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

Put with chosen so that . Put for . We make the -dependent choice of ; where is a large parameter and is fixed, but sufficiently large depending on the dimension (here ). Put . Let be equal to 1 for . For large enough, put

Here is a continuous real valued function on , tending to 0 at infinity.

I need to show that

for some constant .

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