The Mehler-Fock-Clifford transform and pseudo-differential operator on function spaces

Abstract

New Index transform with Mehler-Fock type kernel consisting of the cone function is introduced, named as Mehler-Fock-Clifford (MFC) transform. Some basic properties are studied. Convolution and translation operators are defined and studied. Also their MFC-transform and estimates in Lebesgue space are obtained. Moreover, test function spaces $\mathcal{G}_\alpha$ and $\mathcal{F}_\alpha$ are defined and it is shown MFC-transform is continuous linear mapping from $\mathcal{G}_\alpha$ into $\mathcal{F}_\alpha$. Further pseudo-differential operators (p.d.o.) is defined and its another integral representation is obtained. Further its continuity property and special case are investigated.