suitable geodesic interpolations of probability measures, obtaining as a by product a new estimate for transport densities and a new integral inequality in terms of Wasserstein distances and norms of gradients. This last inequality was conjectured in a paper by S. Steinerberger.