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I know what it means for a pseudodifferential operator $A\in\Psi(\mathbb{R}^n)$ to be elliptic at a point $(x,\xi)\in T^*\mathbb{R}^n$: the principal symbol of $A$ is non-vanishing at the point.

But what does it mean for a Fourier Integral Operator to be elliptic? Is there an analogous notion of principal symbol for an FIO?

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The principal symbol of an FIO $A$ is a section in a line bundle (Maslov tensor half-density) over the canonical relation $C\subset T^* Y\times T^* X$ associated with $A$. See Chapter 25 in volume IV of Hörmander's ALPDO for the symbol calculus. An FIO is called elliptic if its symbol is invertible. However, ellipticity does not imply the existence of parametrix without additional assumptions on $C$. If $C$ is the graph of a symplectomorphism then ellipticity implies the existence of a parametrix which is associated with $C^{-1}$, the inverse relation of $C$. Pseudodifferential operators are FIOs associated with (the graph of) the identity map.

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