$\Psi$DOs with radial symbols and spaces of type $G$

Abstract

We investigate various $G$ type spaces on $\R^d_+$ andtheir relations with the Gelfand-Shilov $S$ type spaces on $\R^{2d}$ through the mapping $w:\R^{2d}\rightarrow \mathbb R^d_+$, $w(x,\xi)=(2x_1^2+2\xi_1^2,\dots,2x_d^2+2\xi_d^2)$. 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.