Decay estimates of functions through singular extensions of vector-valued Laplace transforms

Let X be a Banach space and let f ∈ L ∞ ( R + ; X ) whose Laplace transform extends analytically to some region containing i R ∖ { 0 } , possibly having a pole at the origin. In this paper, we give estimates of the decay of certain slight suitable modification of f in terms of the growth of its Laplace transform along the imaginary axis. This technique is applied to obtain decay estimates of smooth orbits of bounded C 0 -semigroups whose infinitesimal generators have an arbitrary finite boundary spectrum. These results are close to those given recently by C.J.K. Batty and T. Duyckaerts.