\(G\)-type spaces of ultradistributions over \(\mathbb {R}^d_+\) and the Weyl pseudo-differential operators with radial symbols (Q2412807)

The authors reconsider the $G$-type spaces introduced by \textit{A. J. Duran} [Math. Nachr. 153, 145--167 (1991; Zbl 0796.46026)].\par They can be described as variant of the well-known Gelfand-Shilov spaces in the half real line. The authors propose here a generalization to $\mathbb{R}^d_+=(0,+\infty)^d$ and give a detailed analysis, concerning topological properties and modified fractional powers of the partial Hankel-Clifford transform, main tool in the definition of the spaces. As a consequence of the topological structure, the authors prove the nuclearity of the $G$-spaces, in the frame of the corresponding ultradistributions. A very interesting application concerns pseudodifferential operators with radial symbols defined by a $G$-test function or $G$-ultradistributions. They are proved to be bounded in the Gelfand-Shilov spaces.