Abstract
Let K be a compact subgroup of automorphisms of ℝn. We formulate and prove an analogue of Miyachi’s theorem for the semi-direct product K ⋉ ℝn. This allows us to solve the sharpness problems in the theorem of Cowling-Price and in the Lp − Lq analogue of Morgan theorem for any compact extension of ℝn. These upshots are proved using the representations theory and the Plancherel formula for the group Fourier transform.
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