Let
$$K(x) := \lim_{R \to \infty}{\int_{\mathbb{R}^d}{e^{2\pi i \xi \cdot x}\frac{\xi_j}{|\xi|}\phi(\xi/R)d\xi}}$$
Where $\phi$ is a $C^\infty_c(\mathbb{R}^d,[0,1])$ such that $\phi$ is $1$ when $|x| \leq 1/2$ and $\phi = 0$ when $|x|>1$, you can assume $\phi$ is radial if you want
I'm trying to show that $K(x)$ exists whenever $x \neq 0$, on the notes I'm reading it says "it's easy to see" but to me it doesn't seem easy to see at all.
$K$ should represent the integral kernel of the Riesz Transform $R_j$, maybe one could define the Riesz Transform in another way, bypassing this calculation, for example using the hormander mikhlin multiplier theorem, but I'm trying to do this in the same way my professor does.
I tried to use integration by parts, as if one lets $\xi = R\eta$ then gets
$$K(x) = \lim_{R \to \infty}{R^d\int_{\mathbb{R}^d}{e^{2\pi i R\xi \cdot x}\frac{\xi_j}{|\xi|}\phi(\xi)d\xi}}$$
and by Riemann-Lebesgue the latter integral tends to $0$, I tried to use integration by parts to find an explicit order of convergence but I didn't managed to handle the derivatives of $\frac{\xi_j}{|\xi|}$