  We develop a statistical inference method for an optimal transport map
between distributions on real numbers with uniform confidence bands. The
concept of optimal transport (OT) is used to measure distances between
distributions, and OT maps are used to construct the distance. OT has been
applied in many fields in recent years, and its statistical properties have
attracted much interest. In particular, since the OT map is a function, a
uniform norm-based statistical inference is significant for visualization and
interpretation. In this study, we derive a limit distribution of a uniform norm
of an estimation error for the OT map, and then develop a uniform confidence
band based on it. In addition to our limit theorem, we develop a bootstrap
method with kernel smoothing, then also derive its validation and guarantee on
an asymptotic coverage probability of the confidence band. Our proof is based
on the functional delta method and the representation of OT maps on the reals.
