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

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

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

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

I need to show that

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

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

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