G -type spaces of ultradistributions over 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)(R-+(d)), alpha gt = 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) (R-+(d)), alpha gt = 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)(R-+(d)), alpha gt = 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.