In recent years there has been a surge of interest in polynomial decay properties of solutions to various PDEs, most notably damped wave equations. Such polynomial decay rates can be studied effectively using the framework of semigroup theory, by exploiting a recently established link between polynomial decay properties of a -semigroup and fine spectral properties of the generator of the semigroup.
I will explain this link using the recently developed theory of operator-valued Fourier multipliers. I will also present several new results which allow one to obtain optimal decay rates for orbits of a -semigroup given fine asymptotics of the resolvent of its generator.
This is joint work with Mark Veraar (Delft University of Technology), and with David Seifert (University of Oxford) and Reinhard Stahn (TU Dresden).