The Kontorovich-Lebedev-Clifford transform

Abstract

In this paper, we have defined a new integral transform, Kontorovich-Lebedev-Clifford transform (KLC-transform), by slight modification in the modified Bessel function of second kind. Translation and convolution operator associated to the KLC-transform are defined and some basic and useful preliminary results have been discussed. Besides, norm estimates for the translation and convolution operator are investigated. Function spaces $\mathcal{F}_\alpha$ and $\mathcal{G}_\alpha$ are defined and studied the continuity of translation and convolution operator on these spaces. Finally, pseudo-differential operator (p.d.o.) associated with the KLC-transform is defined and continuity of p.d.o. from the function space $\mathcal{G}_\alpha$ into $\mathcal{F}_\alpha$ is discussed. Also an integral representation of p.d.o. is given and further it is estimated in Lebesgue space. At the end a special case for p.d.o. is discussed