Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y This is homework so no answers please
Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y
where $|p|< \frac{1}{2}$.
Any strategies? 
Attempt:
I will type as I go, I am just curious to see strategies suggested. 
I tried the usual ones i.e. Holder, Jensens, Minkowski, linear interpolation.
Thanks

The result is false for $p=1/2$, since $1/|x|$ is not integrable at $0$.
If $p<1/2$, you only have to worry about integrability at $\infty$. Use the Mean Value Theorem. You will find the condition $p>-1/2$.
