A Class of Pseudo-Differential Operators Associated with Bessel Operators

A class of pseudo-differential operators (p.d.o.), generalizing Bessel differential operator d2/dx2 + (1 − 4μ2)/(4x2), is defined. Symbol classes Hm and Hm0 are introduced. It is shown that p.d.o.′s associated with symbols belonging to these classes are continuous linear mappings of the Zemanian space Hμ into itself. An integral representation of p.d.o.′s is obtained. Using Haimo′s theory of the Hankel convolution it is shown that p.d.o.′s satisfy a certain L1 - norm inequality.