A pair of Barut-Girardello type transforms and allied pseudo-differential operators

The main objective of this paper is to generalize the Barut-Girardello transform (BG-transform) in two different ways as a pair of Barut-Girardello type transforms (BG-type transforms). Certain differential operators related to it are introduced and then continuity of the differential operators as well as both BG-type transforms on Zemanian type function spaces are discussed. Translation and convolution operators associated with BG-type transforms are introduced and obtained their estimates in Lebesgue space. Moreover, pseudo-differential operators involving BG-type transforms are defined and discussed their continuity on Zemanian type function spaces. At the end, some applications have been discussed.