$C^{\infty}$ function represented by the diverging integral

Question

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 $$ K(x,y) = \int e^{i(x-y)\xi} a(x,y,\xi){\,\rm d}\xi $$ 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?