  We prove decay rates for a vector-valued function $f$ of a non-negative real
variable with bounded weak derivative, under rather general conditions on the
Laplace transform $\hat{f}$. This generalizes results of Batty-Duyckaerts
(2008) and other authors in later publications. Besides the possibility of
$\hat{f}$ having a singularity of logarithmic type at zero, one novelty in our
paper is that we assume $\hat{f}$ to extend to a domain to the left of the
imaginary axis, depending on a non-decreasing function $M$ and satisfying a
growth assumption with respect to a different non-decreasing function $K$. The
decay rate is expressed in terms of $M$ and $K$. We prove that the obtained
decay rates are essentially optimal for a very large class of functions $M$ and
$K$. Finally we explain in detail how our main result improves known decay
rates for the local energy of waves on exterior domains.
