Weyl transforms associated with Laguerre functions

AbstractWe consider two partial differential operators D1 and D2 having eigenfunctions defined with Laguerre functions. We give the harmonic analysis associated with D1 and D2. We define and study the Wigner transform associated with D1 and D2, and we prove for this transform an inversion formula. Next we consider classes of symbols which allows us to define the Weyl transform associated with D1 and D2. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we study criterions in term of symbols for the boundedness and compactness of this Weyl transform