On the Pseudo-differential operators associated with Bessel transform and the properties of them

Pseudo differential type operators p(x; D) and Bessel type differential operator ?_(?,ß) have agreat value in harmonic analysis. They are an important area of review in recent years; and havebeen studied by such mathematicians as G. Altenburg, J. J. Betankor, M. Belhadj, R. S. Pathak,S. Pathak, A. Prasad, S. Zaidman, A. H. Zemanian, D. T. Haimo, P. K. Pandey, Ryusuke Numata,Micheal Taylor, and Vagif S. Guliyev.This thesis consists of three chapters. Definitions and properties of Pseudo differential operatorson R^n presented in the first chapter. In the second chapter Bessel differential equation, Besselfunction and its properties are given; in the last section Pseudo differential operator p(x;D) in termsof a symbol is de_ned and inverse Hankel transform of this symbol is defined. It is shown that thePseudo differential operator is bounded in certian Sobolev type space associated with Hankel transform.The symbol classes H_0^m and H_m are presented. It is shown that Pseudo differential typeoperator associated with symbols belonging to these classes are continuous linear mappings of theZemanian space H_(?,ß) into itself. Integral representation for Pseudo differential type operator h_(?,ß,a)is given. It is shown that Pseudo differential type operators satisfy L^1 norm inequality.