Remarks on the parametrized symbol calculus

Abstract:

In his paper, L. Hörmander has used the Weyl calculus to study the Fourier integral operator theory. In the present paper, the author considers the correspondences ${W_\tau }$, $\tau \in R$ ($R$ is the set of the real numbers), which mean the standard correspondence of symbol and operator if $\tau = 0$, and the correspondence of Weyl type if $\tau = 1/2$, and shows the explicit asymptotic formula which describes the deviation of ${W_\sigma }{\left ( {{W_\tau }} \right )^{ - 1}}$ from the automorphisms as Lie algebra, and makes some remarks on the above formula.