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!