## Cutoff regularization

I would like some help to obtain a certain estimate, see below.

Put $K(x)=C_ne^{-x^2/2}$ with $C_n>0$ chosen so that $\int _{\mathbb{R}^n}K(x)\,dx=1$. Put $K_\lambda (x)=\lambda ^{-n}K(\lambda ^{-1}x)$ for $\lambda >0$. We make the $x$-dependent choice of $\lambda$; $\lambda (R,x)=R\langle R^{-1}x \rangle ^{-N_0}$ where $R\ge 1$ is a large parameter and $N_0$ is fixed, but sufficiently large depending on the dimension $n$ (here $\langle x \rangle =\sqrt{1+|x|^2}$). Put $K_R(x,y)=K_{\lambda (R,x)}(x-y)$. Let $\chi \in C_0 ^\infty (B(0,2);[0,1])$ be equal to 1 for $|x|<1$. For $R$ large enough, put

$V_0(x)=\int _{\mathbb{R}^n} K_R(x,y)[1-\chi (R^{-1}x)]V_1(y)\,dy.$

Here $V_1$ is a continuous real valued function on $\mathbb{R}^n$, tending to 0 at infinity.

I need to show that

$|V_1(x)-V_0(x)|\le C \langle x \rangle ^{-n-1},$

for some constant $C=C(R)>0$.

I would greatly appreciate it if someone could shed some light on how to do this. Thank you!