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Transactions of the American Mathematical Society

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The order and symbol of a distribution

Author: Alan Weinstein
Journal: Trans. Amer. Math. Soc. 241 (1978), 1-54
MSC: Primary 58G15; Secondary 46G05, 58C35
MathSciNet review: 492288
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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.

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Article copyright: © Copyright 1978 American Mathematical Society