We study behavior in time of tails of solutions to the Boltzmann equation without an angular cutoff, by examining the generation and propagation in time of $L^1$ and $L^\infty$ exponentially weighted estimates and the relation between them. For this purpose, we introduce Mittag-Leffler moments, which can be understood as a generalization of exponential moments. We show how the singularity rate of the angular kernel affects the order of tails that can be shown to propagate in time (by this method). This is based on joint works with Alonso, Gamba, Pavlovic and Gamba, Pavlovic.