Let $K(x,y)= \Omega(x-y/|x-y|)/|x-y|^d$ for $x, y\in \bf R ^d$, $x \neq y$.
I want to show that $K$ is a singular kernel provided that $\Omega$ is Holder continuous on $\bf S^{d-1}$. (I am currently reading Prof. Terry Tao's lecture notes on Harmonic analysis, and he says that $K$ is indeed a singular kernel without any proof.)
To acheive this goal, I need to show that there exists a $0< \sigma \leq 1$ such that $K(x,y')-K(x,y)=O(\frac{|y-y'|^\sigma }{|x-y|^{d+\sigma}})$ whenever $|y-y'| \leq \frac1 2 |x-y|$, which doesn't seem to be obvious for me, since the formulas get quickly messy when I try to estimate the left-hand-side.
Any help will be appreciated. Thank you for reading.