Introduction to Pseudodifferential Operators

The pseudodifferential operators provide a unified treatment of differential and integral operators. They are based on the intensive use of the Fourier transformation F(3.1.12) and its inverse F⁻¹ = F^*(3.1.14). The linear pseudodifferential operators can be characterized by generalized Fourier multipliers, called symbols. The class of pseudodifferential operators form an algebra, and the operations of composition, transposition and adjoining of operators can be analyzed by algebraic calculations of the corresponding symbols.

Moreover, this class of pseudodifferential operators is invariant under diffeomorphic coordinate transformations. As linear mappings between distributions, the pseudodifferential operators can also be represented as Hadamard's finite part integral operators, whose Schwartz kernels can be computed explicitly from their symbols or as integro–differential operators and vice versa. For elliptic pseudodifferential operators, we construct parametrices. For elliptic differential operators, in addition, we construct Levi functions and also fundamental solutions if they exist. Since the latter provide the most convenient basis for boundary integral equation formulations, we also present a short survey on fundamental solutions.