The Pseudo-Differential Type Operator and Hankel Type Transformation Associated with Bessel Type Operator

In this paper the generalized pseudo-differential type operator P(x,D), associated with the Bessel operator d²/dx² + (1-4α²-4β²+8αβ)/4x² is defined. The symbol class H^m_ρ,δ is introduced. It has been shown that the pseudo-differential type operator associated with a symbol belonging to this class is a continuous linear mapping of the Zemanian space H_α,β into itself. An integral representation of pseudo-differential type operator is obtained. Using Hankel type convolution,L^p_μ,a-norm continuity of the pseudo-differential type operator is proved.