An Invariant Approach to Symbolic Calclus foe Pseudodifferential Operators on Manifolds

Abstract

In this work we present a symbolic calculus for DOs on a smooth manifold M based on a suitable notion of the global phase function. In previous literature on the topic, either local coordinates or connections have been used to define the phase functions, symbols Sm ; and oscillatory integrals defining DOs. Traditionally the condition 0 1 &#x100000; < 1 (or at least 0 < 1 and > 1=2) is assumed on the type of the symbol. On the contrary, we rely on a fully analytical notion of the global phase function ' over (TMn f0g) M. We use ' to define full symbols of DOs and develop a new version of symbolic calculus on smooth manifolds. All basic results from classical theory remain true under the condition 0 1 &#x100000; 2 < 1 (or at least 0 < 1 and > 1=3). In particular we obtain global formulae for the compositions and adjoints of DOs. Applications are given to elliptic DOs, boundedness on Sobolev spaces and functional calculus for elliptic DOs.

Date of Award	2015
Original language	English
Awarding Institution	King's College London