G-type spaces of ultradistributions over $$\mathbb {R}^d_+$$ 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 \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)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 \(G^\alpha _\alpha (\mathbb R^d_+)\), \(\alpha \ge \)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 \(G_\alpha ^\alpha (\mathbb {R}^d_+)\), \(\alpha \ge \)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.