I'm reading the article Brownian distance covariance and stumbled upon a equality I can't seem to derive myself. We are first presented with the following lemma:
and after stating this lemma, the authors follow up with a remark in which my problem lies:
Now the first two integrals are rather simple to show using the above lemma but the third iterated principal value integral is where my troubles begin.
I have tried multiple approaches so far, two of them being to rewrite the nominator: $$ 1-\exp(it^\intercal x + i s^\intercal y) = (1-\exp(it^\intercal x)) + (1-\exp(is^\intercal y)) -(1-\exp(it^\intercal x))(1-\exp(is^\intercal y)) $$ but this approach yields nothing since the second term yields a the inner principal value integral of $\infty$ and vice versa for the first term when evaluating the outer principal value integral. Another approach I tried was to write \begin{align*} 1-\exp(it^\intercal x + i s^\intercal y) =& 1- \cos(t^\intercal x + s^t y) -i\sin (t^\intercal x + s^\intercal y) \\ =& 1- \cos t^\intercal x \cos s^\intercal y + \sin t^\intercal x \sin s^\intercal y -i (\sin t^\intercal x \cos s^\intercal y + \cos t^\intercal x \sin s^\intercal y) \\ =&(1-\cos t^\intercal x) + (1- \cos s^\intercal y)-(1-\cos t^\intercal x)(1-\cos s^\intercal y) \\ &+ \sin t^\intercal x \sin s^\intercal y -i (\sin t^\intercal x \cos s^\intercal y + \cos t^\intercal x \sin s^\intercal y) \\ \end{align*} In this expression the third term integrates to the negative value of the wanted solution, so its close but not the wanted expression (even if I ignore the sign, I can't get the other terms to yield a zero integral).
Am approaching this is the wrong way or is there something I have missed? Any solutions/hints would be appreciated.
There is a problem with the last equality. Consider a set $A\subset\mathbb{R^q}$ where $\left<s,Y \right>$ is bounded away from integer multiples of $2\pi$. For $t$ in a small enough neighborhood of the origin, call it $B$, the numerator is bounded away from $0$. So the singularity is not canceled out. Hence, for $s\in A$ the integral
\begin{equation*} \int_B \frac{1-\exp\{i\left<t,X \right>+ i\left<s,Y \right> \}} {|t|_p^{1+p} |s|_q^{1+q}} dt \end{equation*} is not defined (or infinite). This implies \begin{equation*} \int_{\mathbb{R}^p} \frac{1-\exp\{i\left<t,X \right>+ i\left<s,Y \right> \}} {|t|_p^{1+p} |s|_q^{1+q}} dt \end{equation*} is not defined (or infinite).
The inner integral must be defined a.e. for convergence, and this is not the case.