Wells 'Differential Analysis on Complex Manifolds' page 127

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How does the first equation for $Qu(x)$on this page follow from the defining equation (3.10) on the previous page. This is from the section on pseudodifferential operators in chapter 4. I'm starting to think that there might be a typo somewhere.

I've tried some direct computations but can't see exactly how the two formulas are equal.

Hopefully someone who's read this chapter can show me what's going on.

Edit 1: What I'm talking about is $\displaystyle{Qu(x):=(2\pi)^{-n}\int {e^{i(\xi,x-z)}q(x,\xi,z)u(z)dzd\xi}}$. with $u$ and $q$ satisfying some 'nice' properties, for instance, compact support in the $z$, $x$ variables (c.f. page 126 of Wells' book).

This is supposedly equal to $\displaystyle{\int{e^{i(\xi,x)}\hat{q}(x,\xi,\xi-\eta)\hat{u}(\eta)d\eta d\xi}}$ with the fourier transform for $q$ being taken with respect to the third variable.

I've tried to show this equality for a while without success and I'm hoping someone can tell me if it is indeed true (in which case I will try even harder) or if there's something that needs to be changed in either of the equations.

Thanks for reading the question.

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The equality of the two formulas can be seen using the identity $\hat{fg}=\hat{f}*\hat{g}$ for 'Schwarz' functions, in particular for functions with compact support. See, for example, chapter 7 in Rudin's 'Functional Analysis'.