Take $X = \mathbb{R}^n$, denote by $\|x\|$ the standard Euclidean norm and consider the space of $L^2$ functions over $X$ in the Lebesgue measure. I am trying to show that the sequence $$I_s:=\sup_{f\in L^2(X):\|f\|_{L^2}\leq 1}\int_X|f(x)|^2|e^{-is/\|x\|}-1|^2\,\mathrm dx$$ does not converge to zero as $s\to 0$, where $i$ is the imaginary unit $i^2 = -1$. That is, I would like to conclude that e.g. $$ \liminf_{s\to 0} \sup_{f\in L^2(X):\|f\|_{L^2} \leq 1}\int_X|f(x)|^2|e^{-is/\|x\|}-1|^2\,\mathrm dx>0$$
Failure of $I_s$ to converge to zero must happen around the origin, but so far I have not been able to either find some suitable $L^2$ function with which $I_s = \text{constant $C > 0$}$ for all $s > 0$ or deduce some identity of the norm $|e^{-is/\|x\|}-1|^2$ which would make this proof easier. Namely, as $|e^{-is/\|x\|}-1|^2 = 2(1 - \cos(s/\|x\|))$, it is quite hard to try to integrate the function over some neighbourhood $B(0,r)$, $r>0$. How should I try to approach this problem?