I'm trying to show that a Dini-continuous function $\Omega\in L^1(\mathbb{S}^{n-1})$ satisfies the Hörmander condition, i.e. $$\sup_{y\ne0}\int_{|x|\ge2|y|}|K_\Omega(x-y)-K_\Omega(x)|dx:=A_2<\infty,$$ where $K_\Omega(x):=\frac{\Omega\left(\frac{x}{|x|}\right)}{|x|^n}.$ This is ex. 5.4.5 on Loukas Grafakos, Classical Fourier analysis.

I've shown that a sufficient condition for this to be true is that $\exists\ C_0<\infty$ indipendent on $y$ such that $$\int_{|x|\ge2|y|}\left|\frac{1}{|x-y|^n}-\frac{1}{|x|^n}\right|dx\le C_0,$$ but I can't manage to motivate this statement. I'm trying to use that the condition $|x|\ge2|y|$ implies: $$1)\left|\frac{x-y}{|x-y|}-\frac{x}{|x|}\right|\le2\left|\frac{y}{x}\right|$$ $$2)|x-y|^n\ge\left(\frac{|x|}{2}\right)^n.$$

Can someone help me with this?

I'm trying to show that a Dini-continuous function $\Omega\in L^1(\mathbb{S}^{n-1})$ satisfies the Hörmander condition, i.e. $$\sup_{y\ne0}\int_{|x|\ge2|y|}|K_\Omega(x-y)-K_\Omega(x)|dx:=A_2<\infty,$$ where $K_\Omega(x):=\frac{\Omega\left(\frac{x}{|x|}\right)}{|x|^n}.$ This is ex. 5.4.5 on Loukas Grafakos, Classical Fourier analysis.

I've shown that a sufficient condition for this to be true is that $\exists\ C_0<\infty$ indipendent on $y$ such that $$\int_{|x|\ge2|y|}\left|\frac{1}{|x-y|^n}-\frac{1}{|x|^n}\right|dx\le C_0,$$ but I can't manage to motivate this statement. I'm trying to use that the condition $|x|\ge2|y|$ implies: $$\left|\frac{x-y}{|x-y|}-\frac{x}{|x|}\right|\le2\left|\frac{y}{x}\right|\label{1}\tag{1}$$ $$2)|x-y|^n\ge\left(\frac{|x|}{2}\right)^n.\label{2}\tag{2}$$

Can someone help me with this?