  In this paper, we consider various tensorial estimates in geometric
Besov-type norms on a one-parameter foliation of surfaces with evolving
geometries. Moreover, we wish to do this with only very weak control on these
geometries. Several of these estimates were established in previous works by S.
Klainerman and I. Rodnianski, but in very specific settings. A primary
objective of this paper is to significantly simplify and make more robust the
proofs of the estimates. Another goal is to generalize these estimates to more
abstract settings. In upcoming papers (joint with S. Alexakis), we will apply
these estimates in order to study truncated null cones in an Einstein-vacuum
spacetime extending to infinity. This analysis will then be used to study and
to control the Bondi mass and the angular momentum under minimal conditions.
