Global regularity of Weyl pseudo-differential operators with radial symbols in each phase-space variable

We analyse a class of pseudo-differential operators in the Gelfand–Shilov setting whose Weyl symbols are radial in each phase-space variable separately. Namely, the symbols are of the form a_ϑ(x,ξ ):= a(2x_1^2+2ξ _1^2,… ,2x_d^2+2ξ _d^2), where a is a measurable function on ℝ^d_+:={r∈ℝ^d | r_j>0, j=1,… ,d} and has Gelfand–Shilov L^p -growths. We prove that the action of these pseudo-differential operators on a Gelfand–Shilov ultradistribution f can be given by a series of Hermite functions with coefficients that are explicitly computed in terms of the Laguerre coefficients of a and the Hermite coefficients of f . As a consequence, we give a characterisation of the functions a in terms of the growths of their Laguerre coefficients for which the Weyl quantisation of a_ϑ are globally Gelfand–Shilov regular.