  We prove a quantified Tauberian theorem for functions under a new kind of
Tauberian condition. In this condition we assume in particular that the Laplace
transform of the considered function extends to a domain to the left of the
imaginary axis, given in terms of an increasing function $M$ and is bounded at
infinity within this domain in terms of a different increasing function $K$.
Our result generalizes a result of Batty, Borichev and Tomilov (2016). We also
prove that the obtained decay rates are 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 in odd-dimensional
exterior domains.
