interpretation of a singular integral There is a post on MSE about a principal value integral in this paper. It has not received much attention even with a bounty, and since it concerns a published paper, I believe this is a better forum for the topic. 
The integral in question is the third of this remark

and it is based on this lemma

I believe there is a problem with the integral as I pointed out in my answer on MSE. My question is the following: Is there a problem with the integral, or is there an interpretation that makes it valid, possibly some finite part interpretation?
 A: This is definitely wrong. The two first integrals are defined as principal values, allowing to cancel the sine parts, and conclude with the lemma (actually, as OP already knows, the lemma itself do not need to be expressed as principal value). But for the third one, it is not even clear how a principal value could be defined. Since one needs to bound away both $t$ and $s$ from zero, I would go for integrating over $D_\epsilon = \left(\mathbb{R}^p \setminus (\epsilon B_p \cup \epsilon^{- 1}B_p^{c}) \right) \times \left(\mathbb{R}^q \setminus (\epsilon B_q \cup \epsilon^{- 1}B_q^{c}) \right)$, where $B_p$ and $B_q$ are the appropriate unit balls. But now set $Y = 0$, so that for $X \in \mathbb{R}^p$ and $\epsilon > 0$,
$$\int_{D_\epsilon}
\frac{1 - \exp(\mathrm{i}\left<t,X\right>)}
     {\Vert t \Vert^{p+1} \Vert s \Vert^{q+1}} \mathrm{d}t \mathrm{d}s =
\int_{\mathbb{R}^p \setminus (\epsilon B_p \cup \epsilon^{- 1}B_p^{c})}
\frac{1}{\Vert s \Vert^{q+1}}
\int_{\mathbb{R}^p \setminus (\epsilon B_p \cup \epsilon^{- 1}B_p^{c})}
\frac{1 - \cos(\left<t,X\right>)}
     {\Vert t \Vert^{p+1}} \mathrm{d}t \mathrm{d}s.$$
But then by virtue of the lemma, with $X \neq 0$, the inner integral (over $t$) is bounded away from zero for $\epsilon$ small enough, and thus the above quantity diverges. This is far from being equal to zero, as the third equality would imply.
A: I think the "finite part interpretation" is what makes these formulas valid, including the third one: it amounts to computing the distributional Fourier transform of the "pseudo-function" (and tempered distribution) Pf$|x|^{-n-1}$ (in $\mathbb R^n$), which is proportional to $|\xi|$, with the constant given. (The finite part of $\int\int |s|^{-p-1}|t|^{-q-1}\ ds\ dt$ vanishes, obviously, so that the third "integral", while diverging, has a finite part which is exactly minus the product of Fourier transforms of two such pseudo-functions).
I agree with Hugo Raguet that lemma 1 wrongly calls upon a principal value, since the integrand is integrable.
Fourier transforms of pseudo-functions Pf$|x|^\alpha$, $\alpha\in\mathbb R$, may be found e.g. in Schwartz's Théorie des distributions, formula (VII, 7; 13) and (VII, 7; 14), p. 257 and 258 in my copy of 1973.
