The Spaces of Ultradistributions Over ℝ+d and the Weyl Calculus of Pseudo-Differential Operators

Abstract

In this paper we introduce classes of ultradifferentiable functions and the corresponding ultradistributions on ℝ+d, i.e. Gαβ(ℝ+d) and (Gαβ(ℝ+d))′, α,β gt 0, respectively. We give their characterisation through the Laguerre coefficients estimate. Furthermore, we define the modified fractional power of the partial Hankel-Clifford transform and show that this transformation is a topological isomorphism on Gαα(ℝ+d), α≥1. We also investigate the boundedness of the Weyl pseudo-differential operators with symbols from the previously introduced spaces.