@AminIdelhaj There's about a claim here about bounds of a convolution. Suppose $K: \mathbb{R}^n \rightarrow \mathbb{C}$ is measurable. Suppose $K$ vanished outside of $B_R(0)$ and inside of $B_{\epsilon}(0)$. Suppose
$(1)\ |K| \le A |x|^{-n}$,
$(2)$ The Hörmander condition $\int_{\mathbb{R}^n \setminus B_{2r}(0)} |K(x) - K(x - z)| dx \le A$ for all $z \in B_r(0)$.
$(3)$ The cancellation condition $\int_{B_r(0)} K(x) dx = 0$ for all $r > 0$.
One of the claims is that you could get the bound $\Big|\int_{B_{\frac{1}{|\xi|}}(0)} K(x) e^{-2 \pi i x \xi} dx \Big| \le C(n) A$, *independent of $\e…