Let $J_0(x)=J(x)=\sum^{\infty}_{p=0} \frac{(-1)^p}{(p!)^2}(\frac{x^2}{4})^{2p} $ be the function Bessel of order 0. Take a cut-off function $k\in C^{\infty}_{0}(\Bbb{R})$ such that $k(x)=1$ on $]-1,1[$ and $k(x)=0$ if $|x|>=2$.
Define $I_n=\int^\infty_0 |J(nt)k(t)|^2 t^5 dt$.
My question is can we say that $\lim_{n\to\infty} I_n=0$.
Note: It is well known that $|J(x)|<=1$ for all $x\in\Bbb{R}$