A general quantified Ingham-Karamata Tauberian theorem

Abstract: We provide a general quantified Ingham-Karamata Tauberian theorem with a flexible one-sided Tauberian condition under several types of boundary behavior for the Laplace transform. Our results in particular improve a theorem by Stahn, removing a vexing restriction on the growth of the Laplace transform. Improving existing optimality results, we also show that the obtained quantified rate is optimal in almost all cases.