A pair of Kontorovich-Lebedev type transforms and associated pseudo-differential operators

In this paper, we have introduced Kontorovich-Lebedev type transform (KL-type transform) in two variables form, independent of index and further its two classifications are given. Differential operators associated to it is being introduced and further its continuity as well as of KL-type transforms on Zemanian type function spaces are discussed. Moreover, translation and convolution operator involving KL-type transform are given and in Lebesgue space their estimates are deduced. Furthermore, Pseudo-differential operators associated to KL-type transforms are defined and on Zemanian type function spaces their continuity has been studied. Ultimately, an application of it has been shown by solving an integral equation.