  We study polynomial and exponential stability for $C_{0}$-semigroups using
the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier
multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup
in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent.
Using this characterization we derive new polynomial decay rates which depend
on the geometry of the underlying space. We do not assume that the semigroup is
uniformly bounded, our results depend only on spectral properties of the
generator. As a corollary of our work on polynomial stability we reprove and
unify various existing results on exponential stability, and we also obtain a
new theorem on exponential stability for positive semigroups.
