Optimal individual stability estimates for $C_0$-semigroups in Banach spaces

In a previous paper we proved that the asymptotic behavior of a $C_0$-semigroup is completely determined by growth properties of the resolvent of its generator and geometric properties of the underlying Banach space as described by its Fourier type. The given estimates turned out to be optimal. The method of proof uses complex interpolation theory and reflects the full semigroup structure. In the present paper we show that these uniform estimates have to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is lacking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferred from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain the optimal uniform estimates mentioned above from individual ones. Concerning Hardy-abscissas, individual orbits and their local resolvents behave as badly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a $C_0$-semigroup itself and the resolvent of its generator where a simple dichotomy holds true.