Pseudodifferential Operators as Integral Operators

All of the integral operators with nonintegrable kernels are given in terms of computable Hadamard's partie finie, i.e. finite part integrals [117], which can be applied to problems in applications (Guiggiani [114], Schwab et al [274]).

In this chapter, we discuss the interpretation of pseudodifferential operators as integral operators. In particular, we show that every classical pseudodifferential operator is an integral operator with integrable or nonintegrable kernel plus a differential operator of the same order as that of the pseudodifferential operator in case of a nonnegative integer order. In addition, we also give necessary and sufficient conditions for integral operators to be classical pseudodifferential operators in the domain. Symbols and admissible kernels are closely related based on the asymptotic expansions of the symbols and corresponding pseudohomogeneous expansions of the kernels as examined by Seeley [279]. The main theorems in this context are Theorems 7.1.1, 7.1.6 and 7.1.7 below.