Γ-Pseudo-Differential Operators and H-Polynomials

Abstract

Let P be a linear operator, P: $S\left( {\mathbb{R}^d } \right) \to S\left( {\mathbb{R}^d } \right)$, with extension to a map from $S'\left( {\mathbb{R}^d } \right)$ to $S'\left( {\mathbb{R}^d } \right)$. According to Definition 1.3.8 we say that P is globally regular if, for any $f \in S\left( {\mathbb{R}^d } \right)$, all the solutions $u \in S'\left( {\mathbb{R}^d } \right)$ of the equation Pu = f belong to $S\left( {\mathbb{R}^d } \right)$. In particular then, all the solutions $u \in S'\left( {\mathbb{R}^d } \right)$ of the equation Pu = 0 belong to $S\left( {\mathbb{R}^d } \right)$. An important tool for deducing global regularity, when P is a pseudo-differential operator, is given by Theorem 1.3.6, namely: the existence of a left parametrix $\tilde P$ of P, i.e., $\tilde PP = I + R$ where R: $S'\left( {\mathbb{R}^d } \right) \to S\left( {\mathbb{R}^d } \right)$, implies global regularity for P, as well as precise estimates in generalized Sobolev spaces, cf. Proposition 1.5.8. Besides, the simultaneous existence of a right parametrix gives Fredholmness, cf. Theorem 1.6.9.