It is generally well known that the Fourier-Laplace transform converts a linear constant coefficient PDE P(D)u=f on Rn to an equation P(ξ)u˜ (ξ)=f˜ (ξ), for the transforms u˜, f˜ of u and f, so that solving P(D)u=f just amounts to division by the polynomial P(ξ). The practical application was suspect, and ill understood, however, until theory of distributions provided a basis for a logically consistent theory. Thereafter it became the Fourier-Laplace method for solving initial-boundary problems for standard PDE. We recall these facts in some detail in see's 1-4 of ch.0.

The technique of pseudodifferential operator extends the Fourier-Laplace method to cover PDE with variable coefficients, and to apply to more general compact and noncompact domains or manifolds with boundary. Concepts remain simple, but, as a rule, integrals are divergent and infinite sums do not converge, forcing lengthy, often endlessly repetitive, discussions of ‘finite parts’ (a type of divergent oscillatory integral existing as distribution integral) and asymptotic sums (modulo order −∞).

Of course, pseudodifferential operators (abbreviated ψdo's) are (generate) abstract linear operators between Hilbert or Banach spaces, and our results amount to ‘well-posedness’ (or normal solvability) of certain such abstract linear operators. Accordingly both, the Fourier-Laplace method and theory of ψdo's, must be seen in the context of modern operator theory.

To this author it always was most fascinating that the same type of results (as offered by elliptic theory of ψdo;'s) may be obtained by studying certain examples of Banach algebras of linear operators. The symbol of a ψdo has its abstract meaning as Gelfand function of the coset modulo compact operators of the abstract operator in the algebra.