A groupoid approach to pseudodifferential operators

Abstract

We give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural ${\mathbb{R}}_{+}^{\times }$-action. Specifically, we show that a properly supported semiregular distribution on $M\times M$ is the Schwartz kernel of a classical pseudodifferential operator if and only if it extends to a smooth family of distributions on the range fibres of the tangent groupoid which is homogeneous for the ${\mathbb{R}}_{+}^{\times }$-action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Finally, we show that with the appropriate generalization of the tangent bundle, the same definition applies without change to define pseudodifferential calculi on arbitrary filtered manifolds, in particular the Heisenberg calculus.

The orginal article has been split into two parts. This part contains the construction of pseudodifferential calculi. A construction of the relevant tangent groupoid will appear in a separate article. Information on the convolution algebra of fibred distributions on a groupoid has been reduced to a summary. For more details, see earlier versions

Cited by in corpus (5)

• Spectral Theory in a Twisted Groupoid Setting: Spectral Decompositions, Localization and Fredholmness
• Convolution bialgebra of a Lie groupoid and transversal distributions
• Fredholm conditions and index for restrictions of invariant pseudodifferential operators to isotypical components
• Automatic continuity of transversal distributions
• On pseudodifferential operators on filtered and multifiltered manifolds