Asymptotics of Fourier and Laplace transforms in weighted spaces of analytic functions

We study the asymptotics of the Fourier transform in weighted Hardy spaces of analytic functions in the upper half-plane, and of the Laplace transform in weighted spaces of entire functions of zero exponential type.
The results are applied to two closely related problems posed by Dyn'kin: we find the asyniptotics of the depth of zero for flat functions in non-quasianalytic Denjoy–Carleman classes, and of the exact majorant in a version of the Carleman–Levinson–Sjöberg $\log$-$\log$-theorem.