Introduction

Abstract

To give an introduction to the contents of the book, let us consider initially the basic models to which our pseudo-differential calculus will apply, namely the linear partial differential operators with polynomial coefficients in ℝ^d

$$ P = \sum {c_{\alpha \beta } x^\beta D^\alpha } , x \in \mathbb{R}^d , c_{\alpha \beta } \in \mathbb{C}, $$

(I.1)

where in the sum (α,β) ∈ ℕ^d × ℕ^d run over a finite subset of indices. A natural setting for P is given by the Schwartz space $S\left( {\mathbb{R}^d } \right)$ and its dual $S'\left( {\mathbb{R}^d } \right)$. These spaces are invariant under the action of the Fourier transform

$$ Fu\left( \xi \right) = \hat u\left( \xi \right) = \int {e^{ - ix\xi } u\left( x \right)d--x, } with d--x = \left( {2\pi } \right)^{ - \frac{d} {2}} dx. $$

(I.2)

Note also that the conjugation $FPF^{ - 1}$ gives still an operator of the form (I.1).