Slowly oscillating solutions of Cauchy problems with countable spectrum

Let u be a bounded slowly oscillating mild solution of an inhomogeneous Cauchy problem, u̇(t) = Au(t) + f(t), on ℝ or ℝ+, where A is a closed operator such that σap(A) ∩ iℝ is countable, and the Carleman or Laplace transform of f has a continuous extension to an open subset of the imaginary axis wit...