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### Convergence of an oscillatory integral

Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral:
$$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$
I need to show that $I_f(t)$ is finite, ...

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### can we generalize the cauchy principal value and hadamard's finite part for multiple integrals $ n>1 $

I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one
My question is when can we (or if negative answer why can not ) generalize the ...