= Oscillatory tail estimate for a singular kernel
{title2=$\left|\int_{r<|x|<R}K(x)e^{-2\pi i x\cdot\xi}\,dx\right|\leq\frac12\int_{|x|>r}|K(x-y)-K(x)|\,dx+C_dA|y|/r$}
For $\xi\ne0$ set $y=\xi/(2|\xi|^2)$, and take $R>r\geq4|y|$. Translation by $y$ reverses the oscillatory exponential's sign. Comparing the original annulus with its translate gives twice the integral as an integrable kernel difference plus boundary-shell errors. The shell errors are bounded by $C_dA|y|(r^{-1}+R^{-1})$. The displayed estimate follows by enlarging the constant. Its right-hand side tends to zero as $r\to\infty$, proving convergence of the Fourier integral's improper far tail without absolute integrability of $K$ itself.
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