Pseudo-differential operators involving Hankel transforms (Q1378399)

The authors investigate the pseudo-differential operators \(H(x,D)\) and \(L(x,D)\) associated with Hankel transforms. \(H(x,D)\) is a generalization of certain pseudo-differential operator \(h\), studied by \textit{R. S. Pathak} and \textit{P. K. Pandey} [J. Math. Anal. Appl. 196, No. 2, 736-747 (1995; Zbl 0843.35145)] while \(L(x,D)\) is its adjoint. Using the notation and terminology introduced by \textit{S. Zaidman} [Ann. Mat. pura appl., IV. Ser. 92, 345-399 (1972; Zbl 0271.35066)] and \textit{A. H. Zemanian} [Generalized integral transformations, New York (1968; Zbl 0181.12701)] the authors establish that under specified conditions the pseudo-differential operators \(H(x, D)\) and \(L(x, D)\) behave like linear operators from \(L (R)\) into \(L (R)\). In addition, they develop a variety of product and commutator properties.