A groupoid approach to pseudodifferential calculi

In this paper we give an algebraic/geometric characterization of the classical pseudodifferential operators on a smooth manifold in terms of the tangent groupoid and its natural <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mo>×</m:mo> </m:msubsup> </m:math> {R×₊} -action. Specifically, a properly supported semiregular distribution on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>×</m:mo> <m:mi>M</m:mi> </m:mrow> </m:math> {M× 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 fibers of the tangent groupoid that is homogeneous for the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> <m:mo>×</m:mo> </m:msubsup> </m:math> {R×₊} -action modulo smooth functions. Moreover, we show that the basic properties of pseudodifferential operators can be proven directly from this characterization. Further, 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.