The Spaces of Ultradistributions Over \(\boldsymbol {\mathbb {R}^d_+}\) and the Weyl Calculus of Pseudo-Differential Operators

In this paper we introduce classes of ultradifferentiable functions and the corresponding ultradistributions on \(\mathbb {R}_+^d\) , i.e. \(G_\alpha ^\beta (\mathbb {R}_+^d)\) and \((G_\alpha ^\beta (\mathbb {R}_+^d))'\) , \(\alpha ,\beta >0\) , 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 \(G_\alpha ^\alpha (\mathbb {R}_+^d)\) , \(\alpha \geq 1\) . We also investigate the boundedness of the Weyl pseudo-differential operators with symbols from the previously introduced spaces.