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

Abstract

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 forma(& thetasym;)(x,xi) := a(2x(1)(2) + 2 xi(2)(1), . . ., 2x(d)(2) + 2 xi(2)(d)),where a is a measurable function on R-+(d):= {r is an element of R-d | r(j) gt 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 Her-mite 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(& thetasym;) are globally Gelfand-Shilov regular.