  Under reasonable working assumptions including the polynomial boundedness,
one proves the well-known Cerulus-Martin lower bound on how fast an elastic
scattering amplitude can decrease in the hard-scattering regime. In this paper
we consider two non-trivial extensions of the previous bound. (i) We generalize
the assumption of polynomial boundedness by allowing amplitudes to
exponentially grow for some complex momenta and prove a more general lower
bound in the hard-scattering regime. (ii) We prove a new lower bound on elastic
scattering amplitudes in the Regge regime, in both cases of polynomial and
exponential boundedness. A bound on the Regge trajectory for negative momentum
transfer squared is also derived. We discuss the relevance of our results for
understanding gravitational scattering at the non-perturbative level and for
constraining ultraviolet completions. In particular, we use the new bounds as
probes of non-locality in black-hole formation, perturbative string theory,
classicalization, Galileons, and infinite-derivative field theories, where both
the polynomial boundedness and the Cerulus-Martin bound are violated.
