Reverse Translation and Convolution Estimates in the Framework of the Mehler-Fock Transform

In this paper, reverse translation and convolution estimates in the frame of ${\mu }^{th}$ order Mehler-Fock transform are obtained. Continuity properties of these operators on the test function space ${\mathfrak{H}}_{l,r,n}$ have been discussed. Moreover, Sobolev space $S_{r,p}^{N,\alpha }$ has been introduced and again continuity properties of these operators in this Sobolev space have been discussed.