$G$-type Spaces of Ultradistributions over $\mathbb{R}^d_+$ and the Weyl Pseudo-differential Operators with Radial Symbols

Abstract: The first part of the paper is devoted to the $G$-type spaces i.e. the spaces $G^\alpha_\alpha (\mathbb R^d_+)$, $\alpha\geq 1$ and their duals which can be described as analogous to the Gelfand-Shilov spaces and their duals but with completely new justification of obtained results. The Laguerre type expansions of the elements in $G^\alpha_\alpha(\mathbb R^d_+)$, $\alpha\geq 1$ and their duals characterise these spaces through the exponential and sub-exponential growth of coefficients. We provide the full topological description and by the nuclearity of $G_\alpha^\alpha(\mathbb{R}^d_+)$, $\alpha\geq 1$ the kernel theorem is proved. The second part is devoted to the class of the Weyl operators with radial symbols belonging to the $G$-type spaces. The continuity properties of this class of pseudo-differential operators over the Gelfand-Shilov type spaces and their duals are proved. In this way the class of the Weyl pseudo-differential operators is extended to the one with the radial symbols with the exponential and sub-exponential growth rate.