Abstract. We show the existence of ``large" weak orbits of $C_0$-semigroups with generators satisfying natural spectral assumptions. We give also certain applications of our results to harmonic analysis and discuss related results.

AMS Subject Classification (1991): 47D03, 47A10, 42A38

Keyword(s): weak orbit, $C_0$-semigroup, domination, decay, Fourier transform

Received December 6, 2012. (Registered under 108/2012.)