Basic calculus of Fourier integral operators and pseudodifferential operators

We shall introduce several symbol classes, which contain non-analytic symbols, and define pseudodifferential operators and Fourier integral operators. Product formulas of Fourier integral operators will be given in Sections 2.4 and 2.5, which are indispensable to our treatment. We shall prove pseudolocal properties of pseudodifferential operators in Section 2.6. As a simple application of the product formulas ( and the symbol calculus) we shall construct parametrices of elliptic pseudodifferential operators in Section 2.8.