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There is a theorem (see Treves: "Introduction to Pseudodifferential and Fourier integral operators") that states that the kernel of any pseudodifferential operator, i.e. the distribution $$ K(x,y) = \int e^{i(x-y)\xi} a(x,y,\xi)d\xi $$ is the $C^{\infty}$ function on the complement to the diagonal of $\Omega \times \Omega$, $\Omega \subset \mathbb{R}^n$. But what it means if the integral diverges?

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If we consider an operator $Au (x,y) = \int e^{i(x-y)\xi} a(x,y,\xi) d\xi$ then $K(x,y)$ may be considered as it's continuous extension to the class of tempered functions. – Nimza Feb 19 '12 at 18:29

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