We will now show that the subset M K,ψ,x,r of U K,x,r consisting of those y satisfying both inequalities (4.6.5) sup
ℓ,j,i |G ℓ,j,i,K,ψ (x, r)| C ≤ sup ℓ,j |f ℓ,j,K,ψ (x, y)| C ≤ sup ℓ |f ℓ,K,ψ (x, y)| C
has big volume in the sense that (4.6.6) Vol(U K,x,r ) ≤ q d+1 K Vol(M K,ψ,x,r ). Once this is proved, we are done for our original family (f ℓ ) ℓ by replacing d with d + 1 while keeping the **data** of the ϕ, G ℓ,j,i , and h ℓ,j,i .