  In the present paper, we prove that a lower bound on the $1$-weighted Ricci
curvature is equivalent to a convexity of entropies on the Wasserstein space.
Based on such characterization, we provide some interpolation inequalities such
as the Pr'ekopa-Leindler inequality, the Borel-Branscamp-Lieb inequality, and
the Brunn-Minkowski inequality under the curvature bound.
