The order and symbol of a distribution

Abstract:

A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point $({\chi ,\xi })$ of the cotangent bundle $T^{\ast }X$. If $X = \textbf {R}^n$, the order of g at $({0,\xi })$ is the growth order as $\tau \to \infty$ of the distributions ${g^\tau }(x) = {e^{ - i\sqrt \tau \langle x,\xi \rangle }}g\left ( {x /\sqrt \tau } \right )$ ; if the order is less than or equal to N, the N-symbol of g is the family ${g^\tau }$ modulo $O({{\tau ^{N - 1/2}}})$. It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hörmander.