# Γ-Pseudo-Differential Operators and H-Polynomials

• Fabio Nicola
• Luigi Rodino
Part of the Pseudo-Differential Operators book series (PDO, volume 4)

## 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.

## Keywords

Asymptotic Expansion Harmonic Oscillator Fredholm Operator Standard Quantization Principal Symbol
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Fabio Nicola
• 1
• Luigi Rodino
• 2
1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly
2. 2.Dipartimento di MatematicaUniversità di TorinoTorinoItaly