Oscillatory tail estimate for a singular kernel ($\left|{\int }_{r<|x|<R}K(x)e^{-2\pi ix\cdot \xi }dx\right|\le \frac{1}{2}{\int }_{|x|>r}|K(x-y)-K(x)|dx+C_{d}A|y|/r$)

For $\xi \ne 0$ set $y=\xi /(2|\xi {|}^2)$, and take $R>r\ge 4|y|$. 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 $C_{d}A|y|(r^{-1}+R^{-1})$. 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.