I'm trying to show that given any $y\in \mathbb{R}^d$ and $(d/2)-1<p<d/2$, the function $f_y$ defined by $$ f_y(x)=|x-y|^{-p}-|x|^{-p} $$ is in $L^2(\mathbb{R}^d)$.
I've managed this to show that the function is integrable in any bounded neighborhood of $0$ and $y$, so by continuity, it is integrable in any compact subset of $\mathbb{R}^d$. I have troubles showing it is integrable in sets of the form $\{x\in\mathbb{R}^d\mid |x|>c\} $ for $c>0$. As functions of the form $g(x)=|x|^\alpha$ are integrable in this kind of sets if and only if $\alpha<-d$, and $-2p-2<-d$, I tried to show that the limit $$ \lim\limits_{x\to\infty} \frac{f_y(x)^2}{|x|^{-2p-2}} $$ exists and is finite, but I couldn't show this even in the simplest case $d=1$.
So could you please give me a hint as how to proceed here? Any help is appreciated. Thanks in advance