We investigate various G type spaces on R d + and their relations with the Gelfand-Shilov S type spaces on R 2d through the mapping w : R 2d → R d + , w(x, ξ) = (2x 2 1 + 2ξ 2 1 ,. .. , 2x 2 d + 2ξ 2 d). Sufficient conditions for the hypoellipticity of symbols originating from the coordinante radial symbols in G type spaces are also given. Two open problems explained in the introduction are posed.