Γ-Pseudo-Differential Operators and H-Polynomials

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


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.


Asymptotic Expansion Harmonic Oscillator Fredholm Operator Standard Quantization Principal Symbol 
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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

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

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