1132 CHAPTER 33. FOURIER ANALYSIS IN Rn

Let φ (x)≡ ψ (x)g(x)−1. Then

∞

∑k=−∞

φ

(2kx)=

∞

∑k=−∞

ψ(2kx)

g(2kx)= g(x)−1

∞

∑k=−∞

ψ

(2kx)= 1

for each x ̸= 0. This proves the lemma.Now define

ρm (x)≡m

∑k=−m

ρ (x)φ

(2kx), γk (x)≡ ρ (x)φ

(2kx).

Let t > 0 and let |y| ≤ t. Consider the problem of estimating∫|x|≥2t

∣∣F−1γk (x−y)−F−1

γk (x)∣∣dx. (33.3.26)

In the following estimates, C (a,b, · · · ,d) will denote a generic constant depending only onthe indicated objects, a,b, · · · ,d. For the first estimate, note that since |y| ≤ t, 33.3.26 is nolarger than

2∫|x|≥t

∣∣F−1γk (x)

∣∣dx = 2∫|x|≥t

∣∣F−1γk (x)

∣∣ |x|−L |x|L dx

≤ 2(∫|x|≥t|x|−2L dx

)1/2(∫|x|≥t|x|2L ∣∣F−1

γk (x)∣∣2 dx

)1/2

Using spherical coordinates and Plancherel’s theorem,

≤C (n,L) tn/2−L(∫|x|2L ∣∣F−1γk (x)

∣∣2 dx)1/2

≤C (n,L) tn/2−L(∫

∑nj=1

∣∣x j∣∣2L ∣∣F−1γk (x)

∣∣2 dx)1/2

≤C (n,L) tn/2−L(

∑nj=1∫ ∣∣∣F−1DL

j γk (x)∣∣∣2 dx

)1/2

=C (n,L) tn/2−L(

∑nj=1∫

Sk

∣∣∣DLj γk (x)

∣∣∣2 dx)1/2

(33.3.27)

whereSk ≡

[x :2−2−k < |x|< 22−k

], (33.3.28)

a set containing the support of γk. Now from the definition of γk,∣∣DLj γk (z)

∣∣= ∣∣∣DLj

(ρ (z)φ

(2kz))∣∣∣.

By Lemma 33.3.4, this is no larger than

C (L,n,φ)C0 |z|−L. (33.3.29)

It follows, using polar coordinates, that the last expression in 33.3.27 is no larger than

C (n,L,φ ,C0) tn/2−L(∫

Sk

|z|−2L dz)1/2

≤C (n,L,φ ,C0) tn/2−L· (33.3.30)