Abstract
Let K be a compact subgroup of automorphisms of a"e (n) . We formulate and prove an analogue of Miyachi's theorem for the semi-direct product K a <parts per thousand a"e (n) . This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the L (p) - L (q) analogue of Morgan theorem for any compact extension of a"e (n) . These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform.